Topology-Transparent Scheduling in Mobile Ad Hoc Networks ...kcleung/papers/journals... · The powerful MPR capability allows us to assign more codes to a user, since a collision

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

Topology-Transparent Scheduling in Mobile Ad HocNetworks With Multiple Packet

Reception CapabilityYiming Liu, Member, IEEE, Victor O. K. Li, Fellow, IEEE, Ka-Cheong Leung, Member, IEEE, and

Lin Zhang, Member, IEEE

Abstract—Recent advances in the physical layer have enabledwireless devices to have multiple packet reception (MPR) capa-bility, which is the capability of decoding more than one packet,simultaneously, when concurrent transmissions occur. In this pa-per, we focus on the interaction between the MPR physical layerand the medium access control (MAC) layer. Some random ac-cess MAC protocols have been proposed to improve the networkperformance by exploiting the powerful MPR capability. However,there are very few investigations on the schedule-based MAC pro-tocols. We propose a novel m-MPR-l-code topology-transparentscheduling ((m, l)-TTS) algorithm for mobile ad hoc networkswith MPR, where m indicates the maximum number of concur-rent transmissions being decoded, and l is the number of codesassigned to each user. Our algorithm can take full advantageof the MPR capability to improve the network performance.The minimum guaranteed throughput and average throughput ofour algorithm are studied analytically. The improvement of our(m, l)-TTS algorithm over the conventional topology-transparentscheduling algorithms with the collision-based reception model islinear with m. The simulation results show that our proposedalgorithm performs better than slotted ALOHA as well.

Index Terms—Medium access control (MAC), multiple packetreception (MPR), topology-transparent scheduling (TTS).

I. INTRODUCTION

T RADITIONALLY, medium access control (MAC) algo-rithms assume the collision-based reception model, in

which a packet failure (equivalently collision) happens, if morethan one interfering neighbor of a user transmit simultane-ously. Thus, the fundamental goal of the conventional MACalgorithms is to improve the throughput by resolving colli-sions. In random access MAC protocols, each user is allowedto transmit probabilistically with the goal of minimizing thecollision probability. In schedule-based MAC protocols, each

Manuscript received July 11, 2013; revised January 9, 2014 and May 28,2014; accepted September 6, 2014. Date of publication September 17, 2014;date of current version November 7, 2014. This work was supported in partby the Research Grants Council of Hong Kong under Grant HKU 714310E.The associate editor coordinating the review of this paper and approving it forpublication was A. A. Abouzeid.

Y. Liu is with the China Academy of Electronics and Information Technol-ogy, Beijing 100041, China (e-mail: [email protected]).

V. O. K. Li and K.-C. Leung are with the Department of Electrical andElectronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong(e-mail: [email protected]; [email protected]).

L. Zhang is with the Department of Electronic Engineering, TsinghuaUniversity, Beijing 100084, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2014.2358644

user is scheduled to transmit in order to avoid collisions.However, this collision-based reception model does not reflectthe capabilities of the advanced wireless transceivers nowadays.Advanced signal processing technologies have enabled wirelesstransceivers to correctly decode multiple packets transmittedsimultaneously, referred to as the multiple packet reception(MPR) capability. Various technologies can be used to enablethe MPR capability, and can be categorized into two differentgroups, namely, training-based algorithms and so-called blindinterference cancellation algorithms.

In training-based algorithms such as space-time coding andmultiuser detection (MUD) with successive interference can-cellation (SIC) [20], the knowledge of propagation channelsand signal waveforms is assumed to be available. However, thisis difficult to obtain and may be impractical in mobile ad hocnetworks. In blind interference cancellation algorithms [11],[22], [34], the signal of interest is modulated by a known ampli-tude or phase variation. This allows the intended receiver to es-timate and suppress the interfering sources without knowing thechannel states. These blind interference cancellation algorithmscan separate the colliding packets and handle the near-far effect[22] without assuming the knowledge of channel, renderingmakes them suitable for mobile ad hoc networks. However, theyrequire a higher computational complexity in signal process-ing during the packet decoding phase. Yet, the complexity isacceptable. It has been shown in [22] that the complexity ofthe blind interference cancellation algorithm increases linearlywith the increasing m, where m is the maximum number ofconcurrent transmissions being decoded. Thus, we assume thatthe underlying MPR capability in this paper is achieved by thiskind of blind interference cancellation algorithms.

In this paper, we focus on studying the cross layer interactionbetween the MPR physical and MAC layers, rather than thetechnologies that enable the MPR capability. Specifically, wefocus on the throughput in the MAC layer, defined as thenumber of successful transmission. Under the m-MPR model,in which the receiver can correctly decode at most m pack-ets simultaneously, we propose an m-MPR-l-code topology-transparent scheduling ((m, l)-TTS) algorithm. Time is dividedinto equal-sized time slots, grouped into frames. In each frame,a user is allocated some transmission slots based on the as-signed codes. In the traditional topology-transparent schedulingalgorithms with the collision-based reception model, each useris assigned a unique code to achieve the maximum throughput.

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LIU et al.: TOPOLOGY-TRANSPARENT SCHEDULING IN MOBILE AD HOC NETWORKS WITH MPR CAPABILITY 5941

The powerful MPR capability allows us to assign more codesto a user, since a collision does not necessarily lead to a packetfailure with the MPR capability. In our (m, l)-TTS algorithm,we assign l transmission codes to each user due to the fact thatthe increase in the number of packet failures is much less thanthat in the transmission time slots, resulting from the m-MPRcapability. We believe that the difference between the proposedalgorithm and the traditional topology-transparent schedulingalgorithms is nontrivial. Firstly, in our algorithm, the optimalframe structure that maximizes the throughput is totally differ-ent from that in the traditional algorithms. Secondly, assigningmultiple codes (polynomials) to a user introduces a new prob-lem, namely, we have to assign multiple codes (polynomials)in such a way to avoid coincidences among different codes(polynomials).1 We detail the assignment of multiple codesin Section III. In addition, we show that the improvement ofour (m, l)-TTS algorithm with the m-MPR capability over theconventional topology-transparent scheduling algorithms withthe collision-based reception model is linear with increasingm. The simulation results show that our proposed algorithmperforms better than random access protocols, namely, slottedALOHA with MPR.

A. Motivation and Related Work

Several random access MAC protocols have been proposed[16], [17], [21] to improve the network performance by takingadvantage of the powerful MPR capability. The random accessapproaches cannot provide deterministic delay and through-put bounds. That is why most networks offering throughputand delay guarantees, say, the tactical networks, such as theEnhanced Position Location Reporting System (EPLRS) [19]and the Joint Tactical Radio System (JTRS) [26], implementdeterministic, schedule-based protocols, such as time divisionmultiple access (TDMA).

The related work on schedule-based protocols can be cate-gorized into two different groups, namely, topology-dependentand topology-transparent, based on whether the detailed net-work connectivity information is required. Existing topology-dependent approaches focus on finding a minimum-length,conflict-free schedule based on the detailed network topology.This problem is proved to be NP-complete [2], [13], [23].More importantly, re-computation and information exchangesare required to maintain accurate network topology informationand distribute the new schedules when the network topol-ogy changes. Distributed topology-dependent scheduling algo-rithms are introduced in [25]. However, it has been shown that ittakes several minutes to obtain and distribute the updated sched-ules [25], which is too long in mobile networks, especially forreal-time traffic. Thus, the robustness and effectiveness of thesetopology-dependent scheduling algorithms are undermined inlarge, highly dynamic, wireless mobile ad hoc networks.

