A Hybrid Multicast Scheduling Algorithm for Single-Hop WDM Networks

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1 A Hybrid Multicast Scheduling Algorithm for Single-Hop WDM Networks Hwa-Chun Lin and Chun-Hsin Wang Abstract This paper shows that, for single-hop WDM networks, a multicast scheduling algorithm which always tries to partition a multicast transmission into multiple unicast or multicast transmissions may not always produce lower mean packet delay than a multicast scheduling algorithm which does not partition multicast transmissions. The performance of a multicast scheduling algorithm may depend on the traffic conditions and the availability of the channel resource in the network. A hybrid multicast scheduling algorithm that can produce good performance for wide ranges of the traffic conditions and the availability of the channel resource in the network is proposed. Depending on the average utilizations of the data channels and the receivers, the proposed hybrid multicast scheduling algorithm dynamically chooses to employ a multicast scheduling algorithm which always tries to partition multicast transmissions or a multicast scheduling algorithm which does not partition multicast transmissions. Extensive simulations are performed to study the performance of the proposed hybrid algorithm. Our simulation results show that the proposed hybrid algorithm produces lower mean packet delay for wide ranges of the load, the maximum multicast group size, the percentage of unicast traffic, and the number of data channels in the network compared with a multicast scheduling algorithm which always tries to partition multicast transmissions and a multicast scheduling which does not partition multicast transmissions. Keywords Single-hop WDM networks, multicast scheduling, star coupler I. INTRODUCTION The demands for networks with high bandwidth and efficient multicast mechanisms are increasing. For examples, network applications such as video conferences, video on demands, image distributions, and etc. require high bandwidth and multi-destination communications. In the future, it is predicted that per user bandwidth demand will be approximately 1 Gb/s [1], [2]. The vast bandwidth of a fiber (more than THz [3], [4]) can be divided into a lot of high-speed channels using the WDM technology. Each of the channels is capable of operating at the peak rate of an electronic interface. The WDM network considered in this paper is a WDM star coupler network consisting of a number of network nodes connected via optical fibers to a passive star coupler as shown in Fig. 1. Each node is equipped with one or more fixed or tunable transmitters and one or more fixed or tunable receivers. The passive star coupler is able to combine all input optical signals and broadcast the combined signal to all outputs. In single-hop WDM networks, a data packet can be transmitted from one node to another only when one of the transmitters of the source node and one of the receivers of the destination node are tuned to the same wavelength; i.e., a data packet is The authors are with the Department of Computer Science, National Tsing Hua University, Hsinchu 043, Taiwan, R.O.C., hclin, chwang@cs.nthu.edu.tw. This research was supported in part by the National Science Council, Taiwan, R.O.C., under grant NSC E and in part by the Ministry of Education, Taiwan, R.O.C., under the Program for Promoting Academic Excellence of Universities, G89-E-FA Node 2 Node 1 Passive Star Coupler A pair of optical fibers Node 3 Node 4 Fig. 1. A star coupler network..... Node N transmitted from one node to another without going through intermediate nodes. The wavelengths at which the nodes communicate with each other are referred to as channels. Since the number of channels may be less than the number of nodes and two or more nodes may want to send data packets to the same destination node, coordination among nodes that wish to communicate with each other is required. Many access protocols for coordinating data transmissions have been proposed in the literature. These scheduling algorithms can be classified into three categories, namely, random-access based [5], pre-allocation based [6-9], and reservation based scheduling algorithms [10-22]. A number of these scheduling algorithms are for multicast transmissions [5], [7], [8], [], [], [21]. The multicast scheduling algorithms proposed in [7] and [] schedule multicast packets such that the receivers of all destination nodes must tune to the same channel at the same time. If the receivers of one or more destination nodes are not available, transmission of the data packet will be delayed. The multicast protocols proposed in [5], [8], [], and [21] allow a multicast transmission to be partitioned into multiple unicast or multicast transmissions. Modiano [5] proposed several random-access based multicast scheduling algorithms by combining two schemes for transmitting packets and three schemes for receiving packets. One of the schemes for transmitting packets transmits a packet continuously until it is received by all of its destination nodes. The other scheme for transmitting packets introduces random delays between retransmissions of the same packet. When two or more packets are transmitted to the same destination node at the same time, the receiver at the destination node needs to select one of the packets to receive. Three schemes for selecting one of the packets to receive were considered. The first scheme selects one of the packets randomly. The second scheme selects a packet based on the order that they were transmitted.

