5940 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 11, NOVEMBER 2014

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1 5940 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 11, NOVEMBER 014 Topoogy-Transparent Scheduing in Mobie Ad Hoc Networks With Mutipe Packet Reception Capabiity Yiming Liu, Member, IEEE, VictorO.K.Li,Feow, IEEE, Ka-Cheong Leung, Member, IEEE, and Lin Zhang, Member, IEEE Abstract Recent advances in the physica ayer have enabed wireess devices to have mutipe packet reception (MPR) capabiity, which is the capabiity of decoding more than one packet, simutaneousy, when concurrent transmissions occur. In this paper, we focus on the interaction between the MPR physica ayer and the medium access contro (MAC) ayer. Some random access MAC protocos have been proposed to improve the network performance by expoiting the powerfu MPR capabiity. However, there are very few investigations on the schedue-based MAC protocos. We propose a nove m-mpr--code topoogy-transparent scheduing ((m, )-TTS) agorithm for mobie ad hoc networks with MPR, where m indicates the maximum number of concurrent transmissions being decoded, and is the number of codes assigned to each user. Our agorithm can take fu advantage of the MPR capabiity to improve the network performance. The minimum guaranteed throughput and average throughput of our agorithm are studied anayticay. The improvement of our (m, )-TTS agorithm over the conventiona topoogy-transparent scheduing agorithms with the coision-based reception mode is inear with m. The simuation resuts show that our proposed agorithm performs better than sotted ALOHA as we. Index Terms Medium access contro (MAC), mutipe packet reception (MPR), topoogy-transparent scheduing (TTS). I. INTRODUCTION TRADITIONALLY, medium access contro (MAC) agorithms assume the coision-based reception mode, in which a packet faiure (equivaenty coision) happens, if more than one interfering neighbor of a user transmit simutaneousy. Thus, the fundamenta goa of the conventiona MAC agorithms is to improve the throughput by resoving coisions. In random access MAC protocos, each user is aowed to transmit probabiisticay with the goa of minimizing the coision probabiity. In schedue-based MAC protocos, each Manuscript received Juy 11, 013; revised January 9, 014 and May 8, 014; accepted September 6, 014. Date of pubication September 17, 014; date of current version November 7, 014. This work was supported in part by the Research Grants Counci of Hong Kong under Grant HKU E. The associate editor coordinating the review of this paper and approving it for pubication was A. A. Abouzeid. Y. Liu is with the China Academy of Eectronics and Information Technoogy, Beijing , China (e-mai: iuyiming09@tsinghua.org.cn). V. O. K. Li and K.-C. Leung are with the Department of Eectrica and Eectronic Engineering, The University of Hong Kong, Pokfuam, Hong Kong (e-mai: vi@eee.hku.hk; kceung@ieee.org). L. Zhang is with the Department of Eectronic Engineering, Tsinghua University, Beijing , China (e-mai: inzhang@tsinghua.edu.cn). Coor versions of one or more of the figures in this paper are avaiabe onine at Digita Object Identifier /TWC user is schedued to transmit in order to avoid coisions. However, this coision-based reception mode does not refect the capabiities of the advanced wireess transceivers nowadays. Advanced signa processing technoogies have enabed wireess transceivers to correcty decode mutipe packets transmitted simutaneousy, referred to as the mutipe packet reception (MPR) capabiity. Various technoogies can be used to enabe the MPR capabiity, and can be categorized into two different groups, namey, training-based agorithms and so-caed bind interference canceation agorithms. In training-based agorithms such as space-time coding and mutiuser detection (MUD) with successive interference canceation (SIC) [0], the knowedge of propagation channes and signa waveforms is assumed to be avaiabe. However, this is difficut to obtain and may be impractica in mobie ad hoc networks. In bind interference canceation agorithms [11], [], [34], the signa of interest is moduated by a known ampitude or phase variation. This aows the intended receiver to estimate and suppress the interfering sources without knowing the channe states. These bind interference canceation agorithms can separate the coiding packets and hande the near-far effect [] without assuming the knowedge of channe, rendering makes them suitabe for mobie ad hoc networks. However, they require a higher computationa compexity in signa processing during the packet decoding phase. Yet, the compexity is acceptabe. It has been shown in [] that the compexity of the bind interference canceation agorithm increases ineary with the increasing m, where m is the maximum number of concurrent transmissions being decoded. Thus, we assume that the underying MPR capabiity in this paper is achieved by this kind of bind interference canceation agorithms. In this paper, we focus on studying the cross ayer interaction between the MPR physica and MAC ayers, rather than the technoogies that enabe the MPR capabiity. Specificay, we focus on the throughput in the MAC ayer, defined as the number of successfu transmission. Under the m-mpr mode, in which the receiver can correcty decode at most m packets simutaneousy, we propose an m-mpr--code topoogytransparent scheduing ((m, )-TTS) agorithm. Time is divided into equa-sized time sots, grouped into frames. In each frame, a user is aocated some transmission sots based on the assigned codes. In the traditiona topoogy-transparent scheduing agorithms with the coision-based reception mode, each user is assigned a unique code to achieve the maximum throughput IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

2 LIU et a.: TOPOLOGY-TRANSPARENT SCHEDULING IN MOBILE AD HOC NETWORKS WITH MPR CAPABILITY 5941 The powerfu MPR capabiity aows us to assign more codes to a user, since a coision does not necessariy ead to a packet faiure with the MPR capabiity. In our (m, )-TTS agorithm, we assign transmission codes to each user due to the fact that the increase in the number of packet faiures is much ess than that in the transmission time sots, resuting from the m-mpr capabiity. We beieve that the difference between the proposed agorithm and the traditiona topoogy-transparent scheduing agorithms is nontrivia. Firsty, in our agorithm, the optima frame structure that maximizes the throughput is totay different from that in the traditiona agorithms. Secondy, assigning mutipe codes (poynomias) to a user introduces a new probem, namey, we have to assign mutipe codes (poynomias) in such a way to avoid coincidences among different codes (poynomias). 