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HOL Blocking analysis

HOL Blocking analysis. based on: Broadband Integrated Networks by Mischa Schwartz. A basic switch - crossbar. O(n 2 ) switching elements Simple control E.g., FIFO buffers at inputs. head of line blocking – simple upper bound. Assume nxn switch with uniform distribution of destination

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HOL Blocking analysis

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

  2. A basic switch - crossbar • O(n2) switching elements • Simple control • E.g., FIFO buffers at inputs

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

  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, andmore.

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

  7. PGF Examples • Poisson distribution • Geometric distribution

  8. PGF Examples • Bernoulli distribution • Binomial distribution

  9. 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

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

  11. Finding p(0)  is the utilization

  12. The buffer statistics

  13. Home assignment • Show that for Poisson arrivals

  14. 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.

  15. 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.

  16. 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)

  17. Finding the utilization, 0

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