In order to overcome the aforementioned limitations inmobile ad hoc networks, topology-transparent scheduling al-gorithms have been proposed [5], [7], [14], [18]. In thesealgorithms, no updates on network connectivity information are

1A coincidence of two polynomials is defined in Section II-B.

required. In topology-transparent scheduling algorithms, eachnode is assigned multiple time slots for transmission in eachframe. It is guaranteed that there is at least one conflict-freeslot per frame. Chlamtac and Farago [7] developed a topology-transparent algorithm that guarantees at least one collision-free time slot in each frame time, but the performance is evenworse than the conventional TDMA in some cases. Ju andLi [18] proposed another algorithm to maximize the minimumguaranteed throughput. However, only unicast communicationis considered. Cai et al. [5] proposed a broadcast schedulingalgorithm, modified Galois field design (MGD), which sendsthe same message multiple times during one frame time in orderto guarantee exactly one successful broadcast transmission perframe.

However, there are few schedule-based MAC algorithmswith MPR, such as TDMA. The conventional TDMA protocols[2], [9], [13], [23], [25] construct transmission schedules toavoid collisions and maximize the throughput. This does notmatch the characteristics of MPR as it allows collisions. Webelieve that this is the inherent reason why there is much lessinterest in the idea of designing schedule-based TDMA pro-tocols with MPR. However, we find that topology-transparentscheduling algorithms [5], [7], [14], [18] are suitable forthe MPR capability. In the conventional topology-transparentscheduling algorithms with collision-based reception, each useris assigned a unique transmission schedule or code. Althoughthe transmission schedules are not collision-free, each user isguaranteed to have at least one conflict-free transmission slotper frame.

Most existing MAC protocols with the MPR capability arerandom access algorithms. Slotted ALOHA was extensivelystudied in [16], [17] and can significantly improve networkperformance. A multiple access protocol based on receiver-controlled transmission was proposed in [21]. Although thiswork pointed out that it is necessary to redesign the MACprotocol so as to make the best use of the MPR capability,it can only be used in predefined topologies and cannot beextended to an arbitrary ad hoc network. Another randomaccess protocol was proposed in [33]. It takes advantage ofthe MPR capability to support users with different quality ofservice requirements. Some work [6], [8], [33] focused oncarrier sense multiple access (CSMA) with the MPR capability.However, it has been shown in [8] that the throughput gain ofCSMA over slotted ALOHA is very limited, especially whenthe MPR capability increases. All these CSMA algorithms aredesigned only for wireless local area networks (WLANs) andcannot be applied in multihop ad hoc networks, since it hasbeen shown that CSMA suffers from serious instability andunfairness issues in such networks [31]. Reference [28] studiedthe effect of the MPR capability on the network capacity andenergy efficiency. A survey paper on MPR for wireless randomaccess networks can be found in [20]. Reference [27] proposeda polynomial-time heuristic algorithm to solve the joint routingand scheduling problem with the MPR capability. It is similarto our work, as its scheduling algorithm can provide throughputand delay guarantees. However, it is a centralized algorithmwhich needs to know all the detailed network connectivity andtraffic information, making it impractical in mobile networks.


Fig. 1. The frame structure.

In this paper, we propose a schedule-based scheduling algo-rithm, which is oblivious to the network changes in mobilenetworks and takes advantage of the powerful MPR capability.Thus, the proposed algorithm can provide throughput and delayguarantees in mobile networks with the MPR capability.

The remainder of this paper is organized as follows. InSection II, we present our system model and some definitionsand theorems which will be used in the following sections.Section III presents the details of our proposed algorithm. Theaverage throughput and packet delay of the proposed algorithmare also derived. Extensive simulations are conducted to eval-uate the performance of our proposed algorithm in Section IV.The effect of inaccuracies in the estimation of network param-eters such as the maximum node degree and the number ofnetwork nodes on the performance of the proposed algorithmis also investigated. We conclude in Section V.

II. SYSTEM MODEL

A. Network Model

A mobile ad hoc network consisting of N nodes can berepresented by a graph G(V,E). V is the set of all networknodes labeled from 1 to N and E is the set of all edges. If Nodev is within the interference range of Node u, an edge denotedby (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 ofits interference neighbors. The maximum node degree Dmax isdefined as Dmax = max

u∈VD(u). We assume that Dmax is much

smaller than the number of nodes N and remains constant whilethe network topology changes [9]. In practice, empirical datamay be used to estimate Dmax. In addition, Dmax is alwayspessimistically estimated to ensure that the actual number ofinterfering neighbors does not exceed the estimate. We showthat the performance of our algorithm is insensitive to theaccuracy in the estimation of Dmax in Section IV. Thus, itis unnecessary to use distributed online method to obtain themaximum node degree when the network operates.

We focus on TDMA networks. Time is divided into equal-sized frames. Each frame is further divided into p subframes,each of which consists of p synchronized, equal-sized timeslots, where p is a prime. The frame structure is shown inFig. 1. Fig. 1 demonstrates some transmission time slots of anarbitrary node i, in which an “A” is located. The transmissiontime slots are determined by one of its time slot allocationfunctions, which is defined and discussed later. Synchronizationcan be achieved by Global Positioning System (GPS) or thesynchronization algorithms such as the one proposed in [12].

The MPR capability can be modeled by an n× n MPRmatrix C [16], [17], [21], [27], [33], where Cn,i is the con-

ditional probability that i packets are correctly decoded giventhat n packets are transmitted simultaneously by the nodes inthe interference range of a receiving node.

The MPR matrix is a function of many parameters, suchas the channel conditions, the modulation technologies, andthe signal processing technologies, and thus can take manydifferent forms. In this paper, we assume a generally usedstrong m-MPR model [16], [17], [21], [27], [33]. In this case,a user can correctly decode all the packets if the number ofconcurrent transmissions within its interference range is notgreater than m, and none of the packets if the number ofconcurrent transmissions exceeds m. That is,

Ci,i = 1, (1)

if i = 1, 2, . . . ,m, and

Cn,i = 0, (2)

otherwise. The conventional collision-based reception modelcan be modeled by the 1-MPR matrix. We also investigateanother m-MPR model achieved by the blind interferencecancellation algorithm using polynomial phase-modulatingsequences [22] in Section IV.

We assume that the transmission channel is error-free. Thus,the transmission from Node u to Node v fails when: 1) Node vis also transmitting, or 2) there are more than m− 1 other nodesin v’s interference range transmitting simultaneously. Note thatthe transmission from u to v fails when v is also transmitting.This is because the self-interference of v is much stronger thatthe received signal from u, leading to the fact that v with theMPR capability cannot decode a packet from u. This problemcan be solved by applying different full-duplex technologies[4]. However, it is beyond the scope of this work.

B. Definitions and Theorems

In the following, we present some definitions and theoremsused throughout the paper.

Definition 1 [18]: A polynomial with degree k mod p can be

expressed as f(x) =k∑

i=0

aixi(mod p).

