Pipelined Transmission Scheduling in All-Optical TDM/WDM Rings

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1 Pipelined ransmission Scheduling in All-Optical DM/WDM Rings Xijun Zhang and Chunming Qiao Department of ECE, SUNY at Buffalo, Buffalo, NY 460 fxz, Abstract wo properties of optical transmissions, namely, unidirectional propagation and predictable propagation delay, make it possible to pipeline packet transmissions in alloptical networks. In this paper, we study the problem of scheduling All-to-All Personalized Communication (AAPC) in unidirectional DM/WDM rings with pipelined transmissions, which can achieve a much higher bandwidth utilization than non-pipelined transmissions. For a given number of wavelengths, K, and number of transmitter-receiver pairs per node,, the theoretical lower bound (LB) on the schedule lengths is derived, and scheduling methods which can achieve near optimal results are proposed for three different cases, namely, = K, < K and =, respectively. Keywords: DM/WDM, Ring, AAPC, Scheduling, Pipelined transmissions. Introduction All-optical interconnection networks are suitable for high speed communications, mainly due to their large bandwidth. o fully take advantage of all-optical networks, their unique characteristics need to be exploited. Specifically, in order to fully utilize the large bandwidth of optic fibers, two multiplexing techniques, namely, ime Division Multiplexing (DM) and Wavelength Division Multiplexing(WDM), have been studied. DM/WDM hybrid access schemes have also been proposed by many researchers [,4,7,9]. Furthermore, two properties of optical transmissions, namely, unidirectional propagation and predictable propagation delay, make it possible to pipeline packet transmissions in multihop networks, which can achieve a much higher bandwidth utilization than non-pipelined transmissions. Pipelining messages in DM optical communications in multiprocessor arrays was studied in [8,0]. Due to the lack of sophisticated optical logic devices, communications in all-optical networks are normally carried out in a circuit-switching fashion to avoid conversions of data streams between electronic and optical signals at intermediate nodes. In addition, in order to eliminate the overhead involved in establishing all-optical paths his research is supported in part by a grant from NSF under contract number MIP at runtime, compiled communication techniques can be applied to schedule pre-determined communication patterns. One such pattern is All-to-All Personalized Communication (AAPC), which is the densest that can be imposed on an interconnection network. It requires each of the N nodes to send a unique message to each of the other N? nodes, and arises in many important parallel computing algorithms. Furthermore, for an arbitrary dense communication pattern, it is usually more beneficial to use a highly tuned AAPC schedule than attempting to find a specific one for the required communication [, 6]. he length of the schedule for carrying out AAPC is thus an important performance measure of a communication system. In this paper, we will investigate the problem of scheduling pipelined AAPC transmissions in DM/WDM rings. Ring is a fairly common communication network architecture for local area networks (LAN) as well as multiprocessor systems. Early WDM networks to be deployed are likely to be rings, as in several recent testbeds [3, 3]. Work related to scheduling transmissions in rings has been reported in [, 6, ], in which only one transmitter-receiver pair per node was assumed and non-pipelined transmissions were used. he effects of having multiple transmitterreceiver pairs per node on the schedule in bidirectional WDM rings have been considered only recently in []. he work in this paper differs from them in that we consider the problem of scheduling pipelined AAPC transmissions for any given number of wavelengths, K, and any number of transmitter-receiver pairs per node, K, in unidirectional DM/WDM rings. he rest of this paper is organized as follows. Section briefly describes the system and states the basic assumptions. Section 3 studies the theoretical lower bound (LB) on the schedule length for each of the following three cases: = K, < K and =, and also gives an upper bound (UB) achieved by the proposed scheduling methods. Finally, Section 4 concludes this paper. System description We consider an N-node ring in which the nodes are numbered from 0 to N?, and each link is multiplexed with K wavelengths. Each node has fully tunable transmitterreceiver pairs, or alternatively, a multi-wavelength (K-

