Performance analysis of replicated Banyan switches under mixed traffic patterns

The per presents performance evaluations of unbuffered R-replicated banyan switches [1] on th e hypothesis of mixed traffic patterns muted by these, involving packet video flows embedded in a uniform traffic environment. These video flows are assumed of SSSD (Single-Source-to-Single-Destination) type and generated through two-layer video coding [2] : the first layer carries basic image information, the second enhancement information . Bit rate attributes of the layers are strictly dependent on the kind of image bein g coded (still or moving, rough or detailed) and on the coding technique applied : we refer to experimental measures for two-layer video coders already presented in literature [2] . A traffic management policy is adopted to distribute input traffic to the R banyan networks [1] in order to exploit replication in relation to video traffic characteristics . It is assumed that two virtual channel s are activated at call establishment for the first and second layer video provided the SSSD channel con figuration inside the switch does not cause colliding . Inside the switch, each virtual channel is furtherly spli t over R/2 planes, according to a random sharing policy [1] so that a cell is transmitted with the same 2/ R probability on each of the R/2 switch planes involved in the virtual channel to which the cell belongs . Due to contention situations that may occur in the banyan switching elements when two cells nee d the same output link, a random or priority based choice should be made for contention resolution . Two different priorities are associated to cells belonging to the two layers, the highest being assigned to firs t layer cells carrying basic video information . The probability q that a video cell wins a contention is relate d to these priorities and is assumed as a parameter, tuneable in relation to the quality and resolution desire d [2] . Let us refer to a generic banyan plane of an NxN R-replicated switch with Ns=log 2 N stages, links in each stage numbered from) to N and stages from 0 to Ns-1 . The analytical model developed for R-replicated networks under uniform traffic [1] is extended here to evaluate performance under non-unifor m traffic . Unlike [I], load and priority can be different from link to link, depending on the kind of information routed: thus a matrix representation of these variables is necessary . Two matrixes, L and T, describe , respectively, the switching plane topology and the SSSD channel paths through it . L is an Nx(Nrl) matrix whose generic element l(i j) indicates to which output link of stage j the input link i of stage (j+1) i s connected. This matrix can be calculated by means of the relationships given in [1] . T is a binary Nx(Ns+1) matrix whose generic element t(id) is equal to 1 when input link i at stage j (for 0 ~ j N,—l ) or output link i (for j = N,) belongs to an SSSD virtual channel . Due to the presence of SSSI) channel s the packet presence probability is no longer equal for all links of the same stage . Therefore, other tw o matrixes, 4 and 4, have been introduced . 4 is an Nx(Ns+1) matrix, whose element zJi j) is the cel l presence probability for uniform link i at stage j input, for 0 ~ j ~ N, e 1, and z a(i,N,) is the cell presence probability for uniform output link i . 4 is the analogy of Z,, for SSSD channel links . In particular ;(i 3 O ) and z„(i 3 O) represent cell presence probability on switch input links for uniform and SSSD channels , respectively . It is assumed that the probability of a cell arrival in a time slot is a constant and that tim e slots are independent. It is worth underlining that this scheme can be generalized to take more kinds o f traffic into account by simply increasing the number of Z matrixes . Elements of matrix 4 and 4 can be calculated by iterating relations, having fixed the values z,(i 3 O) and zIi 3 O) for i=1 . .N, obtained from the input load by suitable division in relation to the traffic sharin g policy adopted . By indicating with p,, the uniform traffic load, with p, the first layer video load and p,,2 the second layer video load, we have zw(i 3 O)a zJi3O)= 2 pv1 IR for R/2 planes loaded by first layer video and z,(i3O)= 2 p,,2JR for R/2 planes loaded by second layer video . Cell presence probabilities o n switch outputs are obtained from the sum of the independent contributions of the planes from which cells may arrive . The performance figures considered are network throughput and cell loss probability : evaluation s are performed by distinguishing SSSD video channels and uniform channels and, among these, by givin g particular attention to the channels in the worst congestion conditions . Throughput coincides for each banyan network with the value of cell presence probability on output links . By indicating with S, and S,, two vectors whose generic elements s(k), (k=1 . .N), are the values of throughput for uniform and video traffic respectively on output link k, we hav e