Definition 2: A coincidence of two polynomials with degreek mod p is defined as the root of the difference of these twopolynomials. That is, if fu(j)− fv(j) = 0, j is the coincidenceof fu(x) and fv(x), where j = 0, 1, . . . , p− 1.

Theorem 1 [18]: There are at most k coincidences of twoarbitrary different polynomials 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 slot allocation function (TSAF) group consist-ing of l unique TSAFs, i.e., polynomials with degree k mod p,namely, {f j

v (x)}, where j = 1, 2, . . . , l. The polynomials areassigned to each node in such a way that any two polynomialsof an arbitrary node has no coincidence. The detailed methodof assigning polynomials is discussed in the next section.

This TSAF group is used to calculate the allocated trans-mission time slots in one frame for each node. For an arbitrarynode v, it transmits in l time slots {f j

v (i)} in Subframe i,


Fig. 2. The network topology.

Fig. 3. The illustration of definitions.

where i = 0, 1, . . . , p− 1 and j = 1, 2, . . . , l. Thus, eachnode transmits in lp time slots in one frame. We refer tothese lp allocated transmission time slots of an arbitrary nodev as its time slot location vector (TSLV), TSLV (v), i.e.,TSLV (v) = (f1

v (0), . . . , flv(0), f

1v (1), . . . , f

lv(1), . . . , f

1v (p−

1), . . . , f lv(p− 1)).

Theorem 2: There are at most kl2 coincidences of twoarbitrary nodes u and v, each of which is assigned l differentpolynomials with degree k mod p.

Proof: According to Theorem 1, there are at most k coin-cidences of an arbitrary polynomial of Node u and an arbitrarypolynomial of Node v. Each node has l unique polynomials.Thus, there are at most kl2 coincidences of two arbitrary nodesu and v. �

Definition 4: Given the m-MPR model, the number ofpacket failures for a transmission from an arbitrary node v toits destination u, Nf , is defined as the number of assignedtransmission slots as specified in TSLV (v), in which Node ualso transmits or there are more than m− 1 other neighbors ofNode u other than v also transmitting.

Before proceeding, we use Figs. 2, 3, and Table I to illustratethe aforementioned definitions of l, polynomial, coincidence,and packet failure in detail. Considering the network shown inFig. 2, Node 1 is transmitting to Node 2 and Nodes 3, 4, 5 arethe interfering neighbors of Node 2. We assume that m = 2,k = 1, p = 5, and l = 2. That is, each node is assigned twopolynomials with degree 1 mod p = 5, as listed in Table I.Accordingly, the transmission time slots of each node are shownin Fig. 3. Consider the transmission from Node 1 to Node 2. A“C” in Subframe i (i = 0, 1, . . . , 4) of Node 1 indicates thati is a coincidence of Node 1’s and other nodes’ polynomials.Similarly, an “F” in Subframe i (i = 0, 1, . . . , 4) of Node 1indicates a packet failure for a transmission from Node 1 toNode 2. A packet failure for the transmission from Node 1 toNode 2 happens in Subframe i if i is a coincidence of Node 1’sand Node 2’s polynomials or more than m− 1 polynomials

TABLE IASSIGNMENT OF POLYNOMIALS

assigned to Nodes 3, 4, and 5 have the same coincidence i withthose of Node 1 s polynomials.2

Theorem 3: Given that the m-MPR model and each node isassigned l unique polynomials with degree k,

Nf ≤ Nmaxf = min(lp, S), (3)

where

S = kl2 +

⌊(Dmax − 1)kl2

m

⌋. (4)

Proof: Each node has at most Dmax neighbors. Thus, lpolynomials of an arbitrary node v has at most (Dmax − 1)kl2

coincidences with the l(Dmax − 1) polynomials of the otherDmax − 1 interfering neighbors of its destination u. Recall thatif there are more than m− 1 out of these Dmax − 1 interferingneighbors of u also transmitting, the transmission from v to ufails. This is equivalent to throwing (Dmax − 1)kl2 balls into lpbins. The maximum number of bins, in which there are at leastm balls, is

⌊(Dmax−1)kl2

m

⌋. Thus, the second term in the right

hand side of (6) represents the maximum number of time slotsof v, in which there are more than m− 1 neighbors of u otherthan v itself also transmitting. The first term in the right handside of (6) represents the maximum number of time slots of v, inwhich its destination u also transmits. We also note Nf cannotbe larger than lp, i.e., the number of assigned slots. TakingDmax = 10, k = 1, l = 2, and m = 5 as an example, we knowthat an arbitrary node v has at most kl2 = 4 coincidences withone of the other interfering neighbors of its destination u. Thus,Node u has at most (Dmax − 1)kl2 = 36 coincidences with theother Dmax − 1 interfering neighbors of Node u. Note that ifand only if more than m− 1 = 4 out of these 36 coincidencesare located in one transmission slot of Node v, the transmissionof Node v in this slot fails. Thus, 36 coincidences can lead toup to

⌊(Dmax−1)kl2

m

⌋=

⌊(10−1)22

5

⌋= 7 such slots. �

In order to investigate how tight the bound in Theorem 3is, we run simulations as follows. Given that k = 1, p = 23,Dmax = 10, m is set to two and four, and l varies from oneto ten, in each simulation, we randomly select a polynomialwith degree k mod p for Node v and then Dmax other dif-ferent such polynomials for Node u and its Dmax − 1 otherinterfering neighbors from the remaining p2 − 1 polynomials,and calculate the value of Nf . For each value of l, we simulate100 runs. As shown in Fig. 4, the bound of Nf is relativelyloose, especially when the values of m and l are large. This iswhat we prefer to see, since it implies that the average numberof packet failure is typically less than the bound we get.

2A packet failure results from one or more coincidences. However, a coinci-dence does not necessarily leads to a packet failure.


Fig. 4. The tightness of the bound in Theorem 3.

III. PROPOSED ALGORITHM

Consider a TDMA network with N nodes with the m-MPRcapability and the maximum node degree is Dmax. Each nodetransmits according to its TSLV, which is determined by itsassigned TSAF group consisting of l unique polynomials withdegree k mod p (TSAFs).

To avoid coincidences between any two out of l polynomialsdistributed to an arbitrary node, the assignment can be madeas follows. Note that there are a total of pk+1 polynomials. Ithas been proved that the maximum degree of the polynomialsequals one (k = 1) for most cases [18]. Therefore, withoutloss of generality, we assume that k = 1 in the subsequentdiscussion for simplicity. For the case of k = 1, there arep2 polynomials f(x) = ax+ b, where a = 0, 1, . . . , p− 1 andb = 0, 1, . . . , p− 1. We divide these polynomials into p groupsaccording to the value of a, each of which consists of ppolynomials. Each node i selects a group of l polynomialsas its assigned TSAFs for its exclusive use only. Node i se-

lects fi,j(x) =

(⌈i

� pl �

⌉− 1

)x+ bi,j , where j = 1, 2, . . . , l

and bi,j =[mod

(i,⌊pl

⌋+ 1

)− 1

]l + j − 1. Note that the pa-

rameters p and l are known to each node. Thus, each nodeselects its l polynomials in a distributed manner according toits identification number. Taking N = 3, l = 2, and p = 3 asan example, the assigned polynomials to Node i are fi,1(x) =(i− 1)x and fi,2(x) = (i− 1)x+ 1. As the number of nodesin the network (N) grows, more codes are required. Thus, alarger value of parameter p is chosen to satisfy that the codesassigned to one node are orthogonal to each other. That is, thedesign parameter p is chosen such that each node can find its lpolynomials, which will be discussed later.

