HOL Blocking analysis

1. HOL Blocking analysis based on: Broadband Integrated Networks by Mischa Schwartz.

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

3. PGF Examples • Poisson distribution • Geometric distribution

4. PGF Examples • Bernoulli distribution • Binomial distribution

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

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

8. Finding p(0)  is the utilization

9. The buffer statistics

10. Home assignment • Show that for Poisson arrivals

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.

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

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[Bim]=02/2(1-0)

14. Finding the utilization, 0