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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 based on: Broadband Integrated Networks by Mischa Schwartz.
A basic switch - crossbar • O(n2) 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 • 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)
head of line blocking – alternative calculation • The success probability of an input port selection:
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.
PGF – Probability Generating Functions • Let a be a random variable. • The PGF is defined by • moment generation
PGF Examples • Poisson distribution • Geometric distribution
PGF Examples • Bernoulli distribution • Binomial distribution
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
M/D/1 Queue k k+1 k-1
Finding p(0) is the utilization
Home assignment • Show that for Poisson arrivals
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.
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.
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)
