Delay-Constrained Optimal Link Scheduling in Wireless Sensor Networks

4564 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 9, NOVEMBER 2010

Delay-Constrained Optimal Link Scheduling inWireless Sensor Networks

Qing Wang, Dapeng Oliver Wu, Senior Member, IEEE, and Pingyi Fan, Senior Member, IEEE

Abstract—We consider the optimal link scheduling problem inwireless sensor networks. The optimal link scheduler under ourconsideration is intended to assign time slots to different usersto minimize channel usage subject to constraints on data rate,delay bound, and delay bound violation probability; we studythe problem under fading channels and a signal-to-interference-plus-noise-ratio (SINR)-based interference model. To the best ofour knowledge, this problem has not been studied previously.We use the effective capacity model to formulate the optimallink scheduling as a mixed-integer optimization problem. We firstdiscuss a simple case, namely, the scheduling with a fixed powerallocation, and then extend to the case with variable transmitpower. Moreover, because the mixed-integer optimization problemis NP-hard, we propose a computationally feasible column-gener-ation-based iterative algorithm to search for a suboptimal solutionto the problem. Finally, we design a medium access control (MAC)protocol to implement our optimal link scheduling strategy inpractical wireless networks. Simulation results demonstrate thatour proposed scheme achieves a larger throughput, a larger ad-mission region, and a higher power efficiency than a benchmarktime-division multiple-access (TDMA) system.

Index Terms—Column generation, delay constraint, effectivecapacity (EC), link scheduling.

I. INTRODUCTION

TYPICALLY, an optimal link scheduling problem in sensornetworks or ad hoc networks is intended to schedule time

slots and possibly transmit powers for multiple users so that acertain criterion (such as throughput, spatial reuse, or fairnessindex) is optimized. The key issue in the design of link schedul-ing is how to model and mitigate/avoid interference. In the liter-ature, there are two channel models to characterize interference,i.e., 1) the protocol interference model (PrIM) or disk modeland 2) the physical model [1]. In the disk model, which is alsocalled the collision model, an intended receiver, for example,node i, is interfered by an unintended transmitter, for example,node j, if the distance between node i and node j is less than

Manuscript received November 30, 2009; revised May 21, 2010; acceptedSeptember 17, 2010. Date of publication September 27, 2010; date of currentversion November 12, 2010. This work was supported in part by the NationalScience Foundation under Grant CNS-0643731, by the Office of Naval Re-search under Grant N000140810873, by the The National Natural ScienceFoundation of China (NSFC)/Research Grants Council (RGC) of Hong KongJoint Research Scheme under Grant 60831160524, and by the open researchfund of the National Mobile Communications Research Laboratory, SoutheastUniversity, China. The review of this paper was coordinated by Dr. T. Taleb.

Q. Wang and P. Fan are with the Department of Electronic Engineering,Tsinghua University, Beijing 100084, China (e-mail: [email protected]; [email protected]).

D. O. Wu is with the Department of Electrical and Computer Engineering,University of Florida, Gainesville, FL 32611 USA (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/TVT.2010.2080695

a fixed distance called the interference range. In the physicalmodel, node k can directly communicate with node i if the re-ceived signal-to-interference-plus-noise ratio (SINR) at node iexceeds a specified threshold. While the physical model is moreaccurate, it is also more difficult to solve the link schedulingproblem under the SINR constrains, particularly in the case ofdynamic power adaptation, where each node is allowed to varyits transmit power to reduce interference on other links.

