A topology-transparent MAC scheduling algorithm with ... - Springer Link

J Control Theory Appl 2011 9 (1) 106–114 DOI 10.1007/s11768-011-0233-x

A topology-transparent MAC scheduling algorithm with guaranteed QoS for multihop wireless network Chaonong XU 1 , Yongjun XU 2 , Zhiguang WANG 1 , Haiyong LUO 2 1.Department of Computer Science and Technology, China University of Petroleum, Beijing 102249, China; 2.Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China

Abstract: Due to its character of topology independency, topology-transparent medium access control (MAC) scheduling algorithm is very suitable for large-scale mobile ad hoc wireless networks. In this paper, we propose a new topologytransparent MAC scheduling algorithm, with parameters of the node number and the maximal nodal degree known, our scheduling algorithm is based on a special balanced incomplete block design whose block size is optimized by maximizing the guaranteed throughput. Its superiority over typical other scheduling algorithms is proven mathematically with respect to the guaranteed throughput, the maximal transmission delay, and also the minimal transmission delay. The effect of inaccuracy in the estimation of the maximal nodal degree on the guaranteed throughput is deduced mathematically, showing that the guaranteed throughput decreases almost linearly as the actual nodal degree increases. Further techniques for improving the feasibility of the algorithm, such as collision avoidance, time synchronization, etc., are also discussed. Keywords: Topology-transparent; Topology-independent; Balanced incomplete block design; Time-division multiple access

1

Introduction

An ad hoc wireless network consists of a great number of nodes, which exchange information with each other through wireless communication without the help of fixed infrastructure. Recently, many applications of ad hoc wireless networks, such as video monitoring, are suggested. These applications require quality of service (QoS) for wireless communication. One prerequisite for QoS provisioning in ad hoc wireless network is the design of medium access control (MAC) protocol with QoS support. MAC protocols in wireless network fall into two categories [1]. One is the contention-based MAC protocol, with the well-known carrier sense multiple access/collision avoidance (CSMA/CA) protocol as its representative. Due to the possibility of infinite transmission delay caused by wireless collisions, the contention-based MAC protocol is unfit for real-time multimedia applications. The other is the scheduling-based MAC protocol, where each node is assigned a set of time slots in which the transmitting right is granted for the node. By arranging colliding transmissions into different time slots, wireless collision can be eliminated. Therefore, the scheduling-based MAC protocol is potentially more suitable for wireless multimedia network because of its trait of guaranteed throughput and transmission delay. In multihop wireless networks, the key problem of the scheduling-based MAC protocol is how to assign time slots to nodes effectively. Its solution depends greatly on the feature of wireless interference. On one hand, wireless interference prevents neighbor nodes from simultaneous transmissions; on the other hand, simultaneous transmissions are possible if nodes are geographically distant enough. By

exploring the above features of wireless interference, spatial time division multiple access (STDMA)-based scheduling protocols have been developed. In STDMA-based scheduling protocols, only collision-free transmissions should be allocated into the same time slot, while collided transmissions are not. STDMA-based scheduling protocols can be further divided into two subcategories: topology-dependent and topology-transparent. The first subcategory is dependent on the information of network topology. Based on a specific interference model, such as the primary interference model or the k-hop interference model, a wireless network can be abstracted into a directed or undirected graph. Therefore, the STDMA-based scheduling problem can be transformed into the coloring problem or the maximal independent sets problem of the graph. Whatever it is, a uniform topology graph has to be set up either within each node in distributed manner or in the sink node in centralized manner. Obviously, it is very difficult to be implemented in a large-scale mobile wireless network. Furthermore, with additional information such as the number of queued packets, the optimal STDMA-based scheduling problem can be transformed into a linear programming problem (LP) or a nonlinear programming problem (NLP), although the complexity of the problem is also high. For example, determining transmission scheduling with optimal throughput has been proven to be NP-complete [2]. It is obviously unrealistic for wireless nodes with limited computing capability. The second subcategory, the topology-transparent STDMA-based scheduling protocol, is independent of the network topology. Therefore, it is fully distributed and less complex and very fit for the large-scale mobile wireless

