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1 All-to-All Broadcast in Broadcast-and-Select WDM Networs with Tunale Devices of Limited Tuning Ranges Hongsi Choi 1 Hyeong-Ah Choi 1 and Lionel M. Ni 2 1 Department of Electrical Engineering and Computer Science The George Washington University Washington DC Department of Computer Science Michigan State University East Lansing MI fhongsi choig@seas.gwu.edu ni@cps.msu.edu Astract In this paper we consider the all-to-all roadcast prolem in optical roadcast star networs using Wavelength Division Multiplexing. Our networ model assumes that receivers are xed-tuned and transmitters are tunale such that optical lasers assigned to transmitters have limited access to the networ andwidth; hence each nodemust e equipped with multiple optical lasers and/or multiple optical lters in order to maintain a single-hop networ. This paper is primarily concerned with single-hop networs in which each node is assigned a single optical lter. Lower ounds are rst estalished on the numer of lasers per each node and the minimum schedule length and a schedule achieving the minimum schedule length is presented. The results are applicale to aritrary tuning delays aritrary numers of wavelength channels and optical lasers' aritrary tuning ranges. Networ models with optical devices having limited tuning ranges have not yet een considered in connection with transmission schedules and this is the rst wor in this new direction. This wor was supported in part y NSF grant MIP

2 1 Introduction AWavelength Division Multiplexed (WDM) optical networ with a roadcast-star architecture has found applications in local area networs with very high speed information transfer rates. In such a networ n nodes are connected through a passive n n roadcast star. Each node consists of a transmitter and a receiver at least one of which is tunale across the networ andwidth. The numer of wavelength channels is typically limited to a small numer (such as 1) due to various physical layer issues such as interchannel crosstal optical amplier andwidth optical lter resolution etc. This maes it necessary to share awavelength channel among many networ nodes. An important parameter that aects the performance of optical roadcast networs is the tuning delay which is the amount of time it taes an optical device (a tunale transmitter or a tunale receiver) to tune from one wavelength to another. Tuning delay limits the networ throughput as nodes spend time in tuning which could otherwise e used in useful information transfer. Tunale transmitters can e made using tunale lasers while tunale receivers can e realized using tunale lters. There are dierent inds of tunale lasers and wavelength ltering can e achieved y various mechanisms. Current tunale laser or lter technology severely limits either the tuning speed or the tuning range that can e achieved y tunale devices. Consequently a single tunale laser (or a single tunale lter) only covers a surange of the networ andwidth. To accommodate the entire networ andwidth an array of tunale lasers assigned to each transmitter or an array of tunale lters assigned to each receiver may erequired to partition the andwidth. Apparently there is a tradeo etween tuning speed and tuning range. The tuning speed must e proportional to the numer of channels through which a laser or a lter must tune. For instance let denote the tuning delay for a laser (or a lter) to change its tune from one to another under the condition that it can only resolve dierent channels. It should e then that is an non-decreasing function in [9]. The pacet transmissions scheduling prolem has een studied [ ] extensively under the assumption that receivers are xed-tuned and transmitters are tunale to the full range of networ andwidth. The eects of tuning delay in the case of random trac and small tuning delay was considered in [2]. A special deterministic trac was considered in [12] and [1] and various scheduling algorithms and performance ounds were otained. The complexity analysis of the prolem was rst addressed in [13]( transmission is called non-preemptive if thr source node is not involved in any other transmission until it completes its current transmission and preemptive otherwise ) where the non-preemptive version of the prolem is shown to e NP-complete in the wea sense; hence leaving an open question on the existence of any possile pseudo-polynomial time algorithms to nd an optimum solution. The result in [4] completely rules out this possiility where oth preemptive and non-preemptive versions of the prolem are shown to e NP-complete in the strong sense. Several scheduling heuristics for aritrary trac demands are presented in [3] ased on the framewor of load alancing across the channels. The results in [6] show that the well-nown list scheduling algorithm [14] produces a schedule whose schedule length is ounded y twice the optimum schedule length in the worst-case while the numerical 2