Now, we discuss the formulations of the optimal linkscheduling problem and the solution space. The optimal linkscheduling problem can be formulated by optimizing the totalnetwork throughput or spatial reuse, the uniform throughput[1], the max–min fairness [2], the minimum potential delay fair-ness, and the proportional fairness [3], possibly subject to someconstraints on power, channel resources, and/or quality of ser-vice (QoS). The solution under the disk model is usually basedon a link-contention graph [4] or a conflict graph [1]. The ideais to find the maximum independent sets so that the nodes in amaximum independent set can simultaneously transmit withoutcausing collision while potential interfering users are allotteddisjoint time slots (i.e., they belong to different maximumindependent sets). Such a scheduling problem is equivalent tothe well-known graph coloring problem [2]. In [5], the authorscalled the problem of determining a minimum-length schedulethat satisfies given link demands in a wireless network and issubject to the SINR constraints as Max-SIR-Matching problem.Since the graph coloring problem and the Max-SIR-Matchingproblem are both NP-hard, people seek heuristic algorithms orpolynomial-time approximation algorithms [6]. Hence, underthe disk model, interference is avoided by assigning potentialinterfering users disjoint time slots.

On the other hand, the optimal scheduling problem underthe physical model is a mixed-integer program [7], [8] sincethe transmit power is continuous while independent sets arediscrete. In [9], an optimal scheduling problem under the phys-ical model is formulated as the minimum-length schedulingproblem (MLSP) subject to traffic demand of each link. Acolumn-generation-based algorithm was proposed to solve theproblem under fixed transmit power. The authors also extendedtheir algorithm to the case with variable transmit power.

Different from existing works that use either the disk modelor the physical model, in this paper, we use the effectivecapacity (EC) technique [10] to quantify the effect of interfer-ence on system performance, which we call EC model. Sincewe consider fading channels, the received SINR is a randomvariable (i.e., a stochastic process). Hence, it is possible to useless transmit power to reach the same distance, resulting inresource efficiency. In addition, different from existing works,

0018-9545/$26.00 © 2010 IEEE

WANG et al.: DELAY-CONSTRAINED OPTIMAL LINK SCHEDULING IN WIRELESS SENSOR NETWORKS 4565

we consider statistic delay performance, i.e., the triplet of datarate, delay bound, and delay bound violation probability. Ourintention is to leverage time diversity in fading to achieveresource efficiency. Since the EC model captures the effect oftime diversity in fading channels, we will use the EC model inthe design of optimal scheduling. Note that both the physicalmodel and the EC model are based on SINR.

The rest of this paper is organized as follows: We discusssome related works in Section II. In Section III, to make the ECmodel clearer, we review some basic concepts and importantresults in the theory of EC. In Section IV, we describe ournetwork model and formulate the link scheduling problem forthe fixed power case. In Section V, we develop a column-generation-based solution to the optimal scheduling problem.In Section VI, we study the link scheduling problem for thevariable power case. In Section VII, we conduct performanceanalysis for the optimal link scheduler. Section VIII presentsour design of a medium access control (MAC) protocol toimplement the optimal link scheduling scheme. Section IXshows the simulation results. Section X concludes this paper.

II. RELATED WORKS

The link scheduling problem has been a very hop topicin wireless multihop networks. Its key idea is to find atime-division multiple-access (TDMA) schedule that satisfiesrequirements such as admissible link rates, fairness, powerefficiency, robustness of routing, and interference constraints.To achieve some of these demands, people have studied jointpower control (or routing) and link scheduling problems overthe last two decades. Next, we review some representativerelated works as follows.

As a first attempt, Hajek and Sasaki [11] presented a cen-tralized polynomial-time algorithm to find a minimum-lengthschedule in wireless networks given the link traffic require-ments. The basic idea is to represent the network by an undi-rected graph, but interference constraints were not considered.The analysis of the interference case was studied under themodel of conflict graph [12]. In this model, arcs in a conflictgraph connect nodes, which represent the links (in the wirelessnetwork) that cannot simultaneously transmit due to mutualinterference. For the joint routing and link scheduling problem,it can be formulated as a graph coloring problem, since wirelesscontentions can be modeled by conflict graphs, and moreover,coloring on a conflict graph is equivalent to finding a setof independent sets with appropriate cardinality [13], whichleads to a conflict-free schedule [14], [15]. In [16], the authorsstudied link scheduling under a PrIM with fixed transmissionpower. They used the graph coloring method based on a linearprogramming formulation to find a flow route whose achievedthroughput is at least a constant fraction of the optimum. In[17], the authors proposed a versatile framework for the jointdesign of routing and link scheduling under the formulationof constrained linear programming problems for wireless meshnetworks (WMNs) with a predefined system hierarchy, in whichMesh Routers form the backbone that can physically covera large region using wireless multihop communication. Othersimilar works include [18], in which the proposed linear pro-

