PERFORMANCE MODELLING OF COMPUTER SYSTEMS AND COMPUTER NETWORKS

PERFORMANCE MODELLING OF

COMPUTER SYSTEMS AND COMPUTER

NETWORKS

Ramon Puigjaner

Universitat de les Illes Balears

Palma, Spain

[email protected]

LANC 2007. San José, Costa Rica. October 2007..

LANC 2007. San José, Costa Rica. October 2007.2

OUTLINE

INTRODUCTION

CONCEPT OF QUEUE

CONCEPT OF QUEUEING NETWORK

NUMERICAL TECHNIQUES

EXACT ANALYTICAL SOLUTIONS

APPROXIMATE ANALYTICAL SOLUTIONS

SIMULATION TECHNIQUES


INTRODUCTION

What is the performance of a Computer Network?

Performance is how a software is using a hardware when they are serving a load.


INTRODUCTION

This definition considers the three elements intervening in a system:

The load that is externally defined. The hardware to be used. The basic software that controls the hardware.


INTRODUCTION: Performance measures

The performance of a Computer Network is not a unique value but a set of them to take into account the heterogeneous composition of such kind of systems. External performance measures

o response time

o throughput (flow through the system)

o loss rate


INTRODUCTION: Performance measures

Internal performance measures

o mean queue length

o device utilisation (% of busy time)

o overlap

o overhead (operating system utilisation, paging, etc.)


INTRODUCTION: Performance tools

Measuring Monitors Logs Hardware probes Software probes

Modelling

Benchmarking


INTRODUCTION: Performance tools

Measuring

Modelling Queuing networks Petri nets Markov chains

Benchmarking Workload modelling


INTRODUCTION: Measuring

Measuring is the technique to be used when system is installed and running. It is used to verify whether the performance requirements are met or not.


INTRODUCTION: Modelling

A model is an abstract mathematical representation of the system behaviour in steady state. It is the appropriate technique when the computer network, partially or totally, does not exist. Main existing techniques are: Petri nets

o Better suited to represent synchronisation mechanismso Solving techniques may be either numerical (based on

Markov chains) or simulation.


INTRODUCTION: Modelling

Queuing networkso Better suited to represent customer-server mechanisms

o Solving techniques may be either analytical (closed form formulae) or numerical (based on Markov chains) or simulation.

Markov chainso High abstraction level

o Solving techniques are most frequently numerical.


OUTLINE

INTRODUCTION

CONCEPT OF QUEUE







CONCEPT OF QUEUE

Queue: A customer that arrives and finds the server busy joins the queue

Service mechanism: It consists of one or more servers that give service to the customers from the queue


CONCEPT OF QUEUE

Customer source characteristics finite or infinite distribution of inter-arrival times between

consecutive customer arrivals customer service request

Service station characteristics queue number and capacity server number server capacity service discipline queue policy


CONCEPT OF QUEUE

Single queue with single server

Single queue with single server with state dependent capacity

m(k )


CONCEPT OF QUEUE

Single queue with multiple servers


CONCEPT OF QUEUE

Multi-server with no queue


CONCEPT OF QUEUE

Infinite server

.

.

.


OUTLINE

INTRODUCTION

CONCEPT OF QUEUE








A queuing network is nothing else but a collection of single queues, which are arbitrarily interconnected.

A queuing network is an oriented graph that has in each node a server of some type.

The time in traversing the network is spent in the nodes and the arcs are traversed in a null time.



Queuing networks may be either open or closed. In an open queuing network, customers arrive from

outside, circulate through the nodes, and finally they depart from the network.

In a closed queuing network, there is a fixed number of customers constantly circulating through the nodes. Neither departures from the network nor arrivals to the network are allowed.

It is possible to have a queuing network which is both open and closed. Such a network is known as a mixed network.


EXAMPLES OF OPEN QUEUING NETWORKS

Tandem configuration



Tree-like configuration


EXAMPLES OF CLOSED QUEUING NETWORKS

Cyclic network (closed tandem configuration)



Arbitrary configuration



Central server model

C P U

D isk 2

D isk 3

D isk 4

D isk 1

A rriv a ls

E x its




C P U

D isk 2

D isk 3

D isk 4

D isk 1

A rriv a ls

E x its




C P U

D isk 2

D isk 3

D isk 4

D isk 1

A rriv a ls

E x its

T e rm in a ls


EXAMPLES OF MIXED QUEUING NETWORKS

T e rm in a ls

C e n tra lsy s te mT ra n sa c tio n s

C o n v e rsa tio n a lta sk s



Observations Each node can have any of the single-node

characteristics described above.

In order to specify the queuing network we need to

provide information concerning the routing; that is

to specify how a customer chooses the next node

when it leaves the current node. This routing can be

deterministic, probabilistic, function of the state,

etc.



How to set-up a queuing network model? The notion of customer

o Typically a customer may be a piece of software in a

computer system, an information packet in a packet-

switched environment, a phone call in a circuit-

switched environment, etc.

o Customer classes will be defined if there are

differences in the resource consumption or in the

routing across the network



How to set-up a queuing network model? The notion of node

o A node is a service mechanism that may be a hardware component or a piece of software or a combination of both, e.g. a CPU, a disk, a memory module, a bus, a trunk, a switching node, etc.

o Each service mechanism has a buffer (the queue), where customers wait until they are served. The buffer capacity is finite; that is, they can accommodate a finite number of customers. However, if a finite buffer has low probability of being full, then it can be assumed as infinite.



How to set-up a queuing network model? Collecting information

o Once customers and server have been identified, it is

necessary to characterise service time distributions at

each node, routing probabilities and inter-arrival time

distributions.

o In many cases, this information can be compiled from

raw data (technical information, measurements, etc.);

in other cases it is based on an educated guess.



Solution techniques for queuing networks

To study the steady state behavior of a network

the following techniques can be used:

Analytic solutions

Numerical techniques

Simulation techniques


OUTLINE

INTRODUCTION

CONCEPT OF QUEUE








The behaviour of a queuing network can be described in terms of linear equations (known as the steady-state Kolmogorov equations). These equations can be solved numerically to obtain the solution.



To highlight this approach, let us consider the following two-node closed queuing network

Let us assume that: there are 5 customers in the system. µ1 and µ2 are the service rates.

both services are exponentially distributed.

µ1 2µ



The state of the system is described by (n1, n2), that there are the number of customers in each queue.

The numerical analysis approach involves the following steps: Generation of all feasible states. Setting-up the rate matrix. Solving the steady state equations.



Generation of all feasible states.

The states for our example are:

(5,0) (4,1) (3,2) (2,3) (1,4) (0,5)



Setting-up the rate matrix. This matrix which contains all the transitions and

their associated rates between each pair of states.(5,0) (4,1) (3,2) (2,3) (1,4) (0,5)

(5,0) * µ1

(4,1) µ2 * µ1

(3,2) µ2 * µ1

(2,3) µ2 * µ1

(1,4) µ2 * µ1

(0,5) µ2 *



Setting-up the rate matrix.

Let us refer to the this matrix as Q.

All blanks are assumed to be zero.

Each diagonal element marked with * is equal to

the negative sum of the off-diagonal elements of

the same row.



Solving the steady state equations. Let p(n1, n2) be the steady-state probability that the

system is in state (n1, n2) and P the row vector of these probabilities. To determine them we must solve the following system of equations:

P x Q = 0

together with the condition

1

21

21 n,n

n,np



Solving the steady state equations. From the knowledge of these probabilities we can

determine performance measures such as:o Server utilisation:

r1 = p(5,0) + p(4,1) + p(3,2) + p(2,3) + p(1,4):

r2 = p(4,1) + p(3,2) + p(2,3) + p(1,4) + p(0,5)

o Throughputs:

l1 = r1 x µ1

l2 = r2 x µ2

o Queue lengths:

N1 = 5p(5,0) + 4p(4,1) + 3p(3,2) + 2p(2,3) + p(1,4)

N2 = p(4,1) + 2p(3,2) + 3p(2,3) + 4p(1,4) + 5p(0,5)



Solving the steady state equations. Advantages/disadvantages

o There are packages, like QNAP2, that automatically set-up the rate matrix Q, solve it to find the P vector and give the performance results. Other packages give the vector P if the user is able to create the matrix Q.

o This numerical technique gives the exact solution. There are also approximated solutions in some cases in order to reduce the amount of computation.

o The approach is limited to cases where the number of states is not very large.

o In queuing networks, quite often, the rate matrix is sparse. In this cases, one can analyse larger systems by using compact storage techniques.


OUTLINE

INTRODUCTION

CONCEPT OF QUEUE








An analytical solution means that we can obtain the probabilities of the steady steady by the application of a closed formula.

This formula will obviously be a function of the parameters of the system.

Quite often an analytic solution is so complicated that we can not evaluate it "on the back of an envelope". In fact, one might need to write a fairly sophisticated program.



A certain class of queuing networks has an

analytic solution, known as a product-form

solution because the steady state probability has

the form of the product of the state probabilities

of each node.

Its solution can be easily evaluated.



Product-form networks have been proved to be

very useful in computer and communication

systems performance modelling.

Also, there are a lot of queuing networks which

do not have product-form solutions. These

networks are analysed approximately.



The BCMP theorem It is the general theorem concerning queuing

networks with product-form solutions.

Let us consider a BCMP queuing network with: N nodes arbitrarily linked. Multiple classes of customers Probabilistic routing External arrivals with state-dependent rates Different service mechanisms



The BCMP theorem: Classes of customers Customers are grouped in different classes. Each

class has its own service characteristics at each node and its own routing probabilities. A class may be open or closed.

Thus, a BCMP network, in its most general form, can be seen as consisting of several open classes and closed classes of customers.



The BCMP theorem: Classes of customers It is possible that upon departure from a node a

customer may change of class. A superclass or a

subchain is the set of classes among those the

customers can change.

The use of classes of customers provides the

modeller with a lot of modelling flexibility.



The BCMP theorem: Classes of customersExample 1. A queuing network model of a multiprogramming system

C P U

D isk 2

D isk 3

D isk 4

D isk 1

A rriv a ls

E x its



The BCMP theorem: Classes of customersExample 1. A queuing network model of a multiprogramming system The total number of customers constantly

circulating through the system reflects the degree of multiprogramming. Implicitly it is assumed that when a job completes its execution and departs from the system, another job takes its place. That is, there is always at least one job waiting to get into the multiprogramming environment. This rather simplistic model captures the main features of a multiprogramming system.



The BCMP theorem: Classes of customersExample 1. A queuing network model of a multiprogramming system Let us see, how can make this model more useful

by introducing classes. We can introduce different classes for different types of jobs, i. e.:

o Class 1: Interactive jobso Class 2: Short batch jobso Class 3: Medium batch jobso Class 4: Long batch jobs

Also, in a multiprogramming system, a process originally classified in some class may be changed to another one.



The BCMP theorem: Classes of customers

Example 1. A queuing network model of a multiprogramming system Features as service requirements, visit rates to each

node, class change, etc. are captured through the use of classes.

However, the BCMP theorem is limited as it does not allow other features, such as priorities among classes.



The BCMP theorem: Classes of customersExample 2. Queuing network of a packet-switching system

1

2

3

4




Example 2. Queuing network of a packet-switching system A packet is represented by a customer in the

queuing network and each logical end-to-end connection is represented by a class. This allows us to assign a different routing to each class, and, if need be, different service times at each node.




Example 2. Queuing network of a packet-switching system So, we consider the following classes:

o Class 1: packets arrive at node 1, go to node 2, then to node 3 and then they depart from the system.

o Class 2: packets arrive at node 1, go to node 3, then to node 4 and then they depart from the system.

o Class 3: packets arrive at node 2, go to node 3 and then they depart from the system.

o etc.



The BCMP theorem: Arrival processes

If we have open classes of customers, one needs to

specify how these customers arrive from outside. In

general, the rate of arrivals is allowed to be state-

dependent, i. e. it can be an arbitrary function of the

number of customers in the system.



The BCMP theorem: Arrival processes Single arrival stream

o All external arrivals come from a single stream. When

a customer arrives to the network, it joins node i as

class r with probability pi,r.

o The inter-arrival times must be exponentially

distributed. The rate of arrivals may be constant or it

may be dependent upon the total number of customers

in the network.



The BCMP theorem: Arrival processes

One arrival stream per class

o Each open class has its own arrival stream. A new

arrival of class r joins node i with probability pi.

o The inter-arrival times must be exponentially

distributed. The rate of class r arrivals may be constant

or it may be dependent upon the total number of class r

customers in the network.




Example 2. Queuing network of a packet-switching system Window-flow control allows only up to a pre-

specified number of packets in the system. Any additional packets are forced to wait in an input queue.

In order to model a sliding-window flow-control scheme, we need to model the input queue. However, the BCMP theorem does not provide such features.




Example 2. Queuing network of a packet-switching system We can consider in some way the window-flow

control by making the arrival process of customers state-dependent. That is, arrivals will occur as long as the total number of customers of some class is less than some threshold. When it becomes equal to this value, the arrival stream will be turned off. The arrival stream will start again when a customer of the considered class departs from the network.




Example 2. Queuing network of a packet-switching system In this way, we make sure that the total number of

customers of each class does not exceed its threshold.

However, this is done by introducing the erroneous assumption that no arrivals occur during the time the window is full.



The BCMP theorem: Service mechanisms Type 1

o State dependent exponentially distributed service times.

o Class independent service time distribution

o FIFO irrespective of classes.

o Single server

Type 2o Class and state dependent Coxian distributed service

times.

o Processor sharing (PS) discipline (or RR, round robin)

o Single servers



The BCMP theorem: Service mechanisms Type 3

o Class-dependent Coxian distributed service times

o Infinite servers

Type 4

o Class and state dependent Coxian distributed service

times

o Pre-emptive server LIFO queue (PI)

o Single server




Example 1. A queuing network model of a multiprogramming system In this model it makes sense to assume that the

CPU node is a type 2 node, i. e. customer are processor-shared, while the peripheral (disks 1 to 4) are type 1. If we assume an interactive system we would represent the terminals by means of a type 3 node.



The BCMP theorem Assumption that the system reaches a steady state with

state probabilities p(S)

Balance equations:

Normalising equation

'

S toS' from going of rate'

from rateexit

S

Sp

SSp

1S

Sp



The BCMP theorem

G normalisation constant to obtain the addition the probabilities of all states is equal to 1.

o If the system is closed, the number of states is finite and the problem is numerical

o If the system is open, the number of states is infinite and the problem is analytical

N

iii SgSd

GSp

1

1



The BCMP theorem d(S) is a function that if the network is closed its value

is 1 and if the network is open its value is

if the arrival rate depends on M(S),

if the arrival rate to each subchain depends on M(S,Ek)

1

0

SM

i

i

K

k

ESM

ik

k

i1

1,

0



The BCMP theorem The expressions of gi(Si) are

o If the station is of type 1

o If the station is of type 2 or 4

c

c

m

i

C

c

mic

iciii e

mmSg

m

1

!

1!

1

cm

ic

icC

c iciii

e

mmSg

m

1 !

1!



The BCMP theorem The expressions of gi(Si) are

o If the station is of type 3

cm

ic

icC

c icii

e

mSg

m

1 !

1



The BCMP theorem: Solving of a BCMP

queuing network

The expression that gives the probability that the

system is in a precise state is quite complicated.

However it is still possible to write down the

solution in form of product of terms, each term

consisting of parameters related to the node.



The BCMP theorem: Solving of a BCMP queuing network In the case of closed networks, as the number of

states is finite, it is necessary to compute a normalising constant.

o There are various algorithms to do that, such as the convolution algorithm and the mean value analysis.

o These algorithms are available through various network analysers, such as QNAP2, BEST-1 and RESQ. The user simply specifies the network characteristics, and the package produces the solution.



The BCMP theorem: Solving of a BCMP queuing network In the case of open networks, as the number of

states is infinite, it is necessary to compute the result of a series.

o This computation is only possible for specific combinations of node characteristics.

o As for closed networks, these algorithms are available through various network analysers, such as QNAP2, BEST-1 and RESQ. The user simply specifies the network characteristics, and the package produces the solution.



The BCMP theorem: Case studies

Transactional system The case proposed is a throughput input system

with two types of transactions (messages arriving to the system and requiring some process) that have different arrival frequency, CPU consumption and profile (number of accesses to the disks), but the same mean service time to each disk (but different from disk to disk).




Transactional system We assume that these transactions are executed

concurrently on the computer system and that the conflicts in the its execution are due to the access to the same servers (CPU and disks) but not to any kind of synchronisation or use of critical objects.



The BCMP theorem: Case studiesTransactional system

C P U

D isk 2

D isk 3

D isk 4

D isk 1

A rriv a ls

E x its



The BCMP theorem: Transactional system 1 /DECLARE/ QUEUE CPU,DISC(4),ENTRADA1,ENTRADA2; 2 REAL PROF1(4)=(2,1.5,1,0.5); 3 REAL PROF2(4)=(1.5,2,3,3.5); 4 REAL TR1,TR2; 5 CLASS C1,C2; 6 INTEGER I; 7 /STATION/ NAME=CPU; 8 SCHED=PS; 9 SERVICE(C1)=CST(8.52); 10 SERVICE(C2)=CST(12.); 11 TRANSIT(C1)=DISC,PROF1,OUT,1; 12 TRANSIT(C2)=DISC,PROF2,OUT,1;



The BCMP theorem: Transactional system

13 /STATION/ NAME=DISC; 14 TRANSIT=CPU; 15 /STATION/ NAME=DISC(1); 16 SERVICE=EXP(23.); 17 /STATION/ NAME=DISC(2); 18 SERVICE=EXP(22.); 19 /STATION/ NAME=DISC(3); 20 SERVICE=EXP(21.); 21 /STATION/ NAME=DISC(4); 22 SERVICE=EXP(20.);



The BCMP theorem: Transactional system 23 /STATION/ NAME=ENTRADA1; 24 TYPE=SOURCE; 25 SERVICE=EXP(1000./7.); 26 TRANSIT=CPU,C1; 27 /STATION/ NAME=ENTRADA2; 28 TYPE=SOURCE; 29 SERVICE=EXP(1000./3.); 30 TRANSIT=CPU,C2; 31 /CONTROL/ CLASS=ALL QUEUE;



The BCMP theorem: Transactional system 32 /EXEC/ BEGIN 33 PRINT; 34 SOLVE; 35 TR1:=MCUSTNB(CPU,C1); 36 TR2:=MCUSTNB(CPU,C2); 37 FOR I:= 1 STEP 1 UNTIL 4 DO 38 BEGIN 39 TR1:=TR1+MCUSTNB(DISC(I),C1); 40 TR2:=TR2+MCUSTNB(DISC(I),C2); 41 END; 42 TR1:=TR1/MTHRUPUT(ENTRADA1); 43 TR2:=TR2/MTHRUPUT(ENTRADA2); 44 PRINT("RESPONSE TIME OF CLASS C1 =",TR1); 45 PRINT("RESPONSE TIME OF CLASS C2 =",TR2); 46 END;



The BCMP theorem: Transactional system - CONVOLUTION METHOD ("CONVOL") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.05 *0.7538 * 3.062 * 40.83 *0.7500E-01* *(C1 )* 8.520 *0.3578 * 1.454 * 34.61 *0.4200E-01* *(C2 )* 12.00 *0.3960 * 1.609 * 48.75 *0.3300E-01* * * * * * * * * DISC 1 * 23.00 *0.4255 *0.7406 * 40.03 *0.1850E-01* *(C1 )* 23.00 *0.3220 *0.5605 * 40.03 *0.1400E-01* *(C2 )* 23.00 *0.1035 *0.1802 * 40.03 *0.4500E-02* * * * * * * * * DISC 2 * 22.00 *0.3630 *0.5699 * 34.54 *0.1650E-01* *(C1 )* 22.00 *0.2310 *0.3626 * 34.54 *0.1050E-01* *(C2 )* 22.00 *0.1320 *0.2072 * 34.54 *0.6000E-02* * * * * * * *



The BCMP theorem: Transactional system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.3360 *0.5060 * 31.63 *0.1600E-01* *(C1 )* 21.00 *0.1470 *0.2214 * 31.63 *0.7000E-02* *(C2 )* 21.00 *0.1890 *0.2846 * 31.63 *0.9000E-02* * * * * * * * * DISC 4 * 20.00 *0.2800 *0.3889 * 27.78 *0.1400E-01* *(C1 )* 20.00 *0.7000E-01*0.9722E-01* 27.78 *0.3500E-02* *(C2 )* 20.00 *0.2100 *0.2917 * 27.78 *0.1050E-01* * * * * * * * * ENTRADA1 * 142.9 * 1.000 * 1.000 * 142.9 *0.7000E-02* * * * * * * * * ENTRADA2 * 333.3 * 1.000 * 1.000 * 333.3 *0.3000E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 385.1 RESPONSE TIME OF CLASS C2 = 857.5 47 /END/




Conversational system We assume that these programs are executed

concurrently on the computer system and that the

conflicts in the its execution are due to the access to

the same servers (CPU and disks) but not to any

kind of synchronisation or use of critical objects.