3 results otained through the simulation show much etter average-case performance. Other heuristics are proposed in [6] ased on the technique for edge-coloring ipartite multigraphs. An interesting special case of the general scheduling prolem is the all-to-all roadcast in which every transmitter/receiver pair has a single pacet to e transferred. Results in [12] present upper and lower ounds to the minimum schedule length in all-to-all roadcast. Their upper and lower ounds show that all-to-all scheduling can e done within a factor 2 of the minimum schedule length. In [6] it is shown that the upper ound presented in [12] is in fact a lower ound to the optimum schedule length; hence proving the optimality of the algorithm. In the wor of [12] and [6] the numer of wavelength channels availale in the networ is assumed to e an integer divisor of the numer of nodes in the networ. The all-to-all roadcast prolem was revisited in [7] without this assumption; universal lower ounds on the schedule length were otained and three distinct scheduling algorithms were developed to meet the corresponding lower ounds. As a consequence the all-to-all roadcast prolem is resolved for almost every setting of parameters. The only exception is a small suspace of the parameter space in which the algorithms perform within a factor 13=12 of the lower ound. In this paper we consider WDM optical networs with a roadcast-star architecture such that each node in the networ is assigned a single transmitter and a single receiver. Each transmitter is equipped with multiple tunale lasers and each receiver is equipped with a single xed-tuned lter. In our model lasers may access to only limited numer of channels i.e. the numer of channels to which a laser can tune may emuch smaller than the numer of channels that the networ provide. We consider the all-to-all roadcast prolem without any assumption on the tuning range of tunale lasers. The contriution of this paper is twofold. First the pacet transmissions scheduling prolem has not yet een addressed without maing an assumption on the tuning range of optical devices; consequently this is the rst paper in this new direction. The second contriution of the wor is the development of an algorithm that provides an optimal all-toall roadcast. The organization of the paper is as follows. Section 2 species the system model to e studied. Lower ounds are developed in Section 3 and scheduling algorithms are presented in Section 4. Concluding remars are given in Section 5. 2 System Model Consider a WDM roadcast-and-select networ with n nodes and wavelength channels ; ; 1availale in the networ. We assume that transmitters are tunale and receivers are xed-tuned; each transmitter t i is equipped with tunale lasers l i ; ;li 1 and each receiver r j is equipped with xed-tuned lters f j ; 1. A laser can tune to ;fj dierent wavelength channels j ; j+1; ; j+1 for an aritrary integer j ( j + 1). For a laser to tune from i to j (ji j +1j) a tuning delay (expressed in units of pacet durations) is required. In this paper we tae to e an aritrary real valued parameter. 3

4 In the all-to-all roadcast prolem each ofthen 2 transmitter-receiver pairs has one xed length pacet to transfer. We normalize the time such that the pacets have unit-length. We also assume that each schedule requires an initial tuning phase of duration efore pacet transmission can occur. 3 Lower Bounds In the all-to-all roadcast prolem each transmitter needs to transmit to all of n receivers. Since each laser can tune to only dierent channels it implies that each transmitter must e equipped with at least = lasers if each receiver is assigned a single xed-tuned lter. Similarly if each transmitter is equipped with a single tunale laser each receiver must e equipped with = xed-tuned receivers. In general if each transmitter has tunale lasers and each receiver has xed-tuned lters it must e that =. THEOREM 1 Let = is at least and = 1. Then the length of an all-to-all transmission schedule maxf + n2 ; + n 2 +(1)( n )g where each transmitter is assigned tunale lasers and each laser can resolve dierent wavelength channels with tuning time. Proof Let T denote the length of an optimal all-to-all transmission schedule when each opt transmitter is assigned = tunale lasers assuming that each laser can resolve dierent wavelength channels with tuning time. Each of the wavelengths can carry at most one pacet at any time and there are n 2 pacets to e transferred. This implies that any all-to-all transmission schedule requires at least n2 pacet durations. This together with the initial tuning time gives a lower ound T opt + n2. In the following we show that T opt + n2 +(1)( n ) During the initial tuning phase (i.e. from time to time nopacet transmission can occur. Hence from time to time +( n2 1) the numer of completed transmissions is at most ( n2 during each pacet duration. This implies that there exists a transmitter say t such that 1) since at most one transmission can e done per wavelength channel the numer of transmissions completed from t is at most n n. Since n it implies that t has completed at most n 1 transmissions. This in turn implies that there exists a laser say l ( i 1) assigned to t i such that the numer of transmissions completed using l is at most ( n 1)=(= i ) i.e. at most n 1. Note that each wavelength is shared receivers and each laser is supposed resolve dierent wavelength channels. y n The aove discussion implies that the clearance time of laser l is at least time i 1 +time 2 + time 3 where time 1 = +( n2 1) is the initial time period during which at most n 1 4