2 The third scheme selects the packet which has the smallest number of remaining intended destinations. The results in [5] show that the combination of introducing random retransmission delay and selecting the packets with the smallest number of remaining destinations has the best performance. Pre-allocation based multicast scheduling algorithms can be found in [7], [8]. Rouskas and Ammar [7] proposed a multidestination communication protocol for single-hop WDM networks. In their network model, each node is equipped with one fixed transmitter and one tunable receiver. On each channel, time is divided into frames and each frame is further subdivided into slots that are globally synchronized over all channels. They proposed several adaptive multicast protocols based on one-to-one schedule [9]. In each frame, a number of slots selected from the one-to-one schedule are allocated to each of the nodes for multicast purpose. These slots are called multicast slots. In a multicast slot, the owner of the slot may transmit to a group of nodes, the rest of nodes may transmit according to the one-to-one schedule. Three adaptive protocols were proposed to determine the membership of the multicast groups. In the pre-allocation based scheduling algorithm proposed in [8], the slots are pre-allocated for unicast purpose. When a multicast packet arrives at a node, a multicast distance is used to determine whether the multicast packet should be transmitted as a single multicast packet or multiple unicast packets. This information along with the multicast group of this packet is broadcast to all other nodes via a control channel. When the information for the multicast packet is received by all of the nodes, all of the nodes run the same scheduling algorithm to modify the pre-allocated slots to accommodate the multicast packet. Reservation based multicast scheduling algorithms can be found in [], [], [21]. Borella and Mukherjee [] proposed a reservation-based multicast protocol for single-hop WDM networks. In their network model, each node is assumed to be equally distant from the star coupler such that the propagation delays between all pairs of nodes are identical. Each node is equipped with Ø tunable transmitters and Ö tunable receivers for data transmissions and receptions. In addition, each node also has one fixed transmitter and one fixed receiver on a control channel. On the control channel, time is divided into control frames and each control frame is broken into control slots. In a control frame, there is one control slot for each of the nodes. Data packets are of fixed length. Each node keeps track of the same system state including the times beyond which each transmitter, receiver, and channel will be free, and the channels that each transmitter and receiver will be tuned to when they become free. Before a data packet transmission, one control packet which identifies the source and destination nodes must be transmitted on the control channel. After a round-trip delay, all of the nodes receive the control packet and execute the same distributed scheduling algorithm to determine the time at which the data packet will be transmitted and the channel which will be used. Then, each node updates its database of the system state. Their simulation results show that the system has good performance when the multicast group size is either very small or very large. Jue and Mukherjee [] proposed three reservation based scheduling algorithms that allow a multicast transmission to be partitioned into multiple unicast or multicast transmissions. The system model used is similar to the model in []. Every node keeps track of the times beyond which each of the transmitters, receivers, and channels will be available. The first algorithm schedules the transmissions forward in time starting from the time when the earliest available receiver becomes free. The second algorithm schedules the transmissions backward in time starting from the time when the latest available receiver becomes free. Similar to the first algorithm, the third algorithm also schedules the transmissions forward in time. The time instance at which the first transmission is scheduled is chosen such that the average receiver waiting time is minimum. Their simulation results show that partitioning multicast transmissions may lead to significant performance improvement. Ortiz, Rouskas and Perros [21] proposed a reservation based multicast protocol, based on the concept of a virtual receiver, a set of physical receivers that behave identically in terms of tuning. In their network model, each node is equipped with one fixed transmitter and one tunable receiver. The multicast transmissions represented by a traffic demand matrix are processed as a batch. By partitioning the set of physical receivers into virtual receivers, the original network with multicast traffic can be transformed into a new network with unicast traffic only. Any existing scheduling algorithm for unicast traffic can be employed to schedule the transmissions of packets. The authors then studied the problem of optimally partitioning the set of physical receivers into virtual receivers. They proved that this problem is NP-complete, and developed four heuristics for this problem. In this paper, we show that a multicast scheduling algorithm which always tries to partition a multicast transmission into multiple unicast or multicast transmissions may not always produce lower mean packet delay than a multicast scheduling which does not partition multicast transmissions. The