1 We detai the assignment of mutipe codes in Section III. In addition, we show that the improvement of our (m, )-TTS agorithm with the m-mpr capabiity over the conventiona topoogy-transparent scheduing agorithms with the coision-based reception mode is inear with increasing m. The simuation resuts show that our proposed agorithm performs better than random access protocos, namey, sotted ALOHA with MPR. A. Motivation and Reated Work Severa random access MAC protocos have been proposed [16], [17], [1] to improve the network performance by taking advantage of the powerfu MPR capabiity. The random access approaches cannot provide deterministic deay and throughput bounds. That is why most networks offering throughput and deay guarantees, say, the tactica networks, such as the Enhanced Position Location Reporting System (EPLRS) [19] and the Joint Tactica Radio System (JTRS) [6], impement deterministic, schedue-based protocos, such as time division mutipe access (TDMA). The reated work on schedue-based protocos can be categorized into two different groups, namey, topoogy-dependent and topoogy-transparent, based on whether the detaied network connectivity information is required. Existing topoogydependent approaches focus on finding a minimum-ength, confict-free schedue based on the detaied network topoogy. This probem is proved to be NP-compete [], [13], [3]. More importanty, re-computation and information exchanges are required to maintain accurate network topoogy information and distribute the new schedues when the network topoogy changes. Distributed topoogy-dependent scheduing agorithms are introduced in [5]. However, it has been shown that it takes severa minutes to obtain and distribute the updated schedues [5], which is too ong in mobie networks, especiay for rea-time traffic. Thus, the robustness and effectiveness of these topoogy-dependent scheduing agorithms are undermined in arge, highy dynamic, wireess mobie ad hoc networks. In order to overcome the aforementioned imitations in mobie ad hoc networks, topoogy-transparent scheduing agorithms have been proposed [5], [7], [14], [18]. In these agorithms, no updates on network connectivity information are 1 A coincidence of two poynomias is defined in Section II-B. required. In topoogy-transparent scheduing agorithms, each node is assigned mutipe time sots for transmission in each frame. It is guaranteed that there is at east one confict-free sot per frame. Chamtac and Farago [7] deveoped a topoogytransparent agorithm that guarantees at east one coisionfree time sot in each frame time, but the performance is even worse than the conventiona TDMA in some cases. Ju and Li [18] proposed another agorithm to maximize the minimum guaranteed throughput. However, ony unicast communication is considered. Cai et a. [5] proposed a broadcast scheduing agorithm, modified Gaois fied design (MGD), which sends the same message mutipe times during one frame time in order to guarantee exacty one successfu broadcast transmission per frame. However, there are few schedue-based MAC agorithms with MPR, such as TDMA. The conventiona TDMA protocos [], [9], [13], [3], [5] construct transmission schedues to avoid coisions and maximize the throughput. This does not match the characteristics of MPR as it aows coisions. We beieve that this is the inherent reason why there is much ess interest in the idea of designing schedue-based TDMA protocos with MPR. However, we find that topoogy-transparent scheduing agorithms [5], [7], [14], [18] are suitabe for the MPR capabiity. In the conventiona topoogy-transparent scheduing agorithms with coision-based reception, each user is assigned a unique transmission schedue or code. Athough the transmission schedues are not coision-free, each user is guaranteed to have at east one confict-free transmission sot per frame. Most existing MAC protocos with the MPR capabiity are random access agorithms. Sotted ALOHA was extensivey studied in [16], [17] and can significanty improve network performance. A mutipe access protoco based on receivercontroed transmission was proposed in [1]. Athough this work pointed out that it is necessary to redesign the MAC protoco so as to make the best use of the MPR capabiity, it can ony be used in predefined topoogies and cannot be extended to an arbitrary ad hoc network. Another random access protoco was proposed in [33]. It takes advantage of the MPR capabiity to support users with different quaity of service requirements. Some work [6], [8], [33] focused on carrier sense mutipe access (CSMA) with the MPR capabiity. However, it has been shown in [8] that the throughput gain of CSMA over sotted ALOHA is very imited, especiay when the MPR capabiity increases. A these CSMA agorithms are designed ony for wireess oca area networks (WLANs) and cannot be appied in mutihop ad hoc networks, since it has been shown that CSMA suffers from serious instabiity and unfairness issues in such networks [31]. Reference [8] studied the effect of the MPR capabiity on the network capacity and energy efficiency. A survey paper on MPR for wireess random access networks can be found in [0]. Reference [7] proposed a poynomia-time heuristic agorithm to sove the joint routing and scheduing probem with the MPR capabiity. It is simiar to our work, as its scheduing agorithm can provide throughput and deay guarantees. However, it is a centraized agorithm which needs to know a the detaied network connectivity and traffic information, making it impractica in mobie networks.

3 594 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 11, NOVEMBER 014 Fig. 1. The frame structure. In this paper, we propose a schedue-based scheduing agorithm, which is obivious to the network changes in mobie networks and takes advantage of the powerfu MPR capabiity. Thus, the proposed agorithm can provide throughput and deay guarantees in mobie networks with the MPR capabiity. The remainder of this paper is organized as foows. In Section II, we present our system mode and some definitions and theorems which wi be used in the foowing sections. Section III presents the detais of our proposed agorithm. The average throughput and packet deay of the proposed agorithm are aso derived. Extensive simuations are conducted to evauate the performance of our proposed agorithm in Section IV. The effect of inaccuracies in the estimation of network parameters such as the maximum node degree and the number of network nodes on the performance of the proposed agorithm is aso investigated. We concude in Section V. II. SYSTEM MODEL A. Network Mode A mobie ad hoc network consisting of N nodes can be represented by a graph G(V,E). V is the set of a network nodes abeed from 1 to N and E is the set of a edges. If Node v is within the interference range of Node u, an edge denoted by (u, v) is in E. We assume that if (u, v) E, (v, u) E. The degree of a node u, D(u), is defined as the number of its interference neighbors. The maximum node degree D max is defined as D max =max D(u). We assume that D max is much u V smaer than the number of nodes N and remains constant whie the network topoogy changes [9]. In practice, empirica data maybeusedtoestimated max. In addition, D max is aways pessimisticay estimated to ensure that the actua number of interfering neighbors does not exceed the estimate. We show that the performance of our agorithm is insensitive to the accuracy in the estimation of D max in Section IV. Thus, it is unnecessary to use distributed onine method to obtain the maximum node degree when the network operates. We focus on TDMA networks. Time is divided into equasized frames. Each frame is further divided into p subframes, each of which consists of p synchronized, equa-sized time sots, where p is a prime. The frame structure is shown in Fig. 1. Fig. 1 demonstrates some transmission time sots of an arbitrary node i, in which an A is ocated. The transmission time sots are determined by one of its time sot aocation functions, which is defined and discussed ater. Synchronization can be achieved by Goba Positioning System (GPS) or the synchronization agorithms such