5 X 1 X x x1 x-1 x1 x11.. x X x x x size = n/ y y y y y y 1-1 Y Y 1... y... y y Y Figure 1: Partition of Receivers and Transmitters transmissions are completed using l time i 2 n ( n 1) is the amount of time to transmit the remaining pacets and time 3 ( 1) is the amount of time that l still has to spend i for tuning. Therefore T time opt 1 + time 2 + time 3 + n2 +(1)( n ). This completes the proof of the theorem. 4 Transmission Schedules In this section we proceed to discuss the construction of transmissions schedules when = = and = 1. Assume that = is an integer. In the following the scheduling algorithm is outlined and a complete description is given in Algorithm 1. Receivers are rst partitioned into = groups Y ; X 1 such that each group has the same size. Receivers in each group is again partitioned into sugroups of the same size. Hence there are receiver sugroups y q;i (for q and i 1) such that each sugroup has n receivers tuned to the same wavelength channel. Transmitters are similarly partitioned into sugroups. Figure 1 illustrate such partitions. Note that each transmitter has = lasers each of which has tuning range. Therefore for any transmitter t i laser l i can tae care of receivers in group Y q for q. Therefore j transmissions from X p to Y q (i.e. transmitters in X p to receivers in Y q ) can e done in such awaythat every transmitter t i in X p will use laser lq. i Suppose we want toschedule transmissions from X p to Y q and from X p to Y q. As there is no conict in wavelength channels etween two groups Y q and Y q a scheduling from X p to Y q can e done independently of a scheduling from X p to Y q. Furthermore such a scheduling can e treated as the scheduling when the system model has tunale transmitter 5

6 a) pq ij X p x p x ps x p-1.. j y.. q(s+i)%.. ) Y q Figure 2: Transmissions corresponding to loc p;q with full tuning range (i.e. channels). Consequently the optimal all-to-all roadcast scheduling algorithm discussed in [12 6] can e applied which results in the schedule length equal to maxf + n2 n 2 ; ( n 1) + n + g: From the aove discussion one can easily come up with a scheduling algorithm such that in phase transmissions from X p to Y p for each p are scheduled simultaneously; in phase 1 transmissions from X p to Y p+1 for each p are scheduled simultaneously; and in general in phase i transmissions from X p to Y p+i for each p are scheduled simultaneously. The length of this schedule is then computed as i;j maxf + n 2 2 ; n ( n 1) + n + g: Figure 3 illustrates phase of this algorithm where the loc diagram laeled p;q denotes i;j transmissions from the jth transmitter in each sugroup x p;s ( s 1) to every receiver in sugroup y q;(s+i) mod.(see Figure 2 for loc diagram p;q.) i;j One can then verify that the schedule length otained from this simple algorithm does not match the lower ound shown in the previous section. The following discussion estalishes a desired scheduling algorithm. Before we descrie our new approach we note the following. In the rst algorithm each transmitter t i utilizes lasers in such a way that once t i starts to use laser l i r it tries to use 6

