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Understanding Queuing Networks: Key Models and Theorems

This chapter delves into the foundational concepts of queuing networks, exploring both open and closed models. Open queuing networks feature jobs arriving from external sources, circulating, and eventually departing, while closed networks involve a fixed population of jobs that continuously circulate. Key topics include the behavior of tandem queues, the state transition diagram for Markov chains, and fundamental theorems such as Burke’s theorem and Jackson's theorem. These models are essential for analyzing system performance in computing and networking environments.

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Understanding Queuing Networks: Key Models and Theorems

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  1. CDA6530: Performance Models of Computers and NetworksChapter 7: Basic Queuing Networks TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA

  2. Open Queuing Network • Jobs arrive from external sources, circulate, and eventually depart

  3. Closed Queuing Network • Fixed population of K jobs circulate continuously and never leave • Previous machine-repairman problem

  4. 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?

  5. 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?

  6. State transition diagram: (N1, N2), Ni=0,1,2,

  7. 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

  8. Example • ¸i: arrival rate at queue i Why? In M/M/1: Why?

  9. T(i): response time for a job enters queue i Why? In M/M/1:

  10. 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.

  11. Closed QNs • Fixed population of N jobs circulating among M queues. • single server at each queue, exponential service times, mean 1/μifor queue i • routing probabilities pi,j, 1 ≤ i, j ≤ M • visit ratios, {vi}. If v1 = 1, then viis mean number of visits to queue i between visits to queue 1 • : throughput of queue i,

  12. Example • Open QN has infinite no. of states • Closed QN is simpler • How to define states? • No. of jobs in each queue

  13. Steady State Solution • Theorem (Gordon and Newell) • For previous example when p1=0.75 , vi?

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