gramming solution was developed to produce a transmissionschedule that is also interference-free while maximizing thesystem throughput. In addition, [15] showed the scaling ofthe average packet delay with respect to the overall load on thenetwork and the chromatic number of the link conflict graph,and in [19], the authors also studied the delay performanceand proposed a linear integer programming formulation forthe link scheduling problem in TDM WMNs under a sink-treetopology and constant bit rate traffic. Meanwhile, fairness prob-lem and power efficiency in link scheduling have been takeninto consideration in algorithm/protocol designs for multihopwireless networks. The max–min fair scheduling was studiedin [2], [20], and [21]. In [22], Hou et al. advocated the use oflexicographical max–min fair rate allocation for the nodes inwireless sensor networks. Almost in the same time, a distributedfair scheduling algorithm with consideration of power controlwas proposed in [23]. Since power efficiency is significant forwireless sensor networks, for the joint power control and linkscheduling problem, ElBatt and Ephremides [14] proposed asimple two-phase heuristic to minimize the total power con-sumption via two alternating phases, i.e., power control in thefirst phase and link scheduling in the second phase. In [24], Be-hzad and Rubin studied a similar problem but focused on how tominimize the schedule length. Furthermore, in [5], Borbash andEphremides showed that the general problem of determininga minimum-length schedule that satisfies given link demandsin a wireless network and SINR constraints is NP-hard. Afterthis work, although many polynomial-time approximation al-gorithms were proposed, the MLSP of computing the true opti-mal solution still remains open. However, these approximationalgorithms can be used in practice. Moreover, [25] examinedjoint link scheduling and power control with the objective of agood tradeoff between throughput and fairness. The problemwas first formulated as a mixed-integer linear program, andan effective polynomial-time heuristic algorithm was given.It sought for a transmission schedule and power assignmentleading to a maximum throughput subject to the maximumpower and interference constraints in each time slot. In [26],the authors showed that optimal nonpreemptive link scheduling(NPLS) problems are generally NP-hard and are probablyharder to solve than link scheduling without such a constraint.To tackle the problem, a low-complexity list-link scheduling al-gorithm based on the graph model was proposed to approximatethe optimal NPLS by carefully constructing the link orderinglist. Reference [27] presented a method that finds conflict-freeTDMA schedules with minimum scheduling delay; the authorsdevised an algorithm to seek for the transmission order withminimum delay on overlay tree topologies and used it witha modified Bellman–Ford algorithm to find minimum delayschedules in polynomial time.

Different from existing works, first, for the fading casediscussed in this paper, we use the EC technique/model [10]that captures the effect of time diversity in fading channelsto increase the resource efficiency and quantify the effectof interference on system performance, which significantlyextends previous related works. Second, in our joint powercontrol and link scheduling problem, we study the statisticaldelay performance for each link, including the delay bound and


the delay bound violation probability, which has a great impacton the end-to-end delay so it has a more practical use insystem design. Finally, we propose a new distributed protocolto implement the link scheduling based on the column gener-ation algorithm. Meanwhile, to evaluate the efficiency of ourproposed scheme, we compare with some existing approaches,including the works in [19] and [27], which also considereddelay and channel efficiency. In summary, we aim to find theshortest schedule that can achieve the specified link trafficdemands, power efficiency, and QoS requirements, such as linkrates, delay bound, and delay bound violation probability underSINR constraints.

