Final Report Mean Queue Time Study for the Lu_Kumar Network_Nyan Soe Lynn

8/9/2019 Final Report Mean Queue Time Study for the Lu_Kumar Network_Nyan Soe Lynn

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Acad Year(14/15)

PROJECT

NO.

(B004)

MEAN QUEUE TIME STUDY FOR THE LU-KUMAR

NETWORK

ME

ANQUEUETIM

E


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E

MEAN QUEUE TIME STUDY FOR THE LU-KUMAR NETWORK

SUBMITTED

BY

NYAN SOE LYNN


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NTU Final Year Project Report on:Mean queue time study for the Lu-Kumar network

Table of Contents

Table of Contents ..................................................................................................................... i

ABSTRACT ............................................................................................................................ iv

ACKNOWLWDGEMENTS .................................................................................................... v

Table of Figures ...................................................................................................................... vi

List of Tables ......................................................................................................................... vii

CHAPTER 1 INTRODUCTION ....................................................................................... 1

1.1 Introduction .............................................................................................................. 1

1.2 Objectives ................................................................................................................. 1

1.3 Scope of Work ......................................................................................................... 2

1.4 Organization ............................................................................................................. 3

CHAPTER 2 LITERATURE REVIEW ............................................................................ 4

2 1 Q i h 4


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CHAPTER 3 METHODOLOGY ...................................................................................... 9

3.1 Stability region of the Lu-Kumar network ............................................................... 9

3.1.1 Development of Simulation model .................................................................. 9

3.1.2 Parameter of the simulation model, deterministic case .................................. 18

3.1.3 Manual book-keeping ..................................................................................... 19

3.1.4 Parameters of the simulation model, uniform distribution arrival time ........ 19

3.2 Stability, Instability and their relationship ............................................................. 20

3.2.1 Design of Experiment for the Lu-Kumar network started with single inter-

arrival time of 0.9 ........................................................................................................... 20

3.2.2 Design of Experiment for the Lu-Kumar network started with double inter-

arrival time of 0.9 ........................................................................................................... 21

3.3 Virtual Stations in the Lu-Kumar Network ............................................................ 22

3.3.1 Development of Simulation model ................................................................ 22

3.3.2 Parameters of the simulation model ............................................................... 22

3 3 3 P t f i t l t ti 22


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4.4.1 Relationship between virtual stations and MM1 queue ............................... 101

4.5 Merged Queue at Service Centre B of the Lu-Kumar network ............................ 104

CHAPTER 5 CONCLUSIONS AND RECOMMENDATION ........................................... 106

5.1 CONCLUSIONS .................................................................................................. 106

5.1.1 Stability Region of the Lu-Kumar network .................................................. 106

5.1.2 Stability, Instability and their relationship of the Lu-Kumar network ......... 106

5.1.3 Virtual Stations in the Lu-Kumar network ................................................... 107

5.1.4 Merged Queue at Service Centre B of the Lu-Kumar network .................... 107

5.2 LEARNING ......................................................................................................... 107

5.3 RECOMMENDATION FOR FUTURE WORK ................................................. 107

5.3.1 Stability, Instability and their relationship of the Lu-Kumar network ......... 107

5.3.2 Virtual Stations in the Lu-Kumar network ................................................... 108

5.3.3 Merged Queue at Service Centre B of the Lu-Kumar network .................... 108

REFRENCES ....................................................................................................................... 109


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ABSTRACT

Manufacturing system, communication network, silicon wafer manufacturing plant, internet

service provider domain and freeway system are usually modelled with multi-class queuing

network. The aim of this project is to deeply understand the nature of the Lu-Kumar

network. The queue time of the Lu-Kumar network is hard to predict due to the mutual

blocking of virtual stations in the network.

In most of the literature, the stability of a network is discussed using a fluid model. But in

this project, the stability of a network is investigated by mean of manual book-keeping as

well as simulation model with deterministic inter-arrival time and service time. The

condition, for which the network will come back to the stable state when there is a small

disturbance at an initial state, is also examined.

To understand the nature of virtual stations in a network, the network is simulated with

MatEvent simulation model in which Poisson distribution is used for inter-arrival time and

Exponential distribution is used for service times. Moreover, two buffers at service centre B

of the Lu-Kumar network are merged into one queue to understand the blocking nature of the

network and the queue time is compared with MM1 queue time.

From the book-keeping and simulation result, it is found that the stability region of the

d t i i ti L K t k i di t It i b t 1 5 ti f ti j b


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ACKNOWLWDGEMENTS

The author would like to extend his gratitude to individuals who have helped and supported

throughout this project.

Firstly, the author would like to express deep appreciation towards Assistant Professor Wu

Kan for his guidance, patience, support, encouragement and willingness to teach and share

the valuable insight from his research and experience.

Secondly, the author also would like to thank the research fellow, Dr. Daniel David

Moskovich for his guidance and assistance towards clearing the doubts regarding with the

Lu-Kumar network and book-keeping.

Thirdly, the author would like to thank his peer, Cheung Wai Lun for his assistance and

helpful discussion over this project.

Last, but not least, the author would like to thank his family, friends and class-mates for their

constant support and encouragement.


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Table of Figures

Figure 1, Basic queuing process............................................................................................... 4

Figure 2, Lu-Kumar re-entrant network. .................................................................................. 7

Figure 3, Entity Generator sub-system. .................................................................................... 9

Figure 4, Entity Sink. ............................................................................................................... 9

Figure 5, Lu-Kumar network simulation model..................................................................... 11

Figure 6, Simulation model of service centre A..................................................................... 12

Figure 7, Simulation model of service centre B. .................................................................... 13

Figure 8, Queue and timer sub-system. .................................................................................. 14

Figure 9, Service time attribute block. ................................................................................... 15

Figure 10, Set Attribute block and its parameter. .................................................................. 16

Figure 11, Output switch block and its parameter. ................................................................ 17

Figure 12, Path Combiner block and its parameter. ............................................................... 17

Figure 13, Data acquisition subsystem. .................................................................................. 18

Figure 14, Simulation model of merged queue at station B. .................................................. 26

Figure 15, Total queue time of first 30th samples of stable system. ....................................... 28

Figure 16, Total queue time of first 30 samples of unstable system. ..................................... 29

Fi 17 WIP th i f t f l 2 d l 4 f th t bl t k ith IA 1 0 40


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List of Tables

Table 1 Variation of inter-arrival time for following entities ................................................ 20

Table 2 Variation of service time of virtual stations .............................................................. 22

Table 3 Variation of service time of all service centres ......................................................... 23

Table 4 Combination of m2 and m3 variation ....................................................................... 24

Table 5 Book Keeping for inter-arrival time =0.8 ................................................................. 31

Table 6 Book Keeping for inter-arrival time =1.0 ................................................................. 32

Table 7 Book Keeping for inter-arrival time= 1.1 ................................................................. 41

Table 8 Book Keeping for unstable network with inter-arrival time=1.2...............................40

Table 9 Book Keeping for stable network with inter-arrival time=1.2...................................45

Table 10 Book-keeping for inter-arrival time=0.9_1.50_1.50_1.50_... ................................. 64

Table 11 Book-keeping for inter-arrival time=0.9_1.40_1.40_1.40_... ................................. 65

Table 12 Book-keeping for inter-arrival time =0.9_1.30_1.30_1.30_... ................................ 68



Table 15 Book-keeping for inter-arrival time =0.9_1.205_1.205_1.205_... .......................... 74

Table 16 Book-keeping for inter-arrival time =0.9_1.201_1.201_1.201_... .......................... 76

T bl 17 B k k i f i t i l ti 0 9 0 9 1 201 1 201 1 201 80


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CHAPTER 1INTRODUCTION

1.1 Introduction

Queuing networks are used to model manufacturing systems, communication networks,

silicon wafer manufacturing plants, internet service provider domains, freeway systems or

customers waiting scenarios at ATM at the bank. Recent research is focused on multi-classqueuing network [1]. In multi-class queuing network, customers served at same stations have

different requirement and follow different paths. Such networks are used to model the silicon

wafer manufacturing facilities in which wafers follow a deterministic route and revisit the

same station multiple times because same tools and same stations would be needed at

different stages of production for the same wafer. Multiclass queuing networks of this type

are called re-entrant line queuing networks. [1, 2]

There are very high numbers of steps in a silicon wafer manufacturing process and the

economical efficiency is very crucial because of high investment in manufacturing facilities,

the decision could not depends solely on intuition or primitive heuristics. However, the exact

calculation of queuing network is very expensive for practical engineering purposes. Thus,

i l ti f t i t L K t t li t k ill b th t ti i t


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To investigate the behaviour of the Lu-Kumar network when the inter-arrival

time is changed from an instable range to a stable range.

To understand the behaviour of a virtual station in the Lu-Kumar network.

To understand the nature of the queuing system when queues at Service Centre

B of the Lu-Kumar network are merged into one queue.

1.3 Scope of Work

For this study for the Lu-Kumar network, the following assumptions are made

The buffer at each service station is assumed to be infinite.

No pre-emptive interruption within the queuing system during service for every

station.

The travelling time for entities between the stations is instantaneous.

The external inter-arrival times as well as service times of each sever are

independent and identically distributed (iid)

The scope of this FYP is as follow:

Design and develop of Lu-Kumar network model using ‘MatLab SimEvent’.

Verify the model by comparing with the MM1 queue model


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1.4 Organization

This report is divided into five chapters. The scope of the respective chapters is as follows:

Chapter 1: I ntroduction

This chapter highlights the significance of multi-class queuing networks and also introduces

the two service centres Lu-Kumar re-entrant network which is the main focus of this report.

Chapter 2: Literatur e Review

This chapter introduces the basics of queuing theory and the features in the Lu-Kumar

network. The features include:

Stability of the network.

Virtual stations in the network.

Chapter 3: Methodology

This chapter highlights the investigation approach, method of experiments, development ofsimulation models and parameter of simulation models for each case.

Chapter 4: Resul ts, Analysis and Discussion

This chapter shows the result of book-keeping for deterministic condition as well as the result

of simulation result under both deterministic arrival time and uniform distribution arrival time

with deterministic service time. Moreover, it explains the condition where the system will

b k bl f ll di i i i i l d h d i i i


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CHAPTER 2LITERATURE REVIEW

2.1 Queuing theory

2.1.1 Basic queuing process

Customers who require the service enter the queuing system by following queue discipline.

The customers are from calling population or input source. After the customers have acquired

the service from server, they leave the queuing system. The process is show in Fig. 1.Queuing

theory is a branch of Operations Research because the results gathered can be used to make

decisions about the resources (number of servers, type of queue) required to provide a service.


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The time between two consecutive arrivals is defined as inter-arrival time (1/λ). Usually the

inter-arrival time is assumed to be Poisson process in which customers’ arrival to the queuing

system is random but at a certain fixed mean rate.

Sometimes there are unusual assumptions for arrival of customers’ behaviour. For example, if

queue is too long, the customers decide not to enter the system. Such behaviour is called

balking.

2.1.3 Queue

The queue is where customers wait before being served. Queues are called infinite or finite

depending on whether the capacity is infinite or finite. If the upper bound of the queuing

system is small enough to fill with some inter-arrival frequency, the finite queue must be

assumed.

2.1.4 Queue Discipline

The queue discipline is defined as the order or discipline in which customers are selected for

service. And it refers to scheduling policy in Lu-Kumar network. For example, First Buffer

First Serve (FBFS) or Last Buffer First Serve (LBFS).

