2. Networks of queues

3. Subdividing an Ethernet with switches

4. Merging two Poisson streams

5. Prob{event in stream 1 is in h } =   h Prob{event in stream 2 is in h    h  Prob{some event is in h} = Prob{stream 1 event} + Prob{stream 2 event} =  h    h = (     h Merged stream acts like a Poisson process with rate    

6. Partitioning two streams Whenever an event (arrival) occurs we divert it into A or B stream according to fixed probabilities, p A and p B. Prob{A event in h} = prob{event in h} × Prob{choosing A stream} = h pp  The A stream acts like a Poisson stream with a rate of events = p A

7. Burke’s Theorem The output of an M/M/k queue (in steady state) is Poisson with same rate as input. Stage 1 Both stages are independent M/M/k queues No. in system: N = N 1 + N 2 Examples: Stage 2

8. Implication for networks All stations can be analysed as independent M/M/k queues Station 1 Station 2 Station 3 External p1p1 p2p2 p3p3

9. A simple feedback queue p = P(message transmitted correctly) Otherwise retransmitted What is effective arrival rate (input) ? M/M/1 p 1-p

10. A packet switched network

11. A general network of queues 12 iK j j 2 N i 1 p 12 p 21 p K2 p 2N p iK p Ki p i1 p 1i p ij p ji qiqi qjqj p ij = P(job finished at station i goes to station j q i = 1 -  p ij = P(job leaves network after station i

12. Jackson’s theorem for networks In the above setup, (a) if ext. arrivals at each station are Poisson (b) if service times are exponential with rates Then each station is M/M/k i with effective arrival rate: K = total no. of stations

13. Jackson’s Theorem The queues in a Jackson network are independent M/M/1 queues. The numbers of packets and the delays in each queue are independent of those in all the other queues.

14. Example