2. April 10, 20001 HOL Blocking analysis based on: Broadband Integrated Networks by Mischa Schwartz.

3. 2 PGF – Probability Generating Functions Let a be a random variable. The PGF is defined by moment generation

4. 3 PGF Examples Poisson distribution Geometric distribution

5. 4 PGF Examples Bernoulli distribution Binomial distribution

6. 5 M/D/1 Queue To analyze: consider a slotted time scale arrivals q cells in queue n cells in system k-1 kk+1

7. 6 M/D/1 Queue k-1 kk+1

8. 7

9. 8 Finding p(0)  is the utilization

10. 9 The buffer statistics

11. 10 Home assignment Show that for Poisson arrivals

12. 11 Remarks Note that E(n)-E(q)=  =1-p(0)  The average number in service is 1·Pr(n≥1)=1-p(0) The time evolution equation for q:  Note that here we cannot simply isolate the terms, we need to be more careful.

13. 12 HOL blocking analysis at steady state Assume NxN switch, destinations are uniformly distributed Packet queues are always full. B i m = number of packets at the end of time slot m that are blocked for input i. A i m = number of packets destined to output i moving to the head of the line during the mth time slot at “free” input queues. F m = number of cells transmitted in time slot m.

14. 13 This is the form of the equation for the number of packets in M/D/1 queue We analyze the operation of a virtual queue. Based on home exercise: E[B i m ]=  0 2 /2(1-  0 )

15. 14 Finding the utilization,  0