Since any two polynomials of an arbitrary node has nocoincidence, each node can transmit in lp time slots per frame.Thus, the throughput G is:

G =lp−Nf

p2, (5)

and the minimum throughput Gmin can be obtained byTheorem 3,

Gmin =lp−Nmax

f

p2. (6)

In order to guarantee that our algorithm works correctly, thefollowing two constraints must be satisfied:⌊p

l

⌋p ≥Nl, (7)

lp >Nmaxf . (8)

Equation (7) must be satisfied to ensure that any two outof l unique polynomials assigned to an arbitrary node has nocoincidence. That is, each node can have lp assigned time slotsfor transmissions. Equation (8) states that the total number oftransmissions of an arbitrary node is larger than the maximumnumber of possible packet failures in one frame. Note that (7),(8) ensure that each node can be assigned l orthogonal codes.

Theorem 4 discusses how to obtain the maximum valueof Gmin.

Theorem 4: Let p1 be the largest prime smaller than or

equal to2Nmax

f

l and p2 be the smallest prime larger than2Nmax

f

l , respectively. The optimal value of p that achieves themaximum value of Gmin is p = arg

{p1,p2}max(Gmin), where

arg{p1,p2}

max(Gmin) satisfies (7), or is the smallest prime satis-

fying (7) otherwise.Proof: In order to find the maximum value of Gmin, we

have to solve (8):

∂Gmin

∂p= 0. (9)

By (6), (8) can be rewritten as

−lp−2 + 2Nmaxf p−3 =0. (10)

p−3(2Nmax

f − lp)=0. (11)

Thus,

p =2Nmax

f

l. (12)

Since ∂2Gmin

∂p2

∣∣∣p=

2Nmaxfl

= − l4

8(Nmaxf

)3< 0, Gmin increases

with p when p <2Nmax

f

l , and decreases with p when p ≥2Nmax

f

l . Since p is a prime such that (7) and (8) have to besatisfied, we can thus prove Theorem 4. �

When m = 1, it degenerates to the collision-based receptionmodel. Our result thus reduces to the one in [18].

Considering the case that the maximum value of Gmin is

achieved when p =2Nmax

f

l , we have:

max(Gmin) =l2

4(kl2 +

⌊(Dmax−1)kl2

m

⌋) . (13)

Thus, we can conclude that the maximum value of Gmin

generally remains the same with varying l. However, the aver-age throughput of our algorithm highly depends on the value ofl, i.e., the number of assigned polynomials distributed to eachnode, that is discussed as follows.


A. Average Throughput

We study the average throughput, defined as the averagenumber of successful transmissions per node per slot, of ouralgorithm. Given that each node is assigned l TSAFs (polyno-mials) with the m-MPR capability, we give some definitionsas follows. Let Xi and Xj

i be the numbers of successful trans-missions of an arbitrary node determined by its i-th TSAF ineach frame and in Subframe j, respectively. That is, i = 1, . . . , land j = 0, 1, . . . , p− 1. Note that the frame length is p2. Theaverage throughput (Ga) can be expressed as follows:

Ga =

E

[l∑

i=1

Xi

]p2

=

E

[l∑

i=1

p−1∑j=0

Xji

]

p2=

l

pE[Xj

i

], (14)

where

E[Xj

i

]= P

(Xj

i = 1). (15)

Consider a transmission from Node u to Node v. Let f iu(x)

be the i-th TSAF assigned to Node u. For i = 1, . . . , l andj = 0, 1, . . . , p− 1, we obtain E[Xj

i ] in the following. Theevent that Xj

i = 1 happens if and only if both A and B happen.A is the event that Node v does not transmit in Slot f i

u(j) inSubframe j. B is the event that there are less than m neighborsof Node v, other than Node u itself, transmitting in Slot f i

u(j)in Subframe j. Suppose that Events A and B are independent(referred to as Assumption 1). We have:

P(Xj

i = 1)= P (A)P (B). (16)

Node v transmits in l different time slots in Subframe j. Weassume that these l slots are randomly selected from Slot 0 toSlot p− 1 in Subframe j (referred to as Assumption 2). Thus,the probability that A happens is:

P (A) = 1−(p−1l−1

)(pl

) . (17)

Let Nr denote the number of ways, given a TSAF of Node u,f iu(x), for selecting l(Dmax − 1) other TSAFs, r out of which

has the same value f iu(j) in Subframe j. There are

(p2−1

l(Dmax−1)

)ways to select l(Dmax − 1) TSAFs from the remaining p2 − 1TSAFs. Thus, the probability that B happens is as follows:

P (B) =

m−1∑r=0

Nr

(p2−1

l(Dmax−1)

) . (18)

Given j, we can categorize the p2 TSAFs into p groupsaccording to their values in Subframe j. That is, Group Gn,where n = 0, 1, . . . , p− 1, contains those TSAFs, the values ofwhich in Subframe j are n. Note that TSAFs over GF (p) areuniformly distributed over {0, 1, 2, . . . , p− 1}. Thus, |Gn| =p, where n = 0, 1, . . . , p− 1. Thus, the value of Nr (r =0, 1, . . . ,m− 1) is given as follows:

Nr =

(p− 1

r

)(p2 − p

l(Dmax − 1)− r

). (19)

Fig. 5. Validation of analytical results on average throughput (20) withsimulation.

Substituting (16)–(19) into (14) and (15), we can obtain theaverage throughput of our algorithm as follows:

Ga =l

p

[1−

(p−1l−1

)(pl

)]⎡⎢⎢⎣

m−1∑r=0

(p−1r

)(p2−p

l(Dmax−1)−r

)(

p2−1l(Dmax−1)

)⎤⎥⎥⎦ . (20)

In order to validate our analytical results, we run simulationswithout making Assumptions 1 and 2. Each simulation datavalue is obtained by averaging the results of 100 simulationruns, and the 95% confidence intervals are shown on each ofthe simulated point. The simulation results demonstrate that thetwo assumptions do not affect the accuracy of our analyticalresults. The simulations were performed on a regular graphmodel [29], in which we put all N nodes around a circle, andconnect each to its Dmax

2 nearest neighbors on either side asneighbors.3 Each node randomly selects one of its neighbors asthe destination of its packets. Given that N = 100 and with the4-MPR capability, we can observe that the simulations matchour analytical results well. In Fig. 5, we can also observe thatthe average throughput increases with increasing l when Dmax

is small, but remains almost the same with l greater than acertain value when Dmax is large. This implies that there existsa threshold value of l, beyond which, the increase in the numberof collisions is almost the same as that in the number of trans-mission slots introduced by assigning more TSAFs to a node.The threshold is larger when Dmax is smaller, and vice versa.Thus, the value of l can be chosen according to this principle.

B. Algorithm Description

Based on the aforementioned discussions, we can proposethe m-MPR-l-code topology-transparent scheduling algorithmwith the m-MPR capability as below:

1) Use Theorem 4 to select the value of p for the givenN , Dmax, and m such that the minimum guaranteedthroughput is maximized.

3If Dmax is odd, nodes which are opposite with each other are consideredas neighbors.