7 B -1 B B -1 L B B L 1 B B -1 B L -1 B Figure 3: Simple approach 7

8 idle period 2 n / Figure 4: Scheduling to y q; for q 1 the same laser until the laser is no longer necessary in the remaining transmissions. In our new approach this restriction will not hold any more. That is while laser l i is spending r time for changing its tune another laser l i may e used to transmit data from transmitter s t i. From this approach each laser will have longer time etween its transmissions on two dierent wavelength channels. This is the ey idea of reducing the schedule time when the schedule is tuning-limited. Figures 4-6 illustrate our new approach. Let's rst loo at Figure 4. Bloc diagram ; ; shows that for each transmitter t i in x ; laser l i tuned to a single wavelength channel iseing used to transmit to y ;. Once the transmissions from l i to y ; have completed t i starts to transmit immediately to y 1; using laser l1 i which is shown in loc diagram ;. ;1 This will continues until laser i i is used to transmit to y 1 schedule from each transmitter to y q;j (for each q loc diagrams. 1;. In Figure 5 the transmission 1) is completely shown using A complete description of the schedule is shown in Figure 6 where Figure 6(a) shows the andwidth-limited case and 6() shows the tuning-limited case. The computation of schedule length in each case is self-explained from the gures where the schedule length corresponding to loc p;q is n= i;j and the schedule length corresponding to loc R j ( j 1) is n 2. It is not dicult to show from Figure 6(a) that the schedule length is + n2 when 8

9 1-1 j j j j j j j j j j1 j1 j1 j1 j1 j j1 j1 j1 2 n / 1-1 j-1 j-1 j j-1 j-1 j j-1 j-1 j-1 Figure 5: Scheduling to y q;j for q 1 idle period ( δ ) R n 2 / R 1 R -1 n 2 / (a) n/ δ 2 n / + n/ δ R R 1 R -1.. n 2 / n 2 / + (-1) δ + (1-)( n 2 / - n/ ) (). Figure 6: Complete Schedule 9

10 andwidth-limited. To compute the schedule length when tuning-limited note that there are locs R j ( j 1) and 1 gaps etween two locs R j and R j+1. The length of each gap should provide the additional tuning times for each laser and we see that the length of the gap is ( n2 n ). This together with the initial tuning time results in the schedule length + n2 +(1)( ( n2 n )) which is equal to + n2 +(1)( n ): The aove discussion estalishes the following theorem where the upper ound is the same as the lower ound presented in the previous section. Therefore our scheduling algorithm is optimum. THEOREM 2 If = = and = 1 then there exists an all-to-all roadcast schedule with schedule length maxf + n2 ; + n 2 +(1)( n )g: 5 Concluding Remars We have studied the prolem of all-to-all roadcast in a roadcast star networ with n nodes wavelength channels and a tuning delay such that each laser is tunale to ( ) dierent wavelength channels. It was shown that the schedule length is at least maxf + n2 ; + n2 +(1)( n )g if each transmitter is equipped with = lasers and each receiver is equipped with a single xed-tuned lter. A scheduling algorithm achieving this lower ound was presented. The signicance of the model considered in this paper is stressed from the current technological limit on the tunale devices while one considers the design of single-hop networs which require the availaility of tunale lasers and/or lters that can tune across the entire networ andwidth. This paper opens up many new directions for further research in transmission schedules and the results otained in this paper should provide insight into the eect of optical devices in WDM networs. References [1] A. Aggarwal A. Bar-Noy D. Coppersmith R. Ramaswami B. Schieer and M. Sudan \Ecient Routing and Scheduling Algorithms for Optical Networs" Tech. Rep. RC IBM Research Report June [2] M. Azizoglu R. Barry and A. Mohtar \Impact of Tuning Delay on the Performance of Bandwidth-Limited Optical Broadcast Networs with Uniform Trac" IEEE Journal on Selected Areas in Communications June