performance of a multicast scheduling algorithm may depend on the traffic conditions (e.g., the load, the maximum multicast group size, the percentage of unicast traffic, and etc.) and the availability of the channel resource in the network. Our goal is to develop a scheduling algorithm that can produce good performance for wide ranges of the traffic conditions (e.g., the load, the maximum multicast group size, the percentage of unicast traffic, and etc.) and a wide range of the availability of the channel resource in the network. A hybrid multicast scheduling algorithm is successfully developed to achieve this goal. Depending on the average utilizations of the data channels and the receivers, the proposed hybrid multicast scheduling algorithm dynamically chooses to employ a multicast scheduling algorithm which always tries to partition multicast transmissions or a multicast scheduling which does partition multicast

3 1 one control frame control slots N-1 N Fig. 2. The structure of a control frame. transmissions. Extensive simulations are performed to study the performance of the proposed hybrid algorithm. Our simulation results show that the proposed hybrid algorithm produces lower mean packet delay for wide ranges of the load, the maximum multicast group size, the percentage of unicast traffic, and the number of data channels in the network compared with a multicast scheduling algorithm which always tries to partition multicast transmissions and a multicast scheduling which does not partition multicast transmissions. The rest of this paper is organized as follows. The system model is described in the next section. The performance of a multicast scheduling algorithm which always tries to partition multicast transmissions and a multicast scheduling which does not partition multicast transmissions are compared in section III. The proposed hybrid multicast scheduling algorithm is given in section IV. The performance of the proposed hybrid algorithm is studied in section V. Finally, some concluding remarks are given in section VI. II. SYSTEM MODEL The WDM network considered in this paper consists of a passive star coupler and Æ nodes. Each node connects to the passive star coupler via a fiber link consisting of a pair of fibers. There are Ï ½ communication channels in the system, where Ï Æ. One of the channels, ¼, is used as control channel which is shared by all nodes. The rest of the channels are data channels which are used for data transmissions. Each node is equipped with one fixed transmitters (FT), one fixed receiver (FR), one tunable transmitter (TT), and one tunable receiver (TR). The fixed transmitters and the fixed receivers are on the control channel. The tunable transmitters and tunable receivers are tunable over all the data channels in the system. The tuning times of the tunable transmitters and the tunable receivers are Ø Ø seconds. On the control channel, time is divided into control frames. Each control frame is subdivided into Æ control slots numbered from ½ to Æ as shown in Fig. 2. One control packet can be transmitted in one control slot. The control channel is accessed in the Time Division Multiple Access (TDMA) fashion. Node can transmit its control packet only at the th control slot within a control frame. A control packet consists of a source address field and a destination address field. The value of the source address field identifies the source node. The identification of a node is encoded in ÐÓ ¾ Æ bits. The destination address field consists of Æ bits. The th bit is set if node is a destination node. The collection of the destination nodes identified in the destination address field is called a multicast group. The length of a control slot,, is Æ ÐÓ ¾ Æ bits. Data packets are of fixed size with length bits. On the data channels, time is divided into data slots. The length of one data slot equals the sum of the tuning time and the data packet transmission time. Data slots over all channels are assumed to be synchronized. It is assumed that propagation delays between all node pairs are identical. For local networks, the assumptions imposed on the propagation delays can be realized by extending the lengths of fibers between the nodes and the passive star coupler or adding appropriate optical delays at the nodes. Let the propagation delays be Ø Ô seconds. The system operates as follows. When a data packet arrives at a node, it is placed in a local arrival queue in the node. If the local arrival queue is not empty, a control packet is sent to all of the nodes for reservation. Once a control packet is transmitted, the associated data packet is moved to a waiting space in the node until it is transmitted to all of its destination nodes. After a propagation delay, the control packet will reach all of the nodes. The same scheduling algorithm is invoked at all nodes to reserve data channel(s) and data slot(s) at which the data packet will be transmitted and received. If the data packet is scheduled such that the destination nodes receive the data packet at different data slots, the data packet has to be transmitted more than once. The source node will transmit the data packet in the reserved data slot(s). The data packet will arrive at some or all of its destination nodes after Ø Ô seconds. The receiver(s) of the destination node(s) should tune to the transmitting wavelength of the source node to receive the data packet. III. PARTITION VERSUS NO PARTITION OF MULTICAST TRANSMISSIONS In this section, we shall first describe two scheduling algorithms. One of