2) Based on the average throughput expression in (20),choose the proper value of l that maximizes the averagethroughput (20) by exhaustive search for the given valuesof N , Dmax, and m.

3) Each node is randomly assigned l different TSAFs, anytwo of which has no coincidence.

4) Each node calculates its TSLV, which consists of lp timeslots for possible transmissions, according to its assignedTSAFs.

5) Each node transmits its data packets at its assignedslots.

Some discussions of the proposed algorithm are as follows.Firstly, the proposed algorithm is rather simple. Steps 1 and 2of the proposed algorithm can be done offline at the networkinitialization phase. The most time-consuming step is Step 2,since it requires exhaustive search of l to achieve the maximumvalue of the average throughput. However, as discussed at theend of Section III-A, there exists a threshold value of l, beyondwhich, the increase in the number of collisions is almost thesame as that in the number of transmission slots introduced byassigning more TSAFs to a node. The threshold is larger whenDmax is smaller, and vice versa. Thus, the number of rounds ofsearch of l is actually limited. We can observe in Fig. 5 thatwe only need to search six values of l (from one to six) toachieve the maximum average throughput when Dmax = 20.Thus, its complexity is rather small. Secondly, the proposedalgorithm is distributed when the network operates, if no newnodes will join the network after the network initializationphase. Before the network operates, a central entity runs Steps 1and 2 of the algorithm and tells the values of p and l to allnodes. After that, the central entity is not necessary anymore.Then, each node determines its transmission time slots in afully distributed manner according to Steps 3 and 4 of thealgorithm. Each node transmits and receives packets accordingto Step 5 and the transmission time slots of each node willnot change when the network operates. Our algorithm workscorrectly when the network topology changes. The reasons areas follows. Dmax is an estimated parameter. As mentioned inSection II, Dmax is always pessimistically estimated accordingto empirical data to ensure that the actual number of interferingneighbors does not exceed the estimate. We also show that thedegradation of the performance of our algorithm is rather smallwhen the estimation of Dmax is not so accurate in Section IV.Moreover, although sometimes some nodes may disappear,leave, and re-join the network, we assume that the total numberof nodes in the network, N , is a known parameter and remainsconstant. This is a general and practical assumption in mostprevious work focusing on wireless networks [5], [7], [9], [13],[15]–[17], [23], [34]. Note that nodes may join the networkafter the network initialization phase in some cases. In suchcases, a central entity is required in our algorithm to maintainwhich codes are used by the existing nodes and which codescan be assigned to the new nodes. When a new node joinsthe network, it communicates with the central entity and getits codes [1]. Thus, our algorithm cannot be considered as afully distributed scheduling algorithm in this case. However,our algorithm cannot be simply considered as a centralizedalgorithm, since there is a significant difference between our

algorithm and the traditional centralized scheduling algorithm[2], [9], [13], [23], [27]. In the traditional centralized schedulingalgorithms, the central entity has to know the detailed networkconnectivity and traffic information other than the informationof each node. When the central entity fails, the algorithmscannot work correctly in mobile networks. In contrast, thecentral entity in our algorithm only needs to maintain whichcodes are used by the existing nodes and which codes can beassigned to the new nodes, which is a much easier task. Whenthe central entity fails, the existing nodes can still transmitcorrectly, although the new nodes may not join the networkcorrectly. Especially in the case that no node will join thenetwork after the network initialization phase, the central entityis not necessary in our algorithm after the initialization phase.We also show that even if the value of N changes as the networkoperates, the degradation of the performance of our algorithmis very small in Section IV.

C. Delay Analysis

In this subsection, we use a discrete time M/G/1 queuingmodel in which the first customer of each busy period receivesexceptional service [30] to approximate the average packetdelay in our algorithm. The simulation results in Section IVshow that the approximation does not affect the accuracy of ouranalysis. Packets arrive at an infinite buffer in a Poisson fashionwith rate λ packets per slot. We assume that arrivals occur justafter the beginning of a slot, departures take place just beforethe end of a slot.

Considering a given node as a server, its service slots arefixed. That is, the distribution of the service time of the packetsarriving when the server is busy (there are other packets queuedor being served) is different from that of the packets arrivingwhen the server is idle (there are no packets queued or beingserved). We assume that the service time distribution of a packetwhich initiates a busy period is B1(x) with first and secondmoments b̄1 and b̄21. The distribution of service time of allsubsequent packets in the same busy period is B2(x) with firstand second moments b̄2 and b̄22, which is independent of B1(x).Note that the service time of the packet that initiates a busyperiod is less than that of the subsequent packets on average.That is, b̄1 < b̄2. Thus, the average delay can be expressed asfollows:

E[W ] = E[W |Idle]P0 + E[W |Busy](1− P0), (21)

where P0 is the probability that the system is empty. Accordingto the results obtained in [30], we have:

P0 =1− λb̄2

1− λ(b̄2 − b̄1). (22)

According to the Pollaczek–Khinchine Formula, the value ofE[W |Busy] can be obtained as follows:

E[W |Busy] =λb̄2

(1 +

b̄22b̄2

2

)2(

1b̄2

− λ) + b̄2. (23)

In the following, we calculate b̄2 = 1μ . μ is the probabil-

ity that the packet can be transmitted successfully in a slot.


Consider a transmission from an arbitrary node u to its neighborv. Given an arbitrary slot t, let C be the event that t ∈ TSLVu,D the event that t ∈ TSLVv and Node v has packets to trans-mit, and E the event that less than m neighbors of Node v otherthan u itself have packets to transmit in Slot t. Thus, we have:

μ = P (C)P (D)P (E). (24)

According to the calculation in Section III-A, we have:

P (C) =l

p, (25)

and

P (D) =

[1−

(p−1l−1

)(pl

)]+

(p−1l−1

)(pl

) (1− ρ), (26)

where ρ is the probability that the node has packets to transmitand ρ = 1− P0.

Moreover, P (E) can be expressed as follows:

P (E) =

m−1∑r=0

Nr +Dmax−1∑r=m

Nrm−1∑i=0

(ri

)ρi(1− ρ)r−i

(p2−1

l(Dmax−1)

) . (27)

Substituting (25)–(27) into (24), we can calculate the value ofμ numerically, although it is difficult to obtain its closed-formexpression. Substituting the value of μ into (23), we can obtainthe value of E[W |Busy].

Note that E[W |Idle] = b̄1. We can obtain the average packetdelay as follows:

E[W ] = b̄1P0 + E[W |Busy](1− P0). (28)

It is difficult, if not impossible, to calculate the values of b̄22and b̄1. However, the detailed calculation of queuing systemparameters is not the focus of this work. Thus, we get the valuesof b̄22 and b̄1 via simulations. Substituting (22)–(27) into (28),we can get the average packet delay.

IV. PERFORMANCE EVALUATION

In this section, we quantitatively compare our m-MPR-l-code topology-transparent scheduling algorithm with theconventional topology-transparent algorithm designed for thecollision-based reception model in [18] (known as Ju’s algo-rithm). We do not include the algorithm in [33] as a comparison,since it is only designed for static networks with star topology.It assumes that a central node knows all the information of othernodes and cannot be applied in mobile ad hoc networks. Allsimulations have been conducted using Matlab.

