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

2. 2 A basic switch - crossbar O(n 2 ) switching elements Simple control  E.g., FIFO buffers at inputs

3. 3 head of line blocking – simple upper bound Assume nxn switch with uniform distribution of destination Probability for an output port not to be selected is  Capacity is bounded by 1-1/e = 0.63 For 2x2 switch the max capacity is 0.75 (tight bound)

4. 4 head of line blocking – alternative calculation The success probability of an input port selection:

5. 5 Dealing with HOL blocking Per-output queues at inputs (VOQ) Arbiter must choose one of the input ports for each output port How to select? Parallel Iterated Matching  inputs tell arbiter which outputs they are interested in  output selects one of the inputs  some inputs may get more than one grant, others may get none  if >1 grant, input picks one at random, and tells output  losing inputs and outputs try again Used in DEC Autonet 2 switch, McKeown’s iSLIP, and more.

6. 6 PGF – Probability Generating Functions Let a be a random variable. The PGF is defined by moment generation

7. 7 PGF Examples Poisson distribution Geometric distribution

8. 8 PGF Examples Bernoulli distribution Binomial distribution

9. 9 M/D/1 Queue To analyze: consider a slotted time scale arrivals q cells in queue n cells in system k-1 kk+1

10. 10 M/D/1 Queue k-1 kk+1

11. 11

12. 12 Finding p(0)  is the utilization

13. 13 The buffer statistics

14. 14 Home assignment Show that for Poisson arrivals

15. 15 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.

16. 16 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.

17. 17 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 )

18. 18 Finding the utilization,  0