2 1 5 S i M h i


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2.1.7 Stability of queuing system

For a queuing system to be stable, the mean arrival rate should be less than the mean service

rate. In other words, the utilization should be less than 1. For the scheduling policies to be

stable, the utilization must be less than 1. When the utilization is greater than 1, the queue will

grow infinitely and the system will be unstable.

2.1.8Little’s Law

Little’s Law represents the relationship between the expected number of customers in the

queuing system and the cycle time (service time plus queuing time) of the system. Little’s

Law is perhaps most widely used in queuing theory

= L=expected number of customers in the queuing system

=expected arrival rate

W=cycle time (service time plus queue time)

2.1.9 Kendall’s Notation

Queuing model are described by Kendall’s notation. Both inter -arrival time and service times

are assumed to be independent and identically distributed.


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2.2 Lu-Kumar re-entrant Network

The Lu-Kumar re-entrant network is shown in Fig. 2.

Figure 2, Lu-Kumar re-entrant network.

m1, m2, m3 and m4 are service time for respective service station. The arrival rate is α1.

The operational policy is that Last Buffer First Serve (LBFS) policy at Station A and First

Buffer First Serve (FBFS) policy at Station B. The priority buffer is highlighted with orange

in each station inside the figure.

2.2.1 Stability of Lu-Kumar re-entrant network

A di [7] d hi i l li h i bl if d l if


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2.2.2 Virtual Stations of Lu-Kumar re-entrant network

Virtual stations affect the global stability of queuing networks as some buffers can never be

served simultaneously even though they are at different stations. [8]

Jobs in buffer 2 and buffer 4 can never be processed simultaneously.

Under the operational policy,

2()4() = 0 ≥ 0

if 2(0)4(0) = 0 . Thus, classes {2, 4} form a virtual station. [9]If v = α(m2 + m4) > 1 , with probability one, the total number of jobs goes to infinity.


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CHAPTER 3

METHODOLOGY

3.1 Stability region of the Lu-Kumar network

3.1.1 Development of Simulation model

The Lu-Kumar network model is designed in the MatLab SimEvent software. The model is as

shown in Fig. 5.The simulation model consists of Service Centre A, Service Centre B, Entity

Generator sub-system and Entity Sink.

Time-based Entity Generator and inter-arrival time block

Fi 3 E i G b


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Service time for m1, m2, m3 and m4 are set at service time 1, 2, 3 and 4 respectively.

For deterministic case, the service time and inter-arrival time are set as constant. But the

model is simulated by m.file to vary the deterministic inter-arrival time according to the

experimental requirements. The Matlab file (‘.m’) to run the Lu-Kumar network with

deterministic inter-arrival time and service time is shown in Appendix A.

For Poisson arrival and exponential service time, the inter-arrival time is varied according to

the utilization of 0.2, 0.5, 0.8, 0.9 and 0.95 respectively to simulate the Lu-Kumar network in

which the effect of the virtual stations to be investigated. But the model is simulated by

m.file. The m.file is written in such a way that the inter-arrival time is varied according to the

utilization of 0.2, 0.5, 0.8, 0.9 and 0.95 respectively and the respective data are collected. The

Matlab file (‘.m’) to run the Lu-Kumar network with Poisson arrival and exponential service

time is shown in Appendix B.

Fig. 6 and 7 show the detail design of Service Centre A and Service Centre B of the Lu-

Kumar network respectively.

Each service centre consists of queue and timer sub-system, data acquisition sub-system,

server sub-system and entity routing sub-system.


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11 | P a g e

Figure 5, Lu-Kumar network simulation model.


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Figure 7, Simulation model of service centre B.


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Queue and timer

The entity waits at the queue if the server is busy. Only FIFO rule is applied for all the queue in

this project. The start timer counts the time when an entity comes inside the queue and the read

timer counts the time when the entity goes outs from the queue so that the waiting time at the

queue can be calculated.

Figure 8, Queue and timer sub-system.

Service time sub-system

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Figure 9, Service time attribute block.

Routing entity sub-system

The routing entity sub-system consists of set attribute block, attribute function block and path

combiner. Each entity is set to the particular attributes at set attribute block which is shown in

Fig. 10. Based on the attributes, output switch will route the entity ethier reloop or exit. The


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Figure 10, Set Attribute block and its parameter.


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Figure 11, Output switch block and its parameter.


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Only first 30 samples are simulated as 30 samples are good enough to determine the stability of

the system. The total queue time for each case is plotted for the first 30 samples.

3.1.3 Manual book-keeping

The deterministic service time for respective service station in the Lu-Kumar is the same as

above.

The manual book-keeping is performed with deterministic inter-arrival time as follow so that the

behaviour of interaction among the servers can be understood in depth as well as the WIP growth

in front of class 2 and class 4 can be investigated.

AT=0.8, 1.0, 1.1, 1.2

3.1.4 Parameters of the simulation model, uniform distribution arrival time

The deterministic service time for each station in the network is the same as above experiment.

m1=0.2, m2=0.6, m3=0.1 and m4=0.6

But the inter-arrival time is set as uniform distribution as follow.

(a) Uniform distribution between 0.8 and 0.90; unif (0.8,0.9)


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3.2 Stability, Instability and their relationship

3.2.1 Design of Experiment for the Lu-Kumar network started with single

inter-arrival time of 0.9

In order to examine the relationship between stability and instability, the experiment is set as

follow.

The deterministic service time for each station is the same as above experiment.

m1=0.2, m2=0.6, m3=0.1 and m4=0.6

The inter-arrival time for 1st entity 0.9 because the system is unstable when inter-arrival time is

exactly 0.9. In order to find out the condition that the system will come back to stable state, the

deterministic inter-arrival time for following entities are varied as shown in Table 1A.

TABLE 1A: VARIATION OF INTER-ARRIVAL TIME FOR THE FOLLOWING ENTITIES

No. I nter-arri val time Value

1. AT 0.85


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3.3 Virtual Stations in the Lu-Kumar Network


The same Lu-Kumar network simulation model is used and it is simulated by using m.file to run

with different parameters. The m.file is shown in Appendix B.

3.3.2 Parameters of the simulation model

Both inter-arrival time and service time are set as the exponential distribution. The exponential

mean of service time for each station is fixed and the exponential mean of inter-arrival time is

varied to achieve the utilization of 0.2, 0.5, 0.8, 0.9 and 0.95 respectively. This can be achieved

by writing the m.file to simulate with different inter-arrival time. 90% confidence interval is used

when analysing the queuing time for each station as well as the total queuing time.

In this experiment, each observation is made up of 32 samples. Each batch of sample consists of

200,000 data points (sample size) and the first 400,000 data points are neglected as a warm up

samples. A warm up is used to allow the system to reach steady-state before the system behaviour

is analysed.

3.3.3 Parameters of virtual stations

To investigate the nature of virtual stations in the Lu-Kumar network, the service time of virtual


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To observe the effect of varying the service time of m1 and m3, the service times are varied as

shown in Table 3.But the total average service time of each station including the virtual stations is

kept constant mean value of 0.8.

TABLE 3: VARIATION OF SERVICE TIME OF ALL SERVICE CENTRES

7 0.0 0.6 0.2 0.0

8 0.0 0.7 0.1 0.0

9 0.0 0.8 0.0 0.0

Combination

B

Station A Station B

m1 m4 m2 m3

1 0.8 0.0 0.8 0.0

2 0.7 0.1 0.7 0.1

3 0.6 0.2 0.6 0.2

4 0.5 0.3 0.5 0.3

5 0.4 0.4 0.4 0.4

6 0.3 0.5 0.3 0.5


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3.4 Merged Queue at Service Centre B of the Lu-Kumar network


The Lu-Kumar network model is updated at service centre B in such a way that buffer at m2 and

buffer at m3 are merged into one more buffer as shown in Fig. 14.

The m.file is written so that the model can be simulated with different parameters. The m.file is

shown in the appendix B.

3.4.2 Parameters of the simulation model

The mean value of exponential service time for m2 and m3 are varied as follow.

TABLE 4: COMBINATION OF M2 AND M3 VARIATION

Combination m2 m3

1. 0 0.8

2. 0.1 0.7

3. 0.2 0.6

4. 0.25 0.55

5. 0.3 0.5

6. 0.35 0.45


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samples. The total queue time is compared to the queue time of MM1 model respectively and

graph of the ratio of total queue time to MM1 queue time vs. the service time variation is plotted.


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26 | P a g e

Figure 14, Simulation model of merged queue at station B.


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CHAPTER 4

RESULTS, ANALYSIS AND DISCUSSION

4.1 Stability region of the Lu-Kumar network

4.1.1 Simulation Result, deterministic inter-arrival time

The stability region of the Lu-Kumar network is identified when both inter-arrival time andservice time are deterministic. The total queue time of first 30 samples with different inter-arrival

time is shown in Fig. 15 and 16.The former shows total queue time of the stable network and the

latter shows the total queue time of the unstable network. The total queue time of unstable

network is increasing with time. Thus, the system is stable when

0.8 ≤


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28 | P a g e

Figure 15, Total queue time of first 30th samples of stable system.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30 35

t o t a l q u e u e t i m e

nth samples

AT=1.6

AT=1.5

AT=1.4

AT=1.3

AT=1.2

AT=1.1

AT=0.8


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29 | P a g e

Figure 16, Total queue time of first 30 samples of unstable system.

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35

t o t a l q u

e u e t i m e

nth samples

AT=0.7

AT=0.69

AT=0.79

AT=0.9

AT=1


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4.1.2 Book Keeping

4.1.2.1 Inter-arrival time =0.8

The manual book keeping for inter-arrival time of 0.8 is shown in Table 5. When the 2 nd entity is

generated, the 1st entity is at m3. After m3 has served 1 st entity and 1st entity has already reached

in front of m4, m1 is still serving 2nd entity. Hence, 1st entity has to wait at buffer 4 until 2nd

entity is finished serving by m1 even though m4 is priority. When m1 has served 2 nd entity, 2nd

entity goes to m2. Thus, m2 serves 2nd entity and m4 serves 1st entity simultaneously even though

m2 and m4 are virtual stations (ghost machines). It is highlighted with blue colour in the table

when virtual stations serve the jobs simultaneously. Thus, this network is stable because virtual

stations work simultaneously and synchronization occurs even though inter-arrival time is 0.8.