2 wavelength) laser array (or multichannel receiver array) so that a node can transmit (or receive) at K different wavelengths at the same time. We study the scheduling problem for the evacuation mode, in which each node has N? packets to be sent to the other N? nodes, one for each node, and afterwards, either no more packets are generated, or the same communication pattern is repeated. It is assumed that all packets are of the same length, and the network is globally synchronized. In addition, the schedule is determined a priori, which includes all necessary information for a specific node to communicate with the other nodes. Once the schedule is made, every node is informed and will follow the schedule to communicate with others. Assume that the propagation delay between two adjacent nodes is d units, the transmission time of one packet is p( d) units, and the tuning delay of transmitter-receivers is negligible. Consider the time it takes for every node to send one packet to another node s hops away, where s N?, and there is only one wavelength. In non-pipelined transmissions, p + s d units are needed for a node to send one packet, and no intermediate nodes can transmit during this period. Hence, at most b N c nodes on the ring can transmit s at the same time, and d N b N e (p + s d) s (p + s d) units s c are needed. However, if packet transmissions are pipelined, all N nodes can transmit at the same time, and thus only p + s d units are needed. herefore, the efficiency of pipelined transmissions will be at least s times higher than that of non-pipelined transmissions (if we ignore the control overhead involved in pipelined transmissions). For example, when N = 8 and s = 4, at most nodes (e.g. 0 and 4) can transmit at the same time with non-pipelined transmissions as shown in Figure (a), and it takes 4 (p + 4 d) units for all 8 nodes to send (and receive) a packet. With pipelined transmissions as shown in Figure (b), only p + 4 d units are needed (a) non-pipelined transmissions (b) pipelined transmissions Figure. (the number above each packet denotes the destination node). In order to implement the pipelined transmissions described above, the packet size has to be small enough so that p d. A large message of size P can be transmitted in multiple (d P d e) packets which results in repetitions of the schedule. Note that when each node has multiple packets to transmit (either to a single destination or to multiple destinations), the transmission of the second (or later) packet can be overlapped with the propagation of the previous one on each wavelength. More specifically, assume that all the packets transmitted previously on a given wavelength (one packet by each node) are propagating on their last hop. While the first bit of these packets is being received by their destinations, each node can start the transmission of another packet. For the above reason, the time for every node to receive all the packets can always be written as p + L s (or d + d P d e L s for large messages), where L s is the schedule length independent of p (as long as p d). In what follows, we will concentrate only on the schedule length L s. In addition, we will ignore the tuning delay of transmitter-receiver pairs (or the on-off delay of the K-wavelength laser array and receiver array), and treat d units as the duration of a time slot. o facilitate our presentation, a connection (or a transmission) is said to have stride s if a packet is to be transmitted to a node s hops away. In order to fully utilize the bandwidth in pipelined transmissions, only the connections with the same stride are scheduled on the same wavelength at the same time (via pipelining). For example, all connections with stride 4 are pipelined as shown in Figure (b). Accordingly, the behavior of every node will be the same, and we only need to consider how to schedule transmissions at one node, unlike in non-pipelined transmissions []. 