Topology-transparent time division multiple access ... - Semantic Scholar

970

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 4, JULY 2003

Topology-Transparent Time Division Multiple Access Broadcast Scheduling in Multihop Packet Radio Networks Zhijun Cai, Student Member, IEEE, Mi Lu, Senior Member, IEEE, and Costas N. Georghiades, Fellow, IEEE

Abstract—Many topology-dependent transmission scheduling algorithms have been proposed to minimize the time-division multiple-access frame length in multihop packet radio networks (MPRNs), in which changes of the topology inevitably require recomputation of the schedules. The need for constant adaptation of schedules-to-mobile topology entails significant problems, especially in the highly dynamic mobile environments. Hence, topology-transparent scheduling algorithms have been proposed, which utilize Galois field theory and Latin squares theory. In this paper, we discuss the topology-transparent broadcast scheduling design for MPRNs. For single-channel networks, we propose the modified Galois field design (MGD) and the Latin square design (LSD) for topology-transparent broadcast scheduling. The MGD obtains much smaller minimum frame length (MFL) than the existing scheme while the LSD can even achieve possible performance gain when compared with the MGD, under certain conditions. Moreover, the inner relationship between scheduling designs based on different theories is revealed and proved, which provides valuable insight. For the topology-transparent broadcast scheduling in multichannel networks, in which little research has been done, the proposed multichannel Galois field design (MCGD) can reduce the MFL approximately times, as compared with the MGD when channels are available. Numerical results show that the proposed algorithms outperform existing algorithms in achieving a smaller MFL. Index Terms—Latin square design (LSD), modified Galois field design (MGD), multichannel Galois field design (MCGD), packet radio networks, time division multiple access (TDMA).

I. INTRODUCTION

A

PACKET radio network consists of a number of geographically dispersed mobile radio nodes that can wirelessly communicate between each other. Each node has limited transmission range; thus, the packets may be relayed over multiple nodes before the destination node is reached. Scheduling of transmissions so that neighboring nodes can successfully exchange information even with the presence of conflict is a fundamental requirement in packet radio networks. It is also one of the central issues in implementing time-division multiple-access (TDMA) protocols and is usable for code-division multiple-access (CDMA) networks, especially when nodes cannot concurrently receive transmissions from all neighbors [1]. Here, we discuss only TDMA networks. In a scheduled access method of Manuscript received February 21, 2002; revised June 19, 2002. This work was supported by the Texas Advanced Technology Program under Grant 0005120327-2001. The authors are with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843 USA. Digital Object Identifier 10.1109/TVT.2002.807634

the TDMA network, time is divided into equal-length frames, which are composed of a number of fixed-length slots. The slot is designed to accommodate the transmission of one fixed-size packet. When nodes communicate, two types of conflicts may occur [1]. The primary conflict refers to the situation when a node transmitting in a particular slot cannot receive any packet in the same slot and vice versa. The secondary conflict refers to the situation when a node cannot receive more than one packet in one slot. In both cases, all packets are rendered useless. In the TDMA scheduling, a node is able to transmit data during each frame to any neighbor in at least one slot without any conflict. In conventional TDMA networks, every node has one preassigned and unique slot during each frame-to-transmit data. For a fully connected network, which means every node is a neighbor of all other nodes, the scheme works well. However, in multihop networks with a large number of nodes, the method will be inefficient while spatial reuse will greatly improve system performance. Much research has been done on how to design transmission schedules to obtain good performance [1]–[12]. Previous studies concentrated on designing fair conflict-free algorithms that maximize the throughput or minimize the frame length based on topology information [5]–[8], [10]–[12]. The problems of determining schedules with maximum throughput and obtaining the minimum frame length (MFL) have been shown to be nameplate (NP) complete [4], [9]. Although good performance can be obtained for these topology-dependent algorithms, the obvious disadvantage is that they are topology dependent, which means that when the topology changes, the previous schedules are expired and new schedules should be generated. The topology-dependent character will generate significant overhead, especially in highly dynamic environments. Therefore, efficiency and robustness are vulnerable in mobile environments. To address the problem, topology-transparent algorithms have been proposed [1]–[3]. By allowing contentions, a topology-independent algorithm was proposed in [1] to guarantee that each node has at least one successful transmission slot to any neighbor in each frame. In [2], further improvements through a similar method were achieved and a topology-transparent algorithm was proposed to maximize the minimum guaranteed throughput. The above algorithms are focused on single-channel networks. If multiple channels are provided, they cannot be applied efficiently and multichannel scheduling designs should be discussed. The existing topology-transparent scheduling algorithms are based on the Galois field theory and the Latin square theory.

0018-9545/03$17.00 © 2003 IEEE

Authorized licensed use limited to: Texas A M University. Downloaded on April 30,2010 at 00:14:59 UTC from IEEE Xplore. Restrictions apply.

CAI et al.: TOPOLOGY-TRANSPARENT TDMA BROADCAST SCHEDULING IN MULTIHOP PACKET RADIO NETWORKS

In this paper, we discuss the topology-transparent broadcast scheduling design for multihop packet radio networks (MPRNs). For single-channel networks, we propose the modified Galois field design (MGD) and the Latin square design (LSD) for topology-transparent broadcast scheduling. The MGD obtains much smaller minimum frame length (MFL) than the existing scheme, while the LSD can even achieve possible performance gain as compared with the MGD, under certain conditions. Moreover, the inner relationship between scheduling designs based on different theories is revealed and proved, which provides valuable insight. For topology-transparent broadcast scheduling in multichannel networks, in which little research has been done, the proposed multichannel Galois field design (MCGD) can reduce the MFL approximately times as compared with the MGD when channels are available. Detailed numerical results show that the proposed algorithms outperform existing algorithms in the sense that they can achieve a much smaller MFL. The rest of the paper is organized as follows. Relevant background material is given in Section II. In Section III, the MGD and LSD for single-channel networks are proposed. Moreover, the inner relationship between scheduling designs based on Galois field theory and Latin square theory is revealed and proved. The MCGD for multichannel networks is proposed in Section IV. Section V presents numerical results and performance analysis and Section VI concludes.