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TABLE 5: BOOK KEEPING FOR INTER-ARRIVAL TIME =0.8

Time Entity m1 (0.2) m2 (0.6) m3 (0.1) m4 (0.6) Out

0 1st 1st (0)

0.2 1st (0.2) 1st (0)

0.8 2nd 2nd (0) 1st (0.6) 1st (0)

0.9 2nd (0.1) 1st (0.1) 1st (0)

1 2nd (0.2) 2nd (0) 1st (0)

1.5 2nd (0.5) 1st (0.5)

1.6 3rd 3rd (0) 2nd (0.6) 2nd (0) 1st (0.6) 1st

1.7 3rd (0.1) 2nd (0.1) 2nd (0)

1.8 3rd (0.2) 3rd (0) 2nd (0)

2.3 3rd (0.5) 2nd (0.5)

2.4 4th 4th (0) 3rd (0.6) 3rd (0) 2nd (0.6) 2nd

2.5 4th (0.1) 3rd (0.1) 3rd (0)

2.6 4th (0.2) 4th (0) 3rd (0)

3.1 4th (0.5) 3rd (0.5)

3.2 5th 5th (0) 4th (0.6) 4th (0) 3rd (0.6) 3rd

3.3 5th (0.1) 4th (0.1) 4th (0)

3.4 5th (0.2) 5th (0) 4th (0)

3.9 5th (0.5) 4th (0.5)

4 6th 6th (0) 5th (0.6) 5th (0) 4th (0.6) 4th


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33 | P a g e

6.9 0.1 4th(0.1),5th(0),6th(0),7th(0) 4th (0)

7 0.1 1 8th 8th(0) 5th(0.1),6th(0),7th(0) 4th (0.1), 5th (0)

7.1 0.1 8th(0) 6th(0.1),7th(0) 4th (0.2), 5th (0), 6th (0)

7.2 0.1 8th(0) 7th(0.1) 4th (0.3), 5th (0), 6th (0), 7th (0)

7.5 0.3 8th(0) 4th (0.6), 5th (0), 6th (0), 7th (0) 4th

8 0.5 1 9th 8th(0),9th(0) 5th (0.5), 6th (0), 7th (0)

8.1 0.1 8th(0),9th(0) 5th (0.6), 6th (0), 7th (0) 5th

8.7 0.6 8th(0),9th(0) 6th (0.6), 7th (0) 6th

9 0.3 1 10th 8th(0),9th(0),10th(0) 7th (0.3)

9.3 0.3 8th(0),9th(0),10th(0) 7th (0.6) 7th

9.5 0.2 8th(0.2),9th(0),10th(0) 8th(0)

9.7 0.2 9th(0.2),10th(0) 8th(0.2),9th(0)

9.9 0.2 10th(0.2) 8th(0.4),9th(0),10th(0)

10 0.1 1 11th 11th(0) 8th(0.5),9th(0),10th(0)

10.1 0.1 11th(0.1) 8th(0.6),9th(0),10th(0) 8th (0)

10.2 0.1 11th(0.2) 9th(0.1),10th(0),11th(0) 8th (0)

10.7 0.5

0

.

7

9th(0.6),10th(0),11th(0) 8th(0),9th(0)

11 0.3 1 12th 12th(0) 10th(0.3),11th(0) 8th(0),9th(0)

11.2 0.2 12th(0.2) 10th(0.5),11th(0),12th(0) 8th(0),9th(0)

11.3 0.1 10th(0.6),11th(0),12th(0) 8th(0),9th(0),10th(0)

11.9 0.6

0

.

9

11th(0.6),12th(0) 8th(0),9th(0),10th(0),11th(0)

12 0.1 1 13th 13th(0) 12th(0.1) 8th(0),9th(0),10th(0),11th(0)

12.2 0.2 13th(0.2) 12th(0.3),13th(0) 8th(0),9th(0),10th(0),11th(0)

12.5 0.3 12th(0.6),13th(0) 8th(0),9th(0),10th(0),11th(0),12th(0)

13 0.5 1 14th 14th(0) 13th(0.5) 8th(0),9th(0),10th(0),11th(0),12th(0)

13.1 0.1 14th(0.1) 13th(0.6) 8th(0),9th(0),10th(0),11th(0),12th(0),13th(0)

13.2 0.1 14th(0.2) 14th(0) 8th(0),9th(0),10th(0),11th(0),12th(0),13th(0)

13.8 0.6 14th(0.6)8th(0),9th(0),10th(0),11th(0),12th(0),13th(0),

14th(0)

13.9 0.18th(0.1),9th(0),10th(0),11th(0),12th(0),

13th(0),14th(0)8th(0)

14 0.1 1 15th 15th(0) 9th(0.1),10th(0),11th(0),12th(0),13th(0),14th(0)

8th(0.1),9th(0)

14.1 0.1 15th(0) 10th(0.1),11th(0),12th(0),13th(0),14th(0) 8th(0.2),9th(0),10th(0)

14.2 0.1 15th(0) 11th(0.1),12th(0),13th(0),14th(0) 8th(0.3),9th(0),10th(0),11th(0)

14.3 0.1 15th(0) 12th(0.1),13th(0),14th(0) 8th(0.4),9th(0),10th(0),11th(0),12th(0)

14.4 0.1 15th(0) 13th(0.1),14th(0)8th(0.5),9th(0),10th(0),11th(0),12th(0),

13th(0)

14.5 0.1 15th(0) 14th(0.1) 8th(0.6),9th(0),10th(0),11th(0),12th(0), 8th


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34 | P a g e

13th(0),14th(0)

15 0.5 1 16th 15th(0),16th(0)9th(0.5),10th(0),11th(0),12th(0),13th(0),

14th(0)

15.1 0.1 15th(0),16th(0)9th(0.6),10th(0),11th(0),12th(0),13th(0),

14th(0)9th

15.7 0.6 15th(0),16th(0) 10th(0.6),11th(0),12th(0),13th(0),14th(0) 10th

16 0.3 1 17th 15th(0),16th(0),17th(0) 11th(0.3),12th(0),13th(0),14th(0)

16.3 0.3 15th(0),16th(0),17th(0) 11th(0.6),12th(0),13th(0),14th(0) 11th

16.9 0.6 15th(0),16th(0),17th(0) 12th(0.6),13th(0),14th(0) 12th17 0.1 1 18th 15th(0),16th(0),17th(0),18th(0) 13th(0.1),14th(0)

17.5 0.5 15th(0),16th(0),17th(0),18th(0) 13th(0.6),14th(0) 13th

18 0.5 1 19th 15th(0),16th(0),17th(0),18th(0),19th(0) 14th(0.5)

18.1 0.1 15th(0),16th(0),17th(0),18th(0),19th(0) 14th(0.6) 14th

18.3 0.2 15th(0.2),16th(0),17th(0),18th(0),19th(0) 15th(0)

18.5 0.2 16th(0.2),17th(0),18th(0),19th(0) 15th(0.2),16th(0)

18.7 0.2 17th(0.2),18th(0),19th(0) 15th(0.4),16th(0),17th(0)

18.9 0.2 18th(0.2),19th(0) 15th(0.6),16th(0),17th(0),18th(0) 15th(0)

19 0.1 1 20th 19th(0.1),20th(0) 16th(0.1),17th(0),18th(0) 15th(0)

19.1 0.1 19th(0.2),20th(0) 16th(0.2),17th(0),18th(0),19th(0) 15th(0)

19.3 0.2 20th(0.2) 16th(0.4),17th(0),18th(0),19th(0),20th(0) 15th(0)

19.5 0.2 16th(0.6),17th(0),18th(0),19th(0),20th(0) 15th(0),16th(0)

20 0.5 1 21st 21st(0) 17th(0.5),18th(0),19th(0),20th(0) 15th(0),16th(0)20.1 0.1 21st(0.1) 17th(0.6),18th(0),19th(0),20th(0) 15th(0),16th(0),17th(0)

20.2 0.1 21st(0.2) 18th(0),19th(0),20th(0),21st(0) 15th(0),16th(0),17th(0)

20.8 0.6 18th(0.6),19th(0),20th(0),21st(0) 15th(0),16th(0),17th(0),18th(0)

21 0.2 1 22nd 22nd(0) 19th(0.2),20th(0),21st(0) 15th(0),16th(0),17th(0),18th(0)

21.2 0.2 22nd(0.2) 19th(0.4),20th(0),21st(0),22nd(0) 15th(0),16th(0),17th(0),18th(0)

21.4 0.2 19th(0.6),20th(0),21st(0),22nd(0) 15th(0),16th(0),17th(0),18th(0),19th(0)

22 0.6 1 23rd 23rd(0) 20th(0.6),21st(0),22nd(0)15th(0),16th(0),17th(0),18th(0),19th(0),

20th(0)

22.2 0.2 23rd(0.2) 21st(0.2),22nd(0),23rd(0)15th(0),16th(0),17th(0),18th(0),19th(0),

20th(0)

22.6 0.4 21st(0.6),22nd(0),23rd(0)15th(0),16th(0),17th(0),18th(0),19th(0),

20th(0),21st(0)

23 0.4 1 24th 24th(0) 22nd(0.4),23rd(0)15th(0),16th(0),17th(0),18th(0),19th(0),

20th(0),21st(0)

23.2 0.2 24th(0.2) 22nd(0.6),23rd(0),24th(0)15th(0),16th(0),17th(0),18th(0),19th(0),

20th(0),21st(0),22nd(0)

23.8 0.6 23rd(0.6),24th(0)15th(0),16th(0),17th(0),18th(0),19th(0),

20th(0),21st(0),22nd(0),23rd(0)

24 0.2 1 25th 25th(0) 24th(0.2)15th(0),16th(0),17th(0),18th(0),19th(0),

20th(0),21st(0),22nd(0),23rd(0)

24.2 0.2 25th(0.2) 24th(0.4),25th(0) 15th(0),16th(0),17th(0),18th(0),19th(0),


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35 | P a g e

20th(0),21st(0),22nd(0),23rd(0)

24.4 0.2 24th(0.6),25th(0)15th(0),16th(0),17th(0),18th(0),19th(0),

20th(0),21st(0),22nd(0),23rd(0),24th(0)

25 0.6 1 26th 26th(0) 25th(0.6)15th(0),16th(0),17th(0),18th(0),19th(0),20th

(0),21st(0),22nd(0),23rd(0),24th(0),25th(0)

25.1 0.1 26th(0.1)

15th(0.1),16th(0),17th(0),18th(0),19th(0),

20th(0),21st(0),22nd(0),23rd(0),24th(0),

25th(0)

15th(0)

25.2 0.1 26th(0.2) 26th(0)

16th(0.1),17th(0),18th(0),19th(0),20th(0),

21st(0),22nd(0),23rd(0),24th(0),25th(0) 15th(0),16th(0)

25.8 0.6 26th(0.6)17th(0),18th(0),19th(0),20th(0),21st(0),

22nd(0),23rd(0),24th(0),25th(0),26th(0)15th(0.6),16th(0) 15th

25.9 0.117th(0.1),18th(0),19th(0),20th(0),21st(0),

22nd(0),23rd(0),24th(0),25th(0),26th(0)16th(0.1),17th(0)

26 0.1 1 27th 27th(0)18th(0.1),19th(0),20th(0),21st(0),22nd(0),

23rd(0),24th(0),25th(0),26th(0)16th(0.2),17th(0),18th(0)

26.1 0.1 27th(0)19th(0.1),20th(0),21st(0),22nd(0),23rd(0),

24th(0),25th(0),26th(0)16th(0.3),17th(0),18th(0),19th(0)

26.2 0.1 27th(0)20th(0.1),21st(0),22nd(0),23rd(0),24th(0),

25th(0),26th(0)16th(0.4),17th(0),18th(0),19th(0),20th(0)

26.3 0.1 27th(0)21st(0.1),22nd(0),23rd(0),24th(0),25th(0),

26th(0)

16th(0.5),17th(0),18th(0),19th(0),20th(0),

21st(0)

26.4 0.1 27th(0) 22nd(0.1),23rd(0),24th(0),25th(0),26th(0)16th(0.6),17th(0),18th(0),19th(0),20th(0),

21st(0),22nd(0)16th

26.5 0.1 27th(0) 23rd(0.1),24th(0),25th(0),26th(0)17th(0.1),18th(0),19th(0),20th(0),21st(0),