III. REVIEW OF EFFECTIVE CAPACITY THEORY

EC [10] is a connection-layer model in which a wireless linkis modeled by two EC functions, respectively: 1) the probabilityof nonempty buffer γ(μ) and 2) the QoS exponent of thisconnection θ(μ). Both of them are functions of the source trafficrate μ. Specifically, the key idea in the theory of EC is that, ifthe source traffic has a communication delay bound of Dmax

and can only tolerate a delay-bound violation probability of εat most, then we need to limit the source data rate to a maximumof μ, where μ is the solution to ε = γ(μ)e−θ(μ)Dmax in whichθ(μ) = μα−1(μ). Here, α(·) is exactly the originally definedfunction of EC, and α−1(·) is the inverse function. We give thedetail as follows:

Let r(t) be the instantaneous channel capacity at time t.Define S(t) =

∫ t

0 r(τ)dτ , which is the service provided by thechannel. Suppose the channel is ergodic and stationary. Then,the EC function of r(t) is defined as

α(u) =−Λ(−u)

u∀u > 0 (1)

where

Λ(−u) = limt→∞

1t

log E[e−uS(t)

].

Thus, if we can derive the EC function α(u) based ondifferent kinds of fading channels, then we can find the QoSexponent function θ(·) according to θ(μ) = μα−1(μ). Finally,associated with the QoS requirement of the source traffic,respectively, the communication delay bound of Dmax anda delay-bound violation probability ε, we can estimate theprobability of nonempty buffer γ(μ) and then tune the sourcerate μ to guarantee its QoS requirement. Now, we can see thatthe EC model is a triplet of data rate, delay bound, and delaybound violation probability, i.e., {μ,Dmax, ε} or another usefulform, i.e., {μ,Dmax, Perr}, which is also derived by the authorsin [10], where Perr is the packet error probability, and therelation between them is given by u = −log Perr/(μ · Dmax).Since the EC model captures the effect of channel fadingon the queueing behavior of the link, we select this modelto formulate and solve the link scheduling problem in thispaper.

IV. FORMULATION OF THE LINK SCHEDULING PROBLEM

FOR THE FIXED POWER CASE

We model a sensor network by a set of N nodes, which isdenoted by set N , and a set of directed links, which is denotedby set E . Assume that a node cannot transmit and receivesimultaneously; a node i can communicate with only one nodej (j �= i) at any time. Assume that for each link {i, j} ∈ E , thetransmitting node i can directly communicate with the receivingnode j with specified QoS (SINR or bit error rate or delay)satisfied. Let Pi(t) be the transmission power for node i attime t, Gij(t) be the gain of the fading channel from node i tonode j, and ηj be the variance of the thermal noise at receiverj. The SINR at receiver j due to transmission from node i isgiven by

SINRij(t) =Pi(t)Gij(t)

ηj +∑

l �=i,j Pl(t)Glj(t). (2)

Assume that each link {i, j} ∈ E has a traffic demand of r(ij)s

bits per second, which needs to be transmitted across the linkwith delay bound D

(ij)max and delay bound violation probability

P(ij)err .Now, we formulate the optimal link scheduling problem.

Consider an SINR-based scheduler where S time slots oflengths {wk} (k = 1, . . . , S and wk ∈ (0, 1]) are used to sched-ule links in E with QoS requirements {r(ij)

s ,D(ij)max, P

(ij)err }.

If there is no traffic demand for a specific link {i, j}, thenr(ij)s = 0. Denote the transmission power of node i in slot k

by P(k)i . Assume that P

(k)i (∀i ∈ N ∀k ∈ {1, . . . , S}) can only

take two values, i.e., 0 and P0. P (k)i = 0 means that node i does

not transmit in slot k, whereas P(k)i = P0 means that node i

transmits in slot k. Clearly, here we use a fixed power allocationscheme. Our optimal scheduling problem is given by

min{wk}

{P

(k)i

} S∑k=1

wk (3)

s.t. P(k)i ∈{0, P0} ∀i∈N ∀k∈{1, . . . , S} (4)

wk∈(0, 1] ∀k∈{1, . . . , S} (5)

αij,{

P(k)i

},{wk}

(u∗

ij

)≥r(ij)

s ∀{i, j}∈E (6)

where

u∗ij =

− log P(ij)err

r(ij)s × D

(ij)max

(7)