as the one proposed in [1]. The MPR capabiity can be modeed by an n n MPR matrix C [16], [17], [1], [7], [33], where C n,i is the conditiona probabiity that i packets are correcty decoded given that n packets are transmitted simutaneousy by the nodes in the interference range of a receiving node. The MPR matrix is a function of many parameters, such as the channe conditions, the moduation technoogies, and the signa processing technoogies, and thus can take many different forms. In this paper, we assume a generay used strong m-mpr mode [16], [17], [1], [7], [33]. In this case, a user can correcty decode a the packets if the number of concurrent transmissions within its interference range is not greater than m, and none of the packets if the number of concurrent transmissions exceeds m. That is, if i =1,,...,m, and C i,i =1, (1) C n,i =0, () otherwise. The conventiona coision-based reception mode can be modeed by the 1-MPR matrix. We aso investigate another m-mpr mode achieved by the bind interference canceation agorithm using poynomia phase-moduating sequences [] in Section IV. We assume that the transmission channe is error-free. Thus, the transmission from Node u to Node v fais when: 1) Node v is aso transmitting, or ) there are more than m 1 other nodes in v s interference range transmitting simutaneousy. Note that the transmission from u to v fais when v is aso transmitting. This is because the sef-interference of v is much stronger that the received signa from u, eading to the fact that v with the MPR capabiity cannot decode a packet from u. This probem can be soved by appying different fu-dupex technoogies [4]. However, it is beyond the scope of this work. B. Definitions and Theorems In the foowing, we present some definitions and theorems used throughout the paper. Definition 1 [18]: A poynomia with degree k mod p can be expressed as f(x) = k a i x i (mod p). i=0 Definition : A coincidence of two poynomias with degree k mod p is defined as the root of the difference of these two poynomias. That is, if f u (j) f v (j) =0, j is the coincidence of f u (x) and f v (x), where j =0, 1,...,p 1. Theorem 1 [18]: There are at most k coincidences of two arbitrary different poynomias with degree k mod p. Proof: The proof can be found in [18]. Definition 3: For a given network G(V,E), each node v is assigned a time sot aocation function (TSAF) group consisting of unique TSAFs, i.e., poynomias with degree k mod p, namey, {fv j (x)}, where j =1,,...,. The poynomias are assigned to each node in such a way that any two poynomias of an arbitrary node has no coincidence. The detaied method of assigning poynomias is discussed in the next section. This TSAF group is used to cacuate the aocated transmission time sots in one frame for each node. For an arbitrary node v, it transmits in time sots {fv j (i)} in Subframe i,

4 LIU et a.: TOPOLOGY-TRANSPARENT SCHEDULING IN MOBILE AD HOC NETWORKS WITH MPR CAPABILITY 5943 TABLE I ASSIGNMENT OF POLYNOMIALS Fig.. Fig. 3. The network topoogy. The iustration of definitions. where i =0, 1,...,p 1 and j =1,,...,. Thus, each node transmits in p time sots in one frame. We refer to these p aocated transmission time sots of an arbitrary node v as its time sot ocation vector (TSLV), TSLV(v), i.e., TSLV(v) =(fv 1 (0),...,fv(0),f v 1 (1),...,fv(1),...,f v 1 (p 1),...,fv(p 1)). Theorem : There are at most k coincidences of two arbitrary nodes u and v, each of which is assigned different poynomias with degree k mod p. Proof: According to Theorem 1, there are at most k coincidences of an arbitrary poynomia of Node u and an arbitrary poynomia of Node v. Each node has unique poynomias. Thus, there are at most k coincidences of two arbitrary nodes u and v. Definition 4: Given the m-mpr mode, the number of packet faiures for a transmission from an arbitrary node v to its destination u, N f, is defined as the number of assigned transmission sots as specified in TSLV(v), in which Node u aso transmits or there are more than m 1 other neighbors of Node u other than v aso transmitting. Before proceeding, we use Figs., 3, and Tabe I to iustrate the aforementioned definitions of, poynomia, coincidence, and packet faiure in detai. Considering the network shown in Fig., Node 1 is transmitting to Node and Nodes 3, 4, 5 are the interfering neighbors of Node. We assume that m =, k =1, p =5, and =. That is, each node is assigned two poynomias with degree 1 mod p =5, as isted in Tabe I. Accordingy, the transmission time sots of each node are shown in Fig. 3. Consider the transmission from Node 1 to Node. A C in Subframe i (i =0, 1,...,4) of Node 1 indicates that i is a coincidence of Node 1 s and other nodes poynomias. Simiary, an F in Subframe i (i =0, 1,...,4) of Node 1 indicates a packet faiure for a transmission from Node 1 to Node. A packet faiure for the transmission from Node 1 to Node happens in Subframe i if i is a coincidence of Node 1 s and Node s poynomias or more than m 1 poynomias assigned to Nodes 3, 4, and 5 have the same coincidence i with those of Node 1 s poynomias. Theorem 3: Given that the m-mpr mode and each node is assigned unique poynomias with degree k, N f Nf max = min(p, S), (3) where S = k (Dmax 1)k +. (4) m Proof: Each node has at most D max neighbors. Thus, poynomias of an arbitrary node v has at most (D max 1)k coincidences with the (D max 1) poynomias of the other D max 1 interfering neighbors of its destination u. Reca that if there are more than m 1 out of these D max 1 interfering neighbors of u aso transmitting, the transmission from v to u fais. This is equivaent to throwing (D max 1)k bas into p bins. The maximum number of bins, in which there are at east m bas, is (Dmax 1)k m. Thus, the second term in the right hand side of (6) represents the maximum number of time sots of v, in which there are more than m 1 neighbors of u other than v itsef aso transmitting. The first term in the right hand side of (6) represents the maximum number of time sots of v,in which its destination u aso transmits. We aso note N f cannot be arger than p, i.e., the number of assigned sots. Taking D max =10, k =1, =, and m =5as an exampe, we know that an arbitrary node v has at most k =4coincidences with one of the other interfering neighbors of its destination u. Thus, Node u has at most (D max 1)k =36coincidences with the other D max 1 interfering neighbors of Node u. Note that if and ony if more than m 1=4out of these 36 coincidences are ocated in one transmission sot of Node v, the transmission of Node v in this sot fais. Thus, 36 coincidences can ead to up to (Dmax 1)k m = (10 1) 5 =7such sots. In order to investigate how tight the bound in Theorem 3 is, we run simuations as foows. Given that k =1, p =3, D max =10, m is set to two and four, and varies from one to ten, in each simuation, we randomy seect a poynomia with degree k mod p for Node v and then Dmax other different such poynomias for Node u and its D max 1 other interfering neighbors from the remaining p 1 poynomias, and cacuate the vaue of N f. For each vaue of, we simuate 100 runs. As shown in Fig. 4, the bound of N f is reativey oose, especiay when the vaues of m and are arge. This is what we prefer to see, since it impies that the average number of packet faiure is typicay ess than the bound we get. A packet faiure resuts from one or more coincidences. However, a coincidence does not necessariy eads to a packet faiure.