We compare our algorithm with the centralized conflict graphcoloring algorithm in [27]. In this conflict graph coloring algo-rithm, a central scheduler knows the detailed network connec-tivity and traffic information, constructs a conflict graph, andapplies the vertex coloring algorithm to compute and distributethe transmission schedule to each node. When the networkconnectivity and traffic information change, we assume thatthe central scheduler can get the updated network information,re-compute, and distribute the updated schedules immediatelywithout introducing any overhead. Thus, this ideal algorithm

is impractical in mobile networks.4 However, our algorithmis independent of topology changes, which can be easily im-plemented in mobile networks. We also compare our algo-rithm with slotted ALOHA with the m-MPR capability [16],[17], although the latter algorithm cannot offer any throughputguarantees.

A. Simulation Setup

We conduct simulations on two graph models, namely, thegeometric model for the average performance and the regu-lar graph model [29] for the worst performance. In the ge-ometric model, we adopt the Gauss-Markov mobility model,which has been shown to be more realistic than the widelyused Random Waypoint model [3]. All nodes are uniformlyand randomly distributed in a region of 1000 m × 1000 minitially. The tuning parameter θ is used to present differentlevels of randomness in the Gauss-Markov model. We setθ = 0.5. Given Dmax, we set the interference range of eachnode RI such that the probability that the number of inter-fering neighbors of an arbitrary node exceeding Dmax, which

isN−1∑

i=Dmax+1

(N−1i

) (πR2I

A

)i (1− πR2

I

A

)N−1−i

, is smaller than

0.05. For example, RI = 140 m if (N,Dmax) = (100, 10). Inthe regular graph model, the degree of each of the N nodesis set to Dmax. In other words, each node has exactly Dmax

interfering neighbors. Thus, the average number of interferingneighbors in the geometric model is less than that in the regulargraph model with the same Dmax.

We apply the optimal frame structure derived in Theorem 4.l is set to different values for different network parameters, asdiscussed later in details. For each simulation point, we conducta simulation run for 100 continuous frames in the geometricmodel5 and 100 randomly generated topologies in the regulargraph model, unless stated otherwise. The 95% confidenceintervals are also drawn in the figures.

B. Simulation Results

1) Effect of Dmax on Average Throughput: Given that N =100, we investigate the performance of our (m, l)-TTS algo-rithm with different Dmax settings from 6 to 40. A largerDmax indicates that the network is denser and there are morepossible conflicts. In the regular graph model, we set l =m+ 2. However, we set l = m+ 6 in the geometric model,since the average number of interfering neighbors under thegeometric model is smaller than that under the regular model.We can see that the average throughput of our algorithm isabout 60% of that of the ideal coloring algorithm. However, theideal coloring algorithm is impractical, since it is a centralizedalgorithm. The cost of schedule recomputation is unacceptablein mobile networks, since the network connectivity and trafficinformation change frequently. As shown in Figs. 6 and 7, wecan see that our algorithm outperforms slotted ALOHA for

4In fact, this ideal algorithm is impractical even in static networks, sincethe traffic load of each node changes as time evolves, resulting in costly re-computation and distribution of new schedules.

5The movement of each node follows the Gauss-Markov mobility model.


Fig. 6. Average throughput against maximum node degree (regular graph model). (a) m = 2. (b) m = 4.

Fig. 7. Average throughput against maximum node degree (geometric model). (a) m = 2. (b) m = 4.

most cases, especially when the Dmax is large. This is differentfrom the case in the collision-based reception model. It has beenshown [24] that the average throughput of topology-transparentscheduling algorithm is not as good as that of slotted ALOHAunder the collision-based reception model. The superiority ofour algorithm in the MPR model implies that our algorithmcan take advantage of the powerful MPR capability better. Wecan observe that the average throughput of the nodes in thegeometric model is better than that of the nodes in the regulargraph model. This is reasonable, since the average node degreeof the geometric model is smaller than that of the regular graphmodel. Moreover, the superiority of our algorithm over slottedALOHA is greater in the geometric model than that in theregular graph model. This is because the nodes cannot selectthe optimal transmission probability in the geometric model,since they do not know the number of its neighbors.

As shown in Figs. 6 and 7, our algorithm dramaticallyoutperforms the conventional topology-transparent schedulingalgorithm. We can see that the average throughput of the con-

ventional topology-transparent scheduling algorithm remainsalmost the same when m increases, implying that it cannot takefull advantage of the MPR capability.

2) Effect of Inaccuracies in Estimating Dmax and N onPerformance: We use the performance ratio as the metric toinvestigate the effect of inaccuracies in the estimation of Dmax

on the average throughput of the proposed algorithm. Theperformance ratio is defined as the average throughput underthe actual Dmax divided by that under the design value ofDmax. Fig. 8 shows the simulation results under the regulargraph model and the geometric model, respectively.

Given that the design parameters (N,Dmax) is (100, 20), westudy the effect of inaccuracies in the estimation of Dmax onthe average throughput of the proposed (m, l)-TTS algorithm,where m is set to two and four. According to Theorem 4,the optimal values of p for different parameters are shown inTable II. By varying the actual maximum node degree from 6to 40, we can observe in Fig. 8(a) that the performance ratiois smaller (larger) than one when the actual maximum node


Fig. 8. The effect of inaccuracies in the estimation of Dmax on the throughput. (a) Regular graph model. (b) Geometric model.

TABLE IIOPTIMAL VALUE OF p

degree is larger (smaller) than the design value of Dmax. Theperformance ratio decreases with increasing actual maximumnode degree. The performance ratio is close to one when theactual maximum node degree is close to the design value(Dmax = 20). As the difference between the actual and designvalues of Dmax increases, the penalty on the network perfor-mance increases (when the actual value of Dmax is larger thanits design value). We can also observe that the performanceratio of (4, 6)-TTS algorithm is more sensitive to the differencebetween the actual and design values of Dmax than that of(2, 4)-TTS algorithm under the regular graph model. This isdue to the fact that the maximum number of packet failuresof (4, 6)-TTS algorithm increases (decreases) faster with theincreasing (decreasing) actual value of Dmax than that of(2, 4)-TTS algorithm. The reason is explained as follows. Theincreased maximum number of packet failures in one frameis 4×4×ΔD

2 = 8ΔD for (m, l) = (2, 4), but 6×6×ΔD

4 = 9ΔD

for (m, l) = (4, 6), where ΔD is the difference between theactual and design values of Dmax. Similar phenomena are alsoexhibited in Fig. 8(b). Comparing Fig. 8(a) and (b), we can seethat the effect of inaccuracies in the estimation of Dmax on theperformance in the geometric model is much smaller than thatin the regular graph model. The average node degree is typicallysmaller than the maximum node degree Dmax in reality. Thus,we conclude that our algorithm is insensitive to the accuracy inthe estimation of Dmax.