Also we assume that the human behaviour in front

of the terminal is different for each class.



The BCMP theorem: Case studiesConversational system

C P U

D isk 2

D isk 3

D isk 4

D isk 1

A rriv a ls

E x its

T e rm in a ls



The BCMP theorem: Conversational system 1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL; 2 REAL PROB1(4)=(2,1.5,1,0.5); 3 REAL PROB2(4)=(1.5,2,3,3.5); 4 REAL TR1,TR2; 5 CLASS C1,C2; 6 INTEGER I,N; 7 /STATION/ NAME=CPU; 8 SCHED=PS; 9 SERVICE(C1)=CST(8.52);10 SERVICE(C2)=CST(12.);11 TRANSIT(C1)=DISC,PROB1,TERMINAL,C1,0.6,TERMINAL,C2,0 ==> .4;12 TRANSIT(C2)=DISC,PROB2,TERMINAL,C1,0.6,TERMINAL,C2,0 ==> .4;



The BCMP theorem: Conversational system 13 /STATION/ NAME=DISC; 14 TRANSIT=CPU; 15 /STATION/ NAME=DISC(1); 16 SERVICE=EXP(23.); 17 /STATION/ NAME=DISC(2); 18 SERVICE=EXP(22.); 19 /STATION/ NAME=DISC(3); 20 SERVICE=EXP(21.); 21 /STATION/ NAME=DISC(4); 22 SERVICE=EXP(20.); 23 /STATION/ NAME=TERMINAL; 24 TYPE=INFINITE; 25 INIT(C1)=N; 26 SERVICE(C1)=EXP(30000.); 27 SERVICE(C2)=EXP(60000.); 28 TRANSIT=CPU; 29 /CONTROL/ CLASS=ALL QUEUE;



The BCMP theorem: Conversational system 30 /EXEC/ FOR N:=150 STEP 150 UNTIL 750 DO 31 BEGIN 32 PRINT; 33 PRINT("NOMBRE D’USUARIS =",N); 34 SOLVE; 35 TR1:=MCUSTNB(CPU,C1); 36 TR2:=MCUSTNB(CPU,C2); 37 FOR I:= 1 STEP 1 UNTIL 4 DO 38 BEGIN 39 TR1:=TR1+MCUSTNB(DISC(I),C1); 40 TR2:=TR2+MCUSTNB(DISC(I),C2); 41 END; 42 TR1:=TR1/MTHRUPUT(TERMINAL,C1); 43 TR2:=TR2/MTHRUPUT(TERMINAL,C2);; 44 PRINT("RESPONSE TIME OF CLASS C1 =",TR1); 45 PRINT("RESPONSE TIME OF CLASS C2 =",TR2); 46 END;



The BCMP theorem: Conversational system NOMBRE D’USUARIS = 150 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 *0.2960 *0.4189 * 14.76 *0.2837E-01* *(C1 )* 8.520 *0.1088 *0.1539 * 12.06 *0.1277E-01* *(C2 )* 12.00 *0.1873 *0.2650 * 16.98 *0.1560E-01* * * * * * * * * DISC 1 * 23.00 *0.1468 *0.1719 * 26.92 *0.6384E-02* *(C1 )* 23.00 *0.9790E-01*0.1146 * 26.92 *0.4257E-02* *(C2 )* 23.00 *0.4894E-01*0.5729E-01* 26.92 *0.2128E-02* * * * * * * * * DISC 2 * 22.00 *0.1326 *0.1528 * 25.33 *0.6029E-02* *(C1 )* 22.00 *0.7023E-01*0.8088E-01* 25.33 *0.3192E-02* *(C2 )* 22.00 *0.6242E-01*0.7188E-01* 25.33 *0.2837E-02*



The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.1341 *0.1546 * 24.22 *0.6384E-02* *(C1 )* 21.00 *0.4469E-01*0.5155E-01* 24.22 *0.2128E-02* *(C2 )* 21.00 *0.8937E-01*0.1031 * 24.22 *0.4256E-02* * * * * * * * * DISC 4 * 20.00 *0.1206 *0.1370 * 22.72 *0.6029E-02* *(C1 )* 20.00 *0.2128E-01*0.2418E-01* 22.72 *0.1064E-02* *(C2 )* 20.00 *0.9930E-01*0.1128 * 22.72 *0.4965E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 149.0 *0.4200E+05*0.3547E-02* *(C1 )*0.3000E+05*0.0000E+00* 63.84 *0.3000E+05*0.2128E-02* *(C2 )*0.6000E+05*0.0000E+00* 85.12 *0.6000E+05*0.1419E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 199.8 RESPONSE TIME OF CLASS C2 = 430.0



The BCMP theorem: Conversational system NOMBRE D’USUARIS = 300 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 *0.5905 * 1.426 * 25.19 *0.5659E-01* *(C1 )* 8.520 *0.2170 *0.5239 * 20.57 *0.2547E-01* *(C2 )* 12.00 *0.3735 *0.9017 * 28.97 *0.3112E-01* * * * * * * * * DISC 1 * 23.00 *0.2929 *0.4134 * 32.46 *0.1273E-01* *(C1 )* 23.00 *0.1953 *0.2756 * 32.46 *0.8490E-02* *(C2 )* 23.00 *0.9761E-01*0.1378 * 32.46 *0.4244E-02* * * * * * * * * DISC 2 * 22.00 *0.2646 *0.3592 * 29.87 *0.1203E-01* *(C1 )* 22.00 *0.1401 *0.1902 * 29.87 *0.6367E-02* *(C2 )* 22.00 *0.1245 *0.1690 * 29.87 *0.5659E-02*



The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.2674 *0.3644 * 28.62 *0.1273E-01* *(C1 )* 21.00 *0.8914E-01*0.1215 * 28.62 *0.4245E-02* *(C2 )* 21.00 *0.1783 *0.2429 * 28.62 *0.8488E-02* * * * * * * * * DISC 4 * 20.00 *0.2405 *0.3162 * 26.30 *0.1203E-01* *(C1 )* 20.00 *0.4245E-01*0.5582E-01* 26.30 *0.2122E-02* *(C2 )* 20.00 *0.1981 *0.2604 * 26.30 *0.9903E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 297.1 *0.4200E+05*0.7074E-02* *(C1 )*0.3000E+05*0.0000E+00* 127.3 *0.3000E+05*0.4245E-02* *(C2 )*0.6000E+05*0.0000E+00* 169.8 *0.6000E+05*0.2830E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 274.9 RESPONSE TIME OF CLASS C2 = 605.0



The BCMP theorem: Conversational system NOMBRE D’USUARIS = 450 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 *0.8763 * 6.438 * 76.66 *0.8399E-01* *(C1 )* 8.520 *0.3220 * 2.366 * 62.60 *0.3780E-01* *(C2 )* 12.00 *0.5543 * 4.072 * 88.16 *0.4619E-01* * * * * * * * * DISC 1 * 23.00 *0.4347 *0.7667 * 40.57 *0.1890E-01* *(C1 )* 23.00 *0.2898 *0.5112 * 40.57 *0.1260E-01* *(C2 )* 23.00 *0.1449 *0.2555 * 40.57 *0.6298E-02* * * * * * * * * DISC 2 * 22.00 *0.3926 *0.6451 * 36.14 *0.1785E-01* *(C1 )* 22.00 *0.2079 *0.3415 * 36.14 *0.9449E-02*

*(C2 )* 22.00 *0.1848 *0.3035 * 36.14 *0.8398E-02*



The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.3968 *0.6564 * 34.74 *0.1890E-01* *(C1 )* 21.00 *0.1323 *0.2188 * 34.74 *0.6300E-02* *(C2 )* 21.00 *0.2645 *0.4376 * 34.74 *0.1260E-01* * * * * * * * * DISC 4 * 20.00 *0.3569 *0.5541 * 31.05 *0.1785E-01* *(C1 )* 20.00 *0.6300E-01*0.9779E-01* 31.05 *0.3150E-02* *(C2 )* 20.00 *0.2939 *0.4563 * 31.05 *0.1470E-01* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 440.9 *0.4200E+05*0.1050E-01* *(C1 )*0.3000E+05*0.0000E+00* 189.0 *0.3000E+05*0.6299E-02* *(C2 )*0.6000E+05*0.0000E+00* 252.0 *0.6000E+05*0.4199E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 561.2 RESPONSE TIME OF CLASS C2 = 1316.



The BCMP theorem: Conversational system NOMBRE D’USUARIS = 600 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 * 1.000 * 93.51 * 975.7 *0.9584E-01* *(C1 )* 8.520 *0.3675 * 34.37 * 796.7 *0.4313E-01* *(C2 )* 12.00 *0.6325 * 59.15 * 1122. *0.5271E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9841 * 45.63 *0.2157E-01* *(C1 )* 23.00 *0.3307 *0.6561 * 45.63 *0.1438E-01* *(C2 )* 23.00 *0.1653 *0.3280 * 45.63 *0.7187E-02* * * * * * * * * DISC 2 * 22.00 *0.4481 *0.8118 * 39.86 *0.2037E-01* *(C1 )* 22.00 *0.2372 *0.4298 * 39.86 *0.1078E-01*

*(C2 )* 22.00 *0.2108 *0.3820 * 39.86 *0.9583E-02*



The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.4528 *0.8276 * 38.38 *0.2156E-01* *(C1 )* 21.00 *0.1510 *0.2759 * 38.38 *0.7189E-02* *(C2 )* 21.00 *0.3019 *0.5517 * 38.38 *0.1437E-01* * * * * * * * * DISC 4 * 20.00 *0.4073 *0.6872 * 33.74 *0.2037E-01* *(C1 )* 20.00 *0.7189E-01*0.1213 * 33.74 *0.3594E-02* *(C2 )* 20.00 *0.3354 *0.5659 * 33.74 *0.1677E-01* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 4987. RESPONSE TIME OF CLASS C2 = 0.1272E+05



The BCMP theorem: Conversational system NOMBRE D’USUARIS = 750 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 * 1.000 * 243.5 * 2541. *0.9584E-01* *(C1 )* 8.520 *0.3675 * 89.49 * 2075. *0.4313E-01* *(C2 )* 12.00 *0.6325 * 154.0 * 2922. *0.5271E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9841 * 45.64 *0.2157E-01* *(C1 )* 23.00 *0.3307 *0.6561 * 45.64 *0.1438E-01* *(C2 )* 23.00 *0.1653 *0.3280 * 45.64 *0.7187E-02* * * * * * * * * DISC 2 * 22.00 *0.4481 *0.8118 * 39.86 *0.2037E-01* *(C1 )* 22.00 *0.2372 *0.4298 * 39.86 *0.1078E-01*

*(C2 )* 22.00 *0.2108 *0.3820 * 39.86 *0.9583E-02*



The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.4528 *0.8276 * 38.38 *0.2156E-01* *(C1 )* 21.00 *0.1510 *0.2759 * 38.38 *0.7189E-02* *(C2 )* 21.00 *0.3019 *0.5517 * 38.38 *0.1437E-01* * * * * * * * * DISC 4 * 20.00 *0.4073 *0.6872 * 33.74 *0.2037E-01* *(C1 )* 20.00 *0.7189E-01*0.1213 * 33.74 *0.3594E-02* *(C2 )* 20.00 *0.3354 *0.5659 * 33.74 *0.1677E-01* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 0.1266E+05 RESPONSE TIME OF CLASS C2 = 0.3252E+05 47 /END/




Communication network There are four sources of variable length messages.

Routing across the net is fixed. All the lines have the same capacity and the transmission is full duplex. We consider negligible the time spent in each node for protocol verifications, routing, etc. The end-to-end traffic is known and we want to determine the response time also end-to-end.



The BCMP theorem: Communication network

1

2

3

4

5

6

7

8

A

B

C

D




to

from A B C D

A 60 80 100

B 75 50 25

C 80 120 40

D 100 150 50



The BCMP theorem: Communication networkA-1-2-4-3-B

A-1-2-4-6-5-C

A-1-2-8-7-D

B-3-4-2-1-A

B-3-4-6-5-C

B-3-4-8-7-D

C-5-6-8-2-1-A

C-5-6-4-3-B

C-5-6-8-7-D

D-7-8-2-1-A

D-7-8-4-3-B

D-7-8-6-5-C




1 /DECLARE/ QUEUE GEN,LINIA(8,8); 2 REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,==> TRDA,TRDB,TRDC; 3 INTEGER I; 4 CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD==> ,CLDA,CLDB,CLDC; 5



The BCMP theorem: Communication network 6 /STATION/ NAME = GEN; 7 TYPE = SOURCE; 8 SERVICE = EXP(60000./930.); 9 TRANSIT = LINIA(1,2),CLAB,60,LINIA(1,2),CLAC,80,L==> INIA(1,2),CLAD,100,10 LINIA(3,4),CLBA,75,LINIA(3,4),CLBC,50,L==> INIA(3,4),CLBD,25,11 LINIA(5,6),CLCA,80,LINIA(5,6),CLCB,120,==> LINIA(5,6),CLCD,40,12 LINIA(7,8),CLDA,100,LINIA(7,8),CLDB,150==> ,LINIA(7,8),CLDC,50;13



The BCMP theorem: Communication network 14 /STATION/ NAME = LINIA; 15 SERVICE = EXP(256.*8./64.); 16 17 /STATION/ NAME = LINIA(1,2); 18 TRANSIT(CLAB) = LINIA(2,4); 19 TRANSIT(CLAC) = LINIA(2,4); 20 TRANSIT(CLAD) = LINIA(2,8); 21 22 /STATION/ NAME = LINIA(2,1); 23 TRANSIT = OUT; 24 25 /STATION/ NAME = LINIA(2,4); 26 TRANSIT(CLAB) = LINIA(4,3); 27 TRANSIT(CLAC) = LINIA(4,6); 28 29 /STATION/ NAME = LINIA(2,8); 30 TRANSIT(CLAD) = LINIA(8,7); 31



The BCMP theorem: Communication network 32 /STATION/ NAME = LINIA(3,4); 33 TRANSIT(CLBA) = LINIA(4,2); 34 TRANSIT(CLBC) = LINIA(4,6); 35 TRANSIT(CLBD) = LINIA(4,8); 36 37 /STATION/ NAME = LINIA(4,2); 38 TRANSIT(CLBA) = LINIA(2,1); 39 40 /STATION/ NAME = LINIA(4,3); 41 TRANSIT = OUT; 42 43 /STATION/ NAME = LINIA(4,6); 44 TRANSIT(CLAC) = LINIA(6,5); 45 TRANSIT(CLBC) = LINIA(6,5); 46 47 /STATION/ NAME = LINIA(4,8); 48 TRANSIT(CLBD) = LINIA(8,7); 49


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 50 /STATION/ NAME = LINIA(5,6); 51 TRANSIT(CLCA) = LINIA(6,8); 52 TRANSIT(CLCB) = LINIA(6,4); 53 TRANSIT(CLCD) = LINIA(6,8); 54 55 /STATION/ NAME = LINIA(6,4); 56 TRANSIT(CLCB) = LINIA(4,3); 57 58 /STATION/ NAME = LINIA(6,5); 59 TRANSIT = OUT; 60 61 /STATION/ NAME = LINIA(6,8); 62 TRANSIT(CLCA) = LINIA(8,2); 63 TRANSIT(CLCD) = LINIA(8,7); 64 65 /STATION/ NAME = LINIA(7,8); 66 TRANSIT(CLDA) = LINIA(8,2); 67 TRANSIT(CLDB) = LINIA(8,4); 68 TRANSIT(CLDC) = LINIA(8,6); 69



The BCMP theorem: Communication network 70 /STATION/ NAME = LINIA(8,2); 71 TRANSIT(CLCA) = LINIA(2,1); 72 TRANSIT(CLDA) = LINIA(2,1); 73 74 /STATION/ NAME = LINIA(8,4); 75 TRANSIT(CLDB) = LINIA(4,3); 76 77 /STATION/ NAME = LINIA(8,6); 78 TRANSIT(CLDC) = LINIA(6,5); 79 80 /STATION/ NAME = LINIA(8,7); 81 TRANSIT = OUT; 82



The BCMP theorem: Communication network83 /EXEC/ BEGIN84 NETWORK(GEN,85 LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8),86 LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6),87 LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5),88 LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4),89 LINIA(8,6),LINIA(8,7));90 PRINT;91 SOLVE;92 TRAB:=MRESPONSE(LINIA(1,2))+93 MRESPONSE(LINIA(2,4))+MRESPONSE(LINIA(4,3));94 TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+95 MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5));96 TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+97 MRESPONSE(LINIA(8,7));98 TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+99 MRESPONSE(LINIA(2,1));



The BCMP theorem: Communication network 100 TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+101 MRESPONSE(LINIA(6,5));102 TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+103 MRESPONSE(LINIA(8,7));104 TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+105 MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1));106 TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+107 MRESPONSE(LINIA(4,3));108 TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+109 MRESPONSE(LINIA(8,7));110 TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+111 MRESPONSE(LINIA(2,1));112 TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+113 MRESPONSE(LINIA(4,3));114 TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+115 MRESPONSE(LINIA(6,5));



The BCMP theorem: Communication network 116 PRINT(TRAB,TRAC,TRAD); 117 PRINT(TRBA,TRBC,TRBD); 118 PRINT(TRCA,TRCB,TRCD); 119 PRINT(TRDA,TRDB,TRDC); 120 END;



The BCMP theorem: Communication network - CONVOLUTION METHOD ("CONVOL") -******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* * * * * * ** GEN * 64.52 * 1.000 * 1.000 * 64.52 *0.1550E-01** * * * * * ** LINIA 2 * 32.00 *0.1280 *0.1468 * 36.70 *0.4000E-02** * * * * * ** LINIA 9 * 32.00 *0.1360 *0.1574 * 37.04 *0.4250E-02** * * * * * ** LINIA 12 * 32.00 *0.7467E-01*0.8069E-01* 34.58 *0.2333E-02** * * * * * ** LINIA 16 * 32.00 *0.5333E-01*0.5634E-01* 33.80 *0.1667E-02** * * * * * ** LINIA 20 * 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02** * * * * * ** LINIA 26 * 32.00 *0.4000E-01*0.4167E-01* 33.33 *0.1250E-02** * * * * * *


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* LINIA 27 * 32.00 *0.1760 *0.2136 * 38.83 *0.5500E-02** * * * * * ** LINIA 30 * 32.00 *0.6933E-01*0.7450E-01* 34.38 *0.2167E-02** * * * * * ** LINIA 32 * 32.00 *0.1333E-01*0.1351E-01* 32.43 *0.4167E-03** * * * * * ** LINIA 38 * 32.00 *0.1280 *0.1468 * 36.70 *0.4000E-02** * * * * * ** LINIA 44 * 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02** * * * * * ** LINIA 45 * 32.00 *0.9600E-01*0.1062 * 35.40 *0.3000E-02** * * * * * ** LINIA 48 * 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02** * * * * * ** LINIA 56 * 32.00 *0.1600 *0.1905 * 38.10 *0.5000E-02** * * * * * ** LINIA 58 * 32.00 *0.9600E-01*0.1062 * 35.40 *0.3000E-02** * * * * * *




******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* LINIA 60 * 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02** * * * * * ** LINIA 62 * 32.00 *0.2667E-01*0.2740E-01* 32.88 *0.8333E-03** * * * * * ** LINIA 63 * 32.00 *0.8800E-01*0.9649E-01* 35.09 *0.2750E-02** * * * * * ********************************************************************

110.1 141.1 105.6 105.2 104.6 102.3 143.3 109.7 106.0 110.5 111.7 106.4




Communication network considering the nodes This case is identical to the previous one but

considering the process at each node.