Received 14 October 2010. The work was partly supported by the National Natural Science Foundation of China (No. 61003307, 60803159, 60873093), the Basic Disciplines Research Foundation of China University of Petroleum, Beijing (No. JCXK-2010-01), the Beijing Municipal Natural Science Foundation (No. 4102059), and the National High Technology Research and Development Program of China (No. 2009AA062802). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2011 

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network. It attracts focuses of many researchers and also is the focus of this paper. Chlamtac proposed a topology-transparent scheduling algorithm based on the polynomial theory on Galois field [3]. It focused on minimizing the frame length. Further researches show that it is equivalent to the constant weight code (p2 , 2(p − k), p) [4, 5], and the orthogonal array OA (t, p, p) [6, 7], where p is a prime or prime power and k is the order of polynomial. Reference [8] proposed a topology-transparent scheduling policy based on orthogonal Latin squares. Reference [9] proved that reference [3] is also equivalent to orthogonal Latin squares with order p when k = 1. In [10], still based on the polynomial theory on Galois field, Ju argued that higher throughput can be achieved at the expense of longer frame length. He proposed an enhancement policy by maximizing the guaranteed throughput. By eliminating the requirement that the parameter p must be a prime or a prime power, reference [4] tries to reduce the frame length with the theory of constant weight codes. Reference [11] designed link scheduling scheme using steiner systems with the objective to maximize the number of nodes, because steiner systems are denser than orthogonal arrays in some cases [12, 13]. The balanced incomplete block design (BIBD) has inherent relationship with the STDMA scheduling problem. Reference [14] used the idea of two known BIBDs, i.e., BIBD(n2 , n, n(n + 1), 1) and BIBD(p2 +p+1, p+1, p2 + p + 1, 1) are used in designing MAC scheduling algorithm. All of the above works mainly focus on how to apply different combinatorial designs to topology-transparent MAC scheduling, while performance comparison between them is urgently lacking. Similar to the above related works, in this paper, given the parameters of the number of wireless nodes N and the maximum nodal degree Dmax , we proposed a new MAC scheduling algorithm based on the theory of combinatorial design. Its superiority to Ju’s algorithm, Chlamtac’s algorithm, and the conventional TDMA algorithm is proven mathematically. Although this paper will present a better choice for topology-transparent MAC scheduling, we are not sure whether it is the optimal topology-transparent scheduling, which is an open problem for more than 40 years [15]. The rest of this paper is organized as follows. In Section 2, we present the network model. Section 3 presents our scheduling algorithm based on      v v−2 BIBD v, h, , . h h−2 In Section 4, the guaranteed throughput, the minimal transmission delay, and the maximal transmission delay are derived and an analytical comparison between our algorithm and Ju’s algorithm is provided. Numerical results that compare our algorithm with Ju’s algorithm, Chlamtac’s algorithm, and the conventional TDMA algorithm are given in Section 5. Further techniques to improve the feasibility of the algorithm are also discussed in Section 6. The last section is conclusions.

2

Network model

A wireless network is modeled by a directed graph G = (V, E), where V is the set of nodes and E is the set of directed links. If node w is within the transmission range of

node u, then a directed edge connecting these two nodes is denoted by (u, w) ∈ E, with u being a neighbor of w. The degree of a node w, i.e., |D(w) = {u|(u, w) ∈ E, u, w ∈ V }| is defined as the number of its neighbors. In this paper, we assume: 1) The maximum nodal degree Dmax , i.e., maxD(w), remains constant when network opw∈V