the algorithms (greedy algorithm) always tries to partition a multicast transmission into multiple unicast or multicast transmissions. The other algorithm (no-partition algorithm) does not partition multicast transmissions. Intuitively, it seems that a scheduling algorithm which always tries to partition a multicast transmission into multiple unicast or multicast transmissions can always produce lower mean packet delay; however, it is not always true. Several examples will be used to explain the reason. Then some simulation results will be given to show that, under some situations, the no-partition algorithm produces significantly lower mean packet delay than the greedy algorithm. A. The status information Before describing the two algorithms, we shall define the status information which will be used in the scheduling algorithms. Let Ø ¼ be the time instant corresponding to the begin-

4 ning of the first data slot following the time when the control packet under consideration is received by all of the nodes. Let Ä be the last data slot following Ø ¼ in which at least one of the transmitters uses one of the data channels to transmit a data packet to one or more receivers; i.e., all of the transmitters are free in the Ä ½µth data slot following Ø ¼. The status of the data transmitters and the data channels in the Ä data slots following Ø ¼ and the status of the data receivers in the Ä data slots following Ø ¼ Ø Ô are maintained in three matrices, Ì,, and Ê respectively: Transmitter status matrix (Ì Ì ½ Æ ½ Ä) ¼ Ø ØÖÒ ÑØØÖ Ó ÒÓ Ö Ò Ø Ø Ø ÐÓØ ÓÐÐÓÛÒ Ø Ì = ¼ ½ Ø ØÖÒ ÑØØÖ Ó ÒÓ Ò Ö ÖÚ Ò Ø Ø Ø ÐÓØ ÓÐÐÓÛÒ Ø ¼ Receiver status matrix (Ê Ê ½ Æ ½ Ä) ¼ Ø ÖÚÖ Ó ÒÓ Ö Ò Ø Ø Ø ÐÓØ ÓÐÐÓÛÒ Ø ¼ Ø Ô Û Ø ÖÚÖ Ó ÒÓ Ò Ê = ÙÐ ØÓ ÖÚ Ø ÔØ Ò ÓÙÐ ØÙÒ ØÓ ÒÒÐ Û Ò Ø Ø Ø ÐÓØ ÓÐÐÓÛÒ Ø ¼ Ø Ô Data channel status matrix ( ½ Ï ½ Ä) ¼ Ø ÒÒÐ Ö Ò Ø Ø Ø ÐÓØ ÓÐÐÓÛÒ Ø ¼ = Ø ÒÒÐ Ò Ö ÖÚ ÓÖ ÒÓ Ò Ø Ø Ø ÐÓØ ÓÐÐÓÛÒ Ø ¼ Two additional matrices, Ê Ê and, which are derived from the receiver status matrix (R) and the data channel status matrix (C) are also maintained: ¼ Ê ½ ¼ ½ Ê ¼ Ê ¼ ¼ ¼ The content of a control packet from node in a control frame can be represented as an Æ ½ column matrix denoted by É É, where É ¼ ÒÓ ÒÓØ ØÒØÓÒ ÒÓ Ó ÒÓ ½ ÒÓ ØÒØÓÒ ÒÓ Ó ÒÓ B. The greedy algorithm This algorithm always tries to partition a multicast transmission into multiple unicast or multicast transmissions and schedules as many destination nodes as possible in the earliest data slot. To find the destination nodes that are available in a data slot, say slot, we first find the destination nodes that are not available in the th data slot. The nodes among the destination nodes which are not available in the th data slot can be found by performing an entry by entry logical Æ operation of column matrices É and Ê ; i.e., É Ê, where the operator represents an entry by entry logical Æ operation of two column matrices and Ê is the th column of matrix Ê. An 1 at the th entry of the result of É Ê means that destination node is not available for receiving a data packet in the th data slot. A 0 in the th entry of the result of É Ê means that either node is not a destination node of node or node is a destination node of node and it is available for receiving a data packet in the th data slot. The nodes among the destination nodes of node which are available in the th data slot can be found by subtracting É Ê from the column matrix É. An 1 in the th entry of the result of (É É Ê ) means that destination node is available for receiving a data packet in the th data slot. Otherwise, destination node is not available for receiving a data packet in the th data slot. The details of the greedy scheduling algorithm for processing a control packet represented by É is given as follows : 1. Find the earliest slot, say, such that (i) the transmitter at the source node is free (i.e., Ì ¼), (ii) there exists a free channel, say, (i.e., ¼), and (iii) É Ê É. If such a data slot can be found, go to step Schedule the data packet in data slot Ä ½ and select a free data channel, say, for it. Add one column (column Ä ½) to the matrices Ì, Ê, Ê,, and. Let =Ä ½. 3. Reserve data slots and the selected data channel in the matrices Ì, Ê, and ; i.e., Ì ½, Ê Ê É É Ê µ, and. Let É É Ê. Update Ê and according to the new Ê and respectively. 4. If É is not a zero matrix, go to step END C. The no-partition algorithm This algorithm schedules the destination nodes of a multicast packet to receive the packet in the same data slot; i.e., the multicast transmission is not partitioned into multiple unicast or multicast transmissions. The details of this scheduling algorithm for processing a control packet represented by É is given as follows: 1. Find the earliest slot, say, such that (i) the transmitter at the source node is free (i.e., Ì ¼), (ii) there exists a free channel, say, (i.e., ¼), and (iii) É Ê ¼. If such a data slot can be found, go to the step Schedule the data packet in data slot Ä ½ and select a free data channel, say, for it. Add one column (column Ä ½) to the matrices Ì, Ê, Ê,, and. Let =Ä ½. 3. Reserve data slots and the selected data channel in the matrices Ì, Ê, and ; i.e., Ì ½, Ê Ê É, and. Update Ê and according to the new Ê and respectively. 4. END