The BCMP theorem: Communication network 1 /DECLARE/ QUEUE GEN,LINIA(8,8),CPU(8); 2 REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,TRD ==> A,TRDB,TRDC; 3 INTEGER I; 4 CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD,CL ==> DA,CLDB,CLDC; 5 6 /STATION/ NAME = GEN; 7 TYPE = SOURCE; 8 SERVICE = EXP(60000./930.); 9 TRANSIT = CPU(1),CLAB,60,CPU(1),CLAC,80,CPU(1),CLAD,1 ==> 00, 10 CPU(3),CLBA,75,CPU(3),CLBC,50,CPU(3),CLBD,2 ==> 5, 11 CPU(5),CLCA,80,CPU(5),CLCB,120,CPU(5),CLCD, ==> 40, 12 CPU(7),CLDA,100,CPU(7),CLDB,150,CPU(7),CLDC ==> ,50,

13



The BCMP theorem: Communication network 14 /STATION/ NAME = LINIA; 15 SERVICE = EXP(256.*8./64.); 16 17 /STATION/ NAME = LINIA(1,2); 18 TRANSIT = CPU(2); 19 20 /STATION/ NAME = LINIA(2,1); 21 TRANSIT = CPU(1); 22 23 /STATION/ NAME = LINIA(2,4); 24 TRANSIT = CPU(4); 25 26 /STATION/ NAME = LINIA(2,8); 27 TRANSIT = CPU(8); 28 29 /STATION/ NAME = LINIA(3,4); 30 TRANSIT = CPU(4); 31



The BCMP theorem: Communication network 32 /STATION/ NAME = LINIA(4,2); 33 TRANSIT = CPU(2); 34 35 /STATION/ NAME = LINIA(4,3); 36 TRANSIT = CPU(3); 37 38 /STATION/ NAME = LINIA(4,6); 39 TRANSIT = CPU(6); 40 41 /STATION/ NAME = LINIA(4,8); 42 TRANSIT = CPU(8); 43 44 /STATION/ NAME = LINIA(5,6); 45 TRANSIT = CPU(6); 46 47 /STATION/ NAME = LINIA(6,5); 48 TRANSIT = CPU(5); 49



The BCMP theorem: Communication network 50 /STATION/ NAME = LINIA(6,4); 51 TRANSIT = CPU(4); 52 53 /STATION/ NAME = LINIA(6,8); 54 TRANSIT = CPU(8); 55 56 /STATION/ NAME = LINIA(7,8); 57 TRANSIT = CPU(8); 58 59 /STATION/ NAME = LINIA(8,2); 60 TRANSIT = CPU(2); 61 62 /STATION/ NAME = LINIA(8,7); 63 TRANSIT = CPU(7); 64 65 /STATION/ NAME = LINIA(8,6); 66 TRANSIT = CPU(6); 67



The BCMP theorem: Communication network 68 /STATION/ NAME = LINIA(8,4); 69 TRANSIT = CPU(4); 70 71 /STATION/ NAME = CPU; 72 SERVICE = EXP(1.); 73 74 /STATION/ NAME = CPU(1); 75 TRANSIT(CLAB,CLAC,CLAD) = LINIA(1,2); 76 TRANSIT(CLBA,CLCA,CLDA) = OUT; 77 78 /STATION/ NAME = CPU(2); 79 TRANSIT(CLAB) = LINIA(2,4); 80 TRANSIT(CLAC) = LINIA(2,4); 81 TRANSIT(CLAD) = LINIA(2,8); 82 TRANSIT(CLBA) = LINIA(2,1); 83 TRANSIT(CLCA) = LINIA(2,1); 84 TRANSIT(CLDA) = LINIA(2,1);

85




86 /STATION/ NAME = CPU(3); 87 TRANSIT(CLBA,CLBC,CLBD) = LINIA(3,4); 88 TRANSIT(CLAB,CLCB,CLDB) = OUT; 89 90 /STATION/ NAME = CPU(4); 91 TRANSIT(CLAB) = LINIA(4,3); 92 TRANSIT(CLAC) = LINIA(4,6); 93 TRANSIT(CLBA) = LINIA(4,2); 94 TRANSIT(CLBC) = LINIA(4,6); 95 TRANSIT(CLBD) = LINIA(4,8); 96 TRANSIT(CLCB) = LINIA(4,3); 97 TRANSIT(CLDB) = LINIA(4,3); 98 99 /STATION/ NAME = CPU(5); 100 TRANSIT(CLCA,CLCB,CLCD) = LINIA(5,6); 101 TRANSIT(CLAC,CLBC,CLDC) = OUT;

102


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 103 /STATION/ NAME = CPU(6); 104 TRANSIT(CLAC) = LINIA(6,5); 105 TRANSIT(CLBC) = LINIA(6,5); 106 TRANSIT(CLCA) = LINIA(6,8); 107 TRANSIT(CLCB) = LINIA(6,4); 108 TRANSIT(CLCD) = LINIA(6,8); 109 TRANSIT(CLDC) = LINIA(6,5); 110 111 /STATION/ NAME = CPU(7); 112 TRANSIT(CLDA,CLDB,CLDC) = LINIA(7,8); 113 TRANSIT(CLAD,CLBD,CLCD) = OUT; 114 115 /STATION/ NAME = CPU(8); 116 TRANSIT(CLAD) = LINIA(8,7); 117 TRANSIT(CLBD) = LINIA(8,7); 118 TRANSIT(CLCA) = LINIA(8,2); 119 TRANSIT(CLCD) = LINIA(8,7); 120 TRANSIT(CLDA) = LINIA(8,2); 121 TRANSIT(CLDB) = LINIA(8,4); 122 TRANSIT(CLDC) = LINIA(8,6); 123



The BCMP theorem: Communication network 124 /CONTROL/ CLASS = ALL QUEUE; 125 /EXEC/ BEGIN 126 NETWORK(GEN, 127 CPU(1 STEP 1 UNTIL 8), 128 LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8), 129 LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6), 130 LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5), 131 LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4), 132 LINIA(8,6),LINIA(8,7)); 133 PRINT; 134 SOLVE; 135 TRAB := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+ 136 MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,4))+ 137 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+ 138 MRESPONSE(CPU(3)); 139 TRAC := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+ 140 MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,4))+ 141 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,6))+ 142 MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,5))+ 143 MRESPONSE(CPU(5));


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 144 TRAD := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+ 145 MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,8))+ 146 MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+ 147 MRESPONSE(CPU(7)); 148 TRBA := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+ 149 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,2))+ 150 MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+ 151 MRESPONSE(CPU(1)); 152 TRBC := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+ 153 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,6))+ 154 MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,5))+ 155 MRESPONSE(CPU(5)); 156 TRBD := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+ 157 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,8))+ 158 MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+ 159 MRESPONSE(CPU(7)); 160 TRCA := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+ 161 MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,8))+ 162 MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,2))+ 163 MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+ 164 MRESPONSE(CPU(1));




165 TRCB := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+ 166 MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,4))+ 167 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+ 168 MRESPONSE(CPU(3)); 169 TRCD := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+ 170 MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,8))+ 171 MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+ 172 MRESPONSE(CPU(7)); 173 TRDA := MRESPONSE(CPU(7))+MRESPONSE(LINIA(7,8))+ 174 MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,2))+ 175 MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+ 176 MRESPONSE(CPU(1)); 177 TRDB := MRESPONSE(CPU(7))+MRESPONSE(LINIA(7,8))+ 178 MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,4))+ 179 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+

180 MRESPONSE(CPU(3));




181 TRDC := MRESPONSE(CPU(7))+MRESPONSE(LINIA(7,8))+

182 MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,6))+

183 MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,5))+

184 MRESPONSE(CPU(5)); 185 PRINT(TRAB,TRAC,TRAD); 186 PRINT(TRBA,TRBC,TRBD); 187 PRINT(TRCA,TRCB,TRCD); 188 PRINT(TRDA,TRDB,TRDC); 189 END;



The BCMP theorem: Communication network- CONVOLUTION METHOD ("CONVOL") -******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* GEN * 64.52 * 1.000 * 1.000 * 64.52 *0.1550E-01** * * * * * ** LINIA 2 * 32.00 *0.1280 *0.1468 * 36.70 *0.4000E-02**(CLAB )* 32.00 *0.3200E-01*0.3670E-01* 36.70 *0.1000E-02**(CLAC )* 32.00 *0.4267E-01*0.4893E-01* 36.70 *0.1333E-02**(CLAD )* 32.00 *0.5333E-01*0.6116E-01* 36.70 *0.1667E-02** * * * * * ** LINIA 9 * 32.00 *0.1360 *0.1574 * 37.04 *0.4250E-02**(CLBA )* 32.00 *0.4000E-01*0.4630E-01* 37.04 *0.1250E-02**(CLCA )* 32.00 *0.4267E-01*0.4938E-01* 37.04 *0.1333E-02**(CLDA )* 32.00 *0.5333E-01*0.6173E-01* 37.04 *0.1667E-02** * * * * * ** LINIA 12 * 32.00 *0.7467E-01*0.8069E-01* 34.58 *0.2333E-02**(CLAB )* 32.00 *0.3200E-01*0.3458E-01* 34.58 *0.1000E-02**(CLAC )* 32.00 *0.4267E-01*0.4611E-01* 34.58 *0.1333E-02** * * * * * *


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* LINIA 16 * 32.00 *0.5333E-01*0.5634E-01* 33.80 *0.1667E-02**(CLAD )* 32.00 *0.5333E-01*0.5634E-01* 33.80 *0.1667E-02** * * * * * ** LINIA 20 * 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02**(CLBA )* 32.00 *0.4000E-01*0.4348E-01* 34.78 *0.1250E-02**(CLBC )* 32.00 *0.2667E-01*0.2899E-01* 34.78 *0.8333E-03**(CLBD )* 32.00 *0.1333E-01*0.1449E-01* 34.78 *0.4167E-03** * * * * * ** LINIA 26 * 32.00 *0.4000E-01*0.4167E-01* 33.33 *0.1250E-02**(CLBA )* 32.00 *0.4000E-01*0.4167E-01* 33.33 *0.1250E-02** * * * * * ** LINIA 27 * 32.00 *0.1760 *0.2136 * 38.83 *0.5500E-02**(CLAB )* 32.00 *0.3200E-01*0.3883E-01* 38.83 *0.1000E-02**(CLCB )* 32.00 *0.6400E-01*0.7767E-01* 38.83 *0.2000E-02**(CLDB )* 32.00 *0.8000E-01*0.9709E-01* 38.83 *0.2500E-02** * * * * * *



The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* LINIA 30 * 32.00 *0.6933E-01*0.7450E-01* 34.38 *0.2167E-02**(CLAC )* 32.00 *0.4267E-01*0.4585E-01* 34.38 *0.1333E-02**(CLBC )* 32.00 *0.2667E-01*0.2865E-01* 34.38 *0.8333E-03** * * * * * ** LINIA 32 * 32.00 *0.1333E-01*0.1351E-01* 32.43 *0.4167E-03**(CLBD )* 32.00 *0.1333E-01*0.1351E-01* 32.43 *0.4167E-03** * * * * * ** LINIA 38 * 32.00 *0.1280 *0.1468 * 36.70 *0.4000E-02**(CLCA )* 32.00 *0.4267E-01*0.4893E-01* 36.70 *0.1333E-02**(CLCB )* 32.00 *0.6400E-01*0.7339E-01* 36.70 *0.2000E-02**(CLCD )* 32.00 *0.2133E-01*0.2446E-01* 36.70 *0.6667E-03** * * * * * ** LINIA 44 * 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02**(CLCB )* 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02** * * * * * *



The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* LINIA 45 * 32.00 *0.9600E-01*0.1062 * 35.40 *0.3000E-02**(CLAC )* 32.00 *0.4267E-01*0.4720E-01* 35.40 *0.1333E-02**(CLBC )* 32.00 *0.2667E-01*0.2950E-01* 35.40 *0.8333E-03**(CLDC )* 32.00 *0.2667E-01*0.2950E-01* 35.40 *0.8333E-03** * * * * * ** LINIA 48 * 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02**(CLCA )* 32.00 *0.4267E-01*0.4558E-01* 34.19 *0.1333E-02**(CLCD )* 32.00 *0.2133E-01*0.2279E-01* 34.19 *0.6667E-03** * * * * * ** LINIA 56 * 32.00 *0.1600 *0.1905 * 38.10 *0.5000E-02**(CLDA )* 32.00 *0.5333E-01*0.6349E-01* 38.10 *0.1667E-02**(CLDB )* 32.00 *0.8000E-01*0.9524E-01* 38.10 *0.2500E-02**(CLDC )* 32.00 *0.2667E-01*0.3175E-01* 38.10 *0.8333E-03** * * * * * ** LINIA 58 * 32.00 *0.9600E-01*0.1062 * 35.40 *0.3000E-02**(CLCA )* 32.00 *0.4267E-01*0.4720E-01* 35.40 *0.1333E-02**(CLDA )* 32.00 *0.5333E-01*0.5900E-01* 35.40 *0.1667E-02** * * * * * *


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* LINIA 60 * 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02**(CLDB )* 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02** * * * * * ** LINIA 62 * 32.00 *0.2667E-01*0.2740E-01* 32.88 *0.8333E-03**(CLDC )* 32.00 *0.2667E-01*0.2740E-01* 32.88 *0.8333E-03** * * * * * ** LINIA 63 * 32.00 *0.8800E-01*0.9649E-01* 35.09 *0.2750E-02**(CLAD )* 32.00 *0.5333E-01*0.5848E-01* 35.09 *0.1667E-02**(CLBD )* 32.00 *0.1333E-01*0.1462E-01* 35.09 *0.4167E-03**(CLCD )* 32.00 *0.2133E-01*0.2339E-01* 35.09 *0.6667E-03** * * * * * ** CPU 1 * 1.000 *0.8250E-02*0.8319E-02* 1.008 *0.8250E-02**(CLAB )* 1.000 *0.1000E-02*0.1008E-02* 1.008 *0.1000E-02**(CLAC )* 1.000 *0.1333E-02*0.1344E-02* 1.008 *0.1333E-02**(CLAD )* 1.000 *0.1667E-02*0.1681E-02* 1.008 *0.1667E-02**(CLBA )* 1.000 *0.1250E-02*0.1260E-02* 1.008 *0.1250E-02**(CLCA )* 1.000 *0.1333E-02*0.1344E-02* 1.008 *0.1333E-02**(CLDA )* 1.000 *0.1667E-02*0.1681E-02* 1.008 *0.1667E-02*


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* CPU 2 * 1.000 *0.8250E-02*0.8319E-02* 1.008 *0.8250E-02**(CLAB )* 1.000 *0.1000E-02*0.1008E-02* 1.008 *0.1000E-02**(CLAC )* 1.000 *0.1333E-02*0.1344E-02* 1.008 *0.1333E-02**(CLAD )* 1.000 *0.1667E-02*0.1681E-02* 1.008 *0.1667E-02**(CLBA )* 1.000 *0.1250E-02*0.1260E-02* 1.008 *0.1250E-02**(CLCA )* 1.000 *0.1333E-02*0.1344E-02* 1.008 *0.1333E-02**(CLDA )* 1.000 *0.1667E-02*0.1681E-02* 1.008 *0.1667E-02** * * * * * ** * * * * * ** CPU 3 * 1.000 *0.8000E-02*0.8065E-02* 1.008 *0.8000E-02**(CLAB )* 1.000 *0.1000E-02*0.1008E-02* 1.008 *0.1000E-02**(CLBA )* 1.000 *0.1250E-02*0.1260E-02* 1.008 *0.1250E-02**(CLBC )* 1.000 *0.8333E-03*0.8401E-03* 1.008 *0.8333E-03**(CLBD )* 1.000 *0.4167E-03*0.4200E-03* 1.008 *0.4167E-03**(CLCB )* 1.000 *0.2000E-02*0.2016E-02* 1.008 *0.2000E-02**(CLDB )* 1.000 *0.2500E-02*0.2520E-02* 1.008 *0.2500E-02*


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* CPU 4 * 1.000 *0.9333E-02*0.9421E-02* 1.009 *0.9333E-02**(CLAB )* 1.000 *0.1000E-02*0.1009E-02* 1.009 *0.1000E-02**(CLAC )* 1.000 *0.1333E-02*0.1346E-02* 1.009 *0.1333E-02**(CLBA )* 1.000 *0.1250E-02*0.1262E-02* 1.009 *0.1250E-02**(CLBC )* 1.000 *0.8333E-03*0.8412E-03* 1.009 *0.8333E-03**(CLBD )* 1.000 *0.4167E-03*0.4206E-03* 1.009 *0.4167E-03**(CLCB )* 1.000 *0.2000E-02*0.2019E-02* 1.009 *0.2000E-02**(CLDB )* 1.000 *0.2500E-02*0.2524E-02* 1.009 *0.2500E-02** * * * * * ** CPU 5 * 1.000 *0.7000E-02*0.7049E-02* 1.007 *0.7000E-02**(CLAC )* 1.000 *0.1333E-02*0.1343E-02* 1.007 *0.1333E-02**(CLBC )* 1.000 *0.8333E-03*0.8392E-03* 1.007 *0.8333E-03**(CLCA )* 1.000 *0.1333E-02*0.1343E-02* 1.007 *0.1333E-02**(CLCB )* 1.000 *0.2000E-02*0.2014E-02* 1.007 *0.2000E-02**(CLCD )* 1.000 *0.6667E-03*0.6714E-03* 1.007 *0.6667E-03**(CLDC )* 1.000 *0.8333E-03*0.8392E-03* 1.007 *0.8333E-03*



The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* CPU 6 * 1.000 *0.7000E-02*0.7049E-02* 1.007 *0.7000E-02**(CLAC )* 1.000 *0.1333E-02*0.1343E-02* 1.007 *0.1333E-02**(CLBC )* 1.000 *0.8333E-03*0.8392E-03* 1.007 *0.8333E-03**(CLCA )* 1.000 *0.1333E-02*0.1343E-02* 1.007 *0.1333E-02**(CLCB )* 1.000 *0.2000E-02*0.2014E-02* 1.007 *0.2000E-02**(CLCD )* 1.000 *0.6667E-03*0.6714E-03* 1.007 *0.6667E-03**(CLDC )* 1.000 *0.8333E-03*0.8392E-03* 1.007 *0.8333E-03** * * * * * ** CPU 7 * 1.000 *0.7750E-02*0.7811E-02* 1.008 *0.7750E-02**(CLAD )* 1.000 *0.1667E-02*0.1680E-02* 1.008 *0.1667E-02**(CLBD )* 1.000 *0.4167E-03*0.4199E-03* 1.008 *0.4167E-03**(CLCD )* 1.000 *0.6667E-03*0.6719E-03* 1.008 *0.6667E-03**(CLDA )* 1.000 *0.1667E-02*0.1680E-02* 1.008 *0.1667E-02**(CLDB )* 1.000 *0.2500E-02*0.2520E-02* 1.008 *0.2500E-02**(CLDC )* 1.000 *0.8333E-03*0.8398E-03* 1.008 *0.8333E-03*


EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* CPU 8 * 1.000 *0.9083E-02*0.9167E-02* 1.009 *0.9083E-02**(CLAD )* 1.000 *0.1667E-02*0.1682E-02* 1.009 *0.1667E-02**(CLBD )* 1.000 *0.4167E-03*0.4205E-03* 1.009 *0.4167E-03**(CLCA )* 1.000 *0.1333E-02*0.1346E-02* 1.009 *0.1333E-02**(CLCD )* 1.000 *0.6667E-03*0.6728E-03* 1.009 *0.6667E-03**(CLDA )* 1.000 *0.1667E-02*0.1682E-02* 1.009 *0.1667E-02**(CLDB )* 1.000 *0.2500E-02*0.2523E-02* 1.009 *0.2500E-02**(CLDC )* 1.000 *0.8333E-03*0.8410E-03* 1.009 *0.8333E-03******************************************************************** 114.1 146.1 109.6 109.2 108.6 106.3 148.4 113.8 110.0 114.6 115.7 110.4

190 191 /END/


OUTLINE

INTRODUCTION

CONCEPT OF QUEUE








Approximate solutions for queuing networks are useful for tackling problems not covered by the product-form solutions as, for instance, priorities class dependent or non-exponential service times at

FIFO stations finite buffers simultaneous resource possession



There are a lot of different types of approximations in

the literature. Basically, they can be grouped in the

following classes of methods:

decomposition-aggregation

diffusion

iterative



Decomposition-aggregation methods The idea behind decomposition is to break up the queuing

network into smaller subsystems, so that each subsystem can be easily analysed in isolation, and then put together these partial solutions, in order to obtain the solution of the queuing network.