3 Pipelined scheduling of AAPC o derive efficient schedules, the values of both K and have to be taken into consideration. Clearly, K. In what follows, we will start with the simplest case where = K, then extend the discussions to two other cases, namely, < K and =, respectively. In each case, we will first give the heoretical Lower Bound (LB) on the schedule length, and then propose a scheduling method to approach the LB. 3. he case where = K When = K, each transmitter (and receiver) is fixed at a distinct wavelength. Since each node has N? packets to send, and each packet takes s slots, where s = ; ; ; N?, respectively, the total time needed for a node to send all its packets is N(N?) slots on a single wavelength. herefore, the LB on the schedule length is, LB( = K) = d N(N?) K e () which may be achieved if all packet transmissions can be uniformly distributed among K wavelengths. Our idea for approaching the LB is to group the transmissions into phases of a fixed length, L p. More specifically, on each wavelength, up to two transmissions with strides s and L p?s can be scheduled in each phase. Figure shows an example schedule with L p = N = 7, where each row corresponds to a wavelength, and each arrowed line segment corresponds to a connection with a specific stride. As

3 illustrated in the figure, all transmitters start transmissions at the beginning of each phase. he starting time of the second transmission within each phase depends on the stride of the first one, and is different on different wavelengths. he schedule of AAPC usually completes in multiple phases. In this example, LB( = K = 4) = 34, and it is achieved in two phases ( L p = 34). λ 4 λ 3 λ λ =K SAR 3 s = END from left to right (or from right to left) to yield different values of x, which is between and +. It turns out that the schedule length is minimized when x = +, or in other words, when the lower part in Figure 3 slides all the way to the right (the proof is omitted). he resulting schedule is illustrated in Figure 4, where an arrow indicates the direction at which the stride increases. x- mk x x+ x+k- x+k x+mk - N- N- N-K N-K- N-mK µ+ x N-mK- Phase Phase Figure. An example (N = 7, = K = 4). Note that in some cases, combining connections with strides s and N? s to form phases with L p = N may not be appropriate. In general, the appropriate value of L p (or how to combine connections with different strides to form phases), depends on N and K as to be discussed next. In order to find an efficient schedule, we first determine how big K needs to be. heorem. At most d N e wavelengths are needed for scheduling pipelined AAPC transmissions when = K. Proof: Since the largest s is N?, the schedule length cannot be shorter than N?. Without loss of generality, assume that the connection with stride N? is scheduled on wavelength. Let (s ; s ; i ) denote that two connections with strides s and s are scheduled on wavelength i, where i K. A straightforward way to schedule the remaining N? connections is (; N? ; ); (; N? 3; 3 );, and so on. his schedule requires d N e wavelengths in total, and the schedule length achieved is N? (the shortest possible). For any N, let N? = mk +, where m = b N? K c 0 and = mod(n? ; K) (where mod() is the modular function). Note that the proof above also implies an optimal schedule for the case where K = d N e (and m = 0). Below, we will propose an efficient schedule for the case where K < d N e, and determine an upper bound on the schedule length. As shown in Figure 3, let the connection with stride x be combined with the connection with stride N? to make the phase length L p = x + N?. It is then natural to combine connections with strides x + ; ; x + mk? with connections with strides N? ; ; N? mk, respectively. hese mk pairs of connections will be scheduled in m phases with K pairs in each phase, and within each phase, one pair on each wavelength. he remaining connections (there are of them) will be scheduled after the mth phase. Note that in creating these mk pairs of connections, it is implied that x + mk? < N? mk (see Figure 3), that is, x +. Intuitively, the lower part in Figure 3 can slide Figure 3. Combining strides to form phases. Note that, for the remaining connections, it does not matter on which wavelength the connection with stride is scheduled as long as it is scheduled separately from the other? connections, which may also be scheduled differently from Figure 4 as long as the sum of the strides of the connections scheduled on each wavelength is no more than. From Figure 4, it is clear that when = K, an upper bound on the schedule length achieved by the proposed scheduling method is: U B( = K) = m(n + ) + () In the next two subsections, we will consider the cases in which < K. In such cases, a node can only transmit simultaneously on wavelengths. ransmissions on the other K? wavelengths have to be delayed, which results in longer schedule lengths. he cases where < K and = will be considered separately since different scheduling methods need to be applied to achieve near optimal schedule. Note that, having more than d N e wavelengths may help reduce the schedule length when < K. However, to make the assumptions on K uniform across different cases, we will only consider K d N e. Ν+µ Ν+µ Ν+µ µ µ+ N- K µ+κ+ N-K- µ+ N- µ+κ+ N-K- µ+κ N-K µ+κ N-k µ +mk N-mK µ µ phase phase phase m µ < Κ Figure 4. A heuristic schedule. 3. he case where < K We first assume K < d N e and hence, m. When < K, in order to use all the wavelengths available, a transmitter has to transmit on different wavelengths in an