II. BACKGROUND A. Network Model homogeneous nodes, an Given an MPRN consisting of identical transmission radius is assumed for all nodes and each node is given a unique identification number (ID). Every node has its neighbors, which are inside its transmission range. Suppose the maximum number of neighbors for any node in the network is bounded by (also known as the maximum degree of the graph abstracted from the network; such degree-bounded topology has been discussed in [13], [14]). The transmission channel is assumed to be error free; hence, reception failure is only due to the two types of conflicts discussed in Section I. For a node (say ) transmitting a packet to one of its neighboring nodes (say ), successful transmission means that receives the packet without any conflict. For a single-channel MPRN, every node is assumed to be equipped with a single transceiver to communicate with other nodes. Moreover, time is divided into equal-length frames, each composed of a number of fixed-length slots. For a multichannel network (suppose the number of channels is ), every node is assumed to be equipped receivers, but only one transmitter. Therefore, every with node can receive the packets on all channels at the same time, but can only transmit one packet at a time. A node cannot transmit the data while receiving the data on the same channel. Furthermore, the time span of each channel is divided into equal-length frames and synchronized with other channels. Each frame is composed of a number of fixed-length slots. How to schedule the transmissions for all the nodes so that a given node is able to transmit data to an arbitrary neighbor in at

971

least one slot without any conflict during each frame is the key issue for the TDMA scheduling. Due to the mobile topology of the MPRNs, topology-dependent scheduling will entail significant problems. Designing topology-transparent scheduling schemes for the MPRNs is the goal of the paper. The scheduling should be made independent of the network topology. Obviously, the simplest way is to assign each node a unique slot during the frame, which will generate a large frame length. However, utilizing the spatial reuse technology and mathematical theories, significant improvement can be achieved. Two types of theories have been utilized in the topology-transparent scheduling design [1]–[3]. One is the Galois-field and the other is the Latin squares [5]. B. Galois Field Here, we briefly review some fundamental concepts of the Galois field [15]. Definition 1: A set on which an operation is defined is called a group if the following properties are satisfied: 1) asso, ; 2) identity ciative contains an element such that , ; for any , there exists another element 3) inverse such that ; and 4) closure for any two ele, is also in . In addition, if , ments then is a communicative group. Definition 2: Let be a set of elements on which two operations called addition and multiplication are defined. is a field if the following conditions are satisfied: 1) forms a communicative group under and the additive identity element is forms a communicative group under labeled “0” and 2) and the identity element is labeled “1”; 3) , . . This finite field is known as Let Galois field and exists only if is a prime or prime power, de, then the elements of noted as GF( ). For example, if GF(5) can be represented by 0, 1, 2, 3, 4 . The operations on GF(5) are addition and multiplication modulo 5. If is a prime power, the element representation should utilize polynomials, which is somewhat more involved [1]. However, the essential idea of our algorithms is independent of the particular representations of the Galois field. distinct polynomials of degree over GF( ). There exist This is because a degree polynomial contains coefficients ( ) and each of them can have one of the values over GF( ). C. Latin Squares All the definitions and theorems (including the proof) of Latin squares can be found in [3], [5]. square Definition 3: A Latin square of order is an array composed of symbols from 1 to such that each symbol appears once in each row and once in each column. Latin squares Definition 4: Two distinct and , where and , are ordered pairs ( , ) are all said to be orthogonal if the different.


972


Fig. 1. The broadcast traffic.

An example of orthogonal Latin squares is illustrated with and given by

More generally, distinct Latin squares are said to form an orthogonal set if every pair of them is orthogonal. Definition 5: If an orthogonal set of Latin squares of order has a size of , i.e., the number of Latin squares in the , it is called complete. orthogonal family is Theorem 1: If there is an orthogonal set of Latin squares . of order , then and , where is a prime number Theorem 2: If and is a positive integer, then there is a complete orthogonal set of Latin squares of order . A construction method of the complete set of Latin squares as the can be found in [5, Theorem 5.2.4]. We define largest size of the orthogonal set for the Latin square with order . For most nonprime powers of , the exact value of re. The topologymains unknown, but should be less than transparent scheduling design in [3] is based on Latin squares. D. Broadcast Traffic In this context, the broadcast traffic is identified as the mediaaccess-control (MAC) layer traffic, i.e., broadcast means that one node attempts to transfer the same data packet to all of its neighboring nodes simultaneously. An example is illustrated in Fig. 1, in which the traffic from is the broadcast traffic (to all of its neighboring nodes). The MAC layer broadcast is one of

the fundamental issues for MPRNs. Suppose a node attempts to broadcast one packet to all its neighbors with the topologytransparent TDMA scheduling utilizing the broadcast address. When multiple neighboring nodes receive a packet successfully in the same slot, which may occur frequently, the replied acknowledgment packet(s) may suffer the secondary conflict, so cannot receive the acknowledgment packet successfully. Hence, the MAC layer acknowledgment cannot be applied to the broadcast traffic. In this case, the way to guarantee that every neighbor of can receive the packet is to transmit the same packet in all of its transmission slots during one frame. (Recall that, in the TDMA scheduling, a node can transmit data to any neighbor without any conflict in at least one slot during one frame.) can broadcast one packet per frame to its neighbors successfully. Therefore, the maximum broadcast throughput is obtained by determining the MFL and, thus, MFL scheduling should be our design goal. III. PROPOSED SINGLE-CHANNEL TOPOLOGY-TRANSPARENT TDMA BROADCAST SCHEDULING For single-channel networks, we propose the MGD and the LSD for topology-transparent broadcast scheduling, which are based on the Galois field theory and the Latin square theory, respectively. Then the inner relationship between the scheduling designs, based on different theories, is revealed and proved. A. Proposed GF-Based Topology-Transparent Broadcast Scheduling Design In the proposed GF-based scheduling design, for a chosen Galois field GF( ) ( must be a prime or prime power), the