What is difficult to do is to find a good way to decompose the network under study into smaller more manageable subsystems and then put the solution together for the whole network.



Decomposition-aggregation methods

Various decomposition procedures have been developed

specifically for analysing queuing networks with particular

features.

Criteria for decomposing can be:

o Parts with different timing behaviour

o Norton theorem: inspired in the Norton theorem for electric circuits


APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem

BCMP network with a non-BCMP node

m1

m2

m3

m4



Decomposition-aggregation: Norton theorem

BCMP network with a non-BCMP node

o Let us assume that one node violate the BCMP assumptions. In our

particular case, node 3 is the culprit.

o Let us assume that we know all the parameters of the network, i. e. the

service times, routing probabilities, etc. Also, for simplicity, we

assume a single class of customers. Let N be the total number of

customers in it.



Decomposition-aggregation: Norton theorem Decomposition step

o "Short-out" node 3 and analyse the queuing network as if this node did not exist. By "short-out" we mean simply that node 3 is removed without changing the incoming and outgoing flows.

o Now, the resulting network is a BCMP network and it can be easily solved. We study this network to compute the throughput along the "short".

o We do this computation assuming that there are k customers in the "shorted" network, where k = 1, 2, ..., N. Let (k) be the throughput along the "short" when there are k customers in it.


APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem

m1

m2

m4

(k )




Aggregation step

o Now, the original network can be reduced to node 3 and another node,

known as the composite node, which represents approximately the

shorted-out network. The service rate of the composite node is (k),

where k is the number of customers in the composite node (state-

dependent service). The total number of customers in the network is

still N.




m3

(k )



Decomposition-aggregation: Norton theorem Aggregation step

o In general, this network is not a BCMP network because of node 3. So, we have to be able to analyse this aggregate system by either a numerical approach, or another approximate method or a simulation method. However, in any case as the system has only two nodes its analysis will be easier.

o Let us assume that we can obtain the queue length distribution of nodes 3 and composite. Then, the queue length distribution of node 3 is an approximation to the queue length distribution of node 3 in the original network.




o What about the queue length distributions of nodes 1, 2 and 4? They can be obtained by combining the results obtained in step 1 with the queue length distribution of the composite node obtained in step 2. To show how to obtain them, let us assume that we want to obtain the queue length of node 1.




o Let pc(k) be the probability that there are n customers in the composite node, where k = 1, 2, ..., N, and q1(n|k) be the probability that there are n customers in the node 1 when there are k customers in the shorted network. Then

o It is also possible to proceed in a similar way when instead of just one non-BCMP node we have a non-BCMP sub-network

N

k

c Nn,kpnqnq1

11 ..., 2, 1,for k|



Decomposition-aggregation: Multiprogramming system

D isk 1

D isk 2

D isk 3

D isk 4

C P U

T e rm in a ls

M e m o rym a n a g e m e n t




D isk 1

D isk 2

D isk 3

D isk 4

C P U



Decomposition-aggregation: Multiprogramming system 1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL,SC;

2 REAL PROB1(4)=(2,1.5,1,0.5);

3 REAL PROB2(4)=(1.5,2,3,3.5);

4 REAL TR,CAP(20);

5 REAL TR1,TR2;

6 CLASS C1,C2;

7 INTEGER I,N,M;

8 /STATION/ NAME=CPU;

9 SCHED=PS;

10 INIT(C1)=N;

11 SERVICE(C1)=CST(8.52);

12 SERVICE(C2)=CST(12.);

13 TRANSIT(C1)=DISC,PROB1,CPU,C1,0.6,CPU,C2,0.4;

14 TRANSIT(C2)=DISC,PROB2,CPU,C1,0.6,CPU,C2,0.4;




15 /STATION/ NAME=DISC; 16 TRANSIT=CPU; 17 /STATION/ NAME=DISC(1); 18 SERVICE=EXP(23.); 19 /STATION/ NAME=DISC(2); 20 SERVICE=EXP(22.); 21 /STATION/ NAME=DISC(3); 22 SERVICE=EXP(21.); 23 /STATION/ NAME=DISC(4); 24 SERVICE=EXP(20.);



Decomposition-aggregation: Multiprogramming system 25 /EXEC/ BEGIN 26 NETWORK(CPU,DISC); 27 FOR N := 1 STEP 1 UNTIL 20 DO 28 BEGIN 29 PRINT; 30 PRINT("FACTOR DE MULTIPROGRAMACIO =",N); 31 SOLVE; 32 CAP(N) := MTHRUPUT(CPU); 33 FOR I:=1 STEP 1 UNTIL 4 DO CAP(N):=CAP(N)-MTHRUP ==> UT(DISC(I)); 34 END; 35 FOR N := 1 STEP 1 UNTIL 20 DO PRINT(N,CAP(N)); 36 END;




FACTOR DE MULTIPROGRAMACIO = 1 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.3566 *0.3566 * 10.43 *0.3418E-01* * * * * * * * * DISC 1 * 23.00 *0.1769 *0.1769 * 23.00 *0.7690E-02* * * * * * * * * DISC 2 * 22.00 *0.1598 *0.1598 * 22.00 *0.7263E-02* * * * * * * * * DISC 3 * 21.00 *0.1615 *0.1615 * 21.00 *0.7690E-02* * * * * * * * * DISC 4 * 20.00 *0.1453 *0.1453 * 20.00 *0.7263E-02* *******************************************************************




FACTOR DE MULTIPROGRAMACIO = 5 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.8852 * 2.552 * 30.08 *0.8484E-01* * * * * * * * * DISC 1 * 23.00 *0.4390 *0.7037 * 36.87 *0.1909E-01* * * * * * * * * DISC 2 * 22.00 *0.3966 *0.6041 * 33.51 *0.1803E-01* * * * * * * * * DISC 3 * 21.00 *0.4009 *0.6137 * 32.15 *0.1909E-01* * * * * * * * * DISC 4 * 20.00 *0.3606 *0.5263 * 29.19 *0.1803E-01* *******************************************************************




FACTOR DE MULTIPROGRAMACIO = 20 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 16.69 * 174.1 *0.9584E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9838 * 45.62 *0.2156E-01* * * * * * * * * DISC 2 * 22.00 *0.4480 *0.8117 * 39.85 *0.2037E-01* * * * * * * * * DISC 3 * 21.00 *0.4528 *0.8275 * 38.38 *0.2156E-01* * * * * * * * * DISC 4 * 20.00 *0.4073 *0.6872 * 33.74 *0.2037E-01* *******************************************************************




1 0.4272E-02 11 0.1191E-01

2 0.6940E-02 12 0.1194E-01

3 0.8680E-02 13 0.1196E-01

4 0.9835E-02 14 0.1197E-01

5 0.1060E-01 15 0.1197E-01

6 0.1111E-01 16 0.1198E-01

7 0.1144E-01 17 0.1198E-01

8 0.1165E-01 18 0.1198E-01

9 0.1179E-01 19 0.1198E-01

10 0.1187E-01 20 0.1198E-01




T erm in a ls




37 38 /STATION/ NAME=TERMINAL; 39 TYPE=INFINITE; 40 INIT(C1)=N; 41 SERVICE(C1)=EXP(30000.); 42 SERVICE(C2)=EXP(60000.); 43 TRANSIT=SC; 44 /STATION/ NAME=SC; 45 SERVICE=EXP(1.); 46 RATE=CAP(1 STEP 1 UNTIL M); 47 TRANSIT=TERMINAL,C1,0.6,TERMINAL,C2; 48 /CONTROL/ OPTION=NRESULT;




49 /EXEC/ BEGIN 50 NETWORK(TERMINAL,SC); 51 PRINT(" TERM FM PRODUCTIVITAT ==> RESPOSTA"); 52 FOR N := 150 STEP 150 UNTIL 750 DO 53 FOR M := 1 STEP 1 UNTIL 20 DO 54 BEGIN 55 SOLVE; 56 PRINT(N,M,MTHRUPUT(SC),MRESPONSE(SC)); 57 END; 58 END;




TERM FM PRODUCTIVITAT RESPOSTA 150 1 0.3478E-02 1124. 150 2 0.3541E-02 358.5 150 3 0.3546E-02 304.1 150 4 0.3547E-02 294.4 150 5 0.3547E-02 292.4 150 6 0.3547E-02 292.0 150 7 0.3547E-02 291.9 150 8 0.3547E-02 291.9 ……. 150 20 0.3547E-02 291.9




TERM FM PRODUCTIVITAT RESPOSTA 300 1 0.4272E-02 0.2822E+05 300 2 0.6735E-02 2545. 300 3 0.7031E-02 668.5 300 4 0.7062E-02 482.4 300 5 0.7070E-02 433.6 300 6 0.7073E-02 417.0 300 7 0.7074E-02 410.8 300 8 0.7074E-02 408.4 300 9 0.7074E-02 407.5 300 10 0.7074E-02 407.2 300 11 0.7074E-02 407.0 300 12 0.7074E-02 407.0 ………. 300 20 0.7074E-02 407.0


APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system

TERM FM PRODUCTIVITAT RESPOSTA 450 1 0.4272E-02 0.6334E+05 450 2 0.6940E-02 0.2284E+05 450 3 0.8679E-02 9847. 450 4 0.9792E-02 3956. 450 5 0.1026E-01 1872. 450 6 0.1040E-01 1270. 450 7 0.1045E-01 1052. 450 8 0.1048E-01 956.3 450 9 0.1049E-01 910.5 450 10 0.1049E-01 887.4 450 11 0.1050E-01 875.5 450 12 0.1050E-01 869.4 450 13 0.1050E-01 866.3 450 14 0.1050E-01 864.7 450 15 0.1050E-01 864.0 450 16 0.1050E-01 863.6 450 17 0.1050E-01 863.4 450 18 0.1050E-01 863.3 450 19 0.1050E-01 863.2 450 20 0.1050E-01 863.2




TERM FM PRODUCTIVITAT RESPOSTA 600 1 0.4272E-02 0.9845E+05 600 2 0.6940E-02 0.4446E+05 600 3 0.8679E-02 0.2713E+05 600 4 0.9835E-02 0.1901E+05 600 5 0.1060E-01 0.1458E+05 600 6 0.1111E-01 0.1199E+05 600 7 0.1144E-01 0.1043E+05 600 8 0.1165E-01 9486. 600 9 0.1179E-01 8910. 600 10 0.1187E-01 8565. 600 11 0.1191E-01 8360. 600 12 0.1194E-01 8241. 600 13 0.1196E-01 8172. 600 14 0.1197E-01 8133. 600 15 0.1197E-01 8111. 600 16 0.1198E-01 8099. 600 17 0.1198E-01 8092. 600 18 0.1198E-01 8088. 600 19 0.1198E-01 8086. 600 20 0.1198E-01 8085.


APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system TERM FM PRODUCTIVITAT RESPOSTA 750 1 0.4272E-02 0.1336E+06 750 2 0.6940E-02 0.6607E+05 750 3 0.8679E-02 0.4441E+05 750 4 0.9835E-02 0.3426E+05 750 5 0.1060E-01 0.2873E+05 750 6 0.1111E-01 0.2549E+05 750 7 0.1144E-01 0.2354E+05 750 8 0.1165E-01 0.2236E+05 750 9 0.1179E-01 0.2164E+05 750 10 0.1187E-01 0.2121E+05 750 11 0.1191E-01 0.2095E+05 750 12 0.1194E-01 0.2080E+05 750 13 0.1196E-01 0.2071E+05 750 14 0.1197E-01 0.2067E+05 750 15 0.1197E-01 0.2064E+05 750 16 0.1198E-01 0.2062E+05 750 17 0.1198E-01 0.2061E+05 750 18 0.1198E-01 0.2061E+05 750 19 0.1198E-01 0.2061E+05 750 20 0.1198E-01 0.2061E+05 59 /END/



Decomposition-aggregation: Sliding window flow control mechanisms

N etw o rk 1

f(S )

e (S )

t j

N etw o rk 2

tf



Decomposition-aggregation: Sliding window flow control mechanisms Step 1: Decomposition

N etw o rk 1

N e tw o rk 2



Decomposition-aggregation: Sliding window flow control

mechanisms

Step 1: Decomposition

o We analyze this queuing network to obtain its throughput when there

are k customers where k = 1, 2, …, C.

o Let (k) be the throughput we obtain when there are k customers in the

network. The final result of this step is a set of values (1), (2), …,

(C).



Decomposition-aggregation: Sliding window flow control mechanisms Step 2: Aggregation

f(S )

e (S )

t j

g(k )




o The arrival process at queue f(S) is Poisson distributed of mean o There are C tokens.

o The inter-arrival times at queue e(S) are exponentially distributed with a rate g(k), where k is the number of outstanding tokens, i. e. C - k is the number of tokens in queue e(S).

o We set g(k) = (k), for k = 1, 2, …, C.

o The state of this system can be described by the tuple (i, j), where i is the number of customers in queue f(S) and j is the number of tokens in queue e(S).




g(C ) g(C ) g(C ) g(C ) g(C -1 ) g(C -2 ) g(C - j) g(1 )

1 ,0 1 ,0 0 ,0 0 ,1 0 ,2 0 ,j 0 ,C



Decomposition-aggregation: Sliding window flow control

mechanisms

Step 2: Aggregation

o This system is identical to an M/M/1 queue with an arrival rate and a state

dependent service rate g(nq) if nq C, and g(C) if nq > C, where nq is the number of

customers in this M/M/1 queue. The random variables i and j are related to nq as

follows:

i = max(0, nq - C)

j = max(0, C - nq)

o The solution of this system can be obtained by a direct application of classical results.




where

= /g(C)

000

000

,pj

j,p

,p,ip

j

i

g

0 1

0 1

0j

jkCj

j

k




o The probability p(0, 0) is chosen so that the addition of the state probabilities is equal to 1:

C

jj

j,p

1

1

1

100




o We obtain the following marginal probabilities for each queue (index 1 is for queue f(S) and 2 for queue e(S)):

Cj,pj

jp

,pp

i,pip

,pp

j

i

0 00

1

000

0 00

1

00110

2

2

1

1



Diffusion Method Based on the assumption that very probably the queues are never

empty. Under this hypothesis for each queue:o the queue length discrete distribution is studied

o this discrete distribution is replaced by a continuous one with the same first two moments (first approximation)

o this continuous probability distribution is described by a diffusion equation

o this equation is solved with the appropriate contour conditions for the steady state

o the continuous probability distribution of the queue length is discretised by means of some heuristic criteria (second approximation)



Diffusion Method To solve a queuing network it is assumed that it will have a

product-form.

If the system is open it is possible to determine for each node the characteristics of the inter-arrival time distribution and those of the service time are assumed to be known.

From the state probabilities, the performance measures can be computed.



Diffusion method: Packet communication network

1

2

3

4

5

6

7

8

A

B

C

D



Diffusion method: Packet communication network

1 /DECLARE/ QUEUE GEN,LINIA(8,8); 2 REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,==> TRDA,TRDB,TRDC; 3 INTEGER I; 4 CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD==> ,CLDA,CLDB,CLDC; 5



Diffusion method: Packet communication network 6 /STATION/ NAME = GEN; 7 TYPE = SOURCE; 8 SERVICE = EXP(60000./930.); 9 TRANSIT = LINIA(1,2),CLAB,60,LINIA(1,2),CLAC,80,L==> INIA(1,2),CLAD,100,10 LINIA(3,4),CLBA,75,LINIA(3,4),CLBC,50,L==> INIA(3,4),CLBD,25,11 LINIA(5,6),CLCA,80,LINIA(5,6),CLCB,120,==> LINIA(5,6),CLCD,40,12 LINIA(7,8),CLDA,100,LINIA(7,8),CLDB,150==> ,LINIA(7,8),CLDC,50;13



Diffusion method: Packet communication network 14 /STATION/ NAME = LINIA; 15 SERVICE = CST(256.*8./64.); 16 17 /STATION/ NAME = LINIA(1,2); 18 TRANSIT(CLAB) = LINIA(2,4); 19 TRANSIT(CLAC) = LINIA(2,4); 20 TRANSIT(CLAD) = LINIA(2,8); 21 22 /STATION/ NAME = LINIA(2,1); 23 TRANSIT = OUT; 24 25 /STATION/ NAME = LINIA(2,4); 26 TRANSIT(CLAB) = LINIA(4,3); 27 TRANSIT(CLAC) = LINIA(4,6); 28 29 /STATION/ NAME = LINIA(2,8); 30 TRANSIT(CLAD) = LINIA(8,7); 31



Diffusion method: Packet communication network 32 /STATION/ NAME = LINIA(3,4); 33 TRANSIT(CLBA) = LINIA(4,2); 34 TRANSIT(CLBC) = LINIA(4,6); 35 TRANSIT(CLBD) = LINIA(4,8); 36 37 /STATION/ NAME = LINIA(4,2); 38 TRANSIT(CLBA) = LINIA(2,1); 39 40 /STATION/ NAME = LINIA(4,3); 41 TRANSIT = OUT; 42 43 /STATION/ NAME = LINIA(4,6); 44 TRANSIT(CLAC) = LINIA(6,5); 45 TRANSIT(CLBC) = LINIA(6,5); 46 47 /STATION/ NAME = LINIA(4,8); 48 TRANSIT(CLBD) = LINIA(8,7); 49


EXACT ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 50 /STATION/ NAME = LINIA(5,6); 51 TRANSIT(CLCA) = LINIA(6,8); 52 TRANSIT(CLCB) = LINIA(6,4); 53 TRANSIT(CLCD) = LINIA(6,8); 54 55 /STATION/ NAME = LINIA(6,4); 56 TRANSIT(CLCB) = LINIA(4,3); 57 58 /STATION/ NAME = LINIA(6,5); 59 TRANSIT = OUT; 60 61 /STATION/ NAME = LINIA(6,8); 62 TRANSIT(CLCA) = LINIA(8,2); 63 TRANSIT(CLCD) = LINIA(8,7); 64 65 /STATION/ NAME = LINIA(7,8); 66 TRANSIT(CLDA) = LINIA(8,2); 67 TRANSIT(CLDB) = LINIA(8,4); 68 TRANSIT(CLDC) = LINIA(8,6); 69



Diffusion method: Packet communication network 70 /STATION/ NAME = LINIA(8,2); 71 TRANSIT(CLCA) = LINIA(2,1); 72 TRANSIT(CLDA) = LINIA(2,1); 73 74 /STATION/ NAME = LINIA(8,4); 75 TRANSIT(CLDB) = LINIA(4,3); 76 77 /STATION/ NAME = LINIA(8,6); 78 TRANSIT(CLDC) = LINIA(6,5); 79 80 /STATION/ NAME = LINIA(8,7); 81 TRANSIT = OUT; 82