22nd(0),23rd(0)

26.6 0.1 27th(0) 24th(0.1),25th(0),26th(0)17th(0.2),18th(0),19th(0),20th(0),21st(0),

22nd(0),23rd(0),24th(0)

26.7 0.1 27th(0) 25th(0.1),26th(0)17th(0.3),18th(0),19th(0),20th(0),21st(0),

22nd(0),23rd(0),24th(0),25th(0)

26.8 0.1 27th(0) 26th(0.1)17th(0.4),18th(0),19th(0),20th(0),21st(0),

22nd(0),23rd(0),24th(0),25th(0),26th(0)

27 0.2 1 28th 27th(0),28th(0)17th(0.6),18th(0),19th(0),20th(0),21st(0),

22nd(0),23rd(0),24th(0),25th(0),26th(0)17th

27.6 0.6 27th(0),28th(0)18th(0.6),19th(0),20th(0),21st(0),22nd(0),

23rd(0),24th(0),25th(0),26th(0)18th

28 0.4 1 29th 27th(0),28th(0),29th(0)19th(0.4),20th(0),21st(0),22nd(0),23rd(0),

24th(0),25th(0),26th(0)

28.2 0.2 27th(0),28th(0),29th(0)

19th(0.6),20th(0),21st(0),22nd(0),23rd(0),

24th(0),25th(0),26th(0) 19th

28.8 0.6 27th(0),28th(0),29th(0)20th(0.6),21st(0),22nd(0),23rd(0),24th(0),

25th(0),26th(0)20th

29 0.2 1 30th 27th(0),28th(0),29th(0),30th(0)21st(0.2),22nd(0),23rd(0),24th(0),25th(0),

26th(0)

29.4 0.4 27th(0),28th(0),29th(0),30th(0)21st(0.6),22nd(0),23rd(0),24th(0),25th(0),

26th(0)21st

30 0.6 1 31st 27th(0),28th(0),29th(0),30th(0),31st(0) 22nd(0.6),23rd(0),24th(0),25th(0),26th(0) 22nd


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36 | P a g e

30.6 0.6 27th(0),28th(0),29th(0),30th(0),31st(0) 23rd(0.6),24th(0),25th(0),26th(0) 23rd

31 0.4 1 32nd 27th(0),28th(0),29th(0),30th(0),31st(0),32nd(0) 24th(0.4),25th(0),26th(0)

31.2 0.2 27th(0),28th(0),29th(0),30th(0),31st(0),32nd(0) 24th(0.6),25th(0),26th(0) 24th

31.8 0.6 27th(0),28th(0),29th(0),30th(0),31st(0),32nd(0) 25th(0.6),26th(0) 25th

32 0.2 1 33rd27th(0),28th(0),29th(0),30th(0),31st(0),32nd(0),

33rd(0)26th(0.2)

32.4 0.427th(0),28th(0),29th(0),30th(0),31st(0),32nd(0),

33rd(0)26th(0.6) 26th

32.6 0.2 27th(0.2),28th(0),29th(0),30th(0),31st(0),32nd(0),33rd(0)

27th(0)

32.8 0.228th(0.2),29th(0),30th(0),31st(0),32nd(0),

33rd(0)27th(0.2),28th(0)

33 0.2 1 34th29th(0.2),30th(0),31st(0),32nd(0),33rd(0),

34th(0)27th(0.4),28th(0),29th(0)

33.2 0.2 30th(0.2),31st(0),32nd(0),33rd(0),34th(0) 27th(0.6),28th(0),29th(0),30th(0) 27th(0)

33.4 0.2 31st(0.2),32nd(0),33rd(0),34th(0) 28th(0.2),29th(0),30th(0),31st(0) 27th(0)

33.6 0.2 32nd(0.2),33rd(0),34th(0) 28th(0.4),29th(0),30th(0),31st(0),32nd(0) 27th(0)

33.8 0.2 33rd(0.2),34th(0) 28th(0.6),29th(0),30th(0),31st(0),32nd(0),33rd(0) 27th(0),28th(0)

34 0.2 1 35th 34th(0.2),35th(0) 29th(0.2),30th(0),31st(0),32nd(0),33rd(0),34th(0) 27th(0),28th(0)

34.2 0.2 35th(0.2)29th(0.4),30th(0),31st(0),32nd(0),33rd(0),34th(0),

35th(0)27th(0),28th(0)

34.4 0.229th(0.6),30th(0),31st(0),32nd(0),33rd(0),34th(0),

35th(0)27th(0),28th(0),29th(0)

35 0.6 1 36th 36th(0) 30th(0.6),31st(0),32nd(0),33rd(0),34th(0),35th(0) 27th(0),28th(0),29th(0),30th(0)

35.2 0.2 36th(0.2) 31st(0.2),32nd(0),33rd(0),34th(0),35th(0),36th(0) 27th(0),28th(0),29th(0),30th(0)

35.6 0.4 31st(0.6),32nd(0),33rd(0),34th(0),35th(0),36th(0) 27th(0),28th(0),29th(0),30th(0),31st(0)

36 0.4 1 37th 37th(0) 32nd(0.4),33rd(0),34th(0),35th(0),36th(0) 27th(0),28th(0),29th(0),30th(0),31st(0)

36.2 0.2 37th(0.2) 32nd(0.6),33rd(0),34th(0),35th(0),36th(0),37th(0)27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0)

36.8 0.6 33rd(0.6),34th(0),35th(0),36th(0),37th(0)27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0)

37 0.2 1 38th 38th(0) 34th(0.2),35th(0),36th(0),37th(0)27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0)

37.2 0.2 38th(0.2) 34th(0.4),35th(0),36th(0),37th(0),38th(0)27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0)

37.4 0.2 34th(0.6),35th(0),36th(0),37th(0),38th(0)27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0)

38 0.6 1 39th 39th(0) 35th(0.6),36th(0),37th(0),38th(0) 27th(0),28th(0),29th(0),30th(0),31st(0),32nd(0),33rd(0),34th(0),35th(0)

38.2 0.2 39th(0.2) 36th(0.2),37th(0),38th(0),39th(0)27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0)

38.6 0.4 36th(0.6),37th(0),38th(0),39th(0)27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0)

39 0.4 1 40th 40th(0) 37th(0.4),38th(0),39th(0)27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0)


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37 | P a g e

39.2 0.2 40th(0.2) 37th(0.6),38th(0),39th(0),40th(0)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0)

39.8 0.6 38th(0.6),39th(0),40th(0)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0)

40 0.2 1 41st 41st(0) 39th(0.2),40th(0)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0)

40.2 0.2 41st(0.2) 39th(0.4),40th(0),41st(0)27th(0),28th(0),29th(0),30th(0),31st(0),32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0)

40.4 0.2 39th(0.6),40th(0),41st(0)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0)

41 0.6 1 42nd 42nd(0) 40th(0.6),41st(0)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0)

41.2 0.2 42nd(0.2) 41st(0.2),42nd(0)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0)

41.6 0.4 41st(0.6),42nd(0)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0),41st(0)

42 0.4 1 43rd 43rd(0) 42nd(0.4)27th(0),28th(0),29th(0),30th(0),31st(0),32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0),41st(0)

42.2 0.2 43rd(0.2) 42nd(0.6),43rd(0)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0),41st(0),

42nd(0)

42.8 0.6 43rd(0.6)

27th(0),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0),41st(0),

42nd(0),43rd(0)

42.9 0.1

27th(0.1),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0),41st(0),

42nd(0),43rd(0)

27th(0)

43 0.1 1 44th 44th(0)28th(0.1),29th(0),30th(0),31st(0),32nd(0),33rd(0),34th(0),35th(0),36th(0),37th(0),

38th(0),39th(0),40th(0),41st(0),42nd(0),

43rd(0)

27th(0.1),28th(0)

43.1 0.1 44th(0)

29th(0.1),30th(0),31st(0),32nd(0),33rd(0),

34th(0),35th(0),36th(0),37th(0),38th(0),

39th(0),40th(0),41st(0),42nd(0),43rd(0)

27th(0.2),28th(0),29th(0)

43.2 0.1 44th(0)30th(0.1),31st(0),32nd(0),33rd(0),34th(0),

35th(0),36th(0),37th(0),38th(0),39th(0),27th(0.3),28th(0),29th(0),30th(0)


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38 | P a g e

40th(0),41st(0),42nd(0),43rd(0)

43.3 0.1 44th(0)

31st(0.1),32nd(0),33rd(0),34th(0),35th(0),

36th(0),37th(0),38th(0),39th(0),40th(0),

41st(0),42nd(0),43rd(0)

27th(0.4),28th(0),29th(0),30th(0),31st(0)

43.4 0.1 44th(0)

32nd(0.1),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0),41st(0),

42nd(0),43rd(0)

27th(0.5),28th(0),29th(0),30th(0),31st(0),

32nd(0)

43.5 0.1 44th(0)

33rd(0.6),34th(0),35th(0),36th(0),37th(0),

38th(0),39th(0),40th(0),41st(0),42nd(0),

43rd(0)

27th(0.6),28th(0),29th(0),30th(0),31st(0),

32nd(0),33rd(0)

27th

43.6 0.1 44th(0)34th(0.1),35th(0),36th(0),37th(0),38th(0),

39th(0),40th(0),41st(0),42nd(0),43rd(0)

28th(0.1),29th(0),30th(0),31st(0),32nd(0),

33rd(0),34th(0)

43.7 0.1 44th(0)35th(0.1),36th(0),37th(0),38th(0),39th(0),

40th(0),41st(0),42nd(0),43rd(0)

28th(0.2),29th(0),30th(0),31st(0),32nd(0),

33rd(0),34th(0),35th(0)

43.8 0.1 44th(0)36th(0.1),37th(0),38th(0),39th(0),40th(0),

41st(0),42nd(0),43rd(0)

28th(0.3),29th(0),30th(0),31st(0),32nd(0),

33rd(0),34th(0),35th(0),36th(0)

43.9 0.1 44th(0)37th(0.1),38th(0),39th(0),40th(0),41st(0),

42nd(0),43rd(0)

28th(0.4),29th(0),30th(0),31st(0),32nd(0),

33rd(0),34th(0),35th(0),36th(0),37th(0)

44 0.1 1 45th 44th(0),45th(0)38th(0.1),39th(0),40th(0),41st(0),42nd(0),

43rd(0)

28th(0.5),29th(0),30th(0),31st(0),32nd(0),

33rd(0),34th(0),35th(0),36th(0),37th(0),

38th(0)

44.1 0.1 44th(0),45th(0) 39th(0.1),40th(0),41st(0),42nd(0),43rd(0)

28th(0.6),29th(0),30th(0),31st(0),32nd(0),

33rd(0),34th(0),35th(0),36th(0),37th(0),

38th(0),39th(0)

28th

44.2 0.1 44th(0),45th(0) 40th(0.1),41st(0),42nd(0),43rd(0)29th(0.1),30th(0),31st(0),32nd(0),33rd(0),34th(0),35th(0),36th(0),37th(0),38th(0),

39th(0),40th(0)

44.3 0.1 44th(0),45th(0) 41st(0.1),42nd(0),43rd(0)

29th(0.2),30th(0),31st(0),32nd(0),33rd(0),

34th(0),35th(0),36th(0),37th(0),38th(0),

39th(0),40th(0),41st(0)

44.4 0.1 44th(0),45th(0) 42nd(0.1),43rd(0)

29th(0.3),30th(0),31st(0),32nd(0),33rd(0),

34th(0),35th(0),36th(0),37th(0),38th(0),

39th(0),40th(0),41st(0),42nd(0)

44.5 0.1 44th(0),45th(0) 43rd(0.1)

29th(0.4),30th(0),31st(0),32nd(0),33rd(0),

34th(0),35th(0),36th(0),37th(0),38th(0),

39th(0),40th(0),41st(0),42nd(0),43rd(0)

44.7 0.2

0

.