5 5944 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 11, NOVEMBER 014 Fig. 4. The tightness of the bound in Theorem 3. III. PROPOSED ALGORITHM Consider a TDMA network with N nodes with the m-mpr capabiity and the maximum node degree is D max. Each node transmits according to its TSLV, which is determined by its assigned TSAF group consisting of unique poynomias with degree k mod p (TSAFs). To avoid coincidences between any two out of poynomias distributed to an arbitrary node, the assignment can be made as foows. Note that there are a tota of p k+1 poynomias. It has been proved that the maximum degree of the poynomias equas one (k =1) for most cases [18]. Therefore, without oss of generaity, we assume that k =1 in the subsequent discussion for simpicity. For the case of k =1, there are p poynomias f(x) =ax + b, where a =0, 1,...,p 1 and b =0, 1,...,p 1. We divide these poynomias into p groups according to the vaue of a, each of which consists of p poynomias. Each node i seects a group of poynomias as its assigned ( TSAFs for its) excusive use ony. Node i seects f i,j (x) = 1 x + b i p i,j, where j =1,,..., and b i,j = [ mod ( i, ) ] p j 1. Note that the parameters p and are known to each node. Thus, each node seects its poynomias in a distributed manner according to its identification number. Taking N =3, =, and p =3as an exampe, the assigned poynomias to Node i are f i,1 (x) = (i 1)x and f i, (x) =(i 1)x +1. As the number of nodes in the network (N) grows, more codes are required. Thus, a arger vaue of parameter p is chosen to satisfy that the codes assigned to one node are orthogona to each other. That is, the design parameter p is chosen such that each node can find its poynomias, which wi be discussed ater. Since any two poynomias of an arbitrary node has no coincidence, each node can transmit in p time sots per frame. Thus, the throughput G is: G = p N f p, (5) arg {p 1,p } and the minimum throughput G min can be obtained by Theorem 3, max p Nf G min = p. (6) In order to guarantee that our agorithm works correcty, the foowing two constraints must be satisfied: p p N, (7) p > Nf max. (8) Equation (7) must be satisfied to ensure that any two out of unique poynomias assigned to an arbitrary node has no coincidence. That is, each node can have p assigned time sots for transmissions. Equation (8) states that the tota number of transmissions of an arbitrary node is arger than the maximum number of possibe packet faiures in one frame. Note that (7), (8) ensure that each node can be assigned orthogona codes. Theorem 4 discusses how to obtain the maximum vaue of G min. Theorem 4: Let p 1 be the argest prime smaer than or max Nf equa to and p be the smaest prime arger than N max f, respectivey. The optima vaue of p that achieves the maximum vaue of G min is p = arg max(g min ), where {p 1,p } max(g min ) satisfies (7), or is the smaest prime satisfying (7) otherwise. Proof: In order to find the maximum vaue of G min,we have to sove (8): G min p By (6), (8) can be rewritten as Thus, Since G min p= p N max f =0. (9) p +N max f p 3 =0. (10) p 3 ( N max f p ) =0. (11) max Nf p = max Nf. (1) = 4 8(N max f ) 3 < 0, G min increases with p when p<, and decreases with p when p N max f. Since p is a prime such that (7) and (8) have to be satisfied, we can thus prove Theorem 4. When m =1, it degenerates to the coision-based reception mode. Our resut thus reduces to the one in [18]. Considering the case that the maximum vaue of G min is achieved when p = max(g min )= max Nf,wehave: ( 4 k + (Dmax 1)k m ). (13) Thus, we can concude that the maximum vaue of G min generay remains the same with varying. However, the average throughput of our agorithm highy depends on the vaue of, i.e., the number of assigned poynomias distributed to each node, that is discussed as foows.

6 LIU et a.: TOPOLOGY-TRANSPARENT SCHEDULING IN MOBILE AD HOC NETWORKS WITH MPR CAPABILITY 5945 A. Average Throughput We study the average throughput, defined as the average number of successfu transmissions per node per sot, of our agorithm. Given that each node is assigned TSAFs (poynomias) with the m-mpr capabiity, we give some definitions as foows. Let X i and X j i be the numbers of successfu transmissions of an arbitrary node determined by its i-th TSAF in each frame and in Subframe j, respectivey. That is, i =1,..., and j =0, 1,...,p 1. Note that the frame ength is p.the average throughput (G a ) can be expressed as foows: [ ] [ ] p 1 E X i E X j i i=1 i=1 j=0 G a = p = p = [ ] p E X j i, (14) where [ ] ) E X j i = P (X j i =1. (15) Consider a transmission from Node u to Node v. Letfu(x) i be the i-th TSAF assigned to Node u. Fori =1,..., and j =0, 1,...,p 1, we obtain E[X j i ] in the foowing. The event that X j i =1happens if and ony if both A and B happen. A is the event that Node v does not transmit in Sot fu(j) i in Subframe j. B is the event that there are ess than m neighbors of Node v, other than Node u itsef, transmitting in Sot fu(j) i in Subframe j. Suppose that Events A and B are independent (referred to as Assumption 1). We have: ) P (X j i =1 = P (A)P (B). (16) Node v transmits in different time sots in Subframe j. We assume that these sots are randomy seected from Sot 0 to Sot p 1 in Subframe j (referred to as Assumption ). Thus, the probabiity that A happens is: ( p 1 ) 1 P (A) =1 ( p. (17) ) Let N r denote the number of ways, given a TSAF of Node u, fu(x), i for seecting (D max 1) other TSAFs, r out of which has the same vaue fu(j) i in Subframe j. There are ( ) p 1 (D max 1) ways to seect (D max 1) TSAFs from the remaining p 1 TSAFs. Thus, the probabiity that B happens is as foows: P (B) = m 1 N r r=0 ( p 1 (D max 1) ). (18) Given j, we can categorize the p TSAFs into p groups according to their vaues in Subframe j. That is, Group G n, where n =0, 1,...,p 1, contains those TSAFs, the vaues of which in Subframe j are n. Note that TSAFs over GF (p) are uniformy distributed over {0, 1,,...,p 1}. Thus, G n = p, where n =0, 1,...,p 1. Thus, the vaue of N r (r = 0, 1,...,m 1) is given as foows: ( )( p 1 N r p ) p =. (19) r (D max 1) r Fig. 5. Vaidation of anaytica resuts on average throughput (0) with simuation. Substituting (16) (19) into (14) and (15), we can obtain the average throughput of our agorithm as foows: [ G a = ( p 1 )] m 1 ( p 1 )( ) p p r (D max 1) r 1 1 ( r=0 p p ( ) p ) 1. (0) (D max 1) In order to vaidate our anaytica resuts, we run simuations without making Assumptions 1 and. Each simuation data vaue is obtained by averaging the resuts of 100 simuation runs, and the 95% confidence intervas are shown on each of the simuated point. The simuation resuts demonstrate that the two assumptions do not affect the accuracy of our anaytica resuts. The simuations were performed on a reguar graph mode [9], in which we put a N nodes around a circe, and connect each to its D max nearest neighbors on either side as neighbors. 3 Each node randomy seects one of its neighbors as the destination of its packets. Given that N = 100 and with the 4-MPR capabiity, we can observe that the simuations match our anaytica resuts we. In Fig. 5, we can aso observe that the average throughput increases with increasing when D max is sma, but remains amost the same with greater than a certain vaue when D max is arge. This impies that there exists a threshod vaue of, beyond which, the increase in the number of coisions is amost the same as that in the number of transmission sots introduced by assigning more TSAFs to a node. The threshod is arger when D max is smaer, and vice versa. Thus, the vaue of can be chosen according to this principe. B. Agorithm Description Based on the aforementioned discussions, we can propose the m-mpr--code topoogy-transparent scheduing agorithm with the m-mpr capabiity as beow: 1) Use Theorem 4 to seect the vaue of p for the given N, D max, and m such that the minimum guaranteed throughput is maximized. 3 If D max is odd, nodes which are opposite with each other are considered as neighbors.