Diffusion method: Packet communication network 83 /EXEC/ BEGIN84 NETWORK(GEN,85 LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8),86 LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6),87 LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5),88 LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4),89 LINIA(8,6),LINIA(8,7));90 PRINT;91 SOLVE;92 TRAB:=MRESPONSE(LINIA(1,2))+93 MRESPONSE(LINIA(2,4))+MRESPONSE(LINIA(4,3));94 TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+95 MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5));96 TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+97 MRESPONSE(LINIA(8,7));98 TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+99 MRESPONSE(LINIA(2,1));



Diffusion method: Packet communication network 100 TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+101 MRESPONSE(LINIA(6,5));102 TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+103 MRESPONSE(LINIA(8,7));104 TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+105 MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1));106 TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+107 MRESPONSE(LINIA(4,3));108 TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+109 MRESPONSE(LINIA(8,7));110 TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+111 MRESPONSE(LINIA(2,1));112 TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+113 MRESPONSE(LINIA(4,3));114 TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+115 MRESPONSE(LINIA(6,5));



Diffusion method: Packet communication network 116 PRINT(TRAB,TRAC,TRAD); 117 PRINT(TRBA,TRBC,TRBD); 118 PRINT(TRCA,TRCB,TRCD); 119 PRINT(TRDA,TRDB,TRDC); 120 END;



Diffusion method: Packet communication network - APPROXIMATE DIFFUSIONS METHOD ("DIFFU") -******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* * * * * * ** GEN * 64.52 * 1.000 * 1.000 * 64.52 *0.1550E-01** * * * * * ** LINIA 2 * 32.00 *0.1280 *0.1374 * 34.35 *0.4000E-02** * * * * * ** LINIA 9 * 32.00 *0.1360 *0.1466 * 34.48 *0.4250E-02** * * * * * ** LINIA 12 * 32.00 *0.7467E-01*0.7765E-01* 33.28 *0.2333E-02** * * * * * ** LINIA 16 * 32.00 *0.5333E-01*0.5482E-01* 32.89 *0.1667E-02** * * * * * ** LINIA 20 * 32.00 *0.8000E-01*0.8348E-01* 33.39 *0.2500E-02** * * * * * ** LINIA 26 * 32.00 *0.4000E-01*0.4083E-01* 32.66 *0.1250E-02** * * * * * *



Diffusion method: Packet communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* LINIA 27 * 32.00 *0.1760 *0.1945 * 35.37 *0.5500E-02** * * * * * ** LINIA 30 * 32.00 *0.6933E-01*0.7190E-01* 33.18 *0.2167E-02** * * * * * ** LINIA 32 * 32.00 *0.1333E-01*0.1342E-01* 32.22 *0.4167E-03** * * * * * ** LINIA 38 * 32.00 *0.1280 *0.1374 * 34.35 *0.4000E-02** * * * * * ** LINIA 44 * 32.00 *0.6400E-01*0.6617E-01* 33.08 *0.2000E-02** * * * * * ** LINIA 45 * 32.00 *0.9600E-01*0.1011 * 33.68 *0.3000E-02** * * * * * ** LINIA 48 * 32.00 *0.6400E-01*0.6617E-01* 33.08 *0.2000E-02** * * * * * ** LINIA 56 * 32.00 *0.1600 *0.1752 * 35.05 *0.5000E-02** * * * * * ** LINIA 58 * 32.00 *0.9600E-01*0.1011 * 33.68 *0.3000E-02** * * * * * *



Diffusion method: Packet communication network******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT ********************************************************************* LINIA 60 * 32.00 *0.8000E-01*0.8343E-01* 33.37 *0.2500E-02** * * * * * ** LINIA 62 * 32.00 *0.2667E-01*0.2703E-01* 32.44 *0.8333E-03** * * * * * ** LINIA 63 * 32.00 *0.8800E-01*0.9222E-01* 33.53 *0.2750E-02******************************************************************** MEMORY USED: 19812 WORDS OF 4 BYTES ( 2.48 % OF TOTAL MEMORY)

103.0 134.5 100.8 100.5 100.3 99.14 135.6 102.8 101.0 103.2 103.8 101.2

121 122 /END/



Iterative methods In this family of methods we establish an iterative computation

of the result from a reasonable conjecture.

They have neither any theoretical justification of the iteration

convergence nor the coincidence of the limit of the iteration with

the theoretical result; however the experience shows that there

are no counter-examples in the normal domain of use of such

methods.



Iterative methods They are useful tools to study cases not covered by exact methods, as,

for instance:o closed queuing networks with FIFO stations that have with non-

exponentially distributed service time

o systems with customers seizing simultaneously more than one server (for example, in a disk input-output there are simultaneous seizing of the disk and the path between disks and memory).

o systems with customers affected of priorities

They are also useful to reduce the computing time in methods derived from the BCMP theorem when they are applied to large dimension systems.



Iterative methods: Conversational system This case is identical to the previous one with an important difference:

o the disk path to memory is shared by several disks and should be taken into account in the modelling process

o the scheduling policy of disks accesses in the control unit is SLTF (Shortest Latency Time First).

As in this case the basic assumption of analytical models is not fulfilled, we are obliged to build an iterative model starting with the assumption that there will not be conflicts in the path; from this assumption it is possible to compute the throughput in the disks. With this data we recompute the path conflict time due to the lost rotations. This time is introduced in the disks service time and we restart the iteration.



Iterative methods: Conversational system 1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL;

2 REAL PROB1(4)=(2,1.5,1,0.5); 3 REAL PROB2(4)=(1.5,2,3,3.5); 4 REAL TR1,TR2; 5 REAL SD(4),VP,VPN,LT,TF,VR=3600,TAU,UC, 6 SK(4)=(13.,12.,11.,10.); 7 REAL EPS; 8 CLASS C1,C2; 9 INTEGER I,N; 10 /STATION/ NAME=CPU; 11 SCHED=PS; 12 SERVICE(C1)=CST(8.52); 13 SERVICE(C2)=CST(12.); 14 TRANSIT(C1)=DISC,PROB1,TERMINAL,C1,0.6,TERMINAL,C2,0 ==> .4; 15 TRANSIT(C2)=DISC,PROB2,TERMINAL,C1,0.6,TERMINAL,C2,0 ==> .4;



Iterative methods: Conversational system

16 /STATION/ NAME=DISC; 17 TRANSIT=CPU; 18 /STATION/ NAME=DISC(1); 19 SERVICE=EXP(SD(1)); 20 /STATION/ NAME=DISC(2); 21 SERVICE=EXP(SD(2)); 22 /STATION/ NAME=DISC(3); 23 SERVICE=EXP(SD(3)); 24 /STATION/ NAME=DISC(4); 25 SERVICE=EXP(SD(4)); 26 /STATION/ NAME=TERMINAL; 27 TYPE=INFINITE; 28 INIT(C1)=N; 29 SERVICE(C1)=EXP(30000.); 30 SERVICE(C2)=EXP(60000.); 31 TRANSIT=CPU; 32 /CONTROL/ CLASS=ALL QUEUE;




33 /EXEC/ BEGIN 34 TAU:=60.*1000./VR; 35 TF:=TAU/10.; 36 LT:=TAU/2.; 37 FOR N:=150 STEP 150 UNTIL 750 DO 38 BEGIN 39 PRINT("NOMBRE D’USUARIS =",N); 40 EPS:=1.; 41 VP:=0.; 42 WHILE EPS>=1.E-4 DO 43 BEGIN 44 FOR I:=1 STEP 1 UNTIL 4 DO SD(I):=SK(I)+LT+VP+ ==> TF; 45 PRINT; 46 SOLVE;


APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system 47 UC:=0.; 48 FOR I:=1 STEP 1 UNTIL 4 DO UC := UC + TF * MTH ==> RUPUT(DISC(I)); 49 VPN:=TAU*UC/(1.-UC); 50 EPS:=ABS(VP-VPN)/VPN; 51 VP:=VPN; 52 END; 53 TR1:=MCUSTNB(CPU,C1); 54 TR2:=MCUSTNB(CPU,C2); 55 FOR I:= 1 STEP 1 UNTIL 4 DO 56 BEGIN 57 TR1:=TR1+MCUSTNB(DISC(I),C1); 58 TR2:=TR2+MCUSTNB(DISC(I),C2); 59 END; 60 TR1:=TR1/MTHRUPUT(TERMINAL,C1); 61 TR2:=TR2/MTHRUPUT(TERMINAL,C2);; 62 PRINT("RESPONSE TIME OF CLASS C1 =",TR1); 63 PRINT("RESPONSE TIME OF CLASS C2 =",TR2); 64 END; 65 END;



Iterative methods: Conversational systemNOMBRE D’USUARIS = 150 - MEAN VALUE ANALYSIS ("MVA") - ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* *(C1 )* 8.520 *0.1088 *0.1539 * 12.06 *0.1277E-01* *(C2 )* 12.00 *0.1873 *0.2650 * 16.98 *0.1560E-01* * * * * * * * * DISC 1 * 23.00 *0.1468 *0.1719 * 26.92 *0.6384E-02* *(C1 )* 23.00 *0.9790E-01*0.1146 * 26.92 *0.4257E-02* *(C2 )* 23.00 *0.4894E-01*0.5729E-01* 26.92 *0.2128E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 149.0 *0.4200E+05*0.3547E-02* *(C1 )*0.3000E+05*0.0000E+00* 63.84 *0.3000E+05*0.2128E-02* *(C2 )*0.6000E+05*0.0000E+00* 85.12 *0.6000E+05*0.1419E-02* * * * * * * * *******************************************************************



Iterative methods: Conversational systemNOMBRE D’USUARIS = 150 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.2960 *0.4188 * 14.76 *0.2837E-01* *(C1 )* 8.520 *0.1088 *0.1539 * 12.05 *0.1277E-01* *(C2 )* 12.00 *0.1872 *0.2649 * 16.98 *0.1560E-01* * * * * * * * * DISC 1 * 23.72 *0.1514 *0.1782 * 27.91 *0.6383E-02* *(C1 )* 23.72 *0.1009 *0.1188 * 27.91 *0.4256E-02* *(C2 )* 23.72 *0.5046E-01*0.5938E-01* 27.91 *0.2127E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 148.9 *0.4200E+05*0.3546E-02* *(C1 )*0.3000E+05*0.0000E+00* 63.83 *0.3000E+05*0.2128E-02* *(C2 )*0.6000E+05*0.0000E+00* 85.11 *0.6000E+05*0.1418E-02* *******************************************************************



Iterative methods: Conversational systemNOMBRE D’USUARIS = 150 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.2960 *0.4188 * 14.76 *0.2837E-01* *(C1 )* 8.520 *0.1088 *0.1539 * 12.05 *0.1277E-01* *(C2 )* 12.00 *0.1872 *0.2649 * 16.98 *0.1560E-01* * * * * * * * * DISC 1 * 23.72 *0.1514 *0.1782 * 27.91 *0.6383E-02* *(C1 )* 23.72 *0.1009 *0.1188 * 27.91 *0.4256E-02* *(C2 )* 23.72 *0.5046E-01*0.5938E-01* 27.91 *0.2127E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 148.9 *0.4200E+05*0.3546E-02* *(C1 )*0.3000E+05*0.0000E+00* 63.83 *0.3000E+05*0.2128E-02* *(C2 )*0.6000E+05*0.0000E+00* 85.11 *0.6000E+05*0.1418E-02******************************************************************** RESPONSE TIME OF CLASS C1 = 204.6 RESPONSE TIME OF CLASS C2 = 439.5




NOMBRE D’USUARIS = 300 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.5905 * 1.426 * 25.19 *0.5659E-01* *(C1 )* 8.520 *0.2170 *0.5239 * 20.57 *0.2547E-01* *(C2 )* 12.00 *0.3735 *0.9017 * 28.97 *0.3112E-01* * * * * * * * * DISC 1 * 23.00 *0.2929 *0.4134 * 32.46 *0.1273E-01* *(C1 )* 23.00 *0.1953 *0.2756 * 32.46 *0.8490E-02* *(C2 )* 23.00 *0.9761E-01*0.1378 * 32.46 *0.4244E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 297.1 *0.4200E+05*0.7074E-02* *(C1 )*0.3000E+05*0.0000E+00* 127.3 *0.3000E+05*0.4245E-02* *(C2 )*0.6000E+05*0.0000E+00* 169.8 *0.6000E+05*0.2830E-02* *******************************************************************




NOMBRE D’USUARIS = 300 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.5902 * 1.424 * 25.17 *0.5657E-01* *(C1 )* 8.520 *0.2169 *0.5233 * 20.56 *0.2546E-01* *(C2 )* 12.00 *0.3733 *0.9007 * 28.95 *0.3111E-01* * * * * * * * * DISC 1 * 24.50 *0.3118 *0.4521 * 35.52 *0.1273E-01* *(C1 )* 24.50 *0.2079 *0.3015 * 35.52 *0.8486E-02* *(C2 )* 24.50 *0.1039 *0.1507 * 35.52 *0.4242E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 297.0 *0.4200E+05*0.7071E-02* *(C1 )*0.3000E+05*0.0000E+00* 127.3 *0.3000E+05*0.4243E-02* *(C2 )*0.6000E+05*0.0000E+00* 169.7 *0.6000E+05*0.2828E-02* *******************************************************************


APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system

NOMBRE D’USUARIS = 300 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.5902 * 1.424 * 25.17 *0.5657E-01* *(C1 )* 8.520 *0.2169 *0.5233 * 20.56 *0.2546E-01* *(C2 )* 12.00 *0.3733 *0.9007 * 28.95 *0.3111E-01* * * * * * * * * DISC 1 * 24.50 *0.3118 *0.4521 * 35.52 *0.1273E-01* *(C1 )* 24.50 *0.2079 *0.3014 * 35.52 *0.8486E-02* *(C2 )* 24.50 *0.1039 *0.1507 * 35.52 *0.4242E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 297.0 *0.4200E+05*0.7071E-02* *(C1 )*0.3000E+05*0.0000E+00* 127.3 *0.3000E+05*0.4243E-02* *(C2 )*0.6000E+05*0.0000E+00* 169.7 *0.6000E+05*0.2828E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 289.4 RESPONSE TIME OF CLASS C2 = 632.8




NOMBRE D’USUARIS = 450 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.8763 * 6.438 * 76.66 *0.8399E-01* *(C1 )* 8.520 *0.3220 * 2.366 * 62.60 *0.3780E-01* *(C2 )* 12.00 *0.5543 * 4.072 * 88.16 *0.4619E-01* * * * * * * * * DISC 1 * 23.00 *0.4347 *0.7667 * 40.57 *0.1890E-01* *(C1 )* 23.00 *0.2898 *0.5112 * 40.57 *0.1260E-01* *(C2 )* 23.00 *0.1449 *0.2555 * 40.57 *0.6298E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 440.9 *0.4200E+05*0.1050E-01* *(C1 )*0.3000E+05*0.0000E+00* 189.0 *0.3000E+05*0.6299E-02* *(C2 )*0.6000E+05*0.0000E+00* 252.0 *0.6000E+05*0.4199E-02* *******************************************************************




NOMBRE D’USUARIS = 450 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.8754 * 6.393 * 76.20 *0.8390E-01* *(C1 )* 8.520 *0.3217 * 2.349 * 62.22 *0.3776E-01* *(C2 )* 12.00 *0.5537 * 4.044 * 87.64 *0.4614E-01* * DISC 1 * 25.33 *0.4781 *0.9129 * 48.36 *0.1888E-01* *(C1 )* 25.33 *0.3188 *0.6086 * 48.36 *0.1259E-01* *(C2 )* 25.33 *0.1594 *0.3043 * 48.36 *0.6292E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 440.5 *0.4200E+05*0.1049E-01* *(C1 )*0.3000E+05*0.0000E+00* 188.8 *0.3000E+05*0.6293E-02* *(C2 )*0.6000E+05*0.0000E+00* 251.7 *0.6000E+05*0.4195E-02* *******************************************************************


APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system

NOMBRE D’USUARIS = 450 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.8754 * 6.393 * 76.20 *0.8390E-01* *(C1 )* 8.520 *0.3217 * 2.349 * 62.22 *0.3776E-01* *(C2 )* 12.00 *0.5537 * 4.044 * 87.64 *0.4614E-01* * * * * * * * * DISC 1 * 25.32 *0.4781 *0.9127 * 48.35 *0.1888E-01* *(C1 )* 25.32 *0.3187 *0.6085 * 48.35 *0.1259E-01* *(C2 )* 25.32 *0.1593 *0.3042 * 48.35 *0.6292E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 440.5 *0.4200E+05*0.1049E-01* *(C1 )*0.3000E+05*0.0000E+00* 188.8 *0.3000E+05*0.6293E-02* *(C2 )*0.6000E+05*0.0000E+00* 251.7 *0.6000E+05*0.4195E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 594.4 RESPONSE TIME OF CLASS C2 = 1376.




NOMBRE D’USUARIS = 600 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 93.51 * 975.7 *0.9584E-01* *(C1 )* 8.520 *0.3675 * 34.37 * 796.7 *0.4313E-01* *(C2 )* 12.00 *0.6325 * 59.15 * 1122. *0.5271E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9841 * 45.63 *0.2157E-01* *(C1 )* 23.00 *0.3307 *0.6561 * 45.63 *0.1438E-01* *(C2 )* 23.00 *0.1653 *0.3280 * 45.63 *0.7187E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* *******************************************************************


APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE D’USUARIS = 600 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 92.66 * 966.8 *0.9584E-01* *(C1 )* 8.520 *0.3675 * 34.05 * 789.5 *0.4313E-01* *(C2 )* 12.00 *0.6325 * 58.61 * 1112. *0.5271E-01* * * * * * * * * DISC 1 * 25.71 *0.5544 * 1.244 * 57.69 *0.2157E-01* *(C1 )* 25.71 *0.3696 *0.8295 * 57.69 *0.1438E-01* *(C2 )* 25.71 *0.1848 *0.4147 * 57.69 *0.7187E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 4997. RESPONSE TIME OF CLASS C2 = 0.1271E+05




NOMBRE D’USUARIS = 750 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 243.5 * 2541. *0.9584E-01* *(C1 )* 8.520 *0.3675 * 89.49 * 2075. *0.4313E-01* *(C2 )* 12.00 *0.6325 * 154.0 * 2922. *0.5271E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9841 * 45.64 *0.2157E-01* *(C1 )* 23.00 *0.3307 *0.6561 * 45.64 *0.1438E-01* *(C2 )* 23.00 *0.1653 *0.3280 * 45.64 *0.7187E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* *******************************************************************




NOMBRE D’USUARIS = 750 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 242.7 * 2532. *0.9584E-01* *(C1 )* 8.520 *0.3675 * 89.18 * 2067. *0.4313E-01* *(C2 )* 12.00 *0.6325 * 153.5 * 2912. *0.5271E-01* * * * * * * * * DISC 1 * 25.71 *0.5544 * 1.244 * 57.69 *0.2157E-01* *(C1 )* 25.71 *0.3696 *0.8295 * 57.69 *0.1438E-01* *(C2 )* 25.71 *0.1848 *0.4147 * 57.69 *0.7187E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02******************************************************************** RESPONSE TIME OF CLASS C1 = 0.1266E+05 RESPONSE TIME OF CLASS C2 = 0.3251E+05


OUTLINE

INTRODUCTION

CONCEPT OF QUEUE








Computer based simulation implies writing a computer

program which depicts the system operations.

Running this program permits us to "imitate" the

system behaviour in a very short time. Thus, we are

able to "observe" the system and collect performance

statistics.



Simulation advantage are: there is no theoretical limitation and it allows us to study

systems not able to be studied by means of analytical or numerical techniques.

It is easy to learn and to apply.

Simulation disadvantages are: the effort (mainly time) to develop and to debug a

simulation program. the execution time, that can be quite long to obtain results

with enough significance.



The complexity of the simulation model depend on the detail of

the system behaviour representation.

To build up simulation models it is convenient to use: modelling languages as QNAP2, RESQ, PAWS, etc., that make easier to

build up queuing network models and that include analytical, numerical

and simulation procedures.

simulation languages, such as SIMSCRIPT, GPSS, SIMULA, etc., that

offer greater simulation capabilities.

high level programming language for building up very detailed models.



