AN INTRODUCTION TO THE OPERATIONAL ANALYSIS OF QUEUING NETWORK MODELS Peter J. Denning, Jeffrey P....
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Transcript of AN INTRODUCTION TO THE OPERATIONAL ANALYSIS OF QUEUING NETWORK MODELS Peter J. Denning, Jeffrey P....
AN INTRODUCTION TO THE OPERATIONAL ANALYSISOF QUEUING NETWORK MODELS
Peter J. Denning, Jeffrey P. Buzen, The Operational Analysis of Queueing Network Models.ACM Computing Surveys 10(3): 225-261, 1978.
Queuing Models Computer systems contain many queues
Ready queue I/O device queues Message queues …
Bottlenecks and queuing delays have a major impact on computer system performance
Queuing models capture this impact
Operational Analysis Operational Analysis
Provides good bounds of performance indices of many computer systems
Only makes verifiable assumptions of behavior of these systems
Is easier to teach and to understand than stochastic analysis
Applications “Back of the envelope” estimations of
system performance Fast and easy
Validation of results of a simulation study Checks for reasonableness of results
Communication with non-specialists
Other queuing models Stochastic models
Also known as Markov models More powerful Often make unrealistic assumptions
about investigated systems Disk service times are not
exponentially distributed! Provide surprisingly good estimates of
computer system performance
Note Stochastic models can also be used to
estimate Availability of computer systems:
fraction of time system will be operational Reliability of computer systems:
probability that the system will never fail over a time interval of duration t
Essential for real-time systems and storage systems
BASIC CONCEPTS
Hypothesis testability All hypotheses made by OA can be
tested by Observing the behavior of actual
systems Over finite time periods
Operational Quantities Can be
Basic Quantities:Directly measured over a finite observation period
Number of arrivals, … Derived Quantities:
Computed from basic quantities Server utilization, mean service
time, …
A single server
Server has its associated queue We will treat the server and its queue
as a single “black box”
Basic Quantities T the length of the observation
period A the number of arrivals during the
observation period B the total amount of busy times
during the observation period C the number of completions
during the observation period
A CB
T
Derived quantities = A/T the arrival rate X = C/T the output rate U = B/T the utilization S = B/C the mean service time
Example Checkout lane:
T = 2 hours A = 31 customers B = 108 minutes C = 30 customers
We have = 15.5 customers/hour …
Utilization law
U = B/T = (C/T )(B/C) = XS
Can compute utilization knowing both Completion rate Mean service time
When you think about it, it is fairly intuitive
Job flow balance For most systems, arrivals and
completions balance each other over any long observation period Systems have finite buffers
We will assume that A = C Can check validity of assumption over
any specific observation period
If A = C then U = S
Steady state If the system has a steady state, we
can define N = average number of tasks in
system R = average residency of tasks in
system
R is also known as the response time
Little’s law If W is the total time spent by all tasks
inside the system over the observation period Measured in requests × time units
Then N = W/T R = W/C
Since W/T = (C/T)(W/C) = XR, N = XRThis is an important result
Explanation
Number of tasks in the system
Time
T
W
N = W/T
0
The green area and thearea delimited by thedashed lines have equalsizes
An example An FBI agent has observed people entering
and leaving a secret meeting Over a period of two hours he has seen 10
people staying an average of 30 minutes each
Having 10 people staying 30 minutes each corresponds to 300 people x minutes
This is the same as having 300/120 = 2.5 people staying over the whole duration of the meeting
NETWORKS OF SERVERS
Network of servers (I)
Arrivals Departures
Open network
Network of servers (II)
Arrivals Departures
Closed network
Operational quantities Over the observation period, we measure
C = the number of job completions Ck = the number of tasks completed by
device k We define
X0 = C/T = the system throughput Xk = Ck/T = the output rate at server k Vk = Ck/C = the visit count at server k
Relations
Ck = VkC The total number of visits of server k is
equal to the number of visits of server k by job completion times the number of job completions
Xk = Vk X0
The output rate of server k is equal to the number of visits of server k by job completion times the job throughput
Principle of job flow balance
For each device i, the output rate Xi is the same as the total input rates to device i
Ai = Ci True for all observation periods long
enough such that |Ai - Ci | << Ci Output rates Xi are then throughputs
System response time (I) We define
Nbar = average number of jobs in the system
nbari = average number of jobs at device i
Nbar = Σi nbari
in
System response time (II) Applying Little’s law, we have
R = Nbar/X0
andnbari = RiXi
which we can rewritenbari = RiViX0
Hence
R = Σi ViRi
Application:Bottleneck analysis
A system has one CPU and one disk drive
It processes transactions such that VCPU = 12 and SCPU = 5ms
VDisk = 11 and SDISK = 8ms
What is the maximum system throughput?