7 5946 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 11, NOVEMBER 014 ) Based on the average throughput expression in (0), choose the proper vaue of that maximizes the average throughput (0) by exhaustive search for the given vaues of N, D max, and m. 3) Each node is randomy assigned different TSAFs, any two of which has no coincidence. 4) Each node cacuates its TSLV, which consists of p time sots for possibe transmissions, according to its assigned TSAFs. 5) Each node transmits its data packets at its assigned sots. Some discussions of the proposed agorithm are as foows. Firsty, the proposed agorithm is rather simpe. Steps 1 and of the proposed agorithm can be done offine at the network initiaization phase. The most time-consuming step is Step, since it requires exhaustive search of to achieve the maximum vaue of the average throughput. However, as discussed at the end of Section III-A, there exists a threshod vaue of, beyond which, the increase in the number of coisions is amost the same as that in the number of transmission sots introduced by assigning more TSAFs to a node. The threshod is arger when D max is smaer, and vice versa. Thus, the number of rounds of search of is actuay imited. We can observe in Fig. 5 that we ony need to search six vaues of (from one to six) to achieve the maximum average throughput when D max =0. Thus, its compexity is rather sma. Secondy, the proposed agorithm is distributed when the network operates, if no new nodes wi join the network after the network initiaization phase. Before the network operates, a centra entity runs Steps 1 and of the agorithm and tes the vaues of p and to a nodes. After that, the centra entity is not necessary anymore. Then, each node determines its transmission time sots in a fuy distributed manner according to Steps 3 and 4 of the agorithm. Each node transmits and receives packets according to Step 5 and the transmission time sots of each node wi not change when the network operates. Our agorithm works correcty when the network topoogy changes. The reasons are as foows. D max is an estimated parameter. As mentioned in Section II, D max is aways pessimisticay estimated according to empirica data to ensure that the actua number of interfering neighbors does not exceed the estimate. We aso show that the degradation of the performance of our agorithm is rather sma when the estimation of D max is not so accurate in Section IV. Moreover, athough sometimes some nodes may disappear, eave, and re-join the network, we assume that the tota number of nodes in the network, N, is a known parameter and remains constant. This is a genera and practica assumption in most previous work focusing on wireess networks [5], [7], [9], [13], [15] [17], [3], [34]. Note that nodes may join the network after the network initiaization phase in some cases. In such cases, a centra entity is required in our agorithm to maintain which codes are used by the existing nodes and which codes can be assigned to the new nodes. When a new node joins the network, it communicates with the centra entity and get its codes [1]. Thus, our agorithm cannot be considered as a fuy distributed scheduing agorithm in this case. However, our agorithm cannot be simpy considered as a centraized agorithm, since there is a significant difference between our agorithm and the traditiona centraized scheduing agorithm [], [9], [13], [3], [7]. In the traditiona centraized scheduing agorithms, the centra entity has to know the detaied network connectivity and traffic information other than the information of each node. When the centra entity fais, the agorithms cannot work correcty in mobie networks. In contrast, the centra entity in our agorithm ony needs to maintain which codes are used by the existing nodes and which codes can be assigned to the new nodes, which is a much easier task. When the centra entity fais, the existing nodes can sti transmit correcty, athough the new nodes may not join the network correcty. Especiay in the case that no node wi join the network after the network initiaization phase, the centra entity is not necessary in our agorithm after the initiaization phase. We aso show that even if the vaue of N changes as the network operates, the degradation of the performance of our agorithm is very sma in Section IV. C. Deay Anaysis In this subsection, we use a discrete time M/G/1 queuing mode in which the first customer of each busy period receives exceptiona service [30] to approximate the average packet deay in our agorithm. The simuation resuts in Section IV show that the approximation does not affect the accuracy of our anaysis. Packets arrive at an infinite buffer in a Poisson fashion with rate λ packets per sot. We assume that arrivas occur just after the beginning of a sot, departures take pace just before the end of a sot. Considering a given node as a server, its service sots are fixed. That is, the distribution of the service time of the packets arriving when the server is busy (there are other packets queued or being served) is different from that of the packets arriving when the server is ide (there are no packets queued or being served). We assume that the service time distribution of a packet which initiates a busy period is B 1 (x) with first and second moments b 1 and b 1. The distribution of service time of a subsequent packets in the same busy period is B (x) with first and second moments b and b, which is independent of B 1(x). Note that the service time of the packet that initiates a busy period is ess than that of the subsequent packets on average. That is, b1 < b. Thus, the average deay can be expressed as foows: E[W ]=E[W Ide]P 0 + E[W Busy](1 P 0 ), (1) where P 0 is the probabiity that the system is empty. According to the resuts obtained in [30], we have: 1 λ P 0 = b 1 λ( b b 1 ). () According to the Poaczek Khinchine Formua, the vaue of E[W Busy] can be obtained as foows: λ b (1+ b ) E[W Busy] = ( b 1 λ b ) + b. (3) In the foowing, we cacuate b = 1 μ. μ is the probabiity that the packet can be transmitted successfuy in a sot.

8 LIU et a.: TOPOLOGY-TRANSPARENT SCHEDULING IN MOBILE AD HOC NETWORKS WITH MPR CAPABILITY 5947 Consider a transmission from an arbitrary node u to its neighbor v. Given an arbitrary sot t,etc be the event that t TSLV u, D the event that t TSLV v and Node v has packets to transmit, and E the event that ess than m neighbors of Node v other than u itsef have packets to transmit in Sot t. Thus, we have: and μ = P (C)P (D)P (E). (4) According to the cacuation in Section III-A, we have: [ P (D) = 1 P (C) = p, (5) )] ( p + ) ( p 1 1 ( p 1 1 ) ( p ) (1 ρ), (6) where ρ is the probabiity that the node has packets to transmit and ρ =1 P 0. Moreover, P (E) can be expressed as foows: P (E) = m 1 r=0 N r + D max 1 r=m N r m 1 i=0 ( p 1 (D max 1) ( r ) i ρ i (1 ρ) r i ). (7) Substituting (5) (7) into (4), we can cacuate the vaue of μ numericay, athough it is difficut to obtain its cosed-form expression. Substituting the vaue of μ into (3), we can obtain the vaue of E[W Busy]. Note that E[W Ide]= b 1. We can obtain the average packet deay as foows: E[W ]= b 1 P 0 + E[W Busy](1 P 0 ). (8) It is difficut, if not impossibe, to cacuate the vaues of b and b 1. However, the detaied cacuation of queuing system parameters is not the focus of this work. Thus, we get the vaues of b and b 1 via simuations. Substituting () (7) into (8), we can get the average packet deay. IV. PERFORMANCE EVALUATION In this section, we quantitativey compare our m-mpr-code topoogy-transparent scheduing agorithm with the conventiona topoogy-transparent agorithm designed for the coision-based reception mode in [18] (known as Ju s agorithm). We do not incude the agorithm in [33] as a comparison, since it is ony designed for static networks with star topoogy. It assumes that a centra node knows a the information of other nodes and cannot be appied in mobie ad hoc networks. A simuations have been conducted using Matab. We compare our agorithm with the centraized confict graph cooring agorithm in [7]. In this confict graph cooring agorithm, a centra scheduer knows the detaied network connectivity and traffic information, constructs a confict graph, and appies the vertex cooring agorithm to compute and distribute the transmission schedue to each node. When the network connectivity and traffic information change, we assume that the centra scheduer can get the updated network information, re-compute, and distribute the updated schedues immediatey without introducing any overhead. Thus, this idea agorithm is impractica in mobie networks. 