erates. Of course, Dmax > 0 is necessary for keeping connectivity. 2) The node number of network is known and remains constant. 3) The transmission channel is error-free and a reception failure is caused only by packet collisions. A packet transmitted from a neighbor of a node, denoted by R, is successfully received by R only if no packet is transmitted from other neighboring nodes simultaneously. 4) All nodes are homogeneous. 5) The transceiver at each node is half-duplex. As a result, a node cannot transmit and receive concurrently. Time is assumed to be synchronized over the network. Furthermore, time is slotted and slots are grouped into frames. In other words, a frame F = {S1 , S2 , · · · , Sb } consists of b consecutive slots. A schedule or slot assignment is given by a set S(w) ⊆ F for every node w, where S(w) consists of time slots in which node w has the right of transmitting in a frame. For convenience, Table 1 lists all definitions of symbols in this paper. Table 1 Definitions of symbols. Notation Dmax

Definition

The maximum nodal degree of network for designing MAC scheduling algorithm;  The real maximum nodal degree of network; Dmax N Node number of network; G The guaranteed throughput of topology-transparent MAC scheduling; T hBIBD The guaranteed throughput of our algorithm; The guaranteed throughput of Ju’s algorithm; T hJu DT BIBD , DT Ju , DT TDMA The maximal transmission delay of our algorithm, Ju’s algorithm, and TDMA; DT BIBD , DT Ju , DT TDMA The minimal transmission delay of our algorithm, Ju’s algorithm, and TDMA;  |Dmax ) T hmin (Dmax The real guaranteed throughput of our algorithm un with scheduling algorithm designed under Dmax der Dmax ; T hmin (Dmax ) The guaranteed throughput of our algorithm under Dmax ;  DT (Dmax |Dmax ) The real maximal transmission delay of our algo with scheduling algorithm derithm under Dmax signed under Dmax ;  |Dmax ) DT (Dmax The real minimal transmission delay of our algo with scheduling algorithm derithm under Dmax signed under Dmax .

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The scheduling algorithm

3.1 Balanced incomplete block design Definition 1 Assume that V is a set of v (v  2) elements called points, a balanced incomplete block design B with parameter (v, h, b, λ) is a collection of b (b > 0) subsets of V , which are called blocks, such that the following conditions are satisfied: 1) Each block consists of exactly h different points, v > h > 0. 2) ∀p, q ∈ V, p = q, they appear simultaneously in exactly λ blocks. λ > 0. Definition 2 ∀p ∈ V , the replication number of point p, which is notated as rp , is the number of blocks that include point p. Lemma 1 ∀p ∈ V, rp = bh/v. Proof Please refer to [16]. Since all points have the same replication numbers, we omit the indication of point, that is, r = rp .      v v−2 Theorem 1 BIBD v, h, , exists, h h−2   v where v  h > 0 and is the number of combinations h of v things taken h at a time. Proof We prove it by constructing a      v v−2 BIBD v, h, , . h h−2 Assume B is the combinations of the v elements in V taken h at a time. Therefore,     v v−2 b = |B| = , λ= . h h−2 Obviously, based on Lemma 1, G = (r − λDmax ) /b. Theorem 1 provides a way of constructing BIBD. Example 1 If V = {0, 1, 2, 3, 4} , B = {B0 = {0, 1, 2} , B1 = {0, 1, 3} , B2 = {0, 1, 4} , B3 = {0, 2, 3} , B4 = {0, 2, 4} , B5 = {0, 3, 4} , B6 = {1, 2, 3} , B7 = {1, 2, 4} , B8 = {1, 3, 4} , B9 = {2, 3, 4}}, then B is a BIBD (5, 3, 10, 3). A BIBD(v, h, b, Dmax ) can be mapped to a scheduling policy. The mapping between them is: 1) A frame consists of b slots. 2) ∀p ∈ V , if p ∈ Bi , then i ∈ S(p). According to Example 1, we can easily have that: S(0) = {0, 1, 2, 3, 4, 5}, S(1) = {0, 1, 2, 6, 7, 8}, S(2) = {0, 3, 4, 6, 7, 9}, S(3) = {1, 3, 5, 6, 8, 9}, S(4) = {2, 4, 5, 7, 8, 9}. Schedule policy S(0), S(1), S(2), S(3), S(4) corresponds to BIBD(5, 3, 10, 3) in Example 1. 3.2 The MAC schedule algorithm      v v−2 Since BIBD v, h, , exists, if v is set h h−2 as the number of wireless nodes N , then the scheduling problem is converted into the problem of determining the optimal value of h. Based on different criterions, such as

