cda6530: performance models of computers and networks chapter 7: basic queuing networks texpoint...
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CDA6530: Performance Models of Computers and Networks
Chapter 7: Basic Queuing Networks
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Open Queuing Network
Jobs arrive from external sources, circulate, and eventually depart
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Closed Queuing Network
Fixed population of K jobs circulate continuously and never leave Previous machine-repairman problem
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Feed-Forward QNs
Consider two queue tandem system
Q: how to model? System is a continuous-time Markov chain (CTMC) State (N1(t), N2(t)), assume to be stable ¼(i,j) =P(N1=i, N2=j) Draw the state transition diagram
But what is the arrival process to the second queue?
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Poisson in ) Poisson out
Burke’s Theorem: Departure process of M/M/1 queue is Poisson with rate λ independent of arrival process.
Poisson process addition, thinning Two independent Poisson arrival processes adding
together is still a Poisson (¸=¸1+¸2) For a Poisson arrival process, if each customer lefts
with prob. p, the remaining arrival process is still a Poisson (¸ = ¸1¢ p)
Why?
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State transition diagram: (N1, N2), Ni=0,1,2,
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For a k queue tandem system with Poisson arrival and expo. service time
Jackson’s theorem:
Above formula is true when there are feedbacks among different queues Each queue behaves as M/M/1 queue in isolation
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Example
¸i: arrival rate at queue i
Why?
Why?
In M/M/1:
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T(i): response time for a job enters queue i
Why?
In M/M/1:
E [T (1)] = 1=(¹ 1 ¡ ¸1) + E [T (2)]=2E [T (2)] = 1=(¹ 2 ¡ ¸2) + E [T (1)]=4
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Extension
results hold when nodes are multiple server nodes (M/M/c), infinite server nodes finite buffer nodes (M/M/c/K) (careful about interpretation of results), PS (process sharing) single server with arbitrary service time distr.
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Closed QNs
Fixed population of N jobs circulating among M queues. single server at each queue, exponential service
times, mean 1/μi for queue i routing probabilities pi,j, 1 ≤ i, j ≤ M visit ratios, {vi}. If v1 = 1, then vi is mean number of
visits to queue i between visits to queue 1
: throughput of queue i,°i
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Example
Open QN has infinite no. of states Closed QN is simpler
How to define states? No. of jobs in each queue
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Steady State Solution
Theorem (Gordon and Newell)
For previous example when p1=0.75 , vi?