Fig. 9 studies the effect of inaccurate estimation of N on theaverage throughput. In Fig. 9, we use (N,Dmax) = (100, 20)as our design values. The corresponding optimal values of p areshown in Table II. By varying the actual number of nodes Nfrom 100 to 800, we can observe in Fig. 9 that the performance

ratio equals one for most cases, indicating that an inaccurateestimation of N has almost no effect on the performance. Thereason is as follows. According to the discussion in Section IIIand (9), our algorithm can support up to

⌊pl

⌋p nodes. Thus,

the corresponding maximum number of nodes in the networkthat our algorithm can support is listed in Table III. The onlyexception happens when the actual number of nodes, in the(4, 6)-TTS algorithm, is 800, which is larger than the maximumnumber of nodes supported. In this case, some nodes maybe assigned the same polynomials. If these nodes with thesame polynomials are within the interference range of eachother, their transmissions fail. However, the situation that theactual value of N (800) is much larger than the designed value(100) may never happen in well-designed networks. Thus, weconclude that the effect of inaccurate estimation of N on theperformance is very small and can be ignored.

3) Supporting Heterogeneous Traffic: Given N , we inves-tigate the performance of our (m, l)− TTS algorithm withdifferent settings on Dmax in heterogeneous networks. Dif-ferent number of codes can be assigned to provide differentQoS for different classes of nodes in heterogeneous networks.Without loss of generality, we suppose N nodes are di-vided to two classes, namely, Class V1 with a higher QoSrequirement and Class V2 with a lower QoS requirement.The number of nodes in Class V1 and V2 is �βN� andN − �βN�, respectively, where 0 < β < 1. Here, we assumethat β = 0.5. l1 and l2 codes are assigned to the nodes inClasses V1 and V2, respectively, where l1 = 2l2. Note that themaximum value of packet failures for a transmission froman arbitrary node in Classes V1 and V2 is Nmax

f,V1= kl21 +⌊

(Dmax−1)kl21m

⌋= 4kl22 +

⌊4(Dmax−1)kl22

m

⌋and Nmax

f,V2=kl1l2 +⌊

(Dmax−1)kl1l2m

⌋= 2kl22 +

⌊2(Dmax−1)kl22

m

⌋, respectively. Thus,

the average minimum guaranteed throughput is as follows:

Ghmin = β

l1p−Nmaxf,V1

p2+ (1− β)

l2p−Nmaxf,V2

p2, (29)


Fig. 9. The effect of inaccuracies in the estimation of N on the throughput. (a) Regular graph model. (b) Geometric model.

TABLE IIIMAXIMUM NUMBER OF NODES SUPPORTED

where the following constraints must be satisfied:⌊p

l1 + l2

⌋p ≥

⌊N

2

⌋, (30)

l1p >Nmaxf,V1

, (31)

l2p >Nmaxf,V2

. (32)

Let p3 be the largest prime smaller than or equal toNmax

f,V1+Nmax

f,V2

1.5l2and p4 be the smallest prime larger than

Nmaxf,V1

+Nmaxf,V2

1.5l2, respectively. Thus, following the proof of

Theorem 4, we obtain the optimal value of p that achievesthe maximum value of Gh

min is p = arg{p3,p4}

max(Gmin), where

arg{p1,p2}

max(Gmin) satisfies (26)–(28), or is the smallest prime

satisfying (26)–(28) otherwise.6

We set N = 100, β = 0.5, and vary Dmax from 6 to 40. l2 =m+ 2, l1 = 2l2, where m is set to two and four. We study theaverage throughput of the nodes in Classes V1 and V2 in thegeometric model. As shown in Fig. 10, our algorithm workswell as well supporting heterogeneous traffic. The simulationresults for the regular graph model follow the same trend, andare thus omitted due to space limitations.

4) Effect of m-MPR Capability on Minimum GuaranteedThroughput and Average Throughput: We study the effect ofthe m-MPR capability on the minimum guaranteed throughputand average throughput under two different m-MPR model,namely, the strong m-MPR model in (3), (4) and the m-MPR

6Our algorithm can also support more than two classes of nodes in het-erogeneous networks by choosing the proper value of p according to theaforementioned discussion.

Fig. 10. Performance of our algorithm supporting heterogeneous traffic.

model in [22] achieved by the blind interference cancellationalgorithm using polynomial phase-modulating sequences (re-ferred to as PPS m-MPR model). In the PPS m-MPR model,Cn,i is given as follows:

Cn,i =Fn,i(C)

Cn, (33)

if n = 1, 2, . . . ,m and i = 0, 1, . . . , n, and Cn,i = 0 otherwise.C is the size of a codebook (the number of polynomial phase-modulating sequences) and Fn,i(C) is the number of ways thatexactly i out of n taken codes are different. Interested readersare referred to (29)–(34) in [22] for the detailed expression ofCn,i. We set C = 32 according to the discussion in [22].

Given that N = 100, we investigate the performance of ouralgorithm with different settings on Dmax from 6 to 40. Thevalue of p is determined according to Theorem 4. For exam-ple, Nmax

f = kl2 + �(Dmax − 1)kl2� = 92 when Dmax = 20,


Fig. 11. The effect of m on the improvement of the minimum guaranteedthroughput.

l = m = 4, and k = 1. Thus, according to (14), p =1Nmax

f

l =2×924 = 46. Recall that p is a prime and we can obtain that

p = 47. As shown in Fig. 11, we can see that the minimumguaranteed throughput of our algorithm can be greatly im-proved with increasing m. When Dmax = 6, the performanceof our algorithm with m = 10 is only four times of that em-ploying the collision-based reception model. However, thereis over eight times increase in the performance with m = 10when Dmax = 40. This indicates that a small value of m issufficient when the network is sparse. When the network isdense, the minimum guaranteed throughput almost increaseswith m linearly. Similar phenomenons are also exhibited inFig. 12. We can see that the increase in the average throughputof our algorithm, compared to that employing the collision-based reception model, is even larger than m times, when thenetwork is dense. Thus, we conclude that our algorithm cantake full advantage of the powerful MPR capability. ComparingFig. 12(a) and (b), we can also observe that, under the samenetwork configuration, the average throughput of our algorithmunder these two m-MPR models, namely, the strong m-MPRmodel and the PPS m-MPR model, is almost the same. Thisvalidates our statement that although the strong m-MPR modelis not the most accurate reflection of the MPR capability inreality, it can be used to offer enough insights of the effect ofthe m-MPR capability in the physical layer on the operationand performance of MAC layer.

5) Packet Delay and End-to-End Performance: Given thatN = 100 and Dmax = 20, we investigate the average packetdelay of our (m, l)-TTS algorithm with different settings of thevalues of λ and m. In order to validate our analytical results,we run simulations under the regular graph model [29], inwhich each node has Dmax interfering neighbors. We run eachsimulation for 50 data frames. In the simulation, we assumethat each queue has infinite capacity and uses First-In-First-Out (FIFO) service discipline. The destination of a packet israndomly chosen from its neighbors. The transmission slotsof each node are assigned deterministically using our (m, l)-

TTS algorithm. We set l = m+ 2. According to Theorem 4,the values of p in the (2, 4)-TTS algorithm and (4, 6)-TTSalgorithm are 83 and 67, respectively. As shown in Fig. 13, ouranalytical results match well with those from our simulations.