4 interleaved way. Specifically, it has to transmit one packet on one wavelength at the beginning of one slot, and then another packet on another wavelength at the beginning of another slot, and so on. Since each transmitter interleaves among d K e wavelengths, d K e? more slots are needed to send a packet on each of the K wavelengths comparing with the case where = K. Intuitively, LB( < K) = LB( = K) + d K e? (3) F (k) S (k) F (k) µ+ N- λ µ+κ+ λmod(k,) λk- λ K µ+κ N-K µ+κ phase phase Figure. Phase when < K. We now consider how to modify the schedule in Figure 4 to approach the LB. First we divide the K wavelengths into d K e groups such that the ith group, where i < d K e, contains the following wavelengths, K?(i?), K?(i?)?,, K?i +, and the last group (i.e. the d K e-th group) contains only mod(k; ) wavelengths, namely, mod(k; ), mod(k; )?,,,. Let the transmitters transmit on the first group of wavelengths at the beginning of each phase, and then on the second group, and so on. Note that there will be two transmissions of complementary strides on each wavelength in each phase. he first transmissions in phase on the wavelengths in group i ( i d K e) will be delayed for i? slots relative to the case where = K (see Figure ). If we denote by D (k) the delay of the first transmission on k in phase, where k K, we have, D (k) = d K? k + e? (4) Note that, as in the case where = K, the second transmission in phase on k can start after the number of slots equal to the stride of the first transmission. Specifically, since the first transmission in phase on k has stride +k, the second transmission will start at D (k)++k. In effect, the schedule in Figure 4 can be simply modified by delaying every transmission on k by D (k) slots. his is possible, or in other words, the resulting schedule is valid, because of the following lemma and theorem. Lemma. In the modified schedule, there are at most transmissions starting at the same time during phase. he formal derivation of LB( < K) is omitted Proof: Let F (k) and S (k) represent the time of the first and second transmissions on k in phase, respectively. Based on the schedule described above, we have, F (k) = D (k) = d K? k + e? () S (k) = D (k) + + k K + k(? ) + = d e +? (6) Note that F () F () F (K) and S () S () S (K). Since max k ff (k)g = F () = d K e? and min kfs (k)g = S () = d K e + > max k ff (k)g, it is guaranteed that F (k) 6= S (k) for any k. In addition, by checking the formula of F (k), we conclude that F (k) has the same value for at most different k values. Similarly, S (k) has the same value for at most ( ) different wavelengths. Unlike the first transmissions, the second transmissions in phase will start on the wavelengths of group d K e first, then group d K e?, and so on, as shown in Figure. In addition, since the sum of the strides of the two transmissions on every wavelength is the same (i.e. N + ), the first transmission on k in phase will also be delayed for D (k) slots relative to the case where = K. Accordingly, phase will undergo the same changes as phase, and so will phases 3, 4 and so on. he net effect is that every transmission on k during these m phases will be delayed by D (k) slots. Based on Lemma and the above discussions, we have the following theorem (the formal proof is omitted), which guarantees that the modified schedule is valid for the first m phases. heorem : In the modified schedule (illustrated in Figure ), at most packets are transmitted and received simultaneously by a node in the first m phases. N-mK+K- µ Κ+ λ λ mod(k,) λ Κ µ Κ µ Κ λ K- 3 λ K N-mK µ phase m Figure 6. Schedule the remaining connections ( < K, K < < K). We now consider how the remaining connections are scheduled. Note that when = 0, there is no remaining connections to schedule, and the schedule is delayed for D () slots because the connection with stride N?