number of polynomials over GF( ) with a maximum degree is ( is a nonnegative integer). If we assign each node a distinct polynomial (1) is the total number of nodes. Divide time is required where into equal length frames. Each frame is further divided into subframes and each subframe is composed of fixed-length slots. Each slots, which implies that the frame length ( ) is node will transmit during one slot, in each subframe, which is termed its transmission slot. Thus, each node has transmission slots during one frame. For a given node, its transmission slots are determined by its assigned polynomial through the following method. Suppose the polynomial assigned to node is . Then the transmission slot for in the th subframe of a frame is the th slot. Based on polyis satisfied, any two nodes nomial theory, if condition have at most conflicts during one frame. (For a detailed proof, please refer to [1] and [2].) Since each node can have at most neighbors, to guarantee that every node can transmit data to any neighbor in at least one slot during one frame, the following equation should be satisfied:

concern. Considering decreases with , for all

973

increases with

and (7)

Thus (8) , the corresponding frame length . Thus, the will be obtained and for all . We should prove

For a certain when

,

by (7) (9) According to the binomial expansion

(2) (10) We are interested in determining the optimal and . which can is the frame length minimize the frame length. Suppose for a given , which is the shortest over various s. We now . Based on determine the optimal , which can minimize (1) and (2), we present the following theorem. is the unique positive root of Theorem 1: Suppose equation

and (11) where Denote

as

is the number of combination. and as

(3) Then or

(12)

(4)

Proof: To prove Theorem 1, we first prove that for all , . Then we prove that for all . Based on (1), (2,) and

,

Since ( is a positive integer)

(5) increases with Consider that , creases with . Then, for all We have

and that

de.

(13) We have (14)

(6) By (12), , the frame length should be . Since For a certain , the frame length . is . and will increase obtained when , . with . Therefore, for all , . Next, we will prove that for all , since should be a nonnegative integer, the proof If . The case when is our is obvious

. Then by (10) and (11), we have (15)

which implies


(16)

974


rithm outperforms the GRAND algorithm is as follows. In the GRAND algorithm, a frame structure is utilized (MFL is frame struconly dependent on ), while in our MGD, a ture is applied (MFL is dependent on and , where can be different from ), which gives the scheduling design more flexand that ibility. Suppose the MFL generated by the MGD is for the same and . We have generated by the GRAND is

TABLE I COMPARISON OF THE MFL

From (9) and (16) (17) , i.e., . Hence, all , if and , . Considering the case that , we , . acknowledge that for all , if is nonincreasing with when . Thus, for Hence, , . all Recall that is a prime number or a prime power. Let be and the minimum prime number or prime power the minimum prime number or prime power . , . Based on Then the above theorem and discussions, we obtain our algorithm, the MGD, to achieve the minimum frame length and to assign the transmission slots to every node as follows. as well as the optimal . 1) Obtain and determine , then and 2) Determine and from . If . If , then and . polynomials are available and the Hence, we know that chosen Galois field is GF . 3) Distribute the polynomials to nodes; each node should calculate its transmission slots during the frame. In the MGD, each node has at least one slot to transmit data to any neighbor during each frame. Compared with the existing algorithm, the GRAND algorithm, [1] which chose the MFL to to achieve the same performance, be our proposed algorithm will always obtain much smaller MFL. means the Table I illustrates some examples. (In Table I, case when every node obtains a unique slot during a frame.) A detailed comparison will be shown in Section V (“M” is short for the MGD; “G” is short for the GRAND). is greater than 0, . ThereSince . When fore, the MFL generated by the GRAND , the MFL will exceed , which is even worse for the traditional TDMA scheduling (i.e., each node has a unique slot , in which MFL during a frame). The extreme case is . Therefore, we acknowledge that the GRAND algorithm should be utilized only when is not large. Note that here we discuss the broadcast traffic only and the MAC layer acknowledgment cannot be applied. The main concern for the system , throughput is the frame length. In the MGD, when in this case). the MFL will be (the MGD will choose , the MGD will choose the best and to When obtain the MFL, which is always much smaller than the MFL from the GRAND algorithm. The reason that the MGD algoTherefore,