maximizing the average throughput, maximizing the guaranteed throughput, or minimizing the average transmission delay, etc., different optimal values of h can be obtained. Our schedule algorithm is based on the idea of maximizing the guaranteed throughput G. Definition 3 The guaranteed throughput is defined as the ratio of the number of guaranteed successful transmissions in one frame to frame length. For the schedule policy that is mapped from BIBD(v, h, b, λ), the maximum of the number of collisions for any node during one frame is λDmax . Based on the definition of guaranteed throughput, G = (r − λDmax )/b. Theorem 2 Given the number of wireless nodes N and the maximum nodal degree Dmax , the maximum (i.e., upper 2 (N − 1 + Dmax ) bound) of G is no more than . 4Dmax N (N − 1) Proof Since G(h) =

(N − 1)h − h(h − 1)Dmax r − λDmax = , b N (N − 1)

(N − 1) − Dmax (2h − 1) dG(h) = , dh N (N − 1) −2Dmax d2 G(h) = < 0. dh2 N (N − 1) Hence, when h =

1 N −1 + is an integer, 2Dmax 2

max G(h) =

2

(N − 1 + Dmax ) . 4Dmax N (N − 1)

(1)

Otherwise, max G(h) 

2

(N − 1 + Dmax ) . 4Dmax N (N − 1)

For the convenience of comparison with other scheduling 2 (N − 1 + Dmax ) is also notated as T hBIBD . algorithms, 4Dmax N (N − 1) Based on Theorems 1 and 2, the algorithm is as follows: 1) For the given N and Dmax , compute hopt where  N −1 1 1 N −1 G(hopt ) = max G( + ), G( + ), 2Dmax 2 2Dmax 2 1  N −1 + ) . G( 2Dmax 2      N −2 N , 2) Construct BIBD N, hopt , hopt − 2 hopt with the hopt , using the way of getting the combinations of the N elements taken hopt at a time. 3) Calculate slot allocation set S (v) based on      N −2 N N, hopt , , hopt − 2 hopt for each node. 4) Each node can transmit its data packets at its assigned time slots.

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4

Performance analysis

4.1 Guaranteed throughput Lemma 2 The guaranteed throughput of Ju’s algorithm [10] is ⎧ 1 1 ⎪ k+1  2kD ⎪ max , ⎨ 4kDmax , if N (2) T hJu = 1 k+1 − kD ⎪ max ⎪ ⎩N , otherwise, 2 N k+1 where k is the order of time slot assignment function (TSAF), and k  0 is an integer. Proof Please refer to [1]. Theorem 3 T hBIBD  T hJu . Proof Based on equations (1) and (2), T hBIBD T h⎧Ju ⎪ (N − 1 + Dmax )2 k 1 ⎪ ⎪ , if N k+1  2kDmax , ⎨ N (N − 1) 2 = 2 (N − 1 + Dmax ) N k+1 ⎪ ⎪ ⎪ , otherwise. 1 ⎩ 4Dmax N (N − 1)(N k+1 − kDmax ) (3) If k = 0, 2 T hBIBD (N − 1 + Dmax ) 4Dmax (N − 1) =  = 1. T hJu 4Dmax (N − 1) 4Dmax (N − 1) If k > 0, 1 Case 1 If N k+1  2kDmax , since Dmax > 0, Theorem 3 obviously comes into existence. Case 2 Otherwise, 2