4 However, our agorithm is independent of topoogy changes, which can be easiy impemented in mobie networks. We aso compare our agorithm with sotted ALOHA with the m-mpr capabiity [16], [17], athough the atter agorithm cannot offer any throughput guarantees. A. Simuation Setup We conduct simuations on two graph modes, namey, the geometric mode for the average performance and the reguar graph mode [9] for the worst performance. In the geometric mode, we adopt the Gauss-Markov mobiity mode, which has been shown to be more reaistic than the widey used Random Waypoint mode [3]. A nodes are uniformy and randomy distributed in a region of 1000 m 1000 m initiay. The tuning parameter θ is used to present different eves of randomness in the Gauss-Markov mode. We set θ =0.5. GivenD max, we set the interference range of each node R I such that the probabiity that the number of interfering neighbors of an arbitrary node exceeding D max, which N 1 ( is N 1 ) ( ) πr i ( ) N 1 i, I i A 1 πr I A is smaer than i=d max For exampe, R I = 140 mif(n,d max ) = (100, 10). In the reguar graph mode, the degree of each of the N nodes is set to D max. In other words, each node has exacty D max interfering neighbors. Thus, the average number of interfering neighbors in the geometric mode is ess than that in the reguar graph mode with the same D max. We appy the optima frame structure derived in Theorem 4. is set to different vaues for different network parameters, as discussed ater in detais. For each simuation point, we conduct a simuation run for 100 continuous frames in the geometric mode 5 and 100 randomy generated topoogies in the reguar graph mode, uness stated otherwise. The 95% confidence intervas are aso drawn in the figures. B. Simuation Resuts 1) Effect of D max on Average Throughput: Given that N = 100, we investigate the performance of our (m, )-TTS agorithm with different D max settings from 6 to 40. A arger D max indicates that the network is denser and there are more possibe conficts. In the reguar graph mode, we set = m +. However, we set = m +6 in the geometric mode, since the average number of interfering neighbors under the geometric mode is smaer than that under the reguar mode. We can see that the average throughput of our agorithm is about 60% of that of the idea cooring agorithm. However, the idea cooring agorithm is impractica, since it is a centraized agorithm. The cost of schedue recomputation is unacceptabe in mobie networks, since the network connectivity and traffic information change frequenty. As shown in Figs. 6 and 7, we can see that our agorithm outperforms sotted ALOHA for 4 In fact, this idea agorithm is impractica even in static networks, since the traffic oad of each node changes as time evoves, resuting in costy recomputation and distribution of new schedues. 5 The movement of each node foows the Gauss-Markov mobiity mode.

9 5948 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 11, NOVEMBER 014 Fig. 6. Average throughput against maximum node degree (reguar graph mode). (a) m =.(b)m =4. Fig. 7. Average throughput against maximum node degree (geometric mode). (a) m =.(b)m =4. most cases, especiay when the D max is arge. This is different from the case in the coision-based reception mode. It has been shown [4] that the average throughput of topoogy-transparent scheduing agorithm is not as good as that of sotted ALOHA under the coision-based reception mode. The superiority of our agorithm in the MPR mode impies that our agorithm can take advantage of the powerfu MPR capabiity better. We can observe that the average throughput of the nodes in the geometric mode is better than that of the nodes in the reguar graph mode. This is reasonabe, since the average node degree of the geometric mode is smaer than that of the reguar graph mode. Moreover, the superiority of our agorithm over sotted ALOHA is greater in the geometric mode than that in the reguar graph mode. This is because the nodes cannot seect the optima transmission probabiity in the geometric mode, since they do not know the number of its neighbors. As shown in Figs. 6 and 7, our agorithm dramaticay outperforms the conventiona topoogy-transparent scheduing agorithm. We can see that the average throughput of the conventiona topoogy-transparent scheduing agorithm remains amost the same when m increases, impying that it cannot take fu advantage of the MPR capabiity. ) Effect of Inaccuracies in Estimating D max and N on Performance: We use the performance ratio as the metric to investigate the effect of inaccuracies in the estimation of D max on the average throughput of the proposed agorithm. The performance ratio is defined as the average throughput under the actua D max divided by that under the design vaue of D max. Fig. 8 shows the simuation resuts under the reguar graph mode and the geometric mode, respectivey. Given that the design parameters (N,D max ) is (100, 0), we study the effect of inaccuracies in the estimation of D max on the average throughput of the proposed (m, )-TTS agorithm, where m is set to two and four. According to Theorem 4, the optima vaues of p for different parameters are shown in Tabe II. By varying the actua maximum node degree from 6 to 40, we can observe in Fig. 8(a) that the performance ratio is smaer (arger) than one when the actua maximum node

10 LIU et a.: TOPOLOGY-TRANSPARENT SCHEDULING IN MOBILE AD HOC NETWORKS WITH MPR CAPABILITY 5949 Fig. 8. The effect of inaccuracies in the estimation of D max on the throughput. (a) Reguar graph mode. (b) Geometric mode. TABLE II OPTIMAL VALUE OF p degree is arger (smaer) than the design vaue of D max.the performance ratio decreases with increasing actua maximum node degree. The performance ratio is cose to one when the actua maximum node degree is cose to the design vaue (D max = 0). As the difference between the actua and design vaues of D max increases, the penaty on the network performance increases (when the actua vaue of D max is arger than its design vaue). We can aso observe that the performance ratio of (4, 6)-TTS agorithm is more sensitive to the difference between the actua and design vaues of D max than that of (, 4)-TTS agorithm under the reguar graph mode. This is due to the fact that the maximum number of packet faiures of (4, 6)-TTS agorithm increases (decreases) faster with the increasing (decreasing) actua vaue of D max than that of (, 4)-TTS agorithm. The reason is expained as foows. The increased maximum number of packet faiures in one frame is 4 4 Δ D =8Δ D for (m, ) =(, 4), but 6 6 Δ D 4 =9Δ D for (m, ) =(4, 6), where Δ D is the difference between the actua and design vaues of D max. Simiar phenomena are aso exhibited in Fig. 8(b). Comparing Fig. 8(a) and (b), we can see that the effect of inaccuracies in the estimation of D max on the performance in the geometric mode is much smaer than that in the reguar graph mode. The average node degree is typicay smaer than the maximum node degree D max in reaity. Thus, we