7

44th(0),45th(0)

29th(0.6),30th(0),31st(0),32nd(0),33rd(0),

34th(0),35th(0),36th(0),37th(0),38th(0),

39th(0),40th(0),41st(0),42nd(0),43rd(0)

29th

45 0.3 1 46th 44th(0),45th(0),46th(0)30th(0.3),31st(0),32nd(0),33rd(0),34th(0),35th(0),36th(0),37th(0),38th(0),39th(0),

40th(0),41st(0),42nd(0),43rd(0)

45.3 0.3 44th(0),45th(0),46th(0)

30th(0.6),31st(0),32nd(0),33rd(0),34th(0),

35th(0),36th(0),37th(0),38th(0),39th(0),

40th(0),41st(0),42nd(0),43rd(0)

30th

45.9 0.6 44th(0),45th(0),46th(0)31st(0.6),32nd(0),33rd(0),34th(0),35th(0),

36th(0),37th(0),38th(0),39th(0),40th(0),31st


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41st(0),42nd(0),43rd(0)

46 0.1 1 47th 44th(0),45th(0),46th(0),47th(0)

32nd(0.1),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0),41st(0),

42nd(0),43rd(0)

46.5 0.5 44th(0),45th(0),46th(0),47th(0)

32nd(0.6),33rd(0),34th(0),35th(0),36th(0),

37th(0),38th(0),39th(0),40th(0),41st(0),

42nd(0),43rd(0)

32nd

47 0.5 1 48th 44th(0),45th(0),46th(0),47th(0),48th(0)

33rd(0.5),34th(0),35th(0),36th(0),37th(0),

38th(0),39th(0),40th(0),41st(0),42nd(0),

43rd(0)

47.1 0.1 44th(0),45th(0),46th(0),47th(0),48th(0)

33rd(0.6),34th(0),35th(0),36th(0),37th(0),

38th(0),39th(0),40th(0),41st(0),42nd(0),

43rd(0)

33rd

47.7 0.6

0

.

7

44th(0),45th(0),46th(0),47th(0),48th(0)34th(0.6),35th(0),36th(0),37th(0),38th(0),

39th(0),40th(0),41st(0),42nd(0),43rd(0)34th

48 0.3 1 49th 44th(0),45th(0),46th(0),47th(0),48th(0),49th(0)35th(0.3),36th(0),37th(0),38th(0),39th(0),

40th(0),41st(0),42nd(0),43rd(0)

48.3 0.3 44th(0),45th(0),46th(0),47th(0),48th(0),49th(0)35th(0.6),36th(0),37th(0),38th(0),39th(0),

40th(0),41st(0),42nd(0),43rd(0)35th

48.9 0.6 44th(0),45th(0),46th(0),47th(0),48th(0),49th(0)36th(0.6),37th(0),38th(0),39th(0),40th(0),

41st(0),42nd(0),43rd(0)36th

49 0.1 1 50th44th(0),45th(0),46th(0),47th(0),48th(0),49th(0),

50th(0)

37th(0.1),38th(0),39th(0),40th(0),41st(0),

42nd(0),43rd(0)


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Figure 17, WIP growth in front of class 2 and class 4 of the unstable network with IA=1.0

0

5

10

15

0 5 10 15 20 25 30 35 40 45

W I P

Time

Class 2 Class 4


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4.1.2.3 Inter-arrival time=1.1

Manual book keeping for inter-arrival time=1.1 is shown in Table 7. This system is stable

because of similar behaviour which is m2 and m4 (ghost machines) serve simultaneously even

though they are virtual stations. It is also highlighted with blue colour when virtual stations work

simultaneously.

TABLE 7: BOOK KEEPING FOR INTER-ARRIVAL TIME= 1.1

Time Entity m1 (0.2) m2 (0.6) m3 (0.1) m4 (0.6) Out

0 1st 1st (0)

0.2 1st (0.2) 1st (0)

0.8 1st (0.6) 1st (0)

0.9 1st (0.1) 1st (0)

1.1 2nd 2nd (0) 1st (0.2)

1.5 2nd (0) 1st (0.6) 1st (0)

1.7 2nd (0.2) 2nd (0)

2.2 3rd 3rd (0) 2nd (0.5)

2.3 3rd (0.1) 2nd (0.6) 2nd (0)

2.4 3rd (0.2) 3rd (0) 2nd (0.1) 2nd (0)

3 3rd (0.6) 3rd (0) 2nd (0.6) 2nd (0)


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7.5 7th (0.1) 7th (0)

7.7 8th 8th (0) 7th (0.2)8.1 8th (0) 7th (0.6) 7th (0)

8.3 8th (0.2) 8th (0)

8.8 9th 9th (0) 8th (0.5)

8.9 9th (0.1) 8th (0.6) 8th (0)

9 9th (0.2) 9th (0) 8th (0.1) 8th (0)

9.6 9th (0.6) 9th (0) 8th (0.6) 8th (0)

9.7 9th (0.1) 9th (0)9.9 10th 10th (0) 9th (0.2)

10.3 10th (0) 9th (0.6) 9th (0)

4.1.2.4 Inter-arrival time=1.2

There are two scenarios for inter-arrival time=1.2 case.

One results in unstable system. This is due to the virtual stations that do not serve the entities

simultaneously. Book-keeping for this unstable system is shown in Table 8. At time=9.8,

machine, m2 just finish serving 8 th entity and simultaneously, machine, m1 also just finish

serving 9th entity. At the moment, 7th entity is already waiting at machine, m3 and thus, 8th entity

will join the queue behind 7th entity at m3 while 9

th entity will be at the entry of m2. Since m2 is

the priority buffer compared to m3, m2 will serve the 9 th entity even though 7th entity is already


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When m3 finished serving 7th entity, m2 will serve the waiting 9th entity while m4 will serve the

incoming 7th entity. This is when each virtual station serves each respective job simultaneously.

The following cycle will continue repeating thereby, synchronizing virtual stations. That results

in a stable network. Inside the book-keeping, it is highlighted with blue colour when virtual

stations work simultaneously. Fig. 19 shows the growth of WIP in front of virtual stations in the

stable network. It can be seen that the maximum WIP for each machine is kept at a constant of

two jobs all the time.

The main objective of above simulation is to study the buffer effects in a marginally stable

system. The buffer effect is found to stabilize a marginally stable system. Please take note that the

buffer effects of subsequent book-keeping have been eliminated.


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TABLE 8: BOOK KEEPING FOR UNSTABLE NETWORK WITH INTER-ARRIVAL TIME =1.2

TimeTime

diffIA Entity m1 (0.2) m2 (0.6) m3 (0.1) m4 (0.6) Out

0 1st 1st(0)

0.2 0.2 1st(0.2) 1st(0)

0.8 0.6 1st(0.6) 1st(0)

0.9 0.1 0.9 1st(0.1) 1st(0)

1.2 0.3 1.2 2nd 2nd(0) 1st(0.3)1.5 0.3 2nd(0) 1st(0.6) 1st

1.7 0.2 2nd(0.2) 2nd(0)

2.3 0.6 1.1 2nd(0.6) 2nd(0)

2.4 0.1 1.2 3rd 3rd(0) 2nd(0.1) 2nd(0)

3 0.6 3rd(0) 2nd(0.6) 2nd

3.2 0.2 0.8 3rd(0.2) 3rd(0)

3.6 0.4 1.2 4th 4th(0) 3rd(0.4)

3.8 0.2 4th(0.2) 3rd(0.6),4th(0) 3rd(0)

4.4 0.6 0.8 4th(0.6) 3rd(0),4th(0)

4.5 0.1 3rd(0.1),4th(0) 3rd(0)

4.6 0.1 1 4th(0.1) 3rd(0.1),4th(0)

4.8 0.2 1.2 5th 5th(0) 3rd(0.3),4th(0)

5.1 0.3 5th(0) 3rd(0.6),4th(0) 3rd5.7 0.6 0.9 5th(0) 4th(0.6) 4th

5.9 0.2 0.8 5th(0.2) 5th(0)

6 0.1 1.2 6th 6th(0) 5th(0.1)

6.2 0.2 6th(0.2) 5th(0.3),6th(0)

6.5 0.3 5th(0.6),6th(0) 5th(0)

7.1 0.6 1.1 6th(0.6) 5th(0),6th(0)

7.2 0.1 1.2 7th 7th(0) 5th(0.1),6th(0) 5th(0)

7.7 0.5 7th(0) 6th(0.1) 5th(0.1),6th(0)

7.8 0.1 7th(0) 5th(0.6),6th(0) 5th

8.4 0.6 1.2 8th 7th(0),8th(0) 6th(0.6) 6th

8.6 0.2 7th(0.2),8th(0) 7th(0)

8.8 0.2 8th(0.2) 7th(0.2),8th(0)

9.2 0.4 0.8 7th(0.6),8th(0) 7th(0)

9.6 0.4 1.2 9th 9th(0) 8th(0.4) 7th(0)

9.8 0.2 9th(0.2) 8th(0.6),9th(0) 7th(0),8th(0)

10.4 0.6 0.8 9th(0.6) 7th(0),8th(0),9th(0)

10.5 0.1 7th(0.1),8th(0),9th(0) 7th(0)

10.6 0.1 8th(0.1),9th(0) 7th(0.1),8th(0)

10.7 0.1 1.1 9th(0.1) 7th(0.2),8th(0),9th(0)


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10.8 0.1 1.2 10th 10th(0) 7th(0.3),8th(0),9th(0)

11.1 0.3 10th(0) 7th(0.6),8th(0),9th(0) 7th

11.7 0.6 0.9 10th(0) 8th(0.6),9th(0) 8th

12 0.3 1.2 11th 10th(0),11th(0) 9th(0.3)

12.3 0.3 10th(0),11th(0) 9th(0.6) 9th

12.5 0.2 10th(0.2),11th(0) 10th(0)

12.7 0.2 0.7 11th(0.2) 10th(0.2),11th(0)

13.1 0.4 0.8 10th(0.6),11th(0) 10th(0)

13.2 0.1 1.2 12th 12th(0) 11th(0.1) 10th(0)

13.4 0.2 12th(0.2) 11th(0.3),12th(0) 10th(0)

13.7 0.3 11th(0.6),12th(0) 10th(0),11th(0)

14.3 0.6 1.1 12th(0.6) 10th(0),11th(0),12th(0)

14.4 0.1 1.2 13th 13th(0) 10th(0.1),11th(0),12th(0) 10th(0)

14.5 0.1 13th(0) 11th(0.1),12th(0) 10th(0.1),11th(0)

14.6 0.1 13th(0) 12th(0.1) 10th(0.2),11th(0),12th(0)

15 0.4 0.6 13th(0) 10th(0.6),11th(0),12th(0) 10th

15.6 0.6 1.2 14th 13th(0),14th(0) 11th(0.6),12th(0) 11th

16.2 0.6 13th(0),14th(0) 12th(0.6) 12th

16.4 0.2 13th(0.2),14th(0) 13th(0)