2

T hBIBD (N − 1 + Dmax ) N k+1 = 1 T hJu 4Dmax N (N − 1)(N k+1 − kDmax ) 1

N × N k+1 N 2kDmax > 4Dmax (N − 1) 4Dmax (N − 1) Nk k = > . 2(N − 1) 2 1) If k  2, Theorem 3 is obviously true. √ 1 2) If k = 1, since N k+1 > 2kDmax is equal to N > 2Dmax , so 2 (N − 1 + Dmax ) T hBIBD √ = T hJu 4Dmax (N − 1)( N − Dmax ) N − 1 + Dmax √ . > 4Dmax ( N − Dmax ) Due to the fact that T hBIBD T hJu √ > 1 = N − 4Dmax N + 4Dmax 2 + Dmax − 1 0 for any N and Dmax , so T hBIBD /T hJu > 1. In conclusion, Theorem 3 is true in all cases. 4.2 Transmission delay Definition 4 The transmission delay under the worst traffic condition is called the maximal transmission delay, and it is defined as the ratio of the frame length to the minimal number of successful transmission slots in one frame. On the contrary, the transmission delay under the ideal traf>

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fic condition is called the minimal transmission delay, and it is defined as the ratio of the frame length to the maximal number of successful transmission slots in a frame. Theorem 4 If the maximal transmission delay and the minimal transmission delay of our algorithm are notated as DT BIBD and DT BIBD , respectively, then DT BIBD = 2Dmax N 4Dmax N (N − 1) . 2 , DT BIBD = N − 1 + D max (N − 1 + Dmax ) Proof Based on Definition 4, since the maximal transmission delay is the reciprocal of the guaranteed throughput, and the minimal transmission delay is the reciprocal of the maximal throughput, the conclusions are obvious. Lemma 3 The minimal transmission delay of Ju’s algorithm is

1 2kDmax , if N k+1  2kDmax , DT Ju = 1 N k+1 , otherwise. Its maximal transmission delay is ⎧ 1 ⎪ if N k+1  2kDmax , ⎨ 4kDmax , 2 DT Ju = N k+1 ⎪ , otherwise. ⎩ 1 N k+1 − kDmax The minimal transmission delay of the ordinary TDMA algorithm is DT TDMA = N. Its maximal transmission delay is DT TDMA = N. Proof Please refer to [1], and they can be proven easily based on Definition 4. Theorem 5 DT BIBD  DT Ju . Proof Based on Theorem 3, T hBIBD  T hJu . Since DT BIBD = 1/T hBIBD , and DT Ju = 1/T hJu , so DT BIBD < DT Ju . Theorem 6 DT BIBD /DT Ju  1, so DT BIBD  DT Ju . Proof Considering that N  Dmax + 1, we can easily prove it with the similar steps of Theorem 3. Theorem 7 DT BIBD < DT TDMA , DT BIBD < DT TDMA . Proof It can be easily proven based on Theorem 5. 4.3 Effects of inaccuracy in Dmax on performances The algorithm is designed based on Dmax , the estimated maximal nodal degree of network. However, in a real wireless network, due to the factors of supply voltage, radio attenuation, etc., the real maximum nodal degree of net may be different from the designated Dmax . work Dmax  , deTherefore, the real guaranteed throughput under Dmax  |Dmax ), must be different from the noted as T hmin (Dmax original designated T hmin (Dmax ). Similarly, the real maximal transmission delay and the real minimal transmis  , denoted as DT (Dmax |Dmax ) and sion delay under Dmax  |Dmax ), respectively, must also be different from DT (Dmax the original designated DT BIBD and DT BIBD . For intuition, DT BIBD is denoted as DT (Dmax ) and DT BIBD is denoted as DT (Dmax ) in this section. Theorem 8  T hmin (Dmax |Dmax ) T hmin (Dmax ) D 2(N + 1) − (N − 1 − Dmax ) max Dmax . = N − 1 + Dmax