5 mk + K? ends last on in phase m. In order to minimize the schedule length, we have to schedule the remaining (where 0 < < K) connections in a specific way based on the value of, which is different from the case where = K. More specifically, when K < < K, we schedule these connections as follow (see Figure 6): (;?; K ), (?; ; K? ),, (K;?K; K? ), (K? ;?; K?? ),, (?K +;?; ), where a vector containing? means that only one connection whose stride is specified by the first parameter is scheduled on the specified wavelength. In this way, the connection which ends last is the one with stride?k, which is scheduled on K? with a delay of D (K? ) slots. When 0 < K, the last connections will be scheduled on ; ; ;, respectively. Accordingly, the connection with stride ends last on with a delay of D () slots. herefore, the overall delay relative to the case where = K is, D = 8 < : D (K? ) K < < K D () 0 < K D () = 0 and based on Eq. (), the modified schedule will have a length of (7) U B( < K) = m(n + ) + + D (8) Note that if K = d N e and m = 0, the N? connections can be scheduled in the same way as the last connections are scheduled in the case where K < < K. In other words, it does not matter whether there is any phase prior to scheduling these connections. In the next subsection, we will consider the case where =, in which the last connections will be scheduled using a heuristic method. We will assume K < d N e only. his is because, when K = d N e, = N?, and the N? connections can be scheduled using the same heuristic method. 3.3 he case when = We may first derive the following by extending Eq. (3), LB( = ) = LB( = K) + K? (9) However, we cannot simply extend the interleaved scheduling method proposed for the case where < K. o see the problem, consider two transmissions with strides + k and + k +, which are to be scheduled as the first transmissions in phase on k and k+, respectively. If we let the (one and only) transmitter transmit on K first, K? next, and so on, the transmissions on k and k+ will start at K? k and K? k?, respectively, and the two packets will have to be received simultaneously (at time slot + K). his in turn, may force the second transmissions on these two wavelengths to start at the same time, which results in an invalid schedule. A possible solution is to reverse the order at which the wavelengths are interleaved. hat is, we will let the transmitter transmit on first, next, and so on (at the beginning of each phase), which results in D (k) = k? (instead of K?k). Since the first transmission starts on first, then on, and so on in each phase, and in addition, the stride of the first transmission increases with k, it is likely that we can start the second transmission on first, then on, and so on in each phase. Formally, we may denote by Fi 0 (k) and S0 i (k) the time of the first and second transmissions on k in phase i, respectively, relative to the beginning of phase i in the case where = K. It turns out that if all the transmissions on k in the first m phases are delayed for D (k) slots, that is, if we let Fi 0 (k) = k?, and S0 i (k) = F i 0 (k) + + (i? )K + k, we can guarantee that at any time during phases through m, there is at most one packet transmitted or received by a node. Specifically, this is because both Fi 0 (k) and S0 i (k) monotonically increase with k, and in addition, the following condition is met: max k ffi 0 (k)g = F i 0 (K) = K? < min k fsi 0 (k)g = S0 i () = + (i? )K +. From the above discussions, it is clear that during phase (i.e. i = ), the condition will also be met, and there will be at most one packet transmitted/received by any node at any time if K? < + (or in other words, > K? ). However, if K?, the condition is not met, and some of the second transmissions in phase will start at the same time as some of the first transmissions in phase. In such a case, the K connections scheduled in phase will have to be rescheduled differently using a heuristic method, just as the remaining connections. In short, as long as > K?, the first m phases (phases through m) outlined in the schedule shown in Figure 4 can be modified by delaying every transmission on k by D (k) slots. However, if K?, only phases through m can be modified this way, and the connections scheduled in phase, together with the remaining connections, have to be scheduled using a heuristic method. One such heuristic method, for example, is to consider the connections in the order of increasing stride, and try to schedule each connection as soon as possible. 3.4 Numerical results Figure 7 plots the LB, as well as the schedule lengths (which are also the upper bounds) achieved by the proposed scheduling methods, as a function of K for three different cases, namely, = K, = and =, respectively, in a ring having 00 nodes. In all three cases, the UBs achieved are quite close to the LBs, and in fact, they are the same at some points. For example, the maximum difference between LB(=) and UB(=) is at most %, 3% and 9% when K N, N and N, respectively. he results also indicate that the LB does not decrease too much as increases 8 4 beyond =, and in addition, the difference between the UB and the LB is the smallest when =. Accordingly, = seems to be a good choice from the cost-effective design point of view. Note that the proposed scheduling methods are useful

