Bernoulli Arrival Process and Random Traffic On–Off Model and Bursty Traffic

INPUT-BUFFERED SWITCHES 52

3.1.2 Traffic Models and Related Throughput Results

This subsection briefly describes how the FIFO service discipline limits the throughput under some different traffic models. Detailed analytical results are described in Section 2.3.1.

3.1.2.1 Bernoulli Arrival Process and Random Traffic

Cells arrive at each input in a slot-by-slot manner. Under Bernoulli arrival process, the probability that there is a cell arriving in each time slot is identical and is independent of any other slot. This probability is referred as the offered load of the input. If each cell is equally likely to be destined for any output, the traffic becomes uniformly distributed over the switch. Consider the FIFO service discipline at each input. Only the HOL cells contend for access to the switch outputs. If every cell is destined for a different output, the switch fabric allows each to pass through to its output. If k HOL cells are destined for the same output, one is allowed to pass through the switch, and the other k y 1 must wait until the next time slot. While one cell is waiting its turn for access to an output, other cells may be queued behind it and blocked from possibly reaching idle outputs. A Markov model can be established to evaluate the saturated throughput when N is small. 1 When N is large, the slot-by-slot number of HOL cells destined for a particular output becomes a Poisson process. As described in Section 2.3.1, the HOL blocking limits the maximum throughput to 0.586 when N ™ ⬁.

3.1.2.2 On–Off Model and Bursty Traffic

In the bursty traffic model, each input alternates between active and idle periods of geometrically distributed duration. During an active period, cells destined for the same output arrive continuously in consecutive time slots. The probability that an active or an idle period will end at a time slot is fixed. Denote p and q as the probabilities for an active and for an idle period, respectively. The duration of an active or an idle period is geometrically distributed as expressed in the following: w x Ž . iy1 Pr An active period s i slots sp 1 y p , i G 1, j w x Ž . Pr An idle period s j slots sq 1 y q , j G 0. Note that it is assumed that there is at least one cell in an active period. An active period is usually called a burst. The mean burst length is then given by ⬁ 1 iy1 b s ip 1 y p s , Ž . Ý p is1 1 Consider the state as one of the different destination combinations among the HOL cells. METHODS FOR IMPROVING PERFORMANCE