Bottleneck analysis (cont’d) Let us compute maximum device
throughputs Maximum XCPU = 1/0.005 = 200 requests/s
Maximum XDisk = 1/0.008 = 125 requests/s Since Xi = Vi X0
Maximum throughput compatible with CPU workload is 200/12 = 16.7 transactions/s
Maximum throughput compatible with disk workload is 125/11 = 11.4 transactions/s
Bottleneck analysis (cont’d) The disk is this the bottleneck
It has highest ViSi product Identifying feature of any bottleneck
device Increasing the system throughput might
require Sharing disk requests with a second disk Increasing the efficiency of the system
I/O buffer
Important
Bottleneck analysis (cont’d) In addition, the maximum throughput of
11.4 transactions/s can only be achieved with a 100% disk utilization It would result in very large queues at
the disk In practice, we want to limit the disk
utilization to 60 to 80%
Systems with terminals
M Terminals
Wholesystem
Interactive response time formula We have
M terminals Think time Z between the completion of
a job and the submission of the next job
Applying Little’s law to the whole system
M = (R + Z ) X0
thenR = M/X0 – Z
Very Important
Problem 1 A system
Can process up to 5 transactions/s Has 60 client workstations Client think time is 5s
Can the system achieve a response time of5 s?
Answer
Applying R = M/X0 – Z, we compute a
lower bound for the response time
Rmin = M/X0,max – Z = 60/5 – 5 =7
Our answer is no
Problem 2 We have
M = 50 terminals Z = 20 s R = 4s
What is the system throughput?
Answer
From R = M/X0 – Z, we have
X0 = (R + Z)/M
Hence X0 = (20 + 4)/50 = 0.48 tasks/s
Problem 3 Compute the response time of a system
knowing the following parameters M = 25 terminals (users) Z = 18 s Vdisk = 20 visits to the single disk per user
interaction Udisk = 0.30 Sdisk = 25 ms
(http://www.owlnet.rice.edu/~elec428/handouts/Op.Analysis.pdf)
Answer
Let us compute first the throughput X0
Since Xk = Uk/Sk
Xdisk = 0.30/0.025 = 12 disk requests/s
Since Xk = Vk X0 , we have X0 = Xk/Vk
X0 = 12/20 = 0.6 interactions/s
The response time is then
R = M/X0 – Z = 25/0.6 – 18 = 23.7s
Demand
Rather than expressing CPU workloads by the product VCPU SCPU , we can use
the total demand of the task
DCPU = VCPU SCPU
Since Xk = Uk/Sk and Xk = Vk X0
UCPU /DCPU = XCPU /VCPU = X0
Problem 4 A computer system has a single disk It processes tasks with an average CPU
demand of 400 ms. Each task requires 60 disk accesses Each disk access takes 10 ms If the CPU utilization is 50%, what are
The system throughput? The disk utilization?
Answer
X0 = UCPU /DCPU = 0.5/0.4 = 1.25 tasks/s
Since Udisk = Xdisk Sdisk and Xdisk = Vdisk X0
Xdisk = 1.25 x 60 = 75 disk requests/s
Udisk = 75 x 0.010 = .75 or 75%
(a rather high disk utilization)
Load–dependent behavior Device service times may depend on
number of jobs at device Disk drives optimize disk accesses
Number of visits to a device may depend on workload Swap device
We did not cover that topic
Resources http://www.owlnet.rice.edu/~elec428/ha
ndouts/Op.Analysis.pdf Offers a concise summary of the
topics that were discussed in class
Peter J. Denning, Jeffrey P. Buzen, The Operational Analysis of Queueing Network Models. ACM Computing Surveys 10(3): 225-261, 1978. Remains the fundamental text on
operational analysis