6 since for a given system where N, K and are fixed, one can always determine m and, and consequently an appropriate schedule to implement. Schedule Length N=00 = = = K UB ( = ) LB ( = ) UB ( = ) LB ( = ) UB ( = K) LB ( = K) Number of Wavelengths (K) Figure 7. Schedule length of AAPC (N = 00). 4 Conclusion In this paper, we have studied the problem of scheduling pipelined AAPC transmissions in unidirectional rings. We have determined the LBs on the schedule length for a given number of wavelengths, K, and number of transmitterreceiver pairs per node,. Scheduling methods which can achieve a near optimal schedule have been proposed. We have also found that, unlike in non-pipelined transmissions, multiple transmitter-receiver pairs ( > ) do not have much effect on the schedule length in pipelined transmissions. In addition, comparing to non-pipelined transmissions, although the formulas for LBs are similar (see []), the schedule length has been reduced dramatically. his is because the unit of measurement has been changed from the number of rounds in non-pipelined transmissions to the number of slots in pipelined transmissions, and each round can last as many as N slots. Accordingly, pipelined transmissions can potentially achieve a throughput that is N times as higher as non-pipelined transmissions. Given that N (N? ) packets (one for each connection of AAPC) can be sent/received within approximately d N(N?) K e slots, the maximum throughput achieved with pipelined transmissions is about K packets per slot, or equivalently, packets per slot per wavelength. his maximum throughput compares favorably with the maximum throughput of tokenpassing based rings where pipelining is also used. In fact, the maximum throughput of a FDDI ring is equivalently packet per slot (per wavelength). References [] S. H. Bokhari. Multiphase complete exchange: A theoretical analysis. IEEE ransactions on Computers, 4():0 9, 996. [] Michael S. Borella and Biswanath Mukherjee. Efficient scheduling of nonuniform packet traffic in a WDM/DM local lightwave network with arbitrary transceiver tuning latencies. In Proceedings of IEEE Infocom, pages 9 37, 99. [3] G. K. Chang, G. Ellinas, J. K. Gamelin, M. Z. Iqbal, and C. A. Brackett. Multiwavelength reconfigurabale WDM/AM/SONE network testbed. IEEE/OSA J. Lightwave ech./ieee JSAC, 4, 996. [4] H.S. Choi, H.-A. Choi, and M. Azizoglu. Optimum transmission scheduling in optical broadcast networks. In Proc. Int l Conference on Communication, pages 66 70, 99. [] A. Elrefaie. Multiwavelength survivable ring network architectures. In Proc. Int l Conference on Communication, pages 4, 993. [6] S. Hinrichs, C. Kosak, D. R. O Hallaron,. M. Stricker, and R. ake. An architecture for optimal allto-all personalized communication. In Proc. Sixth Annual ACM Symposium on Parallel Algorithms and Architecture (SPAA), pages 30 39, 994. [7] S.K. Lee and H.-A. Choi. Optimal transmission scheduling in WDM broadcast-and-select networks with multiple transmitters and receivers. In International Conference on Massively Parallel Processing Using Optical Interconnections (MPPOI), pages , October 996. [8] R. G. Melhem, D.M. Chiarulli, and S. P. Levitan. Space multiplexing of waveguides in optically interconnected multiprocessor systems. he Couputer Journal, 3(4):36 369, 989. [9] G. Pieris and G. Sasaki. Scheduling transmissions in WDM broadcast-and-select networks. IEEE/ACM ransactions on Networking, ():0 0, 994. [0] C. Qiao and R. G. Melhem. ime-division optical communications in multiprocessor arrays. IEEE ransactions on Computers, 4():77 90, 993. [] C. Qiao, X. Zhang, and L. Zhou. Scheduling all-to-all connections in WDM rings. In SPIE Proceedings, All Optical Communication Systems: Architecture, Control and Network Issues, pages 8 9, November 996. [] L. assiulas and J. Joung. Performance measures and scheduling policies in ring networks. IEEE/ACM ransactions on Networking, 3():76 84, 99. [3] H. oba, K. Oda, K. Inoue, K. Nosu, and. Kitoh. An optical FDM based self-healing ring networks employing arrayed-waveguide-grating ADM filters and ED- FAs with level equalizers. IEEE JSAC/JL Special Issue on Optical Networks, 4:800 83, June 996.