16.6 0.2 1 14th(0.2) 13th(0.2),14th(0)

16.8 0.2 1.2 15th 15th(0) 13th(0.4),14th(0)

17 0.2 15th(0.2) 13th(0.6),14th(0),15th(0) 13th(0)

17.6 0.6 0.8 14th(0.6),15th(0) 13th(0),14th(0)

18 0.4 1.2 16th 16th(0) 15th(0.4) 13th(0),14th(0)

18.2 0.2 16th(0.2) 15th(0.6),16th(0) 13th(0),14th(0),15th(0)

18.8 0.6 0.8 16th(0.6) 13th(0),14th(0),15th(0),16th(0)

18.9 0.1 13th(0.1),14th(0),15th(0),16th(0) 13th(0)

19 0.1 14th(0.1),15th(0),16th(0) 13th(0.1),14th(0)

19.1 0.1 15th(0.1),16th(0) 13th(0.2),14th(0),15th(0)

19.2 0.1 1.2 17th 17th(0) 16th(0.1) 13th(0.3),14th(0),15th(0),16th(0)

19.5 0.3 17th(0) 13th(0.6),14th(0),15th(0),16th(0) 13th

20.1 0.6 0.9 17th(0) 14th(0.6),15th(0),16th(0) 14th

20.4 0.3 1.2 18th 17th(0),18th(0) 15th(0.3),16th(0)

20.7 0.3 17th(0),18th(0) 15th(0.6),16th(0) 15th

21.3 0.6 17th(0),18th(0) 16th(0.6) 16th

21.5 0.2 1.1 17th(0.2),18th(0) 17th(0)

21.6 0.1 1.2 19th 18th(0.1),19th(0) 17th(0.1)

21.7 0.1 18th(0.2),19th(0) 17th(0.2),18th(0)

21.9 0.2 19th(0.2) 17th(0.4),18th(0),19th(0)

22.1 0.2 17th(0.6),18th(0),19th(0) 17th(0)

22.7 0.6 1.1 18th(0.6),19th(0) 17th(0),18th(0)


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22.8 0.1 1.2 20th 20th(0) 19th(0.1) 17th(0),18th(0)

23 0.2 20th(0.2) 19th(0.3),20th(0) 17th(0),18th(0)

23.3 0.3 19th(0.6),20th(0) 17th(0),18th(0),19th(0)

23.9 0.6 1.1 20th(0.6) 17th(0),18th(0),19th(0),20th(0)

24 0.1 1.2 21st 21st(0) 17th(0.1),18th(0),19th(0),20th(0) 17th(0)

24.1 0.1 21st(0) 18th(0.1),19th(0),20th(0) 17th(0.1),18th(0)

24.2 0.1 21st(0) 19th(0.1),20th(0) 17th(0.2),18th(0),19th(0)

24.3 0.1 21st(0) 20th(0.1) 17th(0.3),18th(0),19th(0),20th(0)

24.6 0.3 21st(0) 17th(0.6),18th(0),19th(0),20th(0) 17th

25.2 0.6 1.2 22nd 21st(0),22nd(0) 18th(0.6),19th(0),20th(0) 18th

25.8 0.6 21st(0),22nd(0) 19th(0.6),20th(0) 19th

26.4 0.6 1.2 23rd 21st(0),22nd(0),23rd(0) 20th(0.6) 20th

26.6 0.2 21st(0.2),22nd(0),23rd(0) 21st(0)

26.8 0.2 22nd(0.2),23rd(0) 21st(0.2),22nd(0)

27 0.2 23rd(0.2) 21st(0.4),22nd(0),23rd(0)

27.2 0.2 0.8 21st(0.6),22nd(0),23rd(0) 21st(0)

27.6 0.4 1.2 24th 24th(0) 22nd(0.4),23rd(0) 21st(0)

27.8 0.2 24th(0.2) 22nd(0.6),23rd(0),24th(0) 21st(0),22nd(0)

28.4 0.6 0.8 23rd(0.6),24th(0) 21st(0),22nd(0),23rd(0)

28.8 0.4 1.2 25th 25th(0) 24th(0.4) 21st(0),22nd(0),23rd(0)

29 0.2 25th(0.2) 24th(0.6),25th(0) 21st(0),22nd(0),23rd(0),24th(0)

29.6 0.6 25th(0.6) 21st(0),22nd(0),23rd(0),24th(0),25th(0)

29.7 0.1 21st(0.1),22nd(0),23rd(0),24th(0),25th(0) 21st(0)

29.8 0.1 22nd(0.1),23rd(0),24th(0),25th(0) 21st(0.1),22nd(0)

29.9 0.1 1.1 23rd(0.1),24th(0),25th(0) 21st(0.2),22nd(0),23rd(0)

30 0.1 1.2 26th 26th(0) 24th(0.1),25th(0) 21st(0.3),22nd(0),23rd(0),24th(0)

30.1 0.1 26th(0) 25th(0.1) 21st(0.4),22nd(0),23rd(0),24th(0),25th(0)

30.3 0.2 26th(0) 21st(0.6),22nd(0),23rd(0),24th(0),25th(0) 21st

30.9 0.6 0.9 26th(0) 22nd(0.6),23rd(0),24th(0),25th(0) 22nd

31.2 0.3 1.2 27th 26th(0),27th(0) 23rd(0.3),24th(0),25th(0)

31.5 0.3 26th(0),27th(0) 23rd(0.6),24th(0),25th(0) 23rd

32.1 0.6 0.9 26th(0),27th(0) 24th(0.6),25th(0) 24th

32.4 0.3 1.2 28th 26th(0),27th(0),28th(0) 25th(0.3)

32.7 0.3 26th(0),27th(0),28th(0) 25th(0.6) 25th

32.9 0.2 26th(0.2),27th(0),28th(0) 26th(0)

33.1 0.2 27th(0.2),28th(0) 26th(0.2),27th(0)

33.3 0.2 0.9 28th(0.2) 26th(0.4),27th(0),28th(0)

33.5 0.2 26th(0.6),27th(0),28th(0) 26th(0)

33.6 0.1 1.2 29th 29th(0) 27th(0.1),28th(0) 26th(0)

33.8 0.2 29th(0.2) 27th(0.3),28th(0),29th(0) 26th(0)

34.1 0.3 27th(0.6),28th(0),29th(0) 26th(0),27th(0)


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34.7 0.6 1.1 28th(0.6),29th(0) 26th(0),27th(0),28th(0)

34.8 0.1 1.2 30th 30th(0) 29th(0.1) 26th(0),27th(0),28th(0)

35 0.2 30th(0.2) 29th(0.3),30th(0) 26th(0),27th(0),28th(0)

35.3 0.3 29th(0.6),30th(0) 26th(0),27th(0),28th(0),29th(0)

35.9 0.6 1.1 30th(0.6) 26th(0),27th(0),28th(0),29th(0),30th(0)

36 0.1 1.2 31st 31st(0) 26th(0.1),27th(0),28th(0),29th(0),30th(0) 26th(0)

36.1 0.1 31st(0) 27th(0.1),28th(0),29th(0),30th(0) 26th(0.1),27th(0)

36.2 0.1 31st(0) 28th(0.1),29th(0),30th(0) 26th(0.2),27th(0),28th(0)

36.3 0.1 31st(0) 29th(0.1),30th(0) 26th(0.3),27th(0),28th(0),29th(0)

36.4 0.1 31st(0) 30th(0.1) 26th(0.4),27th(0),28th(0),29th(0),30th(0)

36.6 0.2 0.6 31st(0) 26th(0.6),27th(0),28th(0),29th(0),30th(0) 26th

37.2 0.6 1.2 32nd 31st(0),32nd(0) 27th(0.6),28th(0),29th(0),30th(0) 27th

37.8 0.6 31st(0),32nd(0) 28th(0.6),29th(0),30th(0) 28th

38.4 0.6 1.2 33rd 31st(0),32nd(0),33rd(0) 29th(0.6),30th(0) 29th

39 0.6 31st(0),32nd(0),33rd(0) 30th(0.6) 30th

39.2 0.2 31st(0.2),32nd(0),33rd(0) 31st(0)

39.4 0.2 32nd(0.2),33rd(0) 31st(0.2),32nd(0)

39.6 0.2 1.2 34th 33rd(0.2),34th(0) 31st(0.4),32nd(0),33rd(0)

39.8 0.2 34th(0.2) 31st(0.6),32nd(0),33rd(0),34th(0) 31st(0)

40.4 0.6 0.8 32nd(0.6),33rd(0),34th(0) 31st(0),32nd(0)

40.8 0.4 1.2 35th 35th(0) 33rd(0.4),34th(0) 31st(0),32nd(0)

41 0.2 35th(0.2) 33rd(0.6),34th(0),35th(0) 31st(0),32nd(0),33rd(0)

41.6 0.6 34th(0.6),35th(0) 31st(0),32nd(0),33rd(0),34th(0)

42 0.4 1.2 36th 36th(0) 35th(0.4) 31st(0),32nd(0),33rd(0),34th(0)

42.2 0.2 36th(0.2) 35th(0.6),36th(0) 31st(0),32nd(0),33rd(0),34th(0),35th(0)

42.8 0.6 0.8 36th(0.6) 31st(0),32nd(0),33rd(0),34th(0),35th(0),36th(0)

42.9 0.1 31st(0.1),32nd(0),33rd(0),34th(0),35th(0),36th(0) 31st(0)

43 0.1 32nd(0.1),33rd(0),34th(0),35th(0),36th(0) 31st(0.1),32nd(0)

43.1 0.1 33rd(0.1),34th(0),35th(0),36th(0) 31st(0.2),32nd(0),33rd(0)

43.2 0.1 1.2 37th 37th(0) 34th(0.1),35th(0),36th(0) 31st(0.3),32nd(0),33rd(0),34th(0)

43.3 0.1 37th(0) 35th(0.1),36th(0) 31st(0.4),32nd(0),33rd(0),34th(0),35th(0)

43.4 0.1 37th(0) 36th(0.1) 31st(0.5),32nd(0),33rd(0),34th(0),35th(0),36th(0)

43.5 0.1 37th(0) 31st(0.6),32nd(0),33rd(0),34th(0),35th(0),36th(0) 31st

44.1 0.6 0.9 37th(0) 32nd(0.6),33rd(0),34th(0),35th(0),36th(0) 32nd

44.4 0.3 1.2 38th 37th(0),38th(0) 33rd(0.3),34th(0),35th(0),36th(0)

44.7 0.3 37th(0),38th(0) 33rd(0.6),34th(0),35th(0),36th(0) 33rd

45.3 0.6 0.9 37th(0),38th(0) 34th(0.6),35th(0),36th(0) 34th

45.6 0.3 1.2 39th 37th(0),38th(0),39th(0) 35th(0.3),36th(0)

45.9 0.3 37th(0),38th(0),39th(0) 35th(0.6),36th(0) 35th

46.5 0.6 37th(0),38th(0),39th(0) 36th(0.6) 36th

46.7 0.2 1.1 37th(0.2),38th(0),39th(0) 37th(0)


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46.8 0.1 1.2 40th 38th(0.1),39th(0),40th(0) 37th(0.1)

46.9 0.1 38th(0.2),39th(0),40th(0) 37th(0.2),38th(0)