which implies . Next, we provide some examples for further illustrations of different approaches. and . Hence, Example 1: In this example, . From MGD, we can obtain that , , and . While from GRAND, and the optimal the optimal . The frame structures that different approaches utilized are shown in Fig. 2. In the MGD, the utilized polynomial over , , and are . GF(23) is In the GRAND, the utilized polynomial over GF(17) is , , , and are . Assume to generate its transmission slots in the a node utilizes to generate its transmission MGD and a node utilizes ] as the th slot in the th subslots in the GRAND. Denote [ frame. Then, the transmission slots for are [0,5], [1,8], [2,11], [3,14], [4,17], [5,20], [6,0], [7,3] and the transmission slots for are [0,6], [1,15], [2,7], [3,16], [4,8], [5,0], [6,9], [7,1], [8,10], [9,2], [10,11], [11,3], [12,12], [13,4], [14,13], [15,5], [16,14]. The MFL of the MGD is 184, which is much less than the MFL of the GRAND (289). and . Example 2: In this example, . From MGD, we can obtain that , Hence, , and the optimal , while from GRAND, and the optimal . In the MGD, the utilized , , and are polynomial over GF(47) is . In the GRAND, the utilized polynomial , , , and are over GF(37) is . Assume a node utilizes to generate its transmission slots in the MGD and a node utilizes to generate its transmission slots in the GRAND. are [0,17], [1,19], [2,21], Then the transmission slots for [3,23], [4,25], [5,27], [6,29], [7,31], [8,33], [9,35], [10,37], [11,39], [12,41], [13,43], [14,45], [15,0], [16,2], [17,4], and the transmission slots for are [0,8], [1,13], [2,20], [3,29], [4,3], [5,16], [6,31], [7,11], [8,30], [9,14], [10,0], [11,25], [12,15], [13,7], [14,1], [15,34], [16,32], [17,32], [18,34], [19,1], [20,7], [21,15], [22,25], [23,0], [24,14], [25,30], [26,11], [27,31], [28,16], [29,3], [30,29], [31,20], [32,13], [33,8], [34,5], [35,4], [36,5]. The MFL of the MGD is 846, which is much less than the MFL of the GRAND (1369). B. Proposed Latin Squares Topology-Transparent Broadcast Scheduling Design Latin squares have nice properties that can be applied to many applications. In [3], they have been utilized to design the multichannel topology-transparent point-to-point traffic scheduling. In this part, we will propose a Latin squares topology-transparent broadcast scheduling design, termed LSD.



975

Fig. 2. Frame structures for different approaches.

Suppose the maximum size of the orthogonal set of the Latin . Moreover, suppose the time axis squares with order is is composed of equal-length frames and each frame is divided into subframes. Each subframe is further divided into slots. The frame length is (18) The scheduling construction is from these orthogonal sets. BaLatin square from sically, radio units share one common these orthogonal sets and the time-slot-assignment pattern of each of these radio units is represented by one of the distinct symbol patterns in the first rows of a Latin square. The fol, lowing is the assignment scheme. For example, suppose , and node is assigned symbol “2” in the Latin square (refer to Section II-C). Then the transmission slots of are the following: slot 2 in subframe 1, slot 1 in subframe 2, and slot 4 in subframe 3. The different rows in the Latin square represent the subframes ( rows represent subframes) and the symbol pattern represents the transmission slot pattern. From the construction scheme, it is clear that if two nodes obtain the symbol from the same Latin square, their transmission slots will never conflict at any time. If two nodes obtain the symbol from a different Latin square, their transmission slots will have at most one conflict during each frame [3]. Since every node can have at most neighbors, if (19) we can guarantee that each node will have at least one successful transmission slot for any neighbor during each frame. Note that any two nodes cannot share the same symbol from the same Latin square. Otherwise, these two nodes will have conflicts all Latin the time. Further, radio units share one common

square and the maximum size of the orthogonal set of Latin squares with order is . Therefore, we have (20) . By (20), From Section II-C, we know that . By (18) and (19), the MFL for the and LSD is achieved when . Compared with the MGD, the MFL of the LSD is similar to that of the MGD when is 1. However, since is not restricted to be a prime or prime power in the LSD, in some cases the LSD may generate a somewhat , smaller MFL than the MGD. For example, suppose . For the MGD, . Then we can obtain that , , and the optimal . Each frame is composed of 16 subframes and each subframe is composed of 29 slots. The and the utilized MGD will produce the frame length , and are polynomial over GF(29) is . For the LSD, and . Each frame is composed of 16 subframes and each subframe is composed of 25 slots. The frame length is 400, which is less than that of the MGD. is fixed, The disadvantage of the LSD is as follows. If is . For the MGD, since is for the LSD, [1], is . Thus, the MFL of the LSD increases much faster with than that of the MGD. The reason is that the Latin square with order can, at nodes. If is large, in order to most, be utilized by guarantee that no two nodes have the same symbol in the same Latin square, we should obtain more orthogonal Latin squares, which will significantly increase the order of the Latin square as well as the MFL. The small size of the orthogonal Latin square set with order is the main disadvantage of the LSD. and . For the MGD, For example, suppose . Then we can obtain that , , and the


976

Fig. 3.


Group-formation process.

optimal . Each frame is composed of 13 subframes and each subframe is composed of 17 slots. The MGD will produce and the utilized polynomial over the frame length , , and are . GF(17) is and . Each frame is composed of For the LSD, 7 subframes and each subframe is composed of 48 slots. The frame length is 336, which is larger than that of the MGD. For and the frame length is 361. the GRAND, C. Relationship Between Designs Based on Different Theories Through the analysis in Section III-B, it seems that a certain relationship exists between the designs based on Latin squares and those based on GF. The following theorem illustrates the relationship between the GF-based scheduling design and the Latin square scheduling design. Theorem 2: If we restrict to be a prime or a prime power, the transmission maps produced by all the polynomials with degree 1 based on GF( ) are the same as the transmission maps produced by one of the complete orthogonal Latin square sets with order . Proof: First, we consider the case when, during one frame, ). the number of slots is equal to the number of subframes ( In the GF design scheme, the transmission map for a node is generated by its assigned polynomial and one polynomial can generate one transmission map. If two nodes have conflicts in their transmission slots, we say that their transmission maps , the polynomial is represented by have overlaps. For ( ), where ( or groups. In each group, 0). We split the polynomials into but different . Each group the polynomials have the same has polynomials. The group formation is illustrated in Fig. 3. Inside one group, each polynomial can generate one transmission map. Hence, in total transmission maps can be obtained , its transmission map will in one group. For a polynomial th slot as its determine the th subframe, the

