Output-Buffered Switches PERFORMANCE OF BASIC SWITCHES

BASICS OF PACKET SWITCHING 40 Fig. 2.16 The mean waiting time for input queuing with FIFO buffers for the limiting case for N s ⬁.

2.3.2 Output-Buffered Switches

With output queuing, cells are only buffered at outputs, at each of which a Ž . separate FIFO is maintained. Consider a particular i.e., tagged output queue. Define the random variable A as the number of cell arrivals destined for the tagged output in a given time slot. Based on the same assumptions as in Section 2.3.1 on the arrivals, we have k N yk p p N w x a J Pr A s k s 1 y , k s 0, 1, 2, . . . N 2.9 Ž . k ž ž ž k N N Ž . When N ™ ⬁, 2.9 becomes p k e yp w x a J Pr A s k s , k s 0, 1, 2, . . . . 2.10 Ž . k k Denote by Q the number of cells in the tagged queue at the end of the m mth time slot, and by A the number of cell arrivals during the mth time m slot. We have Q s min max 0, Q q A y 1 , b . 2.11 4 Ž . Ž . m m y1 m PEFORMANCE OF BASIC SWITCHES 41 If Q s 0 and A 0, there is no cell waiting at the beginning of the m y1 m mth time slot, but we have A cells arriving. We assume that one of the m arriving cells is immediately transmitted during the mth time slot; that is, a cell goes through the switch without any delay. For finite N and finite b, this can be modeled as a finite-state, discrete-time w x Markov chain with state transition probabilities P J Pr Q s j Q s i i j m m y1 as follows: a q a , i s 0, j s 0, ° 1 a , 1 F i F b, j s i y 1, ~ a , 1 F j F b y 1, 0 F i F j, P s 2.12 Ž . j yiq1 i j N Ý a , j s b, 0 F i F j, m sjyiq1 m ¢ otherwise , Ž . Ž . where a is given by 2.9 and 2.10 for a finite N and N ™ ⬁, respectively. k The steady-state queue size can be obtained recursively from the following Markov chain balance equations: 1 y a y a 1 w x q J Pr Q s 1 s ⭈ q 1 a n 1 y a a 1 k w x q J Pr Q s n s ⭈ q y ⭈ q , 2 F n F b, Ý n n y1 n yk a a k s2 where 1 w x q J Pr Q s 0 s ⭈ b 1 q Ý q rq n s1 n No cell will be transmitted on the tagged output line during the mth time slot if, and only if, Q s 0 and A s 0. Therefore, the switch throughput m y1 m ␳ is represented as ␳ s 1 y q a ⭈ A cell will be lost if, when emerging from the switch fabric, it finds the output buffer already containing b cells. The cell loss probability can be calculated as follows: ␳ w x Pr cell loss s 1 y , p where p is the offered load. BASICS OF PACKET SWITCHING 42 Fig. 2.17 The cell loss probability for output queuing as a function of the buffer size Ž . Ž . b and the switch size N, for offered loads a p s 0.8 and b p s 0.9. PEFORMANCE OF BASIC SWITCHES 43 Ž . Ž . Figure 2.17 a and b show the cell loss probability for output queuing as a function of the output buffer size b for various switch size N and offered loads p s 0.8 and 0.9. At the 80 offered load, a buffer size of b s 28 is good enough to keep the cell loss probability below 10 y6 for arbitrarily large N. The N ™ ⬁ curve can be a close upper bound for finite N 32. Figure 2.18 shows the cell loss performance when N ™ ⬁ against the output buffer size b for offered loads p varying from 0.70 to 0.95. Output queuing achieves the optimal throughput᎐delay performance. Cells are delayed unless it is unavoidable, when two or more cells arriving on different inputs are destined for the same output. With Little’s result, the mean waiting time W can be obtained as follows: b Q Ý nq n s1 n W s s . ␳ 1 y q a Figure 2.19 shows the numerical results for the mean waiting time as a function of the offered load p for N ™ ⬁ and various values of the output buffer size b. When N ™ ⬁ and b ™ ⬁, the mean waiting time is obtained from the M rDr1 queue as follows: p W s . 2 1 y p Ž . Fig. 2.18 The cell loss probability for output queuing as a function of the buffer size b and offered loads varying from p s 0.70 to p s 0.95, for the limiting case of N ™ ⬁. BASICS OF PACKET SWITCHING 44 Fig. 2.19 The mean waiting time for output queuing as a function of the offered load p, for N ™ ⬁ and output FIFO sizes varying from b s 1 to ⬁.

2.3.3 Completely Shared-Buffer Switches