11 Algorithm 1 With Limited Tuning Range Input: n (Let n = l = n2 n ). Output: schedules S 1 S 3 : Case 1: S c (i t ;i r )=smeans that a transmission from transmitter t it to receiver r ir ( i t ;i r <n)is scheduled in time slot s using schedule S c where c =1for l; c =3for l< S c (i t ;i r )=S c (i t n l n it c;i n n r it c) if i n n t n S 1 (i t ;i r )= +i t + i r n ir c +1+ql + n ( ir n n cit S 1 (i t ;i r )=S 1 (i t ;i r ) n2 if S 1 (i t ;i r ) > n2 + Case 2: >l S 3 (i t ;i r )=p(+ )+(a1) n + i r n + n (ir n cit n c) S 3 (i t ;i r )=S 3 (i t ;i r )( n2 S 3 (i t ;i r )> n2 + +(1)( n2 where p = it c; q = ir c; a = i t it c( +1)+1; +1 x n y=(xy)mod n. ir n n c) c +1+(q+1) +(1) +(1)( n2 n )) if n ) 11

12 [3] M. S. Borella and B. Muherjee \Ecient Scheduling of Nonuniform Pacet Trac in a WDM/TDM Local Lightwave Networ with Aritrary Transceiver Tuning Latencies" Proceedings of Infocom '95 pp. 129{137 April [4] H. Choi and H.-A. Choi \Complexity Results on Pacet Transmissions Scheduling Prolem in Broadcast-and-Select WDM Networs" Technical Report Department of Electrical Engineering and Computer Science George Washington University. [5] H. Choi H.-A. Choi M. Azizoglu \Scheduling Transmission in WDM Optical Communication Networs" Proc. IEEE Intl. Conference on Communications - Gateway to Gloalization Seattle WA June [6] H. Choi H.-A. Choi M. Azizoglu \Ecient Scheduling Transmissions in Optical Broadcast Networs" to appear in IEEE/ACM Trans. on Networing Decemer [7] H. Choi H.-A. Choi and M. Azizoglu \On the All-to-All Broadcast Prolem in Optical Networs" to e presented at Infocom'97 April Koe Japan. [8] P. E. Green Fier Optic Networs Englewood Clis New Jersey: Prentice Hall [9] Rajiv Ramaswami \System Issues in Multi-Wavelength Optical Networs" Unpulished manuscript. [1] L. G. Kazovsy C. Barry M. Hicey C. A. Noronha Jr. and P. Poggiolini \WDM Local Area Networs: Wavelength-division multiplexing puts multiple channels on a single optical er and allows dynamic channel allocation." IEEE Letters. May 1992 pp. 8{15. [11] B. Muherjee \WDM-Based Local Lightwave Networs{ Part I: Single-Hop Systems" IEEE Networ vol. 6 pp. 12{27 May [12] G. R. Pieris and G. H. Sasai \Scheduling Transmissions in WDM Broadcast-and- Select Networs" IEEE/ACM Trans on Networing vol. 2 no. 2 pp. 15{11 April [13] G. N. Rousas and V. Sivaraman \On The Design of Optimal TDM Schedules for Broadcast WDM Networs with Aritrary Transceiver Tuning Latencies" Proceedings of Infocom '96 pp. 1217{1224 March [14] E. G. Coman and P. J. Denning Operating Systems Theory. Englewood Clis New Jersey: Prentice Hall

Scheduling Transmissions in WDM Optical Networks. throughputs in the gigabits-per-second range. That is, transmitters transmit data in xedlength

Scheduling Transmissions in WDM Optical Networks Bhaskar DasGupta Department of Computer Science Rutgers University Camden, NJ 080, USA Michael A. Palis Department of Computer Science Rutgers University