transmission slots ( from 0 to ). Since, in one group is identical for all polynomials, two transmission maps generated by any two polynomials in the same group have no overlap. Next, we utilize these transmission maps in one group to form a matrix in the following way. In each group, denote the index of a polynomial as . Hence, the indices for all polynomials in and each polynomial in the one group will be from 0 to same group has a distinct index. For a transmission map generated by the polynomial with index , we set the ( , ) element of the matrix to be if the th slot of the th subframe is one of the transmission slot of the transmission map. We illustrate the maand trix formation process by a simple example. Assume that there are 4 groups with being 1, 2, 3, and 4, respectively ). The matrix formation process for these 4 ( groups is shown in Fig. 4. Since each transmission map has transmission slots and no overlap exists between any two transmission maps in the same group, all the entries of the matrix will be filled. Obviously, such a matrix [denoted as GF( ) matrix] determines the transmission maps generated by the polymatrices [denoted nomials in one group. There are in total as GF( ) matrix set], which determine all the transmission maps . Next, we prove that generated by the polynomials with a GF( ) matrix set is a complete orthogonal Latin square set. If we can prove 1) GF( ) matrix is a Latin square; 2) any two GF( ) matrices are orthogonal, then by Definition 5 in Section II-C, GF( ) matrix set forms a complete orthogonal Latin square set. Proof of 1: For a given GF( ) matrix, suppose it is con. structed by the group that have the polynomials such as For a transmission map generated by the polynomial with index , the transmission slot will be one and only one per subframe. Hence, the index can appear once and only once in each row. Next, we show that one index can appear once and only once in each column. Suppose one index appears twice in the same



Fig. 4.

977

Matrix-formation process.

column, say and should be satisfied:

. Hence, the following equations

From (24), we have (26)

(21)

and from (25), we have

and

(27)

(22) By (26) and (27), we have From (21) and (22), we have

(28) (23)

, Since and are different numbers from 0 to cannot be and is from 1 to . However, over the GF( ), the multiplication of any two nonzero numbers cannot be zero [15]. Hence, (23) is not true, which implies that one index cannot appear twice in the same column. Since all the entries are filled and only indices are available, one index can appear once and only once in each column. By Definition 3 in Section II-C, GF( ) matrix is a Latin square. As an example, in Fig. 4, each matrix generated is a Latin square. Proof of 2: For any two GF( ) matrices, suppose one is constructed by the group that has polynomials such as and the other is constructed by the group that has polynomials . Suppose one index pair appears twice in such as row , column and row , column . Then we have (24) and (25)

, Since and are different numbers from 0 to cannot be , as does . The multiplication of two nonzero elements on GF( ) cannot be zero [15]. Hence, (28) is not true, which implies that every index pair from these two matrices are different. By Definition 4 in Section II-C, any two GF( ) matrices are orthogonal. As an example, in Fig. 4, any two matrices are orthogonal. Therefore, GF( ) matrix set forms a complete orthogonal set. Further, recall that in the Latin square design, every nodes share one matrix and the transmission map of each of these nodes is represented by one of the distinct symbol patterns in the Latin square. If we utilize GF( ) matrix set as the orthogonal Latin square set for the Latin square design, the transmission maps generated by the Latin square design should be the same over GF( ). as that generated by the polynomials with , the transmission maps generated by When based on GF( ) are the same as the the GF design with transmission maps generated by the first rows of the GF( ) matrix set. Theorem 2 means that if is restricted to be a prime or prime power, the design based on Latin squares is only a special case of


978


the design based on GF (when ). Theorem 2 not only illustrates the relationship between the topology-transparent scheduling designs based on different theories, but also provides a simple method to generate the maximum orthogonal set of Latin squares with order , which is different from [5, Theorem 5.2.4]. Due to space limitation, we omit the details here. In [3], a scheduling design based on Latin squares for the unicast traffic was proposed. Based on Theorem 2, we realize that a better performance may be obtained by utilizing the GF-based design scheme, which will be done in future research.

IV. PROPOSED MULTICHANNEL TOPOLOGY-TRANSPARENT TDMA BROADCAST SCHEDULING If multiple channels are available, topology-transparent TDMA broadcast scheduling design remains unknown. In [3], a topology-transparent scheduling design was proposed for the unicast traffic in multichannel networks. However, little research has been done on the topology-transparent scheduling design for the broadcast traffic in multichannel networks to obtain minimum frame length. Suppose the number of channels and every node is equipped with receivers but only one is transmitter. Every node can receive the packets on all channels at the same time, but can only transmit one packet at a time. A node cannot transmit data while receiving data on the same channel and vice versa. The time axis of each channel is divided into frames and synchronized with other channels. Moreover, each frame is composed of a number of fixed-length slots. Although multichannel networks are the natural extension of single-channel networks, the topology-transparent scheduling design is different. In multichannel networks, we should utilize the given channels as much as possible, which is not a concern in single-channel networks. Moreover, every node cannot transmit on more than one channel at a time, since every node is equipped with only one transmitter. A. Proposed MCGD In the MCGD, for a chosen Galois field GF( ), the number is of polynomials over GF( ) with a maximum degree ( is a non-negative integer). Our approach assigns each node a distinct polynomial, which requires the following inequality to hold: (29) The time span is divided into equal-length frames. Each frame subframes with indices from 0 to is composed of and each subframe is composed of slots with . Suppose the polynomial indices from 0 to . Then the transmission map of assigned to node is is generated as follows. During each frame, will transmit in slot of subframe on channel , from 0 to . Thus, every node will transmit in slots during each frame and the frame length . Based on the above scheme, we have the following is results.