47.1 0.2 39th(0.2),40th(0) 37th(0.4),38th(0),39th(0)

47.3 0.2 40th(0.2) 37th(0.6),38th(0),39th(0),40th(0) 37th(0)

47.9 0.6 1.1 38th(0.6),39th(0),40th(0) 37th(0),38th(0)

48 0.1 1.2 41st 41st(0) 39th(0.1),40th(0) 37th(0),38th(0)

48.2 0.2 41st(0.2) 39th(0.3),40th(0),41st(0) 37th(0),38th(0)

48.5 0.3 39th(0.6),40th(0),41st(0) 37th(0),38th(0),39th(0)

49.1 0.6 1.1 40th(0.6),41st(0) 37th(0),38th(0),39th(0),40th(0)

49.2 0.1 1.2 42nd 42nd(0) 41st(0.1) 37th(0),38th(0),39th(0),40th(0)

49.4 0.2 42nd(0.2) 41st(0.3),42nd(0) 37th(0),38th(0),39th(0),40th(0)

49.7 0.3 41st(0.6),42nd(0) 37th(0),38th(0),39th(0),40th(0),41st(0)

50.3 0.6 1.1 42nd(0.6) 37th(0),38th(0),39th(0),40th(0),41st(0),42nd(0)

50.4 0.1 1.2 43rd 43rd(0) 37th(0.1),38th(0),39th(0),40th(0),41st(0),42nd(0) 37th(0)

50.5 0.1 43rd(0) 38th(0.1),39th(0),40th(0),41st(0),42nd(0) 37th(0.1),38th(0)

50.6 0.1 43rd(0) 39th(0.1),40th(0),41st(0),42nd(0) 37th(0.2),38th(0),39th(0)

50.7 0.1 43rd(0) 40th(0.1),41st(0),42nd(0) 37th(0.3),38th(0),39th(0),40th(0)

50.8 0.1 43rd(0) 41st(0.1),42nd(0) 37th(0.4),38th(0),39th(0),40th(0),41st(0)

50.9 0.1 43rd(0) 42nd(0.1) 37th(0.5),38th(0),39th(0),40th(0),41st(0),42nd(0)

51 0.1 0.6 0 43rd(0) 37th(0.6),38th(0),39th(0),40th(0),41st(0),42nd(0) 37th

51.6 0.6 1.2 44th 43rd(0),44th(0) 38th(0.6),39th(0),40th(0),41st(0),42nd(0) 38th

52.2 0.6 43rd(0),44th(0) 39th(0.6),40th(0),41st(0),42nd(0) 39th

52.8 0.6 1.2 45th 43rd(0),44th(0),45th(0) 40th(0.6),41st(0),42nd(0) 40th

53.4 0.6 43rd(0),44th(0),45th(0) 41st(0.6),42nd(0) 41st

54 0.6 1.2 46th 43rd(0),44th(0),45th(0),46th(0) 42nd(0.6) 42nd

54.2 0.2 43rd(0.2),44th(0),45th(0),46th(0) 43rd(0)

54.4 0.2 44th(0.2),45th(0),46th(0) 43rd(0.2),44th(0)

54.6 0.2 45th(0.2),46th(0) 43rd(0.4),44th(0),45th(0)

54.8 0.2 0.8 46th(0.2) 43rd(0.6),44th(0),45th(0),46th(0) 43rd(0)

55.2 0.4 1.2 47th 47th(0) 44th(0.4),45th(0),46th(0) 43rd(0)

55.4 0.2 47th(0.2) 44th(0.6),45th(0),46th(0),47th(0) 43rd(0),44th(0)

56 0.6 45th(0.6),46th(0),47th(0) 43rd(0),44th(0),45th(0)

56.4 0.4 1.2 48th 48th(0) 46th(0.4),47th(0) 43rd(0),44th(0),45th(0)

56.6 0.2 48th(0.2) 46th(0.6),47th(0),48th(0) 43rd(0),44th(0),45th(0),46th(0)

57.2 0.6 47th(0.6),48th(0) 43rd(0),44th(0),45th(0),46th(0),47th(0)57.6 0.4 1.2 49th 49th(0) 48th(0.4) 43rd(0),44th(0),45th(0),46th(0),47th(0)

57.8 0.2 49th(0.2) 48th(0.6),49th(0) 43rd(0),44th(0),45th(0),46th(0),47th(0),48th(0)

58.4 0.6 0.8 49th(0.6) 43rd(0),44th(0),45th(0),46th(0),47th(0),48th(0),49th(0)

58.5 0.1 43rd(0.1),44th(0),45th(0),46th(0),47th(0),48th(0),49th(0) 43rd(0)

58.6 0.1 44th(0.1),45th(0),46th(0),47th(0),48th(0),49th(0) 43rd(0.1),44th(0)

58.7 0.1 45th(0.1),46th(0),47th(0),48th(0),49th(0) 43rd(0.2),44th(0),45th(0)


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58.8 0.1 1.2 50th 50th(0) 46th(0.1),47th(0),48th(0),49th(0) 43rd(0.3),44th(0),45th(0),46th(0)

58.9 0.1 50th(0) 47th(0.1),48th(0),49th(0) 43rd(0.4),44th(0),45th(0),46th(0),47th(0)

59 0.1 50th(0) 48th(0.1),49th(0) 43rd(0.5),44th(0),45th(0),46th(0),47th(0),48th(0)

59.1 0.1 50th(0) 49th(0.1) 43rd(0.6),44th(0),45th(0),46th(0),47th(0),48th(0),49th(0) 43rd


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TABLE 9: BOOK KEEPING FOR STABLE NETWORK WITH INTER-ARRIVAL TIME =1.2

TimeTime


Buffer Server Buffer Server Buffer Server Buffer Server0 1st

0 1st(0)

0 1st(0)

0.2 0.2 1st(0.2)

0.2 1st(0)

0.2 1st(0)

0.8 0.6 1st(0.6)

0.8 1st(0)

0.8 1st(0)

0.9 0.1 0.9 1st(0.1)

0.9 1st(0)

0.9 1st(0)

1.2 0.3 1.2 2nd 1st(0.3)

1.2 2nd(0) 1st(0.3)

1.5 0.3 2nd(0) 1st(0.6) 1st

1.5 2nd(0)

1.7 0.2 2nd(0.2)

1.7 2nd(0)

1.7 2nd(0)2.3 0.6 1.1 2nd(0.6)

2.3 2nd(0)

2.3 2nd(0)


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2.4 0.1 1.2 3rd 2nd(0.1)

2.4 3rd(0) 2nd(0)

2.4 3rd(0) 2nd(0)

3 0.6 3rd(0) 2nd(0.6) 2nd

3 3rd(0)

3.2 0.2 0.8 3rd(0.2)

3.2 3rd(0)

3.2 3rd(0)

3.6 0.4 1.2 4th 3rd(0.4)

3.6 4th(0)

3.6 4th(0)

3.8 0.2 4th(0.2) 3rd(0.6)

3.8 4th(0) 3rd(0)

3.8 4th(0) 3rd(0)

4.4 0.6 0.8 4th(0.6) 3rd(0)

4.4 4th(0) 3rd(0)

4.5 0.1 4th(0) 3rd(0.1)

4.5 4th(0) 3rd(0)

4.5 3rd(0)

4.6 0.1 1 4th(0.1) 3rd(0.1)

4.6 4th(0) 3rd(0.1)

4.8 0.2 1.2 5th 4th(0) 3rd(0.3)

4.8 5th(0) 4th(0) 3rd(0.3)

5.1 0.3 5th(0) 4th(0) 3rd(0.6) 3rd

5.1 5th(0) 4th(0)

5.7 0.6 0.9 5th(0) 4th(0.6) 4th


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5.7 5th(0)

5.9 0.2 0.8 5th(0.2)

5.9 5th(0)

5.9 5th(0)

6 0.1 1.2 6th 5th(0.1)

6 6th(0) 5th(0.1)

6 6th(0) 5th(0.1)

6.2 0.2 6th(0.2) 5th(0.3)

6.2 6th(0)

6.5 0.3 6th(0) 5th(0.6)

6.5 6th(0) 5th(0)

7.1 0.6 1.1 6th(0.6) 5th(0)

7.1 5th(0),6th(0)

7.1 6th(0) 5th(0)

7.2 0.1 1.2 7th 6th(0) 5th(0.1)

7.2 7th(0) 6th(0) 5th(0)

7.2 7th(0) 6th(0) 5th(0)

7.3 0.1 7th(0) 6th(0.1) 5th(0.1)

7.3 7th(0) 6th(0) 5th(0.1)

7.8 0.5 7th(0) 6th(0) 5th(0.6) 5th

7.8 7th(0) 6th(0)

8.4 0.6 1.2 8th 7th(0) 6th(0.6) 6th

8.4 8th(0) 7th(0)

8.6 0.2 8th(0) 7th(0.2)8.6 8th(0) 7th(0)

8.6 8th(0) 7th(0)


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8.8 0.2 8th(0.2) 7th(0.2)

8.8 8th(0) 7th(0.2)

9.2 0.4 0.8 8th(0) 7th(0.6)

9.2 8th(0) 7th(0)

9.6 0.4 1.2 9th 8th(0.4) 7th(0)

9.6 9th(0) 8th(0.4) 7th(0)

9.6 9th(0) 8th(0.4) 7th(0)

9.8 0.2 9th(0.2) 8th(0.6) 7th(0)

9.8 9th(0) 8th(0) 7th(0)

9.9 0.1 9th(0) 8th(0) 7th(0.1)

9.9 9th(0) 7th(0)

9.9 9th(0) 7th(0)

10.5 0.6 0.9 9th(0.6) 8th(0) 7th(0.6) 7th

10.5 9th(0) 8th(0)

10.6 0.1 9th(0) 8th(0.1)

10.6 9th(0) 8th(0)

10.6 9th(0) 8th(0)

10.7 0.1 9th(0.1) 8th(0.1)

10.7 9th(0) 8th(0.1)

10.8 0.1 1.2 10th 9th(0) 8th(0.2)

10.8 10th(0) 9th(0) 8th(0.2)

11.2 0.4 10th(0) 9th(0) 8th(0.6) 8th

11.2 10th(0) 9th(0)

11.8 0.6 10th(0) 9th(0.6) 9th11.8 10th(0)

12 0.2 1.2 11th 10th(0.2)


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12 11th(0) 10th(0)

12 11th(0) 10th(0)

12.2 0.2 11th(0.2) 10th(0.2)

12.2 11th(0) 10th(0.2)

12.2 11th(0) 10th(0.2)

12.6 0.4 0.6 11th(0) 10th(0.6)

12.6 11th(0) 10th(0)

13.2 0.6 1.2 12th 11th(0.6) 10th(0)

13.2 12th(0) 11th(0) 10th(0)

13.2 12th(0) 10th(0)

13.3 0.1 12th(0.1) 11th(0) 10th(0.1)

13.3 12th(0.1) 11th(0) 10th(0)

13.4 0.1 12th(0.2) 11th(0.1) 10th(0)

13.4 12th(0) 11th(0) 10th(0)

13.4 12th(0) 11th(0) 10th(0)

14 0.6 12th(0.6) 11th(0) 10th(0.6) 10th

14 12th(0) 11th(0)

14 12th(0) 11th(0)

14.1 0.1 0.9 12th(0.1) 11th(0.1)