5 Data slot Transmitter of node 2 Transmitter of node 3 Transmitter of node 5 Receiver of node 1 Receiver of node 2 Receiver of node 3 Receiver of node 4 The number of data channels that has been reserved data packet Reserved for other packets source node destination node(s) 1, 2, 3 4 1, 4 Fig. 3. The status before scheduling the three data packets, Æ ¼. D. Examples For each individual packet, the greedy algorithm schedules the packet such that the packet can be received by all of its destination nodes as early as possible. Intuitively, it seems that the greedy algorithm can always produce lower mean packet delay than the no-partition algorithm. However, it is not always true. In the following, we shall use several examples to explain the reason. Before describing the examples, we shall first explain how the data slots are indexed in our examples. For a source node to be able to transmit a data packet to a set of destination nodes successfully in a data slot, say slot, the transmitter at the source node and one of the data channels must be free in data slot and the receivers at the destination nodes must be available in the data slot Ø Ô seconds later due to the propagation delay. For convenience, let the data slot at the destination nodes Ø Ô seconds later be also denoted as. Therefore, we use the same index to denote the th data slot following Ø ¼ at a transmitter and a data channel and the th data slot following Ø ¼ Ø Ô at a receiver. Let the current number of reserved data channels and the current status of some of the transmitters and receivers of a single-hop WDM network with 50 nodes be as shown in Fig. 3. Let È ½, È ¾, and È be the next three data packets which are to be scheduled in sequence. The source nodes and destination nodes of the three packet are shown in the lower part of Fig. 3. Suppose that the network has 50 data channels. The results of scheduling the three data packets in sequence using the greedy and no-partition algorithms are shown in Figs. 4 and 5 respectively. From the figures, we can observe that the mean packet delay of the three data packets is larger when the nopartition algorithm is used to schedule them. The reason is that the no-partition algorithm schedules each data packet in a data slot such that one of the data channels, the transmitter of the source node, and the receivers of all of the destination nodes Fig. 4. Data slot Transmitter of node 2 Transmitter of node 3 Transmitter of node 5 Receiver of node 1 Receiver of node 2 Receiver of node 3 Receiver of node 4 The number of data channels that has been reserved data packet Reserved for other packets source node P1 destination node(s) 1, 2, 3 4 1, 4 The status after scheduling the three data packets using the greedy algorithm, Æ ¼ Ï ¼. Data slot Transmitter of node 2 Transmitter of node 3 Transmitter of node 5 Receiver of node 1 Receiver of node 2 Receiver of node 3 Receiver of node 4 The number of data channels that has been reserved data packet Reserved for other packets source node 5 destination node(s) 1, 2, 3 Fig. 5. The status after scheduling the three data packets using the no-partition algorithm, Æ ¼ Ï. are available. The starting time of such a data slot could be quite far away from the time instant Ø ¼ resulting in large packet delay. On the other hand, the greedy algorithm always tries to partition the set of destination nodes into smaller subsets such that a multicast transmission can be scheduled as soon as possible. However, the result will be different if the network has limited channel resource, e.g., five data channels. With five data channels, the result of scheduling using the no-partition algorithm remains unchanged; i.e., the same as the result in Fig. 5. The result of scheduling using the greedy algorithm becomes the one shown in Fig. 6. Note that the number of data channels that can be reserved is limited to a maximum of five. From Figs. 5 and 6, we can observe that the mean packet delay of the three data packets is larger when the greedy algorithm is used. This is because partitioning a multicast transmission into mul , 4

6 Data slot Transmitter of node 2 Transmitter of node Transmitter of node 5 Receiver of node 1 Receiver of node 2 Receiver of node 3 Receiver of node 4 The number of data channels that has been reserved P Reserved for other packets M = M = M = 5 M = 5 Fig. 6. data packet source node destination node(s) 5 1, 2, , 4 The status after scheduling the three data packets using the greedy algorithm, Æ ¼ Ï Fig. 7. Performance comparison of the greedy and no-partition algorithms, Æ ¼ Ï ¼ Å ½ ½¼. tiple unicast or multicast transmissions requires more channel resource to transmit the multicast packet. When there is no available data channel in a data slot, the packet has to be scheduled in a later data slot although the transmitter of the source node and the receivers of some of the destination nodes are available. E. Performance comparison of the greedy and no-partition algorithms In the rest of this section, the performance of the greedy and no-partition algorithms are compared by extensive simulations. Simulations are conducted based on the following assumptions. The channel bit rate is 1 Gb/s and the speed of the light in the fiber is ¾ ½¼ m/s. The data packet size () is 500 bits. The tuning times of the tunable transmitters and receivers are assumed to be negligible. The length of the fiber link from a node to the passive star coupler is 0.5 km. The round-trip propagation delay (Ø Ô ) over a 0.5 km fiber link is ½¼ seconds. Packets arrive at each of the nodes form a Poisson process with mean arrival rate «packets per data slot. The maximum size of the multicast group for a packet is Å, where ½ Å Æ. The size of the multicast group for a packet is uniformly distributed on the set ½ ¾ Å. The nodes (excluding the source node) have equal probabilities to be in the