Theorem 3: According to our transmission map-generation scheme, any two nodes have at most conflicts during one frame on all channels. Moreover, every node will not transmit on more than one channels in one slot, which means that the scheme is applicable under the restriction of one transmitter per node. , every frame is composed of one Proof: When , the node will subframe. For a node with polynomial , in slot , from 0 to . transmit on channel Consider the difference of two corresponding polynomials and . Suppose the two nodes have assigned to node conflicts during one frame, which means that more than the number of roots of the difference is more than . On the , the other hand, since the degree of all the polynomials is degree of the difference between any two polynomials should . Therefore, the number of the roots of the difference of be any two polynomials cannot exceed . Thus, any two nodes have at most conflicts during one frame on all the channels. Moreover, during each slot every node transmits once and only once. Therefore, every node will not transmit on more than one channels during each slot. , each frame is composed of subframes When and each subframe is composed of indexed from 0 to slots indexed from 0 to . For a given node , we matrix to represent the transmission slots first establish the . Then of a node. Suppose the polynomial assigned to is of the matrix will the entry of be set to 1, while other entries of the matrix will be set to 0. Let the paired entry be referred to as two entries in different matrices with the same indices. For two matrices generated by any two nodes, at most paired entries will have the same value as 1 (the number of roots of the difference between any two polynomials can not exceed ). (node ’s polynomial), For such a matrix generated by smaller matrices with index we divide it into utilizing the division method illustrated from 0 to th matrix will in Fig. 5. The unfilled part of the matrices will be utilized be set to 0. After that, these subframes by as its transmission maps for all the during one frame, i.e., the th matrix will be applied for the th subframe. If the th row, th column of the th matrix is 1, should transmit in subframe , slot on channel , which establishes one-to-one mapping relation between the matrices frame. Since, for two matrices generated by and the any two nodes, at most paired entries will have the same value as 1, any two nodes have at most conflicts during one frame ) during the on all channels. Moreover, given ( from 0 to th slot of all the subframes, every node should transmit once and only once. Therefore, every node will not transmit on more than one channel during each slot. We are interested in determining the optimal and the optimal , which can minimize the frame length. The frame length in the MCGD is . Since , with the increase of , the minimum to satisfy the condition can be decreased, . For any which implies a potential decrease in ( is a positive integer), will remain the same. Since the increase in implies the potential as much as possible (but decrease of , should be close to



979

Fig. 5. Division of the matrix.

Proof: Suppose is the frame length for a given , which is the shortest over various s. From equation set (30) and

TABLE II MFL FOR THE MCGD

(31) Consider . For all

increases with and

decreases with (32)

Then we obtain less than ). The following relations should be satisfied when determining the MFL:

(33) For a certain

, since

, the frame length

(30) (34) where is the number of subframes in one frame. From equation set (30), we have the following results. Theorem 4: Suppose is the unique positive root of

Then the optimal

corresponding to the MFL

is obtained when . and will increase with . Thereby, for all , , which implies of the MCGD should be produced . The MFL can be determined as the minimum by the . from all the frame length produced by When

.


(35)

980


Fig. 6. Frame structures for the MCGD.

Next, we should determine

when

the MFL of the MCGD is

. Since (36)

considering with , for all

increases with , we have

and

decreases (37)

. For a certain Then we obtain , the frame length

, since

(38) for . Hence, Further, consider that should be a prime or a prime power. Suppose the minimum prime (or prime power) greater than is ( from 0 to ). Then the minimum frame is determined by the following: length (39) Based on the above results, the MCGD can be summarized as follows. and determine and 1) Obtain . , 2) Determine and the optimal (the or where is some integer from optimal should be ). If the optimal , choose the Galois field 0 to . Otherwise, choose GF( ) and as GF( ) and . 3) Distribute the polynomials to nodes; each node calculates its transmission slots during one frame. and , will be the same Under the same for both the MGD and the MCGD. The MFL of the , , while MGD is

, , which is , . , the MFL of the MCGD for the -channel Since times smaller than the MFL of the MPRN is approximately MGD for the same MPRN with a single channel. The minimum value of the MFL of the MCGD is 1 and can be achieved when . receivers and transmitIf each node is equipped with ters, it is easy to extend the single-channel MGD algorithm larger broadcast traffic to the multichannel system with times smaller MFL). However, throughput (equivalent to under the condition of one transmitter per node, the MGD cannot be applied to multichannel networks. In this case, the times smaller MFL MCGD can still achieve approximately than the MGD, which is considered a good performance. Moreover, it is the first topology-transparent scheduling scheme for the broadcast traffic in the multichannel MPRNs. Table II illustrates the MFL of the MCGD, which is compared with the MFL of the MGD. Next, we provide an example for further illustration. Assume the number of MSs is 800, the maximum number of neighbors for each node is 7, and the available channels are 5. Hence, . By the MCGD, we can determine that , , , and the optimal . Over each channel, each frame is composed of six subframes, each of which is composed . The frame strucof eight slots. Hence, the MFL is ture is illustrated in Fig. 6. The utilized polynomial over GF(29) , , and are . Assume is to generate its transmission slots in a node utilizes ] as the th slot in the th subthe MCGD and denote [ frame over the th channel. Then the transmission slots for are [0,1,3], [1,2,4], [2,4,0], [3,5,1], [4,0,3], [5,1,4], [6,3,0], [7,4,1]. B. Proposed Multichannel (MC) LSD From Theorem 2, we know that the LSD scheme is a special case of the GF design scheme in which is restricted to



Fig. 7.