14.1 12th(0) 11th(0.1)

14.4 0.3 1.2 13th 12th(0) 11th(0.4)

14.4 13th(0) 12th(0) 11th(0.4)

14.6 0.2 13th(0) 12th(0) 11th(0.6) 11th

14.6 13th(0) 12th(0)15.2 0.6 13th(0) 12th(0.6) 12th

15.2 13th(0)


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15.4 0.2 1 13th(0.2)

15.4 13th(0)

15.4 13th(0)

15.6 0.2 1.2 14th 13th(0.2)

15.6 14th(0) 13th(0.2)

15.6 14th(0) 13th(0.2)

15.8 0.2 14th(0.2) 13th(0.4)

15.8 14th(0)

16 0.2 14th(0) 13th(0.6)

16 14th(0) 13th(0)

16.6 0.6 1 14th(0.6) 13th(0)

16.6 14th(0) 13th(0)

16.7 0.1 14th(0) 13th(0.1) 13th(0)

16.7 14th(0) 13th(0)

16.7 13th(0)

16.8 0.1 1.2 15th 14th(0.1) 13th(0.1)

16.8 15th(0) 14th(0) 13th(0.1)

17.3 0.5 15th(0) 14th(0) 13th(0.6) 13th

17.3 15th(0) 14th(0)

17.9 0.6 1.1 15th(0) 14th(0.6) 14th

17.9 15th(0)

18 0.1 1.2 16th 15th(0.1)

18 16th(0) 15th(0.1)

18.1 0.1 16th(0) 15th(0.2)18.1 16th(0) 15th(0)

18.1 16th(0) 15th(0)


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18.3 0.2 16th(0.2) 15th(0.2)

18.3 16th(0) 15th(0.2)

18.7 0.4 0.7 16th(0) 15th(0.6)

18.7 16th(0) 15th(0)

19.2 0.5 1.2 17th 16th(0.5) 15th(0)

19.2 17th(0) 16th(0.5) 15th(0)

19.2 17th(0) 16th(0.5) 15th(0)

19.3 0.1 17th(0.1) 16th(0.6) 15th(0)

19.3 17th(0.1) 16th(0) 15th(0)

19.4 0.1 17th(0.2) 16th(0) 15th(0.1)

19.4 17th(0) 16th(0) 15th(0)

19.4 17th(0) 16th(0) 15th(0)

19.5 0.1 17th(0) 16th(0.1) 15th(0.1)

19.5 17th(0) 16th(0) 15th(0.1)

20 0.5 17th(0.5) 16th(0) 15th(0.6) 15th

20 17th(0.5) 16th(0)

20.1 0.1 17th(0.6) 16th(0.1)

20.1 17th(0) 16th(0.1)

20.1 17th(0) 16th(0.1)

20.2 0.1 1 17th(0.1) 16th(0.2)

20.2 17th(0) 16th(0.2)

20.4 0.2 1.2 18th 17th(0) 16th(0.4)

20.4 18th(0) 17th(0) 16th(0.4)

20.6 0.2 18th(0) 17th(0) 16th(0.6) 16th20.6 18th(0) 17th(0)

21.2 0.6 18th(0) 17th(0.6) 17th


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21.2 18th(0)

21.4 0.2 1 18th(0.2)

21.4 18th(0)

21.4 18th(0)

21.6 0.2 1.2 19th 18th(0.2)

21.6 19th(0) 18th(0.2)

21.6 19th(0) 18th(0.2)

21.8 0.2 19th(0.2) 18th(0.4)

21.8 19th(0) 18th(0.4)

22 0.2 19th(0) 18th(0.6)

22 19th(0) 18th(0)

22.6 0.6 1 19th(0.6) 18th(0)

22.6 19th(0) 18th(0)

22.7 0.1 0.9 19th(0) 18th(0.1)

22.7 19th(0) 18th(0)

22.7 19th(0) 18th(0)

22.8 0.1 1.2 20th 19th(0.1) 18th(0.1)

22.8 20th(0) 19th(0) 18th(0.1)

23.3 0.5 20th(0) 19th(0) 18th(0.6) 18th

23.3 19th(0)

23.9 0.6 20th(0) 19th(0.6) 19th

23.9 20th(0)

24 0.1 1.2 21st 20th(0.1)

24 21st(0) 20th(0.1)24.1 0.1 21st(0) 20th(0.2)

24.1 21st(0) 20th(0)


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24.1 20th(0)

24.3 0.2 21st(0.2) 20th(0.2)

24.7 0.4 0.7 21st(0) 20th(0.6)

24.7 21st(0) 20th(0)

25.2 0.5 1.2 22nd 21st(0.5) 20th(0)25.2 22nd(0) 21st(0.5) 20th(0)

25.2 22nd(0) 21st(0.5) 20th(0)

25.3 0.1 22nd(0.1) 21st(0.6) 20th(0)

25.3 22nd(0.1) 21st(0) 20th(0)

25.4 0.1 22nd(0.2) 21st(0) 20th(0.1)

25.4 22nd(0) 21st(0) 20th(0)

25.4 22nd(0) 21st(0) 20th(0)

25.5 0.1 22nd(0) 21st(0.1) 20th(0.1)

25.4 22nd(0) 21st(0) 20th(0.1)

26 0.5 0.8 22nd(0.5) 21st(0) 20th(0.6) 20th

26 22nd(0.5) 21st(0)

26.1 0.1 22nd(0.6) 21st(0.1)

26.1 22nd(0) 21st(0.1)

26.1 22nd(0) 21st(0.1)

26.2 0.1 1 22nd(0.1) 21st(0.2)


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Figure 18, WIP growth in front of virtual stations in the unstable network with IA=1.2

0

5

10

15

0 5 10 15 20 25 30 35 40 45

W I P

Time

Class 2 Class 4


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Figure 19, WIP growth in front of virtual stations in the stable network with IA=1.2

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

W I P

Time

Class 2 Class 4


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4.1.3 Simulation Result, uniform distribution inter-arrival time

The simulation result of the Lu-Kumar network is shown in Fig. 20 in which the inter-arrival time

is the uniform distribution and the service time is deterministic. The system is stable when the

uniform distribution of the inter-arrival time is between 0.8 and 0.9. When uniform distribution is

(0.8, 0.9), the probability of falling on two end points is zero. In mathematical form,

(0.8) = 0 & (0.9) = 0 Thus, inter-arrival time will never be 0.9 and the system will not go into unstable state. However,

when the uniform distribution of the inter-arrival time is (0.8, 1.0) or (0.8, 0.905) or even (0.8,

0.9005), the system will finally go into unstable state as there is a probability that the inter-arrival

time becomes 0.9 or more than 0.9. When the inter-arrival time becomes 0.9 or more than 0.9, the

system falls into unstable state and will not come back to the stable state in this case even though

the following inter-arrival time is between 0.8 and 0.9. That leads to the next topic of the

condition in which the Lu-Kumar network will come back to the stable state when the initial

inter-arrival time is 0.9 i.e. there is a small disturbance at initial state of the system.


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Figure 20, Total queue time of first 300 samples with different uniform distribution arrival time.

-50

0

50

100

150

200

250

0 50 100 150 200 250 300 350

T o t a

l Q t i m e

nth sample

AT= uniform dis (0.8,0.9)

AT=uniform dis (0.8,0.905)

AT=uniform dis (0.8,0.9005)

AT= uniform dis (0.8,0.92)

AT=uniform dis (0.8,1)


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4.2 Stability, Instability and their relationship

4.2.1 System started single 0.9 inter-arrival time

After inter-arrival time for 1st entity is 0.9, the system will come back to stable state only when

the inter-arrival time of the following entities is more than 1.2 (excluding 1.2). The respective

book keeping of each experiment is discussed in detail as below.

4.2.1.1 Book-keeping for Inter-arrival time = 0.9_0.85_0.85_0.85…

The system with inter-arrival time =0.9_0.85_0.85_0.85… is not stable as the synchronization

between virtual stations happens only 0.6 sec at 11.35 sec from the start. Only this 0.6sec is not

enough for the system to come back to the stable state.

Book-keeping for inter-arrival time =0.9_0.85_0.85_0.85_... is shown in appendix C because of

limited space inside this main report. It is highlighted with blue colour when virtual stations work

simultaneously i.e. synchronization occurs between virtual stations. The book-keeping is shown

only until 50th entity because of limited space of the report.



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The book-keeping for inter-arrival time=0.9_1.50_1.50_1.50_... is shown in Table 10. This

system is stable because the inter-arrival time becomes equal to the total service time

(0.2+0.6+0.1+0.6=1.5) of the system. When a particular entity comes into the system, another

entity is about to come out of the system. That helps the system to be balanced and the system

stable even though there is a small disturbance at initial state and synchronization between virtual

machines does not occur at all.

TABLE 10: BOOK-KEEPING FOR INTER-ARRIVAL TIME=0.9_1.50_1.50_1.50_...

Timetime


0 1st 1st (0)

0.2 0.2 1st (0.2) 1st (0)0.8 0.6 1st (0.6) 1st (0)

0.9 0.1 0.9 2nd 2nd (0) 1st (0.1) 1st (0)

1.5 0.6 2nd (0) 1st (0.6) 1st

1.7 0.2 2nd (0.2) 2nd (0)

2.3 0.6 2nd (0.6) 2nd (0)

2.4 0.1 1.5 3rd 3rd(0) 2nd (0.1) 2nd (0)


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6.8 0.3 6th(0) 5th(0.6) 5th

7 0.2 6th(0.2) 6th(0)7.6 0.6 1.1 6th(0.6) 6th(0)

7.7 0.1 6th(0.1) 6th(0)

7.9 0.2 1.4 7th 7th(0) 6th(0.2)

8.3 0.4 7th(0) 6th(0.6) 6th

8.5 0.2 7th(0.2) 7th(0)

9.1 0.6 7th(0.6) 7th(0)

9.2 0.1 7th(0.1) 7th(0)9.3 0.1 1.4 8th 8th(0) 7th(0.1)

9.8 0.5 8th(0) 7th(0.6) 7th

10 0.2 8th(0.2) 8th(0)

10.6 0.6 8th(0.6) 8th(0)

10.7 0.1 1.4 9th 9th(0) 8th(0.1) 8th(0)

11.3 0.6 9th(0) 8th(0.6) 8th

11.5 0.2 9th(0.2) 9th(0)12.1 0.6 1.4 10th 10th(0) 9th(0.6) 9th(0)

12.2 0.1 10th(0.1) 9th(0.1) 9th(0)

12.3 0.1 10th(0.2) 10th(0) 9th(0)

12.9 0.6 10th(0.6) 10th(0) 9th(0.6) 9th

13 0.1 10th(0.1) 10th(0)

13.5 0.5 1.4 11th 11th(0) 10th(0.5)


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30.2 0.6 22nd(0.6) 22nd(0)

30.3 0.1 1.4 23rd 23rd(0) 22nd(0.1) 22nd(0)30.9 0.6 23rd(0) 22nd(0.6) 22nd

31.1 0.2 23rd(0.2) 23rd(0)

31.7 0.6 1.4 24th 24th(0) 23rd(0.6) 23rd(0)

31.8 0.1 24th(0.1) 23rd(0.1) 23rd(0)

31.9 0.1 24t

Final Report Mean Queue Time Study for the Lu_Kumar Network_Nyan Soe Lynn

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