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Proof Since  T hmin (Dmax |Dmax )  r − λDmax = b   (N − 1)h − h(h − 1)Dmax  , =  N (N − 1) h=hopt it can be easily obtained based on the definitions of  |Dmax ) and T hmin (Dmax ). T hmin (Dmax Further comparison results are as follows:   1) If Dmax < Dmax , then T hmin (Dmax |Dmax ) > T hmin (Dmax ) . Further,  T hmin (Dmax |Dmax ) T hmin (Dmax ) D 2(N + 1) − (N − 1 − Dmax ) max Dmax = N − 1 + Dmax N − 1 − Dmax D . N − 1 + Dmax D 2(N + 1) − N max D Dmax ≈ 2 − max . In If N  Dmax , N − 1 + Dmax Dmax other word,  T hmin (Dmax |Dmax ) D > 2 − max . T hmin (Dmax ) Dmax Theorem 9 N − 1 + Dmax DT (Dmax |Dmax ) = , D DT (Dmax ) 2(N +1)−(N −1−Dmax ) max Dmax  |Dmax ) DT (Dmax = 1. DT (Dmax ) Proof Since 1  DT (Dmax |Dmax ) =  T hmin (Dmax |Dmax )   b  , =   r − λDmax h=hopt

 DT (Dmax |Dmax ) = DT (Dmax ), it can be easily proven. Theorems 8 and 9 are valid guidelines for setting Dmax if it is hard to be estimated. Under the term that the guaranteed throughput is satisfied, if Dmax is set deliberately larger  , the real minimal than the real maximal nodal degree Dmax  throughput T hmin (Dmax |Dmax ) will be larger than the designated T hmin (Dmax ), with the tradeoff that the real max-

 |Dmax ) is larger than imal transmission delay DT (Dmax the designated DT (Dmax ). On the contrary, if Dmax is set  , the real smaller than the real maximal nodal degree Dmax  minimal throughput T hmin (Dmax |Dmax ) will be smaller than the designated T hmin (Dmax ). However, the real maximal transmission delay Dmax is smaller than DT (Dmax ). 4.4 Utilization rate analysis One drawback of topology-transparent MAC scheduling is the low utilization rate of wireless channel, which is caused by its ignorance of network topology. In the most ideal case, where every transmission is successful, according to Lemma 1 and Theorem 2, the utilization rate of our algorithm is rp 1 hopt 1 + = ≈ . b v 2Dmax 2N Obviously, the best utilization rate is closely related with Dmax . When every node has Dmax neighbor nodes, the utilization rate of the optimal scheduling algorithm is about 1/ (Dmax + 1). Therefore, the performance of our topology-transparent scheduling algorithm is just about 1/2 of that of the optimal scheduling algorithm. In fact, since not every node has Dmax neighbor nodes, the utilization rate of our algorithm must be lower. Similarly, according to Theorem 2, in the most congest case, the performance rate of our algorithm to that of the optimal scheduling algorithm is about 2

(N − 1 + Dmax ) . 4Dmax N (N − 1)(Dmax + 1)

5 Numerical results In this section, we compare the performance of our scheduling algorithm with Chlamtac’s algorithm, Ju’s algorithm, and conventional TDMA algorithm. Table 2 gives a comparison among our algorithm, Ju’s algorithm, Chlamtac’s algorithm, and TDMA. To have an objective comparison, we have the same parameters with reference [1]. We considered four cases where the number of nodes is 121, 256, 800, and 1024, respectively, and Dmax ranges from 1 to 44. Table 2 Comparison among typical scheduling algorithms. Algorithm

Reference

TDMA Chlamtac’s algorithm Ju’s algorithm

Our algorithm

[3, 6, 7] [10]

Main feature Every slot is assigned to only one node. Based on OA (t, p, p) and by minimizing the frame length. Based on OA (t, p, p) and by maximizing the guaranteed throughput. Based on ! !! v v−2 , h h−2 and by maximizing the guaranteed throughput.