multicast group. The performance measure used is the mean packet delay. The packet delay for a packet is measured starting from the time instant when the packet arrives at a node until the packet is transmitted to all of its destination nodes. The 90 percent confidence intervals are calculated. Fig. 7 shows that the greedy algorithm always produces lower mean packet delay than the no-partition algorithm for various maximum sizes of the multicast group when the channel resource is sufficient (i.e., the network has 50 data channels) for the reason described previously. However, when the Fig. 8. M= M = M= 10 M =5 M = Performance comparison of the greedy and no-partition algorithms, Æ ¼ Ï Å ½ ½¼. channel resource is limited, the no-partition algorithm may produce significantly lower mean packet delay than the greedy algorithm as shown in Fig. 8. The number of data channels in the network is five for the results displayed in Fig. 8. This figure shows that, under the condition that the channel resource is limited, the greedy algorithm produces lower mean packet delay than the no-partition algorithm when the load of the network is light. However, when the load of the network is heavy, the greedy algorithm produces higher mean packet delay than the no-partition algorithm because it uses too much of the channel resource. IV. THE PROPOSED HYBRID ALGORITHM From the examples in the previous section, we can make the following important observations: When there are sufficient number of available data channels,

7 the receivers should be utilized as much as possible. When the channel resource becomes a potential bottleneck, the data channels should be used conservatively. The utilizations of the data channels and the receivers are the key factors that determine the performance of a scheduling algorithm. The first observation suggests that the greedy algorithm be used when there are sufficient number of available data channels since the greedy algorithm always tries to partition the set of destination nodes into a number of subsets such that the receivers can be utilized as much as possible. The second observation suggests that the no-partition algorithm be used when the channel resource becomes a potential bottleneck since the no-partition algorithm utilizes the least amount of channel resource; i.e., exactly one data channel per data packet. The above observations motivate us to combine the greedy and nopartition algorithms to become a hybrid scheduling algorithm. Our goal is to develop a scheduling algorithm that can produce good performance for wide ranges of the traffic conditions (e.g., the load, the maximum multicast group size, the percentage of unicast traffic, and etc.) and a wide range of the number of data channels in the network. The critical component of the proposed hybrid scheduling algorithm is how the two algorithms are combined. Since the utilizations of the data channels and the receivers are two important factors, we shall use the two utilization factors to determine how the two algorithms are employed. We define the average utilization factors of the data channels and the receivers in data slots 1 through Ä, and Ö respectively, as follows: Ö È Ä ½ È Ï ½ (1) Ï Ä È Ä È Æ ½ ½ Ê (2) ÆÄ When a data packet is to be scheduled, the current values of the average utilization factors are calculated. Then the following heuristic rules are used to determine whether the greedy algorithm or the no-partition algorithm is employed. The discussion of the heuristic rules includes of the following two cases: 1. Ö Since the average utilization factor of the data channels,, is less than the average utilization factor of the receivers, Ö, it is likely that the channel resource is sufficient if the average utilization factor of the receivers is low to medium. Thus the greedy algorithm is used to schedule the current data packet. If the average utilization factor of the receivers is high, conserving the channel resource and saving some of the data channels for upcoming data packets may not be helpful since most of the receivers are not available in data slots 1 through Ä anyway. Therefore, the greedy algorithm is used to schedule the current data packet such that the packet can be transmitted as soon as possible. In summary, the greedy algorithm is employed in this case. 2. Ö In this case, the discussion includes three situations: low, medium, and high average utilization factor of the data channels. The situations where the average utilization factor of the data channels is low and high will be discussed first. Then the situation where the average utilization factor of the data channels is medium will be discussed. We shall regard the average utilization factor of the data channels as low when it is less than or equal to one third. The average utilization factor of the data channels is regarded as high when it is greater than two third. If the average utilization factor of the data channels is low, the greedy algorithm is used since there is sufficient channel resource. If the average utilization factor of the data channels is high, the no-partition algorithm is employed in order to conserve channel resource. It is possible that the channel resource saved can be used by upcoming data packets since the average utilization factor of the receivers is less than or equal to the average utilization factor of the data channels. When the average utilization factor of the data channels is medium (i.e., it is greater than one third and less than or equal to two third), we have to be more careful in