1. Cheng-Fu Chou, CMLab, CSIE, NTU Open Queueing Network and MVA Cheng-Fu Chou

2. Cheng-Fu Chou, CMLAB, CSIE, NTU P. 2 Jackson Networks  Assume each queue has one or more servers with expo. distributed service time, and Poisson arrival of jobs to the network  If the network has Q queues and if ni is the number of jobs at queue I, then a Jackson network in steady state has the surprising property that –A product of marginal prob., such a network is called product-form or separable

3. Cheng-Fu Chou, CMLAB, CSIE, NTU  Each queue behaves like an independent M/M/K queue or M/G/1 queue  We can use a Markov chain derive an expression for Prob(ni=ki) separately and substitute it into the above eqn.  Note that the arrival process at some queue may not be Poisson P. 3

4. Cheng-Fu Chou, CMLAB, CSIE, NTU Ex P. 4  Consider the open network in Fig. 3.6 with single-server queues A, B, and C with branching prob. p A, p B, p C, respectively.

5. Cheng-Fu Chou, CMLAB, CSIE, NTU  Let p done = 1 – (p A +p B +p C ), then, the number of visits to A has distribution Geometric(p done ).  Let V A, V B, and V C be the expected number of visits to A, B, and C, respetively.  V A =1/ p done, V B = p B V A = p B / p done, and V C = p C / p done  Let l A, l B, abd l C be the job arrival rates at A, B, and C respectively.  l A = l V A, l B = l V B, l C = l V C  After we can get prob(n A =i), prob(n B =j), and prob(n C =h), we get prob(n A =i, n B =j, n C =h) P. 5

6. Cheng-Fu Chou, CMLAB, CSIE, NTU Closed Queueing Network  Closed systems are used for the interactive systems  In a closed network, we can model a set of users submitting requests to a system, waiting for results, then submitting more requests –human users interacting with a system, –threads contending for a lock, –processes blocking for I/O, –networked servers waiting for a response message. P. 6

7. Cheng-Fu Chou, CMLAB, CSIE, NTU Product Forum Queueing Networks  A PFQN consists of a collection of queueing and delay centers. It satisfies the following conditions. –All queueing centers : FCFS, PS, or LCFSPR –Any delay centers –FCFS with exponential distribution –If a FCFS center has multiple service classes, they must all have the same average service time –External arrivals, if any, are Poisson –Routing is state-independent P. 7

8. Cheng-Fu Chou, CMLAB, CSIE, NTU Arrival Theorem  For a separated closed network with N jobs, an arrival at a queue sees a network state that is (distribution-wise) the same as that seen by an outside observer of the same network with N-1 jobs. – P. 8

9. Cheng-Fu Chou, CMLAB, CSIE, NTU Mean Value Analysis  The Iterative Solution Method P. 9

10. Cheng-Fu Chou, CMLAB, CSIE, NTU P. 10

11. Cheng-Fu Chou, CMLAB, CSIE, NTU P. 11