We investigate the end-to-end performance of our algorithmin terms of throughput. The average end-to-end throughput ofscheduling algorithms highly depends on the network topolo-gies and routing protocol applied. The design and implemen-tation of the topology control and routing protocols that canbalance the network loads to optimize the network performanceare out of the scope of this paper. Thus, we use a simple regulargraph model in the simulations, in which each node connectsthe nearest Dmax nodes as its neighbors. Let S and Dest be thesets of source nodes and destination nodes, respectively. 2NSD

out of N nodes are randomly selected. The first NSD nodesbelong to S, and the other NSD ones belong to Dest. A pairof a source S(i) and a destination Dest(i) are called a source-destination (S-D) pair, where i = 1, 2, . . . , NSD. The Bellman-Ford algorithm [10] is used as the routing protocol to calculatethe shortest path between each S-D pair. For each of NSD

source nodes, the arrivals follow a Poisson distribution with rateλ packets per slot. We define the average end-to-end throughputas the average number of packets delivered from sources to theirdestinations per slot. We set (N,Dmax = (100, 20)), (m, l) =(4, 6)), and NSD to be 10, 20, and 50. Varying λ from 0.01to 0.1, we run each simulations for 50 data frames. As shownin Fig. 14, the average end-to-end throughput increases almostlinearly with the arrival rate when λ is small. When the arrivalrate becomes larger, the end-to-end throughput increases moreslowly or remains the same. Comparing Figs. 7(b) and 14, wecan see that when the number of S-D pairs is small, the end-to-end throughput is comparable to the one-hop throughput shownin Fig. 7(b). This is due to the fact that only a small fraction ofnodes have packets to transmit when the number of S-D pairs issmall. Thus, the number of interfering neighbors of each nodehere is much smaller than that in the simulations of Fig. 7(b).With the increasing number of S-D pairs, the average end-to-end throughput decreases, especially when the arrival rate islarge.

V. CONCLUSION

In this paper, we propose a novel m-MPR-l-code topology-transparent scheduling ((m, l)-TTS) algorithm for mobilead hoc networks with the MPR capability. As far as we know,this is the first schedule-based algorithm with MPR, whichis independent of topology changes. Our algorithm is easy toimplement and is oblivious to the network topology changes,because it only needs global parameters such as the number ofnodes and the maximum node degree in the network. Moreover,our algorithm provides the minimum guaranteed throughputto each participating node. The proposed algorithm greatlyimproves the minimum guaranteed throughput and averagethroughput of our algorithm by assigning multiple polynomialsto each node to take full advantage of the powerful MPRcapability. The performance improvement of our (m, l)-TTS al-gorithm over the conventional topology-transparent schedulingalgorithms designed for the collision-based reception model is


Fig. 12. The effect of m on the improvement of the average throughput. (a) Regular graph model. (b) Geometric model.

Fig. 13. Average packet delay.

Fig. 14. Average end-to-end throughput.

almost linear with increasing m. The simulation results showthat our algorithm performs better than slotted ALOHA withMPR as well.

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Yiming Liu (S’14–M’14) received the B.Eng.and Ph.D. degrees in electronic engineering fromTsinghua University, Beijing, China, in 2009 and2014, respectively.

He is an Engineer with the China Academy ofElectronics and Information Technology, Beijing,China. His research interest is in the area of wire-less communications and networking, specifically intransmission scheduling and cross-layer design forwireless multihop networks.

Victor O. K. Li (S’80–M’81–F’92) received theS.B., S.M., E.E., and Sc.D. degrees from MIT,Cambridge, MA, USA, in 1977, 1979, 1980, and1981, respectively, all in electrical engineering andcomputer science. He is a Chair Professor of in-formation engineering and the Head of the Depart-ment of Electrical and Electronic Engineering atThe University of Hong Kong (HKU), Pokfulam,Hong Kong, where he has also served as the Asso-ciate Dean of Engineering and the Managing Direc-tor of Versitech Ltd., which is the technology transfer

and commercial arm of HKU. He served on the board of China.com Ltd., andnow, he serves on the board of Sunevision Holdings Ltd. and Anxin-ChinaHoldings Ltd., which are listed on the Hong Kong Stock Exchange. Previously,he was a Professor of electrical engineering at the University of SouthernCalifornia (USC), Los Angeles, CA, USA, where he was also the Director of theUSC Communication Sciences Institute. Sought by government, industry, andacademic organizations, he has lectured and consulted extensively around theworld. He has received numerous awards, including the PRC Ministry of Edu-cation Changjiang Chair Professorship at Tsinghua University, the U.K. RoyalAcademy of Engineering Senior Visiting Fellowship in Communications, theCroucher Foundation Senior Research Fellowship, and the Order of the BronzeBauhinia Star, Government of the Hong Kong Special Administrative Region,China. He is a Registered Professional Engineer and a Fellow of the Hong KongAcademy of Engineering Sciences, the IAE, and the HKIE.

Ka-Cheong Leung (S’95–M’01) received theB.Eng. degree in computer science from theHong Kong University of Science and Technology,Kowloon, Hong Kong, in 1994 and the M.Sc. degreein electrical engineering (computer networks) andthe Ph.D. degree in computer engineering from theUniversity of Southern California, Los Angeles, CA,USA, in 1997 and 2000, respectively. From 2001 to2002, he was a Senior Research Engineer at NokiaResearch Center, Nokia Inc., Irving, TX, USA. Be-tween 2002 and 2005, he was an Assistant Professor

at the Department of Computer Science, Texas Tech University, Lubbock, TX.Since June 2005, he has been with The University of Hong Kong, Pokfulam,Hong Kong, where he is currently an Assistant Professor in the Department ofElectrical and Electronic Engineering. His research interests include transportlayer protocol design, wireless packet scheduling, congestion control, routing,and vehicle-to-grid (V2G).

Lin Zhang received the B.Sc., M.Sc., and Ph.D.degrees from Tsinghua University, Beijing, China, in1998, 2001, and 2006, respectively. He is currently aVisiting Associate Professor at Stanford University,Stanford, CA, USA, and an Associate Professor atTsinghua University, where he has been happilyteaching selected topics in communication networksand information theory to senior undergraduate andgraduate students. He is a co-author of more than40 peer-reviewed technical papers and five U.S. orChinese patent applications. His current research fo-

cuses on wireless sensor networks, distributed data processing, and informationtheory.

In 2006, he led a 2008 Beijing Olympic Stadium (the “Bird’s Nest”)structural security surveillance project, which deployed more than 400 wirelesstemperature and tension sensors across the stadium’s steel support structureand dome. The system adopted a flexible spectrum sensing and adaptive multi-hop routing algorithm, to overcome strong radio interference and long-distancetransmission channel-fading, and played a critical role in the construction of thestadium. Since then, he has implemented wireless sensor networks in a widerange of application scenarios, including underground mine security, precisionagriculture, and industrial monitoring.

Since 2008, he has been working in close association with CISCO to de-velop a Metropolitan Area Sensing and Operating Network (MASON), whichprovides a smart-city and intelligent-urbanization sensor network system formetropolitan areas and has attracted the interest of several large-sized Chinesecities, including Beijing, Shenzhen, Tianjing, and Chengdu. Recently, he alsohas led two National Science Foundation of China projects, three NationalHigh-Tech Developing (863) projects, and more than ten research projectsfunded primarily by private industry in the area of wireless sensor networks.

Dr. Zhang and his team were the recipient of the IEEE/ACM SenSys 2010Best Demo Awards. In 2004 and 2010, he was a recipient of the ExcellentTeacher Awards from Tsinghua University.

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Topology-Transparent Scheduling in Mobile Ad Hoc Networks ...kcleung/papers/journals... · The powerful MPR capability allows us to assign more codes to a user, since a collision