concude that our agorithm is insensitive to the accuracy in the estimation of D max. Fig. 9 studies the effect of inaccurate estimation of N on the average throughput. In Fig. 9, we use (N,D max ) = (100, 0) as our design vaues. The corresponding optima vaues of p are shown in Tabe II. By varying the actua number of nodes N from 100 to 800, we can observe in Fig. 9 that the performance ratio equas one for most cases, indicating that an inaccurate estimation of N has amost no effect on the performance. The reason is as foows. According to the discussion in Section III and (9), our agorithm can support up to p p nodes. Thus, the corresponding maximum number of nodes in the network that our agorithm can support is isted in Tabe III. The ony exception happens when the actua number of nodes, in the (4, 6)-TTS agorithm, is 800, which is arger than the maximum number of nodes supported. In this case, some nodes may be assigned the same poynomias. If these nodes with the same poynomias are within the interference range of each other, their transmissions fai. However, the situation that the actua vaue of N (800) is much arger than the designed vaue (100) may never happen in we-designed networks. Thus, we concude that the effect of inaccurate estimation of N on the performance is very sma and can be ignored. 3) Supporting Heterogeneous Traffic: Given N, weinves- tigate the performance of our (m, ) TTS agorithm with different settings on D max in heterogeneous networks. Different number of codes can be assigned to provide different QoS for different casses of nodes in heterogeneous networks. Without oss of generaity, we suppose N nodes are divided to two casses, namey, Cass V 1 with a higher QoS requirement and Cass V with a ower QoS requirement. The number of nodes in Cass V 1 and V is βn and N βn, respectivey, where 0 <β<1. Here, we assume that β = and codes are assigned to the nodes in Casses V 1 and V, respectivey, where 1 =. Note that the maximum vaue of packet faiures for a transmission from an arbitrary node in Casses V 1 and V is Nf,V max 1 = k1 + =4k 4(Dmax 1)k + m and Nf,V max =k 1 + =k (Dmax 1)k + m, respectivey. Thus, the average minimum guaranteed throughput is as foows: (Dmax 1)k1 m (Dmax 1)k 1 m G h min = β 1p N max f,v 1 p +(1 β) p N max f,v p, (9)

11 5950 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 11, NOVEMBER 014 Fig. 9. The effect of inaccuracies in the estimation of N on the throughput. (a) Reguar graph mode. (b) Geometric mode. TABLE III MAXIMUM NUMBER OF NODES SUPPORTED where the foowing constraints must be satisfied: p N p, 1 + (30) 1 p>nf,v max 1, (31) p>nf,v max. (3) Let p 3 N max f,v 1 +N max f,v be the argest prime smaer than or equa to 1.5 and p 4 be the smaest prime arger than N max +N max f,v 1 f,v 1.5, respectivey. Thus, foowing the proof of Theorem 4, we obtain the optima vaue of p that achieves the maximum vaue of G h min is p = arg {p 3,p 4 } max(g min ), where max(g min ) satisfies (6) (8), or is the smaest prime arg {p 1,p } satisfying (6) (8) otherwise. 6 We set N = 100, β =0.5, and vary D max from 6 to 40. = m +, 1 =, where m is set to two and four. We study the average throughput of the nodes in Casses V 1 and V in the geometric mode. As shown in Fig. 10, our agorithm works we as we supporting heterogeneous traffic. The simuation resuts for the reguar graph mode foow the same trend, and are thus omitted due to space imitations. 4) Effect of m-mpr Capabiity on Minimum Guaranteed Throughput and Average Throughput: We study the effect of the m-mpr capabiity on the minimum guaranteed throughput and average throughput under two different m-mpr mode, namey, the strong m-mpr mode in (3), (4) and the m-mpr 6 Our agorithm can aso support more than two casses of nodes in heterogeneous networks by choosing the proper vaue of p according to the aforementioned discussion. Fig. 10. Performance of our agorithm supporting heterogeneous traffic. mode in [] achieved by the bind interference canceation agorithm using poynomia phase-moduating sequences (referred to as PPS m-mpr mode). In the PPS m-mpr mode, C n,i is given as foows: C n,i = F n,i(c) C n, (33) if n =1,,...,mand i =0, 1,...,n, and C n,i =0otherwise. C is the size of a codebook (the number of poynomia phasemoduating sequences) and F n,i (C) is the number of ways that exacty i out of n taken codes are different. Interested readers are referred to (9) (34) in [] for the detaied expression of C n,i.wesetc =3according to the discussion in []. Given that N = 100, we investigate the performance of our agorithm with different settings on D max from 6 to 40. The vaue of p is determined according to Theorem 4. For exampe, N max f = k + (D max 1)k =9when D max =0,

12 LIU et a.: TOPOLOGY-TRANSPARENT SCHEDULING IN MOBILE AD HOC NETWORKS WITH MPR CAPABILITY 5951 Fig. 11. The effect of m on the improvement of the minimum guaranteed throughput. 1N max f = m =4, and k =1. Thus, according to (14), p = = 9 4 =46. Reca that p is a prime and we can obtain that p =47. As shown in Fig. 11, we can see that the minimum guaranteed throughput of our agorithm can be greaty improved with increasing m. When D max =6, the performance of our agorithm with m =10is ony four times of that empoying the coision-based reception mode. However, there is over eight times increase in the performance with m =10 when D max =40. This indicates that a sma vaue of m is sufficient when the network is sparse. When the network is dense, the minimum guaranteed throughput amost increases with m ineary. Simiar phenomenons are aso exhibited in Fig. 1. We can see that the increase in the average throughput of our agorithm, compared to that empoying the coisionbased reception mode, is even arger than m times, when the network is dense. Thus, we concude that our agorithm can take fu advantage of the powerfu MPR capabiity. Comparing Fig. 1(a) and (b), we can aso observe that, under the same network configuration, the average throughput of our agorithm under these two m-mpr modes, namey, the strong m-mpr mode and the PPS m-mpr mode, is amost the same. This vaidates our statement that athough the strong m-mpr mode is not the most accurate refection of the MPR capabiity in reaity, it can be used to offer enough insights of the effect of the m-mpr capabiity in the physica ayer on the operation and performance of MAC ayer. 5) Packet Deay and End-to-End Performance: Given that N = 100 and D max =0, we investigate the average packet deay of our (m, )-TTS agorithm with different settings of the vaues of λ and m. In order to vaidate our anaytica resuts, we run simuations under the reguar graph mode [9], in which each node has D max interfering neighbors. We run each simuation for 50 data frames. In the simuation, we assume that each queue has infinite capacity and uses First-In-First- Out (FIFO) service discipine. The destination of a packet is randomy chosen from its neighbors. The transmission sots of each node are assigned deterministicay using our (m, )- TTS agorithm. We set = m +. According to Theorem 4, the vaues of p in the (, 4)-TTS agorithm and (4, 6)-TTS agorithm are 83 and 67, respectivey. As shown in Fig. 13, our anaytica resuts match we with those from our simuations. We investigate the end-to-end performance of our agorithm in terms of throughput. The average end-to-end throughput of scheduing agorithms highy depends on the network topoogies and routing protoco appied. The design and impementation of the topoogy contro and routing protocos that can baance the network oads to optimize the network performance are out of the scope of this paper. Thus, we use a simpe reguar graph mode in the simuations, in which each node connects the nearest D max nodes as its neighbors. Let S and Dest be the sets of source nodes and destination