choosing between the greedy and no-partition algorithms since the channel resource may become a potential bottleneck for the upcoming data packets. First of all, a heuristic rule is used to determine whether the channel resource is currently a potential bottleneck. If yes, the no-partition algorithm is used to schedule the current data packet; otherwise, another heuristic rule is used to determine whether the channel resource will become a potential bottleneck after the current packet is scheduled. If yes, the no-partition algorithm is used to schedule the current packet; otherwise, the greedy algorithm is employed. The two heuristic rules are described as follows: The rule for the current status of the data channels We define a channel flag,, as follows: ØÖÙ if ¾ ½ Ä s.t. Ð ÓØÖÛ È Ï ½ Ï ½ É Æĵ, where É is the content of the control packet associated with the data packet under consideration. If the least amount of increase in the average uti- If there is no available data channel in at least one of the data slots 1 through Ä, the channel flag,, is set to be ØÖÙ indicating that the channel resource could be a potential bottleneck currently. The rule for the future average utilization factor of the data channels Suppose that the packet under consideration is scheduled using the greedy algorithm and that at least one of the data slots 1 through Ä is used, the least amount of increase in the average utilization factor of the data channel is ½ Ï Äµ. The maximum amount of increase in the utilization factor for the receivers is the size of the multicast group of the packet divided by ÆÄ; i.e., È Æ ½

8 lization factor of the data channels is larger than the maximum amount of increase in the utilization factor of the receivers (i.e., Ï Ä È Æ ½ ½ É ), the channel resource may become a potential bottleneck after the data packet is scheduled using the Ï Ä greedy algorithm. Therefore, the greedy algorithm is not used; in other words, the no-partition algorithm is employed. On the other hand, suppose that the packet under consideration is scheduled using the greedy algorithm and that none of the data slots 1 through Ä is used, then the packet must be scheduled in data slot Ä ½. In this case, the result of the scheduling is the same no matter which of the algorithms is used. The details of the proposed hybrid scheduling algorithm for processing a control packet represented by É is given as follows : M =49 M = M =49 M = begin Compute, Ö, and ; if ( Ö ) use the greedy algorithm; if ( Ö ) switch ( ) end. case ( ½ ) : use the greedy algorithm; break; case ( ½ ¾ ) : if ( == true) use the no-partition algorithm; È Æ else if ( ½ ½ É ) use the no-partition Ï Ä Ï Ä algorithm; else use the greedy algorithm; break; case ( ¾ ) : use the no-partition algorithm; break; V. PERFORMANCE OF THE PROPOSED HYBRID SCHEDULING ALGORITHM In this section the performance of the proposed hybrid scheduling algorithm is compared with those of the greedy and no-partition algorithms. The simulation model used is the same as one described previously. The performance measure used is the mean packet delay. The 90 percent confidence intervals are calculated. First of all, we compare the performance of the three algorithms under the condition that there are sufficient number of data channels for all nodes such that the channel resource will never become a bottleneck. Figs. 9 and 10 show the mean packet delays for various maximum multicast group sizes. From the two figures, we can observe that the mean packet delays for the hybrid and greedy algorithms match closely. In fact, our simulation results show that they produce the same mean packet delays. When the number of data channels is limited, the greedy algorithm which always tries to partition a multicast transmission into multiple unicast or multicast transmissions may not always produce better performance than the no-partition algorithm. The proposed hybrid algorithm will dynamically Fig. 9. Performance of the hybrid scheduling algorithm, Æ ¼ Ï ¼ Å ¾¼. M = M = M = 5 M = Fig. 10. Performance of the hybrid scheduling algorithm, Æ ¼ Ï ¼ Å ½ ½¼. choose to use the greedy or no-partition algorithm depending on the average utilizations of the data channels and the receivers. Figs. 11 and 12 show the mean packet delays produced by the three algorithms for the same system with five data channels. From the figures, we can observe that the hybrid algorithm always produces lower mean packet delay than the no-partition and greedy algorithms. Finally, we compare the performance of the three scheduling algorithms under the condition that there are higher percentage of unicast traffic in the system. To increase the percentage of the unicast traffic, we increase the probability of selecting 1 to be the size of a multicast group. The rest of the probability for selecting the multicast group size is uniformly distributed on the set ¾ Å. Let Í be the probability of selecting 1 to be the size of a multicast group. Figs. 13 and 14 show the mean packet delays produced by the three scheduling al-