MFL comparison 1.

be a prime or prime power. Through a similar method as described in Section III-B, the Latin squares can be utilized in the multichannel MPRNs. The frame structure of the MCLSD is the same as the MCGD. nodes will share one common Latin square from the orthogonal set and the transmission pattern of each of these nodes is represented by one of the distinct symbol patterns in the first rows of the Latin square. For a given node , suppose its assigned symbol appears in entry of the square. Then, during each frame, will transmit on th channel during the th subframe, the th slot. Since each symbol will appear times in the first rows of the Latin square, each node will transmit in slots during . Through a simeach frame and the frame length is ilar derivation process, we can obtain the MFL of the MCLSD, . Further, in the MCLSD, there is no which is prime (or prime-power) limitation, by which the MCLSD may achieve possible performance gain as compared with the MCGD under certain conditions. Detailed description and the derivation process of the MCLSD is omitted due to space limitations. V. PERFORMANCE ANLAYSIS AND NUMERICAL RESULTS A. General Performance The proposed TDMA broadcast scheduling designs are topology independent. By being immune to the topology changes, no protocol overhead will be involved due to the dynamic topology, even if nodes move very quickly. Moreover, during each frame, every node is guaranteed to successfully broadcast one packet to all of its neighbors, regardless of the topology change speed and traffic load. The fairness of the proposed scheduling schemes is obvious since, even if the traffic load is very high, no node can monopolize the channel. By determining the MFL, the maximum throughput of the . Since the LSD broadcast traffic can be obtained as scheme can be regarded as a specific case of the GF design scheme when is a prime or prime power and when is not a

981

Fig. 8. MFL comparison 2.

prime or prime power is still unknown, we mainly focus on the performance of the GF-based scheduling designs. B. Performance Analysis of the MGD In the MGD, and are computed during the system initialization. Then the polynomials will be assigned to all the nodes and each node will calculate its transmission slots within the frame. During each frame, a node can broadcast one packet to its neighbors. Hence, the maximum throughput of the MGD is determined by its MFL. Compared with the GRAND, the MFL of the MGD is much smaller, which significantly increases the maximum broadcast traffic throughput. 1) MFL Comparisons With the GRAND: We will compare the MFL generated by the GRAND and that by the MGD in the is fixed while increases following two cases. One is that and the other is that is fixed while increases. The comparisons of the MFL between the GRAND and MGD for and is given in Fig. 7. In each case, increases from 3 to 45. In Fig. 8, the comparisons of the MFL between the and . GRAND and the MGD are displayed for In each case, increases from 100 to 5000. It can be seen that the MFL generated by the MGD is always much smaller than the MFL generated by the GRAND, especially when is not small. When is fixed, with the increase of , the MFL generated by the GRAND increases much more quickly than that generated by the MGD. Furthermore, we can find that the GRAND is more sensitive to the change of and, obviously, the stability of the MGD is much better than the GRAND. In Fig. 7, it is also seen that the MFL of the GRAND drops sharply at some values of . The reason is as follows. and, From [1], we acknowledge that decreases when increases. Since hence, for a certain , with the increase the MFL of the GRAND decreases by 1 at some values of . For example, of , for , which makes the MFL drop sharply.


982

Fig. 9.


S

comparison between the MGD and the GRAND.

Fig. 10. Broadcast-traffic throughput performance.

During the periods that remains unchanged, the MFL inincreases. When is fixed, the MFL of the creases when GRAND increases with in a step-like manner, while the MFL of the MGD increases with more slowly and smoothly. Again, the MFL of the MGD is much smaller than that of the GRAND. 2) Broadcast Throughput Performance: Next, we will analyze the throughput performance of the MGD for the broadcast traffic. In the broadcast mode, since the acknowledgment packets may suffer conflicts, no MAC layer-acknowledgment scheme is assumed. If a node attempts to broadcast a packet to its neighbors, it should transmit the same packet in all of its transmission slots during one frame to guarantee that every neighbor receives at least one packet successfully. Define the system throughput as the average number of packets broad-

Fig. 11.

MFL of the MCGD.

Fig. 12.

S

of the MCGD.

casted successfully per slot in the network and as the maximum system throughput. Moreover, define the service load as the average number of packets arrived per slot in the network and assume that the service load is uniquely distributed to all . Then . the nodes. Suppose the MFL is , the system When the service load becomes less than throughput should be equal to the service load. Otherwise, the . Fig. 9 illustrates system throughput will be equal to between the MGD and the GRAND. Moreover, Fig. 10 illustrates the relationship between the system throughput and the service load. C. Performance Analysis of the MCGD are given, and can be In the MCGD, when , , and determined. Then every node will be assigned a polynomial to determine its transmission map.



Fig. 13.

983

Broadcast traffic throughput performance.

1) MFL of the MCGD: The MCGD is the same as the MGD . For -channel networks, the MFL of the MCGD when is about ( is not large) times smaller than that of the MGD. , the MFL of the MCGD will be 1. The MFL of If the MCGD is determined by , , and . Fig. 11 illustrates given and ( the relationship between the MFL and increases from 1 to 50). It can be seen that given and , is not smooth. Better, can the change of the MFL with and . For example, when be determined for certain , the MFL is the same for from 24 to 33. Thus, . the efficient choice should be 2) Throughput Performance: In the MCGD, . Suppose the service overload is uniquely distributed , to all the nodes. When the service overload is less than the system throughput should be equal to the service overload. . Fig. 12 Otherwise, the system throughput will equal to with , , and . It can illustrates the changes of be seen that, with the increase of , the difference between (throughput of MGD) the throughput of MCGD and becomes larger. The reason is as follows. In the MCGD, with the increase of , the MFL will be closer to its upper bound 1, will be closer to its upper bound, which means that the . After reaches its upper bound, a further increase will not improve . The closer the to its of increases with . Fig. 13 upper bound, the slower the illustrates the system throughput performance with the service overload for given , , and .