The intrinsic problems of any simulation are:

random numbers generation

simulated time management

extraction of statistical estimations of the simulated behaviour

evaluation of the confidence interval of the estimations



Conversational system Identical to the previously studied by analytical

techniques

Influence of the memory management by a multiprogramming factor



Conversational system

D isk 1

D isk 2

D isk 3

D isk 4

C P U

T e rm in a ls

M e m o rym a n a g e m e n t


SIMULATION TECHNIQUESConversational system

1 /DECLARE/ QUEUE CPU,DISC(4),MEM,SMEM,TERMINAL,R(2);

2 REAL PROF1(4)=(2.,3.5,4.5,5.);

3 REAL PROF2(4)=(1.5,3.5,6.5,10.);

4 REAL TR1,TR2;

5 REAL D;

6 CLASS C1,C2;

7 INTEGER I,N,M;

8 REF CUSTOMER C;


SIMULATION TECHNIQUESConversational system

9 /STATION/ NAME=CPU;

10 SCHED=PS;

11 SERVICE(C1)=BEGIN

12 CST(8.52);

13 D := UNIFORM(0., 6.);

14 FOR I := 1 STEP 1 UNTIL 4 DO

15 IF D <= PROF1(I) THEN TRANSIT(DISC(I));

16 C:=R(1).FIRST;

17 WHILE C.FATHER <> CUSTOMER DO C:=C.NEXT;

18 TRANSIT(C,OUT);

19 V(SMEM);

20 IF D <= 5.6 THEN TRANSIT(TERMINAL,C1)

21 ELSE TRANSIT(TERMINAL,C2);

22 END;


SIMULATION TECHNIQUESConversational system 23 SERVICE(C2)=BEGIN 24 CST(12.); 25 D := UNIFORM(0., 11.); 26 FOR I := 1 STEP 1 UNTIL 4 DO 27 IF D <= PROF2(I) THEN TRANSIT(DISC(I)); 28 C:=R(2).FIRST; 29 WHILE C.FATHER <> CUSTOMER DO C:=C.NEXT; 30 TRANSIT(C,OUT); 31 V(SMEM); 32 IF D <= 10.6 THEN TRANSIT(TERMINAL,C1) 33 ELSE TRANSIT(TERMINAL,C2); 34 END; 35 /STATION/ NAME=DISC; 36 TRANSIT=CPU; 37 /STATION/ NAME=DISC(1); 38 SERVICE=EXP(23.); 39 /STATION/ NAME=DISC(2); 40 SERVICE=EXP(22.);



Conversational system

41 /STATION/ NAME=DISC(3);

42 SERVICE=EXP(21.);

43 /STATION/ NAME=DISC(4);

44 SERVICE=EXP(20.);

45 /STATION/ NAME=MEM;

46 SERVICE=BEGIN

47 P(SMEM);

48 TRANSIT(CPU);

49 END;

50 /STATION/ NAME=SMEM;

51 TYPE=SEMAPHORE,MULTIPLE(M);


SIMULATION TECHNIQUESConversational system 52 /STATION/ NAME=TERMINAL; 53 TYPE=INFINITE; 54 INIT(C1)=6*N/10; 55 INIT(C2)=4*N/10; 56 SERVICE(C1)=BEGIN 57 EXP(30000.); 58 TRANSIT(NEW(CUSTOMER),R(1),C1); 59 END; 60 SERVICE(C2)=BEGIN 61 EXP(60000.); 62 TRANSIT(NEW(CUSTOMER),R(2),C2); 63 END; 64 TRANSIT=MEM; 65 /CONTROL/ TMAX=5000000.; 66 CLASS=ALL QUEUE; 67 ACCURACY=ALL QUEUE,ALL CLASS; 68 OPTION=NRESULT;


SIMULATION TECHNIQUESConversational system 69 /EXEC/ BEGIN 70 FOR N:=150 STEP 150 UNTIL 750 DO 71 FOR M:=1 STEP 1 UNTIL 20 DO 72 BEGIN 73 PRINT(" "); 74 PRINT("NUMERO DE USUARIOS =",N); 75 PRINT("FACTOR DE MULTIPROGRAMACION =",M); 76 SIMUL; 77 PRINT("TIEMPO DE RESPUESTA DE LA CLASE C1 =",MRESPONS ==> E(R(1))," +/-"),CRESPONSE(R(1)); 78 PRINT("TIEMPO DE RESPUESTA DE LA CLASE C2 =",MRESPONS ==> E(R(2))," +/-",CRESPONSE(R(2))); 79 PRINT("PRODUCTIVIDAD C1 =",MTHRUPUT(TERMINAL,C1)); 80 PRINT("PRODUCTIVIDAD C2 =",MTHRUPUT(TERMINAL,C2)); 81 PRINT("PRODUCTIVIDAD TOTAL =",MTHRUPUT(TERMINAL)); 82 END; 83 END;


SIMULATION TECHNIQUESConversational system NUMERO DE USUARIOS = 150 FACTOR DE MULTIPROGRAMACION = 1 TIEMPO DE RESPUESTA DE LA CLASE C1 = 1203. +/- 132.0 TIEMPO DE RESPUESTA DE LA CLASE C2 = 1368. +/- 145.4 PRODUCTIVIDAD C1 = 0.2059E-02 PRODUCTIVIDAD C2 = 0.1356E-02 PRODUCTIVIDAD TOTAL = 0.3415E-02

NUMERO DE USUARIOS = 150 FACTOR DE MULTIPROGRAMACION = 2 TIEMPO DE RESPUESTA DE LA CLASE C1 = 292.9 +/- 14.44 TIEMPO DE RESPUESTA DE LA CLASE C2 = 499.6 +/- 23.42 PRODUCTIVIDAD C1 = 0.2131E-02 PRODUCTIVIDAD C2 = 0.1380E-02 PRODUCTIVIDAD TOTAL = 0.3511E-02



Communication network Identical to the previously studied by analytical

techniques


SIMULATION TECHNIQUESCommunication network 1 /DECLARE/ QUEUE INTEGER J; 2 QUEUE GEN,LINIA(8,8),R(4,4); 3 INTEGER DIREC(8,4)=(0,2,2,2, 4 1,4,4,8, 5 4,0,4,4, 6 2,3,6,8, 7 6,6,0,6, 8 8,4,5,8, 9 8,8,8,0, 10 2,4,6,7); 11 12 REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,TR ==> DA,TRDB,TRDC; 13 INTEGER OD(4,4) = ( 0, 60,140,240, 14 315,315,365,390, 15 470,590,590,630, 16 730,880,930,930); 17 INTEGER INIC(4) = (1, 3, 5, 7);


SIMULATION TECHNIQUESCommunication network 18 INTEGER I,K; 19 CUSTOMER INTEGER ORIG,DESTI; 20 REF CUSTOMER C; 21 /EXEC/ FOR I := 1 STEP 1 UNTIL 8 DO FOR K := 1 STEP 1 UNTIL ==> 8 DO 22 LINIA(I,K).J := K; 23 /STATION/ NAME = GEN; 24 TYPE = SOURCE; 25 SERVICE = BEGIN 26 EXP(60000./930.); 27 I:= RINT(1,930); 28 ORIG := 1; 29 DESTI := 1; 30 WHILE (I > OD(ORIG,DESTI)) DO 31 IF DESTI < 4 THEN DESTI := DESTI + 1 32 ELSE 33 BEGIN 34 ORIG := ORIG + 1; 35 DESTI := 1; 36 END; 37 C := NEW(CUSTOMER); 38 TRANSIT(C,R(ORIG,DESTI)); 39 TRANSIT(LINIA(INIC(ORIG),DIREC(INIC(ORIG ==> ),DESTI))); 40 END;



Communication network 42 /STATION/ NAME = LINIA; 43 SERVICE = BEGIN 44 EXP(256.*8./64.); 45 I:=DIREC(J,DESTI); 46 IF I=0 THEN 47 BEGIN 48 C := R(ORIG,DESTI).FIRST; 49 WHILE C <> SON DO C := C.NEXT; 50 TRANSIT(C,OUT); 51 TRANSIT (OUT); 52 END; 53 TRANSIT (LINIA(J,I)); 54 END; 55 56 /CONTROL/ TMAX= 300000; 57 ACCURACY= ALL QUEUE;



Communication network 59 /EXEC/ BEGIN 60 PRINT; 61 SIMUL; 62 TRAB:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+ 63 MRESPONSE(LINIA(4,3)); 64 TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+ 65 MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5)); 66 TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+ 67 MRESPONSE(LINIA(8,7)); 68 TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+ 69 MRESPONSE(LINIA(2,1)); 70 TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+ 71 MRESPONSE(LINIA(6,5)); 72 TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+ 73 MRESPONSE(LINIA(8,7)); 74 TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+ 75 MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1));



Communication network 76 TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+ 77 MRESPONSE(LINIA(8,7)); 78 TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+ 79 MRESPONSE(LINIA(4,3)); 80 TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+ 81 MRESPONSE(LINIA(2,1)); 82 TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+ 83 MRESPONSE(LINIA(4,3)); 84 TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+ 85 MRESPONSE(LINIA(6,5)); 86 PRINT(TRAB,TRAC,TRAD); 87 PRINT(TRBA,TRBC,TRBD); 88 PRINT(TRCA,TRCB,TRCD); 89 PRINT(TRDA,TRDB,TRDC); 90 END;



Communication network ***SIMULATION WITH SPECTRAL METHOD ***

.. TIME = 300000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * * * * * * * * GEN * 64.03 * 1.000 * 1.000 * 64.03 * 4684* * +/- * 1.829 *0.0000E+00*0.0000E+00* 1.829 * * * * * * * * * * LINIA 2 * 30.88 *0.1246 *0.1477 * 36.61 * 1210* * +/- * 2.017 *0.1055E-01*0.1607E-01* 3.188 * * * * * * * * * * LINIA 9 * 31.34 *0.1347 *0.1531 * 35.63 * 1289* * +/- * 1.999 *0.1170E-01*0.1384E-01* 2.597 * * * * * * * * * * LINIA 12 * 31.07 *0.7519E-01*0.8277E-01* 34.20 * 726* * +/- * 2.504 *0.7059E-02*0.1367E-01* 3.444 * * * * * * * * *



Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * LINIA 16 * 34.91 *0.5632E-01*0.6102E-01* 37.82 * 484* * +/- * 4.215 *0.9778E-02*0.1091E-01* 5.125 * * * * * * * * * * LINIA 20 * 32.59 *0.7801E-01*0.8174E-01* 34.15 * 718* * +/- * 2.378 *0.7146E-02*0.8014E-02* 2.579 * * * * * * * * * * LINIA 26 * 34.57 *0.4171E-01*0.4353E-01* 36.08 * 362* * +/- * 3.707 *0.6404E-02*0.6778E-02* 4.122 * * * * * * * * * * LINIA 27 * 32.92 *0.1857 *0.2277 * 40.37 * 1691* * +/- * 1.851 *0.1625E-01*0.2106E-01* 3.039 * * * * * * * * * * LINIA 30 * 35.30 *0.7448E-01*0.8044E-01* 38.12 * 633* * +/- * 3.819 *0.9264E-02*0.1115E-01* 4.429 * *


SIMULATION TECHNIQUESCommunication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * LINIA 32 * 28.69 *0.1243E-01*0.1243E-01* 28.69 * 130* * +/- * 4.742 *0.2918E-02*0.2918E-02* 4.742 * * * * * * * * * * LINIA 38 * 30.73 *0.1345 *0.1547 * 35.35 * 1313* * +/- * 1.734 *0.1065E-01*0.1591E-01* 2.745 * * * * * * * * * * LINIA 44 * 32.35 *0.6922E-01*0.7420E-01* 34.68 * 642* * +/- * 2.659 *0.8011E-02*0.9049E-02* 3.228 * * * * * * * * * * LINIA 45 * 31.92 *0.9321E-01*0.1024 * 35.07 * 876* * +/- * 2.950 *0.8071E-02*0.1099E-01* 3.485 * * * * * * * * * * LINIA 48 * 29.55 *0.6609E-01*0.6892E-01* 30.81 * 671* * +/- * 2.910 *0.6452E-02*0.6856E-02* 2.968 * * * * * * * * *


SIMULATION TECHNIQUESCommunication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * LINIA 56 * 32.19 *0.1548 *0.1929 * 40.11 * 1443* * +/- * 2.196 *0.1281E-01*0.2053E-01* 3.851 * * * * * * * * * * LINIA 58 * 32.63 *0.1008 *0.1104 * 35.71 * 927* * +/- * 2.415 *0.9855E-02*0.1273E-01* 3.771 * * * * * * * * * * LINIA 60 * 33.53 *0.8171E-01*0.8924E-01* 36.62 * 731* * +/- * 2.046 *0.8194E-02*0.9643E-02* 3.040 * * * * * * * * * * LINIA 62 * 35.31 *0.2860E-01*0.2887E-01* 35.64 * 243* * +/- * 5.058 *0.5594E-02*0.5755E-02* 5.120 * * * * * * * * * * LINIA 63 * 34.11 *0.9402E-01*0.1019 * 36.97 * 827* * +/- * 2.188 *0.8583E-02*0.9742E-02* 2.344 * *


SIMULATION TECHNIQUESCommunication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * R 2 *0.0000E+00*0.0000E+00*0.1180 * 110.9 * 319* * +/- *0.0000E+00*0.0000E+00*0.3193E-01* 9.902 * * * * * * * * * * R 3 *0.0000E+00*0.0000E+00*0.1920 * 141.5 * 407* * +/- *0.0000E+00*0.0000E+00*0.2514E-01* 14.12 * * * * * * * * * * R 4 *0.0000E+00*0.0000E+00*0.1836 * 113.8 * 484* * +/- *0.0000E+00*0.0000E+00*0.2303E-01* 8.691 * * * * * * * * * * R 5 *0.0000E+00*0.0000E+00*0.1280 * 106.1 * 362* * +/- *0.0000E+00*0.0000E+00*0.1609E-01* 7.103 * * * * * * * * * * R 7 *0.0000E+00*0.0000E+00*0.8297E-01* 110.1 * 226* * +/- *0.0000E+00*0.0000E+00*0.1289E-01* 9.790 * *


SIMULATION TECHNIQUESCommunication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * R 8 *0.0000E+00*0.0000E+00*0.4081E-01* 94.18 * 130* * +/- *0.0000E+00*0.0000E+00*0.7638E-02* 9.646 * * * * * * * * * * R 9 *0.0000E+00*0.0000E+00*0.2082 * 136.4 * 458* * +/- *0.0000E+00*0.0000E+00*0.2374E-01* 6.981 * * * * * * * * * * R 10 *0.0000E+00*0.0000E+00*0.2370 * 110.8 * 642* * +/- *0.0000E+00*0.0000E+00*0.2296E-01* 6.225 * * * * * * * * * * R 12 *0.0000E+00*0.0000E+00*0.7255E-01* 102.2 * 213* * +/- *0.0000E+00*0.0000E+00*0.7541E-02* 7.903 * * * * * * * * * * R 13 *0.0000E+00*0.0000E+00*0.1787 * 114.3 * 469* * +/- *0.0000E+00*0.0000E+00*0.2146E-01* 8.399 * *


SIMULATION TECHNIQUESCommunication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * R 14 *0.0000E+00*0.0000E+00*0.2829 * 116.2 * 730* * +/- *0.0000E+00*0.0000E+00*0.3170E-01* 6.388 * * * * * * * * * * R 15 *0.0000E+00*0.0000E+00*0.8914E-01* 110.0 * 243* * +/- *0.0000E+00*0.0000E+00*0.1678E-01* 9.066 * * * * * * * * * ******************************************************************* ... END OF SIMULATION ... 111.2 144.0 111.4 105.9 107.3 99.80 137.5 110.4 103.1 111.4 117.1 110.8



Token ring network 8 nodes

Uniform traffic



Token ring network

1 /DECLARE/ QUEUE INTEGER N, M; 2 QUEUE NUS(8),ESP(8),S,R; 3 INTEGER I; 4 REAL TARR; 5 CUSTOMER REAL TSERV; 6 CUSTOMER INTEGER ORIGEN,DESTI; 7 REF CUSTOMER C,D; 8 FLAG SEM; 9 CLASS TOK,MIS; 10



Token ring network

11 /STATION/ NAME = S; 12 TYPE = SOURCE; 13 SERVICE = BEGIN 14 EXP(TARR); 15 ORIGEN := RINT(1,8); 16 DESTI := RINT(1,7); 17 IF DESTI >= ORIGEN THEN DESTI := DESTI + 1; 18 TSERV := EXP(800.); & microsegons 19 C := NEW(CUSTOMER); 20 TRANSIT(C,R,MIS); 21 TRANSIT(ESP(ORIGEN),MIS); 22 END; 23



Token ring network

24 /STATION/ NAME = NUS; 25 TYPE = MULTIPLE(2); 26 SERVICE(TOK) = BEGIN 27 WHILE ESP(N).NB > 0 DO 28 BEGIN 29 D := ESP(N).FIRST; 30 CST(D.TSERV); 31 TRANSIT(D,NUS(M)); 32 UNSET(SEM); 33 WAIT(SEM); 34 END; 35 CST(20); & microsegons 36 TRANSIT(NUS(M)); 37 END;


SIMULATION TECHNIQUESToken ring network

38 SERVICE(MIS) = BEGIN 39 IF N = ORIGEN THEN 40 BEGIN 41 SET(SEM); 42 TRANSIT(OUT); 43 END; 44 IF N = DESTI THEN 45 BEGIN 46 C := R.FIRST; 47 WHILE SON <> C DO C := C.NEXT; 48 TRANSIT(C,OUT); 49 END; 50 CST(1.6); & microsegons 51 TRANSIT(NUS(M)); 52 END; 53 54 /STATION/ NAME = NUS(1); 55 INIT(TOK) = 1;


SIMULATION TECHNIQUESToken ring network 57 /CONTROL/ TMAX = 10000000; CLASS = ALL QUEUE; 58 ACCURACY = ALL QUEUE, ALL CLASS; 59 60 /EXEC/ BEGIN 61 FOR I := 1 STEP 1 UNTIL 8 DO 62 BEGIN 63 NUS(I).N := I; 64 ESP(I).N := I; 65 NUS(I).M := I + 1; 66 END; 67 NUS(8).M := 1; 68 FOR TARR := 100000, 10000, 1000, 800 DO 69 BEGIN 70 PRINT; 71 PRINT ("TEMPS ENTRE ARRIBADES ",TARR,"MICROSEG"); 72 SIMUL; 73 END; 74 END;


SIMULATION TECHNIQUESToken ring network TEMPS ENTRE ARRIBADES 0.1000E+06MICROSEG

***SIMULATION WITH SPECTRAL METHOD ***... TIME = 10000000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB ********************************************************************* NUS 1 * 20.11 *0.6248E-01*0.1250 * 20.11 * 62141** +/- *0.1347 *0.3563E-03*0.7126E-03*0.1347 * **(TOK )* 20.14 *0.6247E-01*0.1249 * 20.14 * 62038** +/- *0.1349 *0.3569E-03*0.7137E-03*0.1349 * **(MIS )* 1.398 *0.7199E-05*0.1440E-04* 1.398 * 103** +/- *-1.000 *0.1365E-05*0.2730E-05*-1.000 * *