BIBD v, h,

In Figs. 1∼4, the guaranteed throughputs for the four cases are shown. Obviously, because the probability of wireless collision always increases with Dmax , the guaranteed throughput decreases with Dmax for all algorithms, except for the ordinary TDMA algorithm. For the above three

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scheduling algorithms, our algorithm has the best performance for all cases. Our algorithm performs better especially when a network is sparse. For example, when N = 121 and Dmax = 5, the guaranteed throughput of our algorithm is about two times better than that of Ju’s algorithm. Just as the conclusions from reference [10], Ju’s algorithm performs better than Chlamtac’s algorithm and the conventional TDMA algorithm, similar result can also be found in Figs. 1∼4. When a network is dense, Chlamtac’s algorithm performs worse than the conventional TDMA algotithm, but our algorithhm and Ju’s algorithm always perform better than the other two algorithms.

Fig. 4 The guaranteed throughput for N = 1024.

Fig. 5∼8 show the transmission delay for the four cases when N = 121, 256, 800, and 1024. In each case, our algorithm has less both minimal and maximal transmission delay than the other three algorithms.

Fig. 1 The guaranteed throughput for N = 121.

Fig. 5 The transmission delay for N = 121.

Fig. 2 The guaranteed throughput for N = 256.

Fig. 6 The transmission delay for N = 256.

Fig. 3 The guaranteed throughput for N = 800.

From Fig. 5, the maximal transmission delay of Ju’s algorithm and Chlamtac’s algorithm can even be larger than that of the conventional TDMA algorithm in some cases, but that of our algorithm is always smaller than those of

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the other three algorithms. The minimal transmission delay of our algorithm is always smaller than those of the other three algorithms. Furthermore, transmission delay of our algorithm increases smoothly when the maximal nodal degree increases, which implies that our algorithm is also stable.

 Fig. 10 T hmin (Dmax |Dmax ) /T hmin (Dmax ) for Dmax = 20 when  varies. the real maximal nodal degree Dmax

From Figs. 9 and 10, it can be seen that when the real  is less than Dmax , larger maximal degree of network Dmax  inguaranteed throughput can be obtained. When Dmax creases, the guaranteed throughput decreases almost linearly.

6 Further improvement for feasibility Fig. 7 The transmission delay for N = 800.

Fig. 8 The transmission delay for N = 1024.

We also investigate the effect of inaccurate Dmax on the guaranteed throughput using Dmax = 10 and Dmax = 20. In Figs. 9 and 10, the x axis is the real maximal nodal degree   and the y axis is the ratios of T hmin (Dmax |Dmax ) Dmax to T hmin (Dmax ), i.e., the ratio of the actual guaranteed throughput to the designated one.

 Fig. 9 T hmin (Dmax |Dmax ) /T hmin (Dmax ) for Dmax = 10 when  varies. the real maximal nodal degree Dmax

The topology-transparent scheduling scheme determines the scheduling policy irrespective of the underlying topology. Thus, it is immune to node mobility. However, for a specific node, collision may still happen in its assigned time slots. Moreover, collision may not happen even if the node tries to transmit in its nonassigned time slots. Combined with the topology-transparent scheduling algorithms, some methods try to overcome the above two shortcomings. These methods can also be used in our algorithm to enhance throughput and delay performance. In [17], a topology-transparent scheduling algorithm assigns time slots for every node. Each node always transmits in its assigned time slot and transmits with probability p in its nonassigned time slots. Obviously, for a specific node, it only utilizes free time slots that are not assigned to it, and possible collision may still happen in its assigned time slots. In [18], every node resolves possible collisions in its assigned time slots by reservation, which is similar to the IEEE 802.11 four-way handshaking process (RTS/CTS/DADA/ACK). Every node utilizes its nonassigned time slots by carrier sense. Simulated results reveal its effectiveness. These proposed methods are also suitable for our algorithm. Another scheme to deal with the collisions in the assigned time slots is to divide a time slot into Dmax + 1 reservation slots and a transmission slot. Every reservation slot is uniquely assigned to a node beforehand, and it will declare in its reservation slot that it will occupy the transmission slot if its preceding reservation slots are all clear. Acknowledgement is a valid way to guarantee reliable wireless unicast. However, simple acknowledgement schemes, such as a reverse acknowledging packet, are unsuitable because collision between acknowledgement packets is inevitable even if data is received correctly [19]. By using rateless forward error correction (RFEC) code as acknowledgement, reference [20] effectively implemented reliable unicast without using an acknowledgement packet. This scheme can also be adopted in our algorithm. Topology-transparent ternary scheduling is proposed to enhance energy efficiency for mobile ad hoc networks by adding a sleeping state for nodes [21], and it can be modeled as the decompositions of the directed complete bipar-