αij,{

P(k)i

},{wk}

(u) =S∑

k=1

wk × αij,P

(k)i

(wk × u) (8)

where

αij,P

(k)i

(u) = limt→∞

−1ut

log E

[e−u∫ t

0W log(1+SINRij,k(τ))dτ

]

u ≥ 0 (9)


where W is the channel bandwidth, and

SINRij,k(t) =P

(k)i (t)Gij(t)

ηj +∑

l �=i,j P(k)l (t)Glj(t)

. (10)

We call the optimal scheduler resulting from (3)–(6) the SINR-EC scheduler under fixed power. Note that (9) is the expressionfor the EC function according to its definition in (1), and(8) is derived from Propositions 1 and 2, which are given asfollows.

Proposition 1: Scaling Law for EC Function: The ECfunction α

ij,P(k)i

,wk(u) for Link {i, j} under the SINR-EC

scheduler with time fraction wk (wk ∈ (0, 1]) and transmissionpower P

(k)i satisfies

αij,P

(k)i

,wk(u) = wk × α

ij,P(k)i

(wk × u) (11)

where αij,P

(k)i

(u) is the EC function for Link {i, j}, as defined

by (9).For the proof of Proposition 1, see Appendix A.Proposition 2: Additivity Law for EC Function: As-

sume that the channel gains of Link {i, j} in differ-ent slots are independent of each other. The EC functionα

ij,{P (k)i

},1{wk}(u) for Link {i, j} under the SINR-EC sched-

uler with time fractions {wk (k = 1, . . . , S)} (wk ∈ (0, 1] ∀k)and corresponding transmission powers {P (k)

i (k = 1, . . . , S)}satisfies

αij,{

P(k)i

},{wk}

(u) =S∑

k=1

wk × αij,P

(k)i

(wk × u). (12)

For the proof of Proposition 2, see Appendix B. In prac-tice, if the correlation between the channel gains in differentslots is small, then we use α

ij,{P (k)i

},{wk}(u) ≈∑S

k=1 wk ×α

ij,P(k)i

(wk × u).If Problem (3) results in a solution with

∑wk ≤ 1, then this

solution or the scheduling process is feasible; otherwise, it isnot feasible since we assume that there is only one channel. TheS time slots could form a superframe, and the same slot patternrepeats in each superframe. We assume admission control isin place, which works as follows: For a new link requestingfor admission, if the corresponding problem (3) results in asolution with

∑wk ≤ 1, then the new link can be accepted

by the admission control. Here, we just give a general idea forthe MAC. It thus leads to a new MAC protocol, which willbe presented in Section VIII. In addition, we can see that ina schedule, a link may be active in one or multiple time slots.In a slot, multiple links may simultaneously satisfy their SINRcondition so that all of these links can be activated; this groupof active links in the same time slot forms a matching [5], [9]. Itcan be shown that allowing multiple active links in a matchingincreases the throughput of the whole network.

If we do not allow interference, i.e., at any time, only onenode is allowed to transmit, then we call the optimal schedulerresulting from (3)–(6) with SINR in (10) replaced by SNR

as No-Interference TDMA (NI-TDMA) scheduler under fixedpower. We will use the NI-TDMA scheduler as a bench-mark to evaluate the performance of our proposed SINR-ECscheduler.

Unfortunately, the optimization problem specified by (3)–(6)is NP-hard. Note that for S = 1, nonfading channels, 0 ≤P

(k)i ≤ P0, and {D(ij)

max = ∞, P(ij)err = 0}, the problem has

been solved for code-division multiple-access (CDMA) cellularsystems [28] using quasi-convexity. To solve (3)–(6), one mayresort to one of the following three methods: 1) column gener-ation [9]; 2) polynomial-time approximation algorithm [6]; and3) branch and price [29]. In this paper, we focus on the columngeneration method. Next, in Section V, we present a column-generation-based algorithm to solve the optimal link schedulingproblem.