SIMULATION TECHNIQUESToken ring network* NUS 2 * 20.07 *0.6237E-01*0.1247 * 20.07 * 62140** +/- *0.8790E-01*0.3143E-03*0.6286E-03*0.8790E-01* **(TOK )* 20.10 *0.6236E-01*0.1247 * 20.10 * 62037** +/- *0.8881E-01*0.3151E-03*0.6302E-03*0.8881E-01* **(MIS )* 1.382 *0.7119E-05*0.1424E-04* 1.382 * 103** +/- *-1.000 *0.1691E-05*0.3381E-05*-1.000 * ** NUS 3 * 20.09 *0.6242E-01*0.1248 * 20.09 * 62140** +/- *0.7185E-01*0.2305E-03*0.4610E-03*0.7185E-01* **(TOK )* 20.12 *0.6241E-01*0.1248 * 20.12 * 62037** +/- *0.7276E-01*0.2315E-03*0.4629E-03*0.7276E-01* **(MIS )* 1.351 *0.6959E-05*0.1392E-04* 1.351 * 103** +/- *-1.000 *0.1862E-05*0.3724E-05*-1.000 * ** NUS 4 * 20.08 *0.6239E-01*0.1248 * 20.08 * 62140** +/- *0.1148 *0.3504E-03*0.7009E-03*0.1148 * **(TOK )* 20.11 *0.6238E-01*0.1248 * 20.11 * 62037** +/- *0.1140 *0.3514E-03*0.7028E-03*0.1140 * **(MIS )* 1.522 *0.7839E-05*0.1568E-04* 1.522 * 103** +/- *-1.000 *0.1646E-05*0.3292E-05*-1.000 * *


SIMULATION TECHNIQUESToken ring network* NUS 5 * 20.19 *0.6274E-01*0.1255 * 20.19 * 62140** +/- *0.1325 *0.3989E-03*0.7978E-03*0.1325 * **(TOK )* 20.22 *0.6273E-01*0.1255 * 20.22 * 62037** +/- *0.1470 *0.3994E-03*0.7988E-03*0.1470 * **(MIS )* 1.382 *0.7119E-05*0.1424E-04* 1.382 * 103** +/- *-1.000 *0.1337E-05*0.2674E-05*-1.000 * ** NUS 6 * 20.11 *0.6249E-01*0.1250 * 20.11 * 62140** +/- *0.1433 *0.4475E-03*0.8951E-03*0.1433 * **(TOK )* 20.14 *0.6248E-01*0.1250 * 20.14 * 62037** +/- *0.1429 *0.4480E-03*0.8960E-03*0.1429 * **(MIS )* 1.398 *0.7199E-05*0.1440E-04* 1.398 * 103** +/- *-1.000 *0.1526E-05*0.3053E-05*-1.000 * ** NUS 7 * 20.11 *0.6249E-01*0.1250 * 20.11 * 62140** +/- *0.1094 *0.3093E-03*0.6186E-03*0.1094 * **(TOK )* 20.15 *0.6249E-01*0.1250 * 20.15 * 62037** +/- *0.9936E-01*0.3094E-03*0.6187E-03*0.9936E-01* **(MIS )* 1.398 *0.7199E-05*0.1440E-04* 1.398 * 103** +/- *-1.000 *0.1483E-05*0.2966E-05*-1.000 * *


SIMULATION TECHNIQUESToken ring network* NUS 8 * 20.17 *0.6268E-01*0.1254 * 20.17 * 62140** +/- *0.1637 *0.4324E-03*0.8647E-03*0.1637 * **(TOK )* 20.20 *0.6267E-01*0.1253 * 20.20 * 62037** +/- *0.1642 *0.4327E-03*0.8654E-03*0.1642 * **(MIS )* 1.367 *0.7039E-05*0.1408E-04* 1.367 * 103** +/- *-1.000 *0.1334E-05*0.2667E-05*-1.000 * ** ESP 1 *0.0000E+00*0.0000E+00*0.9553E-03* 734.8 * 13** +/- *-1.000 *0.0000E+00*0.8521E-03*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.9553E-03* 734.8 * 13** +/- *-1.000 *0.0000E+00*0.8521E-03*-1.000 * ** ESP 2 *0.0000E+00*0.0000E+00*0.7404E-03* 528.8 * 14** +/- *-1.000 *0.0000E+00*0.6695E-03*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.7404E-03* 528.8 * 14** +/- *-1.000 *0.0000E+00*0.6695E-03*-1.000 * ** ESP 3 *0.0000E+00*0.0000E+00*0.8659E-03* 541.2 * 16** +/- *-1.000 *0.0000E+00*0.4874E-03*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.8659E-03* 541.2 * 16** +/- *-1.000 *0.0000E+00*0.4874E-03*-1.000 * *



Token ring network* ESP 4 *0.0000E+00*0.0000E+00*0.7169E-03* 1434. * 5** +/- *-1.000 *0.0000E+00*0.6916E-03*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.7169E-03* 1434. * 5** +/- *-1.000 *0.0000E+00*0.6916E-03*-1.000 * ** ESP 5 *0.0000E+00*0.0000E+00*0.1484E-02* 1060. * 14** +/- *-1.000 *0.0000E+00*0.9771E-03*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.1484E-02* 1060. * 14** +/- *-1.000 *0.0000E+00*0.9771E-03*-1.000 * ** ESP 6 *0.0000E+00*0.0000E+00*0.9811E-03* 754.7 * 13** +/- *-1.000 *0.0000E+00*0.9433E-03*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.9811E-03* 754.7 * 13** +/- *-1.000 *0.0000E+00*0.9433E-03*-1.000 * ** ESP 7 *0.0000E+00*0.0000E+00*0.9871E-03* 759.3 * 13** +/- *-1.000 *0.0000E+00*0.6366E-03*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.9871E-03* 759.3 * 13** +/- *-1.000 *0.0000E+00*0.6366E-03*-1.000 * *


SIMULATION TECHNIQUESToken ring network* ESP 8 *0.0000E+00*0.0000E+00*0.1368E-02* 912.0 * 15** +/- *-1.000 *0.0000E+00*0.1041E-02*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.1368E-02* 912.0 * 15** +/- *-1.000 *0.0000E+00*0.1041E-02*-1.000 * ** S *0.9609E+05* 1.000 * 1.000 *0.9609E+05* 103** +/- *-1.000 *0.0000E+00*0.0000E+00*-1.000 * ** * * * * * ** R *0.0000E+00*0.0000E+00*0.8148E-02* 791.0 * 103** +/- *-1.000 *0.0000E+00*0.2265E-02*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.8148E-02* 791.0 * 103** +/- *-1.000 *0.0000E+00*0.2265E-02*-1.000 * ** * * * * * ********************************************************************... END OF SIMULATION ...


SIMULATION TECHNIQUESToken ring network TEMPS ENTRE ARRIBADES 0.1000E+05MICROSEG

***SIMULATION WITH SPECTRAL METHOD ***... TIME = 10000000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB ********************************************************************* NUS 1 * 21.31 *0.6245E-01*0.1249 * 21.31 * 58612** +/- *0.4077 *0.1228E-02*0.2456E-02*0.4077 * **(TOK )* 21.66 *0.6238E-01*0.1248 * 21.66 * 57608** +/- *0.4775 *0.1230E-02*0.2460E-02*0.4775 * **(MIS )* 1.402 *0.7039E-04*0.1408E-03* 1.402 * 1004** +/- *0.3061E-01*0.5425E-05*0.1085E-04*0.3061E-01* *


SIMULATION TECHNIQUESToken ring network* NUS 2 * 21.70 *0.6360E-01*0.1272 * 21.70 * 58612** +/- *0.5427 *0.1433E-02*0.2866E-02*0.5427 * **(TOK )* 22.06 *0.6353E-01*0.1271 * 22.06 * 57608** +/- *0.5847 *0.1434E-02*0.2868E-02*0.5847 * **(MIS )* 1.381 *0.6935E-04*0.1387E-03* 1.381 * 1004** +/- *0.4022E-01*0.4872E-05*0.9745E-05*0.4022E-01* ** NUS 3 * 21.54 *0.6314E-01*0.1263 * 21.54 * 58612** +/- *0.5847 *0.1705E-02*0.3410E-02*0.5847 * **(TOK )* 21.90 *0.6307E-01*0.1261 * 21.90 * 57608** +/- *0.5530 *0.1706E-02*0.3412E-02*0.5530 * **(MIS )* 1.393 *0.6991E-04*0.1398E-03* 1.393 * 1004** +/- *0.3717E-01*0.4980E-05*0.9959E-05*0.3717E-01* ** NUS 4 * 21.30 *0.6241E-01*0.1248 * 21.30 * 58611** +/- *0.4710 *0.1174E-02*0.2348E-02*0.4710 * **(TOK )* 21.64 *0.6234E-01*0.1247 * 21.64 * 57607** +/- *0.4629 *0.1175E-02*0.2349E-02*0.4629 * **(MIS )* 1.399 *0.7023E-04*0.1405E-03* 1.399 * 1004** +/- *0.3573E-01*0.4258E-05*0.8516E-05*0.3573E-01* *


SIMULATION TECHNIQUESToken ring network* NUS 5 * 21.02 *0.6161E-01*0.1232 * 21.02 * 58611** +/- *0.4159 *0.9359E-03*0.1872E-02*0.4159 * **(TOK )* 21.37 *0.6154E-01*0.1231 * 21.37 * 57607** +/- *0.4241 *0.9370E-03*0.1874E-02*0.4241 * **(MIS )* 1.415 *0.7103E-04*0.1421E-03* 1.415 * 1004** +/- *0.3856E-01*0.7813E-05*0.1563E-04*0.3856E-01* ** NUS 6 * 21.06 *0.6171E-01*0.1234 * 21.06 * 58611** +/- *0.4502 *0.1248E-02*0.2497E-02*0.4502 * **(TOK )* 21.40 *0.6164E-01*0.1233 * 21.40 * 57607** +/- *0.4786 *0.1249E-02*0.2499E-02*0.4786 * **(MIS )* 1.413 *0.7095E-04*0.1419E-03* 1.413 * 1004** +/- *0.3104E-01*0.4489E-05*0.8978E-05*0.3104E-01* ** NUS 7 * 21.39 *0.6269E-01*0.1254 * 21.39 * 58611** +/- *0.4933 *0.1266E-02*0.2532E-02*0.4933 * **(TOK )* 21.74 *0.6262E-01*0.1252 * 21.74 * 57607** +/- *0.3923 *0.1268E-02*0.2537E-02*0.3923 * **(MIS )* 1.410 *0.7079E-04*0.1416E-03* 1.410 * 1004** +/- *0.3545E-01*0.7663E-05*0.1533E-04*0.3545E-01* *


SIMULATION TECHNIQUESToken ring network* NUS 8 * 21.48 *0.6294E-01*0.1259 * 21.48 * 58611** +/- *0.6782 *0.1235E-02*0.2471E-02*0.6782 * **(TOK )* 21.83 *0.6287E-01*0.1257 * 21.83 * 57607** +/- *0.7549 *0.1235E-02*0.2470E-02*0.7549 * **(MIS )* 1.385 *0.6951E-04*0.1390E-03* 1.385 * 1004** +/- *0.3685E-01*0.4594E-05*0.9188E-05*0.3685E-01* ** ESP 1 *0.0000E+00*0.0000E+00*0.1149E-01* 926.7 * 124** +/- *-1.000 *0.0000E+00*0.3546E-02*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.1149E-01* 926.7 * 124** +/- *-1.000 *0.0000E+00*0.3546E-02*-1.000 * ** ESP 2 *0.0000E+00*0.0000E+00*0.1344E-01* 980.8 * 137** +/- *0.0000E+00*0.0000E+00*0.3167E-02* 125.4 * **(MIS )*0.0000E+00*0.0000E+00*0.1344E-01* 980.8 * 137** +/- *0.0000E+00*0.0000E+00*0.3167E-02* 125.4 * ** ESP 3 *0.0000E+00*0.0000E+00*0.1317E-01* 1013. * 130** +/- *0.0000E+00*0.0000E+00*0.3675E-02* 180.5 * **(MIS )*0.0000E+00*0.0000E+00*0.1317E-01* 1013. * 130** +/- *0.0000E+00*0.0000E+00*0.3675E-02* 180.5 * *



Token ring network* ESP 4 *0.0000E+00*0.0000E+00*0.1155E-01* 916.4 * 126** +/- *-1.000 *0.0000E+00*0.3341E-02*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.1155E-01* 916.4 * 126** +/- *-1.000 *0.0000E+00*0.3341E-02*-1.000 * ** ESP 5 *0.0000E+00*0.0000E+00*0.1060E-01* 914.1 * 116** +/- *-1.000 *0.0000E+00*0.3776E-02*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.1060E-01* 914.1 * 116** +/- *-1.000 *0.0000E+00*0.3776E-02*-1.000 * ** ESP 6 *0.0000E+00*0.0000E+00*0.9766E-02* 834.7 * 117** +/- *-1.000 *0.0000E+00*0.2652E-02*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.9766E-02* 834.7 * 117** +/- *-1.000 *0.0000E+00*0.2652E-02*-1.000 * ** ESP 7 *0.0000E+00*0.0000E+00*0.1208E-01* 1015. * 119** +/- *-1.000 *0.0000E+00*0.3174E-02*-1.000 * **(MIS )*0.0000E+00*0.0000E+00*0.1208E-01* 1015. * 119** +/- *-1.000 *0.0000E+00*0.3174E-02*-1.000 * *


SIMULATION TECHNIQUESToken ring network* ESP 8 *0.0000E+00*0.0000E+00*0.1295E-01* 959.5 * 135** +/- *0.0000E+00*0.0000E+00*0.3163E-02* 200.6 * **(MIS )*0.0000E+00*0.0000E+00*0.1295E-01* 959.5 * 135** +/- *0.0000E+00*0.0000E+00*0.3163E-02* 200.6 * ** S * 9958. * 1.000 * 1.000 * 9958. * 1004** +/- * 670.0 *0.0000E+00*0.0000E+00* 670.0 * ** * * * * * ** R *0.0000E+00*0.0000E+00*0.9552E-01* 951.4 * 1004** +/- *0.0000E+00*0.0000E+00*0.9458E-02* 75.77 * **(MIS )*0.0000E+00*0.0000E+00*0.9552E-01* 951.4 * 1004** +/- *0.0000E+00*0.0000E+00*0.9458E-02* 75.77 * ** * * * * * ********************************************************************... END OF SIMULATION ...


SIMULATION TECHNIQUESToken ring network TEMPS ENTRE ARRIBADES 1000. MICROSEG

***SIMULATION WITH SPECTRAL METHOD ***... TIME = 10000000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB ********************************************************************* * * * * * ** NUS 1 * 61.63 *0.6358E-01*0.1272 * 61.63 * 20631** +/- * 6.193 *0.3718E-02*0.7435E-02* 6.193 * **(TOK )* 119.1 *0.6288E-01*0.1258 * 119.1 * 10562** +/- * 20.65 *0.3716E-02*0.7433E-02* 20.65 * **(MIS )* 1.390 *0.6999E-03*0.1400E-02* 1.390 * 10069** +/- *0.9350E-02*0.1776E-04*0.3552E-04*0.9350E-02* *


SIMULATION TECHNIQUESToken ring network* NUS 2 * 63.39 *0.6539E-01*0.1308 * 63.39 * 20631** +/- * 4.534 *0.3894E-02*0.7787E-02* 4.534 * **(TOK )* 122.5 *0.6468E-01*0.1294 * 122.5 * 10562** +/- * 16.47 *0.3904E-02*0.7809E-02* 16.47 * **(MIS )* 1.396 *0.7026E-03*0.1405E-02* 1.396 * 10069** +/- *0.1102E-01*0.2228E-04*0.4456E-04*0.1102E-01* ** NUS 3 * 62.98 *0.6497E-01*0.1299 * 62.98 * 20631** +/- * 5.727 *0.3575E-02*0.7150E-02* 5.727 * **(TOK )* 121.7 *0.6427E-01*0.1285 * 121.7 * 10562** +/- * 29.96 *0.3579E-02*0.7158E-02* 29.96 * **(MIS )* 1.389 *0.6994E-03*0.1399E-02* 1.389 * 10069** +/- *0.9545E-02*0.1971E-04*0.3942E-04*0.9545E-02* ** NUS 4 * 60.30 *0.6220E-01*0.1244 * 60.30 * 20631** +/- * 5.706 *0.4437E-02*0.8874E-02* 5.706 * **(TOK )* 116.4 *0.6149E-01*0.1230 * 116.4 * 10562** +/- * 19.09 *0.4432E-02*0.8865E-02* 19.09 * **(MIS )* 1.409 *0.7095E-03*0.1419E-02* 1.409 * 10069** +/- *0.1115E-01*0.2008E-04*0.4017E-04*0.1115E-01* *


SIMULATION TECHNIQUESToken ring network* NUS 5 * 60.82 *0.6274E-01*0.1255 * 60.82 * 20631** +/- * 7.048 *0.4138E-02*0.8275E-02* 7.048 * **(TOK )* 117.5 *0.6203E-01*0.1241 * 117.5 * 10562** +/- * 19.66 *0.4143E-02*0.8285E-02* 19.66 * **(MIS )* 1.404 *0.7069E-03*0.1414E-02* 1.404 * 10069** +/- *0.1153E-01*0.1863E-04*0.3726E-04*0.1153E-01* ** NUS 6 * 58.71 *0.6056E-01*0.1211 * 58.71 * 20631** +/- * 6.512 *0.4101E-02*0.8203E-02* 6.512 * **(TOK )* 113.3 *0.5985E-01*0.1197 * 113.3 * 10562** +/- * 30.61 *0.4105E-02*0.8210E-02* 30.61 * **(MIS )* 1.411 *0.7102E-03*0.1420E-02* 1.411 * 10069** +/- *0.1168E-01*0.2026E-04*0.4051E-04*0.1168E-01* ** NUS 7 * 57.14 *0.5913E-01*0.1183 * 57.14 * 20630** +/- * 6.698 *0.3162E-02*0.6325E-02* 6.698 * **(TOK )* 110.3 *0.5842E-01*0.1168 * 110.3 * 10561** +/- * 18.34 *0.3164E-02*0.6328E-02* 18.34 * **(MIS )* 1.409 *0.7096E-03*0.1419E-02* 1.409 * 10069** +/- *0.9951E-02*0.2002E-04*0.4005E-04*0.9951E-02* *


SIMULATION TECHNIQUESToken ring network* NUS 8 * 65.03 *0.6708E-01*0.1342 * 65.03 * 20630** +/- * 6.675 *0.3661E-02*0.7321E-02* 6.675 * **(TOK )* 125.7 *0.6638E-01*0.1328 * 125.7 * 10561** +/- * 32.18 *0.3660E-02*0.7320E-02* 32.18 * **(MIS )* 1.390 *0.6998E-03*0.1400E-02* 1.390 * 10069** +/- *0.1438E-01*0.1897E-04*0.3795E-04*0.1438E-01* ** ESP 1 *0.0000E+00*0.0000E+00*0.7598 * 5756. * 1319** +/- *0.0000E+00*0.0000E+00*0.1753 * 1254. * **(MIS )*0.0000E+00*0.0000E+00*0.7598 * 5756. * 1319** +/- *0.0000E+00*0.0000E+00*0.1753 * 1254. * ** ESP 2 *0.0000E+00*0.0000E+00*0.7163 * 5564. * 1285** +/- *0.0000E+00*0.0000E+00*0.1420 * 1172. * **(MIS )*0.0000E+00*0.0000E+00*0.7163 * 5564. * 1285** +/- *0.0000E+00*0.0000E+00*0.1420 * 1172. * ** ESP 3 *0.0000E+00*0.0000E+00*0.7218 * 5436. * 1325** +/- *0.0000E+00*0.0000E+00*0.1340 * 1028. * **(MIS )*0.0000E+00*0.0000E+00*0.7218 * 5436. * 1325** +/- *0.0000E+00*0.0000E+00*0.1340 * 1028. * *



Token ring network* ESP 4 *0.0000E+00*0.0000E+00*0.6485 * 5402. * 1199** +/- *0.0000E+00*0.0000E+00*0.1442 * 1177. * **(MIS )*0.0000E+00*0.0000E+00*0.6485 * 5402. * 1199** +/- *0.0000E+00*0.0000E+00*0.1442 * 1177. * ** ESP 5 *0.0000E+00*0.0000E+00*0.7021 * 5699. * 1232** +/- *0.0000E+00*0.0000E+00*0.1777 * 1054. * **(MIS )*0.0000E+00*0.0000E+00*0.7021 * 5699. * 1232** +/- *0.0000E+00*0.0000E+00*0.1777 * 1054. * ** ESP 6 *0.0000E+00*0.0000E+00*0.6706 * 5635. * 1190** +/- *0.0000E+00*0.0000E+00*0.1593 * 1203. * **(MIS )*0.0000E+00*0.0000E+00*0.6706 * 5635. * 1190** +/- *0.0000E+00*0.0000E+00*0.1593 * 1203. * ** ESP 7 *0.0000E+00*0.0000E+00*0.6491 * 5413. * 1198** +/- *0.0000E+00*0.0000E+00*0.1391 * 1111. * **(MIS )*0.0000E+00*0.0000E+00*0.6491 * 5413. * 1198** +/- *0.0000E+00*0.0000E+00*0.1391 * 1111. * *


SIMULATION TECHNIQUESToken ring network* ESP 8 *0.0000E+00*0.0000E+00*0.6883 * 5200. * 1321** +/- *0.0000E+00*0.0000E+00*0.1291 * 907.1 * **(MIS )*0.0000E+00*0.0000E+00*0.6883 * 5200. * 1321** +/- *0.0000E+00*0.0000E+00*0.1291 * 907.1 * ** S * 992.0 * 1.000 * 1.000 * 992.0 * 10079** +/- * 23.59 *0.0000E+00*0.0000E+00* 23.59 * ** * * * * * ** R *0.0000E+00*0.0000E+00* 5.561 * 5517. * 10069** +/- *0.0000E+00*0.0000E+00* 1.161 * 1063. * **(MIS )*0.0000E+00*0.0000E+00* 5.561 * 5517. * 10069** +/- *0.0000E+00*0.0000E+00* 1.161 * 1063. * ** * * * * * ********************************************************************... END OF SIMULATION ...