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tite graph [22]. Reference [23] examined the connection between topology-transparent ternary scheduling and such nonsleeping scheduling and proposed an algorithm that convert a nonsleeping scheduling policy into a ternary scheduling policy. Therefore, our scheduling policy can also be converted into a ternary scheduling policy easily. Time synchronization is an indispensable component for implementing our algorithm. According to reference [24], our algorithm is of the type of frame synchronized scheduling algorithm, which requires both frame synchronization and slot synchronization. Since very high synchronization precision has been reached [25], time synchronization is not a problem any longer. In CDMA-based wireless network, correct reception can be guaranteed only if the number of secondary interference nodes does not exceed a given limit [26]. Therefore, our algorithm can also be modified to get shorter frame length in the same way as reference [27].

7

Conclusions

Because of its character of topology independency, the topology-transparent scheduling algorithm is very fit for large-scale mobile ad hoc wireless networks. In this paper, we proposed a topology-independent scheduling algorithm based on a balanced incomplete block design whose block size is optimized by maximizing the guaranteed throughput to not only larger guaranteed throughput but also shorter maximal transmission delay and shorter minimal transmission delay get. Compared with Ju’s algorithm, Chlamtac’s algorithm, and the conventional TDMA algorithm, our algorithm performs best. Some methods to enhance its practicability are also suggested. Topology-transparent MAC scheduling codes are essentially the superimposed codes [28]. Therefore, the existence of the optimal superimposed codes and its construction are the keys to the optimal topology-transparent MAC scheduling. Although this paper provided a better choice for topology-transparent MAC scheduling, we are not sure whether it is the optimal superimposed code due to the absence of the optimal superimposed code criterion. It is reasonable to expect that the optimal topologytransparent MAC scheduling will exhibit better throughput characteristics and therefore will be a hot research area in future with the application of mobile ad hoc network (MANET).

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[11]

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Chaonong XU received his B.S. and M.S. degrees in Computer Sciences, in 1998 and 2001, respectively, both from Hefei University of Technology, China, and Ph.D. degree from the Institute of Computing Technology, Chinese Academy of Sciences, in 2007, also in Computer Science. Dr. Xu joined Tsinghua University as a postdoctoral research associate in 2007. He is currently a lecturer in China University of Petroleum, Beijing, China. His research interests include embedded system and wireless sensor networks. E-mail: [email protected].

Zhiguang WANG received his B.S. degree in 1986 from the Inner Mongolia Normal University and M.S. degree in 1997 from Jilin University, China. He is currently a professor in the China University of Petroleum, Beijing, China. His research interests include embedded system and wireless sensor networks. E-mail: [email protected].

Yongjun XU received his B.S. degree in 2001 from Xi’an University of Posts and Telecommunications, China, and Ph.D. degree from the Institute of Computing Technology, Chinese Academy of Sciences, in 2006. He is currently an associate professor at the Institute of Computing Technology, Chinese Academy of Sciences. His research interests include low power system and wireless sensor networks. Email: [email protected].

Haiyong LUO received his B.S. degree in 1989 from Huazhong University of Science and Technology, China, and Ph.D. degree from the Institute of Computing Technology, Chinese Academy of Sciences, in 2008. He is currently an associate professor at the Institute of Computing Technology, Chinese Academy of Sciences. His research interests include pervasive computing and wireless sensor networks. E-mail: [email protected].