nodes, respectivey. N SD out of N nodes are randomy seected. The first N SD nodes beong to S, and the other N SD ones beong to Dest. A pair of a source S(i) and a destination Dest(i) are caed a sourcedestination (S-D) pair, where i =1,,...,N SD. The Beman- Ford agorithm [10] is used as the routing protoco to cacuate the shortest path between each S-D pair. For each of N SD source nodes, the arrivas foow a Poisson distribution with rate λ packets per sot. We define the average end-to-end throughput as the average number of packets deivered from sources to their destinations per sot. We set (N,D max = (100, 0)), (m, ) = (4, 6)), and N SD to be 10, 0, and 50. Varying λ from 0.01 to 0.1, we run each simuations for 50 data frames. As shown in Fig. 14, the average end-to-end throughput increases amost ineary with the arriva rate when λ is sma. When the arriva rate becomes arger, the end-to-end throughput increases more sowy or remains the same. Comparing Figs. 7(b) and 14, we can see that when the number of S-D pairs is sma, the end-toend throughput is comparabe to the one-hop throughput shown in Fig. 7(b). This is due to the fact that ony a sma fraction of nodes have packets to transmit when the number of S-D pairs is sma. Thus, the number of interfering neighbors of each node here is much smaer than that in the simuations of Fig. 7(b). With the increasing number of S-D pairs, the average end-toend throughput decreases, especiay when the arriva rate is arge. V. C ONCLUSION In this paper, we propose a nove m-mpr--code topoogytransparent scheduing ((m, )-TTS) agorithm for mobie ad hoc networks with the MPR capabiity. As far as we know, this is the first schedue-based agorithm with MPR, which is independent of topoogy changes. Our agorithm is easy to impement and is obivious to the network topoogy changes, because it ony needs goba parameters such as the number of nodes and the maximum node degree in the network. Moreover, our agorithm provides the minimum guaranteed throughput to each participating node. The proposed agorithm greaty improves the minimum guaranteed throughput and average throughput of our agorithm by assigning mutipe poynomias to each node to take fu advantage of the powerfu MPR capabiity. The performance improvement of our (m, )-TTS agorithm over the conventiona topoogy-transparent scheduing agorithms designed for the coision-based reception mode is

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Liew, How does mutipe-packet reception capabiity scae the performance of wireess oca area networks? IEEE Trans. Mobie Comput., vo. 8, no. 7, pp , Ju [33] Q. Zhao and L. Tong, A mutiqueue service room MAC protoco for wireess networks with mutipacket reception, IEEE/ACM Trans. Netw., vo. 11, no. 1, pp , Feb [34] Q. Zhao and L. Tong, Semi-bind coision resoution in random access wireess ad hoc networks, IEEE Trans. Signa Process., vo. 48, no. 10, pp , Oct Yiming Liu (S 14 M 14) received the B.Eng. and Ph.D. degrees in eectronic engineering from Tsinghua University, Beijing, China, in 009 and 014, respectivey. He is an Engineer with the China Academy of Eectronics and Information Technoogy, Beijing, China. His research interest is in the area of wireess communications and networking, specificay in transmission scheduing and cross-ayer design for wireess mutihop networks. Victor O. K. Li (S 80 M 81 F 9) received the S.B., S.M., E.E., and Sc.D. degrees from MIT, Cambridge, MA, USA, in 1977, 1979, 1980, and 1981, respectivey, a in eectrica engineering and computer science. He is a Chair Professor of information engineering and the Head of the Department of Eectrica and Eectronic Engineering at The University of Hong Kong (HKU), Pokfuam, Hong Kong, where he has aso served as the Associate Dean of Engineering and the Managing Director of Versitech Ltd., which is the technoogy transfer and commercia arm of HKU. He served on the board of China.com Ltd., and now, he serves on the board of Sunevision Hodings Ltd. and Anxin-China Hodings Ltd., which are isted on the Hong Kong Stock Exchange. Previousy, he was a Professor of eectrica engineering at the University of Southern Caifornia (USC), Los Angees, CA, USA, where he was aso the Director of the USC Communication Sciences Institute. Sought by government, industry, and academic organizations, he has ectured and consuted extensivey around the word. He has received numerous awards, incuding the PRC Ministry of Education Changjiang Chair Professorship at Tsinghua University, the U.K. Roya Academy of Engineering Senior Visiting Feowship in Communications, the Croucher Foundation Senior Research Feowship, and the Order of the Bronze Bauhinia Star, Government of the Hong Kong Specia Administrative Region, China. He is a Registered Professiona Engineer and a Feow of the Hong Kong Academy of Engineering Sciences, the IAE, and the HKIE. Ka-Cheong Leung (S 95 M 01) received the B.Eng. degree in computer science from the Hong Kong University of Science and Technoogy, Kowoon, Hong Kong, in 1994 and the M.Sc. degree in eectrica engineering (computer networks) and the Ph.D. degree in computer engineering from the University of Southern Caifornia, Los Angees, CA, USA, in 1997 and 000, respectivey. From 001 to 00, he was a Senior Research Engineer at Nokia Research Center, Nokia Inc., Irving, TX, USA. Between 00 and 005, he was an Assistant Professor at the Department of Computer Science, Texas Tech University, Lubbock, TX. Since June 005, he has been with The University of Hong Kong, Pokfuam, Hong Kong, where he is currenty an Assistant Professor in the Department of Eectrica and Eectronic Engineering. His research interests incude transport ayer protoco design, wireess packet scheduing, congestion contro, routing, and vehice-to-grid (VG). Lin Zhang received the B.Sc., M.Sc., and Ph.D. degrees from Tsinghua University, Beijing, China, in 1998, 001, and 006, respectivey. He is currenty a Visiting Associate Professor at Stanford University, Stanford, CA, USA, and an Associate Professor at Tsinghua University, where he has been happiy teaching seected topics in communication networks and information theory to senior undergraduate and graduate students. He is a co-author of more than 40 peer-reviewed technica papers and five U.S. or Chinese patent appications. His current research focuses on wireess sensor networks, distributed data processing, and information theory. In 006, he ed a 008 Beijing Oympic Stadium (the Bird s Nest ) structura security surveiance project, which depoyed more than 400 wireess temperature and tension sensors across the stadium s stee support structure and dome. The system adopted a fexibe spectrum sensing and adaptive mutihop routing agorithm, to overcome strong radio interference and ong-distance transmission channe-fading, and payed a critica roe in the construction of the stadium. Since then, he has impemented wireess sensor networks in a wide range of appication scenarios, incuding underground mine security, precision agricuture, and industria monitoring. Since 008, he has been working in cose association with CISCO to deveop a Metropoitan Area Sensing and Operating Network (MASON), which provides a smart-city and inteigent-urbanization sensor network system for metropoitan areas and has attracted the interest of severa arge-sized Chinese cities, incuding Beijing, Shenzhen, Tianjing, and Chengdu. Recenty, he aso has ed two Nationa Science Foundation of China projects, three Nationa High-Tech Deveoping (863) projects, and more than ten research projects funded primariy by private industry in the area of wireess sensor networks. Dr. Zhang and his team were the recipient of the IEEE/ACM SenSys 010 Best Demo Awards. In 004 and 010, he was a recipient of the Exceent Teacher Awards from Tsinghua University.