SIMULATION TECHNIQUESToken ring network TEMPS ENTRE ARRIBADES 800.0 MICROSEG

***SIMULATION WITH SPECTRAL METHOD ***... TIME = 10000000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB ********************************************************************* NUS 1 * 109.5 *0.6738E-01*0.1348 * 109.5 * 12306** +/- * 6.905 *0.3586E-02*0.7173E-02* 6.905 * **(TOK )*0.2511E+05*0.6653E-01*0.1331 *0.2511E+05* 53** +/- *-1.000 *0.3590E-02*0.7179E-02*-1.000 * **(MIS )* 1.388 *0.8504E-03*0.1701E-02* 1.388 * 12253** +/- *0.6208E-02*0.9743E-05*0.1949E-04*0.6208E-02* *


SIMULATION TECHNIQUESToken ring network* NUS 2 * 101.4 *0.6239E-01*0.1248 * 101.4 * 12306** +/- * 5.621 *0.2796E-02*0.5592E-02* 5.621 * **(TOK )*0.2322E+05*0.6154E-01*0.1231 *0.2322E+05* 53** +/- *-1.000 *0.2801E-02*0.5602E-02*-1.000 * **(MIS )* 1.401 *0.8583E-03*0.1717E-02* 1.401 * 12253** +/- *0.6746E-02*0.1243E-04*0.2486E-04*0.6746E-02* ** NUS 3 * 96.70 *0.5950E-01*0.1190 * 96.70 * 12306** +/- * 5.425 *0.3074E-02*0.6148E-02* 5.425 * **(TOK )*0.2213E+05*0.5864E-01*0.1173 *0.2213E+05* 53** +/- *-1.000 *0.3074E-02*0.6148E-02*-1.000 * **(MIS )* 1.409 *0.8635E-03*0.1727E-02* 1.409 * 12253** +/- *0.6752E-02*0.1281E-04*0.2563E-04*0.6752E-02* ** NUS 4 * 102.6 *0.6311E-01*0.1262 * 102.6 * 12306** +/- * 5.116 *0.2576E-02*0.5152E-02* 5.116 * **(TOK )*0.2349E+05*0.6225E-01*0.1245 *0.2349E+05* 53** +/- *-1.000 *0.2576E-02*0.5152E-02*-1.000 * **(MIS )* 1.407 *0.8622E-03*0.1724E-02* 1.407 * 12253** +/- *0.6784E-02*0.1131E-04*0.2261E-04*0.6784E-02* *


SIMULATION TECHNIQUESToken ring network* NUS 5 * 107.5 *0.6615E-01*0.1323 * 107.5 * 12306** +/- * 3.395 *0.1742E-02*0.3485E-02* 3.395 * **(TOK )*0.2464E+05*0.6529E-01*0.1306 *0.2464E+05* 53** +/- *-1.000 *0.1747E-02*0.3494E-02*-1.000 * **(MIS )* 1.399 *0.8573E-03*0.1715E-02* 1.399 * 12253** +/- *0.5035E-02*0.1366E-04*0.2733E-04*0.5035E-02* ** NUS 6 * 97.64 *0.6008E-01*0.1202 * 97.64 * 12306** +/- * 3.719 *0.3193E-02*0.6385E-02* 3.719 * **(TOK )*0.2235E+05*0.5922E-01*0.1184 *0.2235E+05* 53** +/- *-1.000 *0.3198E-02*0.6396E-02*-1.000 * **(MIS )* 1.402 *0.8588E-03*0.1718E-02* 1.402 * 12253** +/- *0.6413E-02*0.1194E-04*0.2389E-04*0.6413E-02* ** NUS 7 * 104.8 *0.6473E-01*0.1295 * 104.8 * 12305** +/- * 4.550 *0.3379E-02*0.6758E-02* 4.550 * **(TOK )*0.2446E+05*0.6387E-01*0.1277 *0.2446E+05* 52** +/- *-1.000 *0.3384E-02*0.6769E-02*-1.000 * **(MIS )* 1.396 *0.8553E-03*0.1711E-02* 1.396 * 12253** +/- *0.4878E-02*0.1412E-04*0.2824E-04*0.4878E-02* *


SIMULATION TECHNIQUESToken ring network* NUS 8 * 103.2 *0.6352E-01*0.1270 * 103.2 * 12305** +/- * 4.519 *0.2117E-02*0.4234E-02* 4.519 * **(TOK )*0.2410E+05*0.6266E-01*0.1253 *0.2410E+05* 52** +/- *-1.000 *0.2115E-02*0.4230E-02*-1.000 * **(MIS )* 1.396 *0.8551E-03*0.1710E-02* 1.396 * 12253** +/- *0.5278E-02*0.1169E-04*0.2338E-04*0.5278E-02* ** ESP 1 *0.0000E+00*0.0000E+00* 20.89 *0.1285E+06* 1622** +/- *0.0000E+00*0.0000E+00* 2.330 *0.1118E+05* **(MIS )*0.0000E+00*0.0000E+00* 20.89 *0.1285E+06* 1622** +/- *0.0000E+00*0.0000E+00* 2.330 *0.1118E+05* ** ESP 2 *0.0000E+00*0.0000E+00* 19.79 *0.1297E+06* 1523** +/- *0.0000E+00*0.0000E+00* 2.213 *0.1044E+05* **(MIS )*0.0000E+00*0.0000E+00* 19.79 *0.1297E+06* 1523** +/- *0.0000E+00*0.0000E+00* 2.213 *0.1044E+05* ** ESP 3 *0.0000E+00*0.0000E+00* 18.60 *0.1274E+06* 1458** +/- *0.0000E+00*0.0000E+00* 1.609 *0.1332E+05* **(MIS )*0.0000E+00*0.0000E+00* 18.60 *0.1274E+06* 1458** +/- *0.0000E+00*0.0000E+00* 1.609 *0.1332E+05* *



Token ring network* ESP 4 *0.0000E+00*0.0000E+00* 18.17 *0.1232E+06* 1474** +/- *0.0000E+00*0.0000E+00* 1.572 *0.1280E+05* **(MIS )*0.0000E+00*0.0000E+00* 18.17 *0.1232E+06* 1474** +/- *0.0000E+00*0.0000E+00* 1.572 *0.1280E+05* ** ESP 5 *0.0000E+00*0.0000E+00* 19.04 *0.1240E+06* 1535** +/- *0.0000E+00*0.0000E+00* 2.000 *0.1414E+05* **(MIS )*0.0000E+00*0.0000E+00* 19.04 *0.1240E+06* 1535** +/- *0.0000E+00*0.0000E+00* 2.000 *0.1414E+05* ** ESP 6 *0.0000E+00*0.0000E+00* 19.32 *0.1273E+06* 1517** +/- *0.0000E+00*0.0000E+00* 2.245 *0.1378E+05* **(MIS )*0.0000E+00*0.0000E+00* 19.32 *0.1273E+06* 1517** +/- *0.0000E+00*0.0000E+00* 2.245 *0.1378E+05* ** ESP 7 *0.0000E+00*0.0000E+00* 20.31 *0.1297E+06* 1561** +/- *0.0000E+00*0.0000E+00* 1.996 *0.1299E+05* **(MIS )*0.0000E+00*0.0000E+00* 20.31 *0.1297E+06* 1561** +/- *0.0000E+00*0.0000E+00* 1.996 *0.1299E+05* *


SIMULATION TECHNIQUESToken ring network* ESP 8 *0.0000E+00*0.0000E+00* 19.63 *0.1249E+06* 1563** +/- *0.0000E+00*0.0000E+00* 1.951 *0.1021E+05* **(MIS )*0.0000E+00*0.0000E+00* 19.63 *0.1249E+06* 1563** +/- *0.0000E+00*0.0000E+00* 1.951 *0.1021E+05* ** S * 811.1 * 1.000 * 1.000 * 811.1 * 12328** +/- * 14.66 *0.0000E+00*0.0000E+00* 14.66 * ** * * * * * ** R *0.0000E+00*0.0000E+00* 155.7 *0.1269E+06* 12253** +/- *0.0000E+00*0.0000E+00* 24.78 *0.2250E+05* **(MIS )*0.0000E+00*0.0000E+00* 155.7 *0.1269E+06* 12253** +/- *0.0000E+00*0.0000E+00* 24.78 *0.2250E+05* ** * * * * * ********************************************************************... END OF SIMULATION ...



Ethernet network 8 nodes

Uniform traffic



Ethernet network

1 /DECLARE/ QUEUE S, EST(8), BUS, R, SF; 2 CUSTOMER INTEGER I; 3 CUSTOMER REAL TSERV, TESP; 4 REAL TARR, SERV, ESP, TEMPS, TPROP, TEMPSC; 5 REF CUSTOMER C,D; 6 LABEL L; 7 FLAG FL, BL; 8 INTEGER CONF = 0; 9



Ethernet network

10 /STATION/ NAME = S; 11 TYPE = SOURCE; 12 SERVICE = BEGIN 13 EXP(TARR); 14 I := RINT(1,8); 15 TSERV := EXP(SERV); 16 IF TSERV < 3.*TPROP THEN TSERV := 3.*TPROP; 17 C := NEW(CUSTOMER); 18 TRANSIT(C,R); 19 TRANSIT(EST(I)); 20 END; 21



Ethernet network

22 /STATION/ NAME = EST; 23 SERVICE = BEGIN 24 L: IF BUS.NB = 0 THEN 25 BEGIN 26 C := NEW(CUSTOMER); 27 C.TSERV := TSERV; 28 UNSET(FL); 29 UNSET(BL); 30 TEMPS := TIME + TPROP; 31 TRANSIT(C, BUS); 32 WAIT(FL);



Ethernet network

33 IF CONF = 0 THEN 34 BEGIN 35 SET(BL); 36 TRANSIT(OUT); 37 END; 38 SET(BL); 39 CONF := 0; 40 TESP := EXP(ESP); 41 IF TESP<2.*TPROP THEN TESP := 2. * TPROP; 42 CST(TESP); 43 GOTO L; 44 END;



Ethernet network

45 IF (TIME < TEMPS) AND (CONF = 0) THEN 46 BEGIN 47 CONF := 1; 48 D := BUS.FIRST; 49 TEMPSC := TEMPS - TIME; 50 CST(TEMPSC); 51 TRANSIT(D, OUT); 52 SET(FL); 53 TESP := EXP(ESP); 54 IF TESP<2.*TPROP THEN TESP := 2. * TPROP; 55 CST(TESP); 56 GOTO L; 57 END;



Ethernet network

58 IF (TIME < TEMPS) AND (CONF = 1) THEN 59 BEGIN 60 TESP := EXP(ESP); 61 IF TESP<2.*TPROP THEN TESP := 2. * TPROP; 62 CST(TESP); 63 GOTO L; 64 END; 65 WAIT(BL); 66 GOTO L; 67 END; 68



Ethernet network

69 /STATION/ NAME = BUS; 70 SERVICE = BEGIN 71 CST(TSERV + 2.*TPROP); 72 C := R.FIRST; 73 WHILE C.FATHER <> FATHER DO C := C.NEXT; 74 TRANSIT(C, OUT); 75 SET(FL); 76 TRANSIT(OUT); 77 END; 78



Ethernet network 79 /STATION/ NAME = SF; 80 INIT = 1; 81 SERVICE = BEGIN 82 SET(FL); 83 SET(BL); 84 TRANSIT(OUT); 85 END; 86 87 /CONTROL/ TMAX = 100000.; ACCURACY = ALL QUEUE; 88 89 /EXEC/ BEGIN 90 TPROP := 0.01; 91 SERV := 0.8; 92 ESP := 0.1; 93 TEMPS := -TPROP; 94 FOR TARR := 5, 2.5, 1.25 DO SIMUL; 95 END;


SIMULATION TECHNIQUESEthernet network ***SIMULATION WITH SPECTRAL METHOD ***

... TIME = 100000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95

*******************************************************************

* NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *

*******************************************************************

* S * 5.073 * 1.000 * 1.000 * 5.073 * 19713*

* +/- *0.7027E-01*0.0000E+00*0.0000E+00*0.7027E-01* *

* EST 1 *0.9595 *0.2390E-01*0.2459E-01*0.9873 * 2491*

* +/- *0.4293E-01*0.1102E-02*0.1180E-02*0.5266E-01* *

* EST 2 *0.9714 *0.2364E-01*0.2425E-01*0.9966 * 2433*

* +/- *0.3115E-01*0.1343E-02*0.1491E-02*0.3891E-01* *

* EST 3 *0.9993 *0.2534E-01*0.2609E-01* 1.029 * 2536*

* +/- *0.4504E-01*0.1101E-02*0.1159E-02*0.4904E-01* *

* EST 4 *0.9660 *0.2350E-01*0.2407E-01*0.9892 * 2433*

* +/- *0.4483E-01*0.1456E-02*0.1518E-02*0.7335E-01* *

* EST 5 *0.9709 *0.2321E-01*0.2356E-01*0.9856 * 2390*

* +/- *0.4695E-01*0.1439E-02*0.1497E-02*0.4730E-01* *



Ethernet network

* EST 6 *0.9677 *0.2312E-01*0.2366E-01*0.9905 * 2389*

* +/- *0.3450E-01*0.9612E-03*0.1328E-02*0.4168E-01* ** EST 7 *0.9455 *0.2308E-01*0.2367E-01*0.9696 * 2441** +/- *0.3795E-01*0.1200E-02*0.1322E-02*0.3643E-01* ** EST 8 *0.9635 *0.2505E-01*0.2576E-01*0.9906 * 2600** +/- *0.3988E-01*0.1589E-02*0.1727E-02*0.4196E-01* ** BUS *0.7994 *0.1624 *0.1624 *0.7994 * 20314** +/- *0.9631E-02*0.2830E-02*0.2830E-02*0.9631E-02* ** R *0.0000E+00*0.0000E+00*0.1956 *0.9924 * 19713** +/- *0.0000E+00*0.0000E+00*0.5210E-02*0.2011E-01* ********************************************************************... END OF SIMULATION ...


SIMULATION TECHNIQUESEthernet network ***SIMULATION WITH SPECTRAL METHOD ***... TIME = 100000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB ********************************************************************* S * 2.523 * 1.000 * 1.000 * 2.523 * 39639** +/- *0.3238E-01*0.0000E+00*0.0000E+00*0.3238E-01* ** EST 1 * 1.190 *0.6058E-01*0.6443E-01* 1.266 * 5091** +/- *0.3643E-01*0.1890E-02*0.2066E-02*0.4254E-01* ** EST 2 * 1.189 *0.5951E-01*0.6372E-01* 1.274 * 5003** +/- *0.3580E-01*0.4234E-02*0.3192E-02*0.4344E-01* ** EST 3 * 1.174 *0.5754E-01*0.6086E-01* 1.242 * 4901** +/- *0.4252E-01*0.2450E-02*0.2627E-02*0.4427E-01* ** EST 4 * 1.191 *0.5883E-01*0.6312E-01* 1.278 * 4937** +/- *0.3555E-01*0.2305E-02*0.2675E-02*0.4592E-01* ** EST 5 * 1.198 *0.5927E-01*0.6323E-01* 1.278 * 4949** +/- *0.4266E-01*0.2633E-02*0.2816E-02*0.4674E-01* *



Ethernet network

* EST 6 * 1.185 *0.5835E-01*0.6266E-01* 1.272 * 4926** +/- *0.3425E-01*0.2123E-02*0.2498E-02*0.4209E-01* ** EST 7 * 1.181 *0.5895E-01*0.6315E-01* 1.265 * 4993** +/- *0.3740E-01*0.2661E-02*0.3207E-02*0.5317E-01* ** EST 8 * 1.182 *0.5717E-01*0.6104E-01* 1.262 * 4837** +/- *0.3262E-01*0.2484E-02*0.3251E-02*0.4343E-01* ** BUS *0.7520 *0.3316 *0.3316 *0.7520 * 44091** +/- *0.1085E-01*0.4436E-02*0.4436E-02*0.1085E-01* ** R *0.0000E+00*0.0000E+00*0.5022 * 1.267 * 39637** +/- *0.0000E+00*0.0000E+00*0.1750E-01*0.2458E-01* ********************************************************************... END OF SIMULATION ...


SIMULATION TECHNIQUESEthernet network ***SIMULATION WITH SPECTRAL METHOD ***... TIME = 100000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95******************************************************************** NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB ********************************************************************* S * 1.257 * 1.000 * 1.000 * 1.257 * 79568** +/- *0.9795E-02*0.0000E+00*0.0000E+00*0.9795E-02* ** EST 1 * 2.045 *0.2052 *0.2709 * 2.699 * 10034** +/- *0.5773E-01*0.8730E-02*0.1612E-01*0.1282 * ** EST 2 * 2.020 *0.2039 *0.2667 * 2.642 * 10096** +/- *0.7101E-01*0.8998E-02*0.1768E-01*0.1390 * ** EST 3 * 2.011 *0.1991 *0.2592 * 2.617 * 9904** +/- *0.6745E-01*0.8805E-02*0.1469E-01*0.1115 * ** EST 4 * 2.028 *0.2000 *0.2602 * 2.637 * 9864** +/- *0.5786E-01*0.6761E-02*0.1179E-01*0.1073 * ** EST 5 * 2.001 *0.2017 *0.2622 * 2.601 * 10079** +/- *0.6415E-01*0.6065E-02*0.1510E-01*0.1624 * *



Ethernet network

* EST 6 * 2.035 *0.1993 *0.2610 * 2.664 * 9797** +/- *0.7039E-01*0.8648E-02*0.1688E-01*0.1333 * ** EST 7 * 2.021 *0.2007 *0.2594 * 2.611 * 9934** +/- *0.6259E-01*0.7496E-02*0.1742E-01*0.2369 * ** EST 8 * 2.014 *0.1985 *0.2571 * 2.608 * 9858** +/- *0.7488E-01*0.7814E-02*0.1403E-01*0.1413 * ** BUS *0.5523 *0.6622 *0.6622 *0.5523 * 119899** +/- *0.5897E-02*0.7369E-02*0.7369E-02*0.5897E-02* ** R *0.0000E+00*0.0000E+00* 2.096 * 2.635 * 79566** +/- *0.0000E+00*0.0000E+00*0.8279E-01*0.1023 * ********************************************************************... END OF SIMULATION ...

PERFORMANCE MODELLING OF COMPUTER SYSTEMS AND COMPUTER NETWORKS

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