VI. CONCLUSION AND FUTURE WORK In this paper, we propose the MGD and the LSD for topologytransparent broadcast scheduling in single-channel networks. The MGD obtains much smaller MFL than the existing scheme, while the LSD can even achieve potential performance gain compared with the MGD. Moreover, the inner relationship between scheduling designs based on different theories is revealed and proved, which provides valuable insight to the scheduling design. For the topology-transparent broadcast scheduling in multichannel networks, in which little research has been done, the proposed MCGD can reduce the MFL approximately times as compared with the MGD when channels are available. Detailed numerical results show that the proposed algorithms outperform existing algorithms in the sense that they can generate much smaller MFL. It is worthwhile to make continuous efforts in this and related research areas. REFERENCES [1] I. Chlamtac and A. Farago, “Making transmission schedules immune to topology changes in multi-hop packet radio networks,” IEEE Trans. Networking, vol. 2, pp. 23–29, Feb. 1994. [2] J. H. Ju and V. O. K. Li, “An optimal topology-transparent scheduling method in multihop packet radio networks,” IEEE Trans. Networking, vol. 6, no. 3, pp. 298–306, June 1998. [3] , “TDMA scheduling design of multihop packet radio networks based on latin squares,” IEEE J. Select. Areas Commun., vol. 17, no. 8, pp. 1345–1352, Aug. 1999. [4] A. Ephremides and T. V. Truong, “Scheduling broadcasts in multihop radio networks,” IEEE Trans. Commun., vol. 38, no. 4, pp. 456–460, Apr. 1990.


984


[5] Latin Squares and Their Applications. New York: Academic, 1974. [6] B. Hajek and G. Sasaki, “Link scheduling in polynomial time,” IEEE Trans. Inform. Theory, vol. 34, pp. 910–917, Sept. 1988. [7] A. M. Chou and V. O. K. Li, “Slot allocation strategies for TDMA protocols in multihop packet radio networks,” in Proc. IEEE INFOCOM’92, Florence, Italy, May 1992, pp. 710–716. [8] K. Hung and T. Yum, “Fair and efficient transmission scheduling in multihop packet radio networks,” in Proc. IEEE GLOBECOM’92, Orlando, FL, Dec. 1992, pp. 6–10. [9] R. Ramaswami and K. K. Parhi, “Distributed scheduling of broadcasts in a radio network,” in Proc. IEEE INFOCOM’89, Ottawa, ON, Canada, Apr. 1989, pp. 497–504. [10] D. Newman and K. Tolly, “Wireless LANs: How far? How fast?,” Data Commun., no. 4 , pp. 77–87, Mar. 1995. [11] S. Kishore, P. Agrawal, K. M. Sivalingam, and J. C. Chen, “MAC layer scheduling strategies during handoff for wireless mobile multimedia networks,” in Proc. IEEE Int.Conf. Pers. Wireless Commun. (ICPWC), Mumbai, India, Dec. 1997, pp. 100–104. [12] D. J. Baker and A. Ephremides, “The architecture organization of a mobile radio network via a distributed algorithm,” IEEE Trans. Commun., vol. COM-29, pp. 1694–1701, Nov. 1981. [13] T. Hou and V. O. K. Li, “Transmission range control in multihop packet radio networks,” IEEE Trans. Commun., vol. COM-34, pp. 38–44, Jan. 1986. [14] L. Hu, “A novel topology control for multihop packet radio networks,” in Proc. IEEE INFOCOM’91, Bal Harbour, FL, Apr. 1991, pp. 1084–1093. [15] Introduction to the Theory of Error-Correcting Codes. New York: Wiley, 1989.

Zhijun Cai (S’99) received the B.S. and M.S. degrees in electrical engineering from University of Science and Technology of China, Anhui, in 1995 and 1998, respectively. He was with Nortel Networks as a System Engineer in 1999. He is currently a graduating Ph.D. Candidate, Department of Electrical Engineering, Texas A&M University, College Station. His advisor is Professor Mi Lu. His research interests include mobile computing, wireless communications, wireless LAN, parallel and distributed computing, media access-control algorithms, quality of service, and routing algorithms. He has published over 20 technical papers in these areas. He has also served as a referee for several conferences and journals.

Mi Lu (SM’94) received the M.S. and Ph.D. degrees in electrical engineering from Rice University, Houston, TX, in 1984 and 1987, respectively. She joined the Department of Electrical Engineering, Texas A&M University, College Station, in 1987 and she is currently a Professor. Her research interests include parallel computing, distributed processing, parallel computer architectures and algorithms, computer networks, computer arithmetic, computational geometry, and VLSI algorithms. She has published over 100 technical papers in these areas and has served as an Associate Editor of the Journal of Computing and Information and the Information Sciences Journal. Dr. Lu was the Stream Chair of the Seventh International Conference of Computing and Information and the Conference Chair of the Fifth and Sixth International Conference on Computer Science and Informatics. She has served on the panels of the National Science Foundation and the 1992 IEEE Workshop on Imprecise and Approximate Computation, as well as on many conference program committees. She is the Chair of 60 research advisory committees for Ph.D. and M.A. students, is a registered professional engineer; and is recognized in Who’s Who in the World—2001 and Who’s Who in America—2002.

Costas N. Georghiades (S’82–M’85–SM’90–F’98) received the B.E. degree with distinction from the American University of Beirut, Beirut, Lebanon, in June 1980, and the M.S. and D.Sc. degrees from Washington University, St. Louis, MO, in May 1983 and May 1985, respectively, all in electrical engineering. Since September 1985, he has been with the Electrical Engineering Department, Texas A&M University, College Station, where he is a Professor and Holder of the Delbert A. Whitaker Endowed Chair. His general interests include the application of information, communication, and estimation theories to the study of communication systems. Dr. Georghiades is a member of Sigma Xi and Eta Kappa Nu and is a registered Professional Engineer in Texas. Over the years, he has served in editorial positions with the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON INFORMATION THEORY, the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, and the IEEE COMMUNICATIONS LETTERS. He has been involved in organizing a number of conferences, including serving as Technical Program Chair for the 1999 IEEE Vehicular Technology Conference and the 2001 Communication Theory Workshop and as Chair of the Communication Theory Symposium within IEEE GLOBECOM’01.
