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CDA6530: Performance Models of Computers and Networks Chapter 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 (N 1 (t), N 2 (t)), assume to be stable  ¼(i,j) =P(N 1 =i, N 2 =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: (N 1, N 2 ), N i =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:

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/μ i for queue i  routing probabilities p i,j, 1 ≤ i, j ≤ M  visit ratios, {v i }. If v 1 = 1, then v i is 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

14 Steady State Solution  Theorem (Gordon and Newell)  For previous example when p1=0.75, v i ?