Cost-effective WDM broadcast-and-select networks for all-to-all transmission schedules
Introduction
Wavelength division multiplexing (WDM) is a promising approach to exploit the enormous bandwidth of optical fibers 1, 2, 3. There are two major classes of WDM networks: single-hop networks (also known as broadcast-and-select networks) and multihop networks. Both classes have their own strength and weakness and a thorough discussion can be found in 1, 2.
In broadcast-and-select networks, a station can send packets to the other station without passing through intermediate nodes (see Fig. 1). For this purpose, either the transmitters or receivers are tunable (i.e., the transmission or reception wavelength can be tuned) [1]. Based on the current device technology, the tuning time is not negligible compared with the packet transmission time 1, 4, 5. Therefore, when we schedule packet transmission in broadcast-and-select networks, we must take the tuning time into account.
Some research have been done on scheduling packet transmission for non-negligible tuning time 4, 5, 6, 7, 8, 9, 10, 11, 12. In particular, the all-to-all transmission scheduling problem is to schedule a packet transmission between every input–output pair. This scheduling problem arises when the network traffic is uniform.
Aggarwal et al. [4] are the first ones to study the effect of tuning time on all-to-all transmission scheduling. They considered a network with N inputs, N outputs, and one transmitter (receiver) per input (output), and they derived a lower bound for the length of the optimal all-to-all transmission schedules. The lower bound indicates that the schedule length is at least proportional to where T is the tuning time.
To reduce the adverse effect of tuning time on the schedule length, Pieris and Sasaki [5] proposed to use t transmitters per input and r receivers per output in order to pipeline transmission/reception with tuning. They considered a network with N inputs and M outputs, and considered two different transmission constraints: when a transmitter is tuning, the other transmitters can or cannot send packets (the latter one is called the tune-transmit-separability constraint [5]; similar constraint could be found in the satellite-switched time division multiple access system [13]). They derived lower bounds for the optimal schedule length and these lower bounds indicate that the schedule length is at least proportional to . In addition, they proposed sub-optimal all-to-all schedules and these schedules are still the shortest ones to the best of our knowledge. For the special case where N=M, t=r=1 and N/k is an integer (where k is the number of wavelength channels), Choi et al. [7] proved that the schedule proposed in [5] is optimal. Unfortunately, the optimal scheduling problem for t, r⩾1 and M≠N is rather difficult and is still an open problem.
In this paper, we enhance Pieris–Sasaki's results [5] in two ways. First, we study a cheaper network configuration to hide the tuning time in all-to-all transmission scheduling. In this configuration, every electronic transmitter (receiver) is connected to l tunable lasers (f fixed-tuned filters), so that we can pipeline transmission/reception with tuning. We show that our network configuration can give nearly the same schedule length as the network studied in [5] at a significantly lower cost and hence our network configuration is more cost-effective. Second, we derive tighter lower bounds for the optimal schedule lengths.
Section snippets
Notation
We consider WDM broadcast-and-select networks with N inputs and M outputs. There are W wavelength channels. We let k (1⩽k⩽W) be the number of channels that are actually used. Each channel is time slotted and each slot can accommodate one packet. The transmitter tuning time is equal to T slots.
A t-frame is defined to be the contiguous time slots in which there is at least one packet transmission in at least one channel, and a d-frame is those contiguous slots without packet transmission. An
Pieris–Sasaki network configuration and results
Every input has t electronic transmitters with one tunable laser per transmitter, and every output has r receivers with one fixed-tuned filter per receiver (see Fig. 2). Based on this network configuration, the lower bounds for the minimal schedule lengths are [5]where . Sub-optimal schedules were proposed in [5] and the lengths of these sub-optimal schedules are clearly upper bounds on the minimal schedule length:
Network configuration
In our network configuration, every input has one electronic transmitter which is connected to l tunable lasers via an electronic switch, and every output has one electronic receiver which is connected to f fixed-tuned filters (see Fig. 3). The tunable lasers are independently controlled, so that when the transmitter is sending a packet via a laser, the other lasers can be tuned simultaneously. Therefore, transmission can be pipelined with tuning. Fig. 4 shows an illustrative example. The
Schedule description
In this section, we describe an all-to-all transmission schedule for our network configuration. Compared with the schedule proposed in [5], the salient feature of our schedule is that no input is required to transmit multiple packets simultaneously and no output is required to receive multiple packets simultaneously. Therefore, our schedule only requires one electronic transmitter per input and one electronic receiver per output.
We propose the following all-to-all transmission schedule. Number
Comparison with different schedules
In this section, we compare the schedule length in our network configuration and in Pieris–Sasaki's network configuration. We consider the case where tr⩽k because only this case was considered in [5]. In our network configuration, if every input has one electronic transmitter with t tunable lasers and every output has one electronic receiver with r fixed-tuned filters, then the schedule length isIn Pieris–Sasaki's network configuration, the schedule length is
Lower bounds for minimal schedule lengths
In [5], Pieris and Sasaki derived lower bounds for the minimal schedule lengths in the network having tunable lasers and tunable filters (see the proof of Theorem 3.4 in [5]). In this section, we consider a generalized network configuration which includes our and Pieris–Sasaki's configuration as its special cases: every input has t electronic transmitters with l tunable lasers per transmitter and r electronic receivers with f tunable filters per receiver. We derive lower bounds for the minimal
Conclusions
In this paper, we enhance the results reported in [5] in two ways. First, we studied an alternative and cheaper network configuration for all-to-all transmission scheduling. In this network configuration, every input has one electronic transmitter with multiple tunable lasers and every output has one electronic receiver with multiple fixed-tuned filters. We demonstrated that this network configuration is more cost-effective to exploit pipelined transmission/reception/tuning to hide the tuning
Gaoxi Xiao received the BSc and MSc degrees in applied mathematics from Xidian University, Xi'an, China, in 1991 and 1994, respectively. He is currently working toward the Ph.D. degree in the Department of Computing, Hong Kong Polytechnic University. His research interests include optical networks and algorithmic techniques.
Yiu-Wing Leung received his BSc and PhD degrees from the Chinese University of Hong Kong in 1989 and 1992, respectively. He was with the Hong Kong Polytechnic University.
References (13)
- B. Mukherjee, WDM-based local lightwave networks part I: Single-hop systems, IEEE Network (1992)...
- B. Mukherjee, WDM-based local lightwave networks part II: Multihop systems, IEEE Network (1992)...
Dense wavelength division multiplexing networks: principles and applications
IEEE J. Select. Areas Commun.
(1990)- A. Aggarwal, A. Bar-Noy, D. Coppersmith, R. Ramaswami, B. Schieber, M. Sudan, Efficient routing and scheduling...
- et al.
Scheduling transmission in WDM broadcast-and-select networks
IEEE/ACM Trans. Networking
(1994) - et al.
Time-wavelength assignment algorithms for high performance WDM star based systems
IEEE Trans. Commun.
(1994)
Cited by (0)
Gaoxi Xiao received the BSc and MSc degrees in applied mathematics from Xidian University, Xi'an, China, in 1991 and 1994, respectively. He is currently working toward the Ph.D. degree in the Department of Computing, Hong Kong Polytechnic University. His research interests include optical networks and algorithmic techniques.
Yiu-Wing Leung received his BSc and PhD degrees from the Chinese University of Hong Kong in 1989 and 1992, respectively. He was with the Hong Kong Polytechnic University. Now, he is an associate professor in the Department of Computing Studies, Hong Kong Baptist University. His research interests include information networks, algorithmic and heuristic techniques, and software technology. He has been a senior member of IEEE since 1996.