9 45 40 M = 49 M= 49 M = 49 M = M = M= M= M= M= Fig. 11. Performance of the hybrid scheduling algorithm, Æ ¼ Ï Å ¾¼ Fig. 13. Performance of the hybrid scheduling algorithm, Æ ¼ Ï Å ½ ½¼ Í ¼. M= M = M= 10 M = M =5 M =5 M= M= 10 M= M= Fig. 12. Performance of the hybrid scheduling algorithm, Æ ¼ Ï Å ½ ½¼ Fig. 14. Performance of the hybrid scheduling algorithm, Æ ¼ Ï Å ½ ½¼ Í ¼. gorithms under the conditions that Í equals to 0.3 and 0.6 respectively. The number of data channels in the network is 5. From the two figures, we can also observe that the no-partition algorithm may produce significantly lower mean packet delay than the greedy algorithm when the channel resource is limited. Our simulation results for the two figures show that the proposed hybrid algorithm always produces the lowest mean packet delays. VI. CONCLUSIONS In this paper, we have shown that, in single-hop WDM networks, a multicast scheduling algorithm which always tries to partition a multicast transmission into multiple unicast or multicast transmissions may not always produce lower mean packet delay than a multicast scheduling algorithm which does not partition multicast transmissions. The reason for this phe- nomenon has been analyzed using several examples. We found that when there are sufficient number of available data channels, the receivers should be utilized as much as possible by partitioning a multicast transmission into multiple unicast or multicast transmissions. On the other hand, when the channel resource becomes a potential bottleneck, the data channels should be used conservatively by not partitioning multicast transmissions. Based on the findings in our analysis, a hybrid multicast scheduling algorithm has been proposed. The goal for developing this hybrid multicast scheduling algorithm is to produce good performance for wide ranges of the traffic conditions (e.g., the load, the maximum multicast group size, the percentage of unicast traffic, and etc.) and a wide range of the number of data channels in the network. The proposed hybrid algorithm dynamically chooses to employ a multicast scheduling

10 algorithm which always tries to partition multicast transmissions (namely, the greedy algorithm) or a multicast scheduling algorithm which does not partition multicast transmissions (namely, the no-partition algorithm) depending on the average utilization factors of the data channels and the receivers. Extensive simulations were performed to compare the performance of the proposed hybrid algorithm with the greedy and no-partition algorithms. Our simulation results showed that the proposed hybrid algorithm produces the lowest mean packet delay for wide ranges of the load, the maximum multicast group size, the percentage of unicast traffic, and the number of data channels in the network. Transmissions in a Single-Hop Optical WDM Network, Proceedings of IEEE ICC, [21] Z. Ortiz, G. N. Rouskas, and H. G. Perros, Scheduling of Multicast Traffic in Tunable-Receiver WDM Networks with Non-Negligible Tuning Latencies, ACM SIGCOMM, [22] A. Smiljanic, An Efficient Channel Access Protocol for an Optical Star Network, Proceedings of IEEE ICC, REFERENCES [1] M. N. Ransom and D. R. Spears, Applications of Public Gigabit Networks, IEEE Network, Mar [2] B. E. Carpenter, L. H. Landweber, and R. Tirler, Where Are We with Gigabits? IEEE Network, Guest Editorial, Mar [3] C. A. Brackett, Dense Wavelength Division Multiplexing Networks: Principles and Applications, IEEE J. Select Areas Commun., vol. 8, pp , Aug [4] P. R. Trischitta and W. C. Marra, Applying WDM Technology to Undersea Cable Networks, IEEE Communication Magazine, pp , Feb [5] E. Modiano, Unscheduled Multicasts in WDM Broadcast-and-Select Networks, Proceedings of IEEE INFOCOM, [6] M. S. Borella, and B. Mukherjee, Efficient Scheduling of Nonuniform Packet Traffic in a WDM/TDM Local Lightwave Network with Arbitrary Transceiver Tuning Latencies, Proceedings of IEEE INFOCOM, [7] G. N. Rouskas and M. H. Ammar, Multidestination Communication Over Tunable-Receiver Single-Hop WDM Networks, IEEE J. Select Areas Commun., vol., no. 3, pp , April [8] W. Y. Tseng and S. Y. Kuo, A Combinational Media Access Protocol For Multicast Traffic In Single-Hop WDM LANs, Proceedings of IEEE GLOBECOM, [9] G. N. Rouskas and M. H. Ammar, Analysis and Optimization of Transmission Schedules for Single-Hop WDM networks, IEEE INFO- COM 93, pp [10] K. M. Sivalingam and P. W. Dowd, A Lightweight MAC Control Protocol for WDM-based Distributed Shared Memory System, Proceedings of IEEE INFOCOM, [11] J. S. Choi and H. H. Lee, A Dynamic Wavelength Allocation Scheme with Status Information for Fixed- and Variable-length Messages, Proceedings of IEEE GLOBECOM, [12] D. Guo, Y. Temini, and Z. Zhang, Scalable High-Speed Protocols for WDM Optical-Star Networks, Proceedings of IEEE INFOCOM, [13] F. Jia and B. Mukherjee, Variable-Length Message Scheduling Algorithms for a WDM Based Local Lightwave Network, Proceedings of IEEE INFOCOM, 1994 [14] D. A. Levine, I. F. Akyildiz, A Reservation and Collision-Free Media Access Protocol for Optical Star Local Area Network, Proceedings of IEEE GLOBECOM, [] M. S. Borella and B. Mukherjee, A Reservation-Based Multicasting Protocol for WDM Local Lightwave Networks, Proceedings of IEEE ICC, 1995, pp [16] R. Chipalkatti, and Z. Zhang, A Hybrid Dynamic Reservation Protocol for an Optical Star Setwork, Proceedings of IEEE GLOBECOM, [17] H. B. Jeon and C. K. Un, Contention-Based Reservation Protocols in Multiwavelength Optical Networks with a Passive Star Topology, IEEE Trans. On Commun., Vol. 43, pp , Nov [18] F. Jia, B. Mukherjee, and J. Iness, Scheduling Variable-Length Message in a Single-Hop Multichannel Local Lightwave Network, IEEE/ACM Transactions on Networking, Vol.3, No. 4, pp , Aug [19] B. Hamidzadeh, M. Maode, and M. Hamdi, Message Sequencing Techniques for On-Line Scheduling in WDM Networks, Proceedings of IEEE GLOBECOM, [] J. P. Jue and B. Mukherjee, The Advantages of Partitioning Multicast

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