V. COLUMN-GENERATION-BASED SOLUTION TO THE

OPTIMAL SCHEDULING PROBLEM

First, we give the basic idea of the column generationalgorithm. Column generation is an iterative algorithm forsolving huge linear or integer programming problems, wherethe number of variables is too large to be considered explicitly.While experience suggests that only a small subset of thesevariables is found in the optimal solution, the rest of thesevariables will be nonbasic and always take a value of zero inthe optimal solution. Therefore, column generation leveragesthis idea by generating only those variables that have the poten-tial to improve the objective function. Consequently, the hugeproblem can be simplified. More specifically, in the columngeneration algorithm, the original problem is decomposed intoa master problem and a subproblem. The master problem andsubproblem could be either linear or integer program, depend-ing on the problem formulation, such as the examples in [9].The strategy of this decomposition procedure is to iterativelyoperate on two separate but easier-to-solve problems. Duringeach iteration, the algorithm tries to determine whether anyvariables exist that have a negative reduced cost (in the caseof minimization problem) and adds the variable with the mostnegative reduced cost to the master problem. Therefore, thekey idea of the column generation algorithm is to sequentiallyimprove the current solution by first solving a subproblem thatidentifies a single new variable (a column) and adding it to themaster problem, then solving the master problem, and repeatingthis process until the algorithm terminates under some user-specified stopping criteria. In particular, the variables or sayingcolumns here are now matchings and elements of the power setof links in our scheduling problem.

With the basic knowledge of the column generation algo-rithm, we next describe how to solve our scheduling problem.We present the main idea as below. Denote M as the power setof E , i.e., M contains all possible combinations of members inE . The master problem is a restriction of the original problem[see (3)–(6)]. The master problem only uses a subset of columnsindexed by s ∈ {1, . . . , |M|}, where |M| is the cardinalityof M. The master problem is first initialized in a random waywith any S ⊂ M that satisfies (4)–(6). For each transmitter


i ∈ {i, j} ∈ S, the transmit power P(k)i (∀k) is equal to P0.

Therefore, the master problem is given by

min{wk}

|S|∑k=1

wk (13)

s.t. wk ∈ (0, 1] ∀k ∈ {1, . . . , |S|} (14)

αij,{

P(k)i

},{wk}

(u∗

ij

)≥ r(ij)

s ∀{i, j} ∈ E . (15)

Since this formulation optimizes over a subset S of all feasiblesolutions, the optimal solution to (13)–(15) provides an upperbound for the original problem [see (3)–(6)].

In each iteration, after the master problem [see (13)–(15)]is solved, if the solution to the master problem also providesthe solution to the original problem [see (3)–(6)], then theprocedure terminates; otherwise, we need to solve a sub-problem, which identifies a new column (independent set)that can improve the current solution. The subproblem forgenerating a new column is formulated as follows: For eachmember Sm ∈ M\S, which refers to the set of all columnsthat are in M but are not in S. The dual variable corre-sponding to (14) is ξij , and the reduced cost m for anycolumn m in the master problem is expressed by m = 1 −∑

{i,j}∈Smξijα

(m)

ij,{P (k)i

},{wk}(u∗

ij)/r(ij)s ; therefore, the subprob-

lem is given by

m = min{ξij}

⎧⎪⎨⎪⎩1 −

∑{i,j}∈Sm

ξijα(m)

ij,{

P(k)i

},{wk}

(u∗

ij

)r(ij)s

⎫⎪⎬⎪⎭ (16)

s.t. P(k)i ∈ {0, P0} ∀i ∈ {i, j} ∈ Sm (17)

where

α(m)

ij,{

P(k)i

},{wk}

(u)

= limt→∞

−1ut

log E

[e−u∫ t

0W log(1+SINRij,m(τ))dτ

]

∀u ≥ 0 (18)

where

SINRij,m(t) =P0Gij(t)

ηj +∑

l �=i,j,&{l,j}∈SmP0Glj(t)

. (19)

If the cost m < 0, then add the column induced by Sm

to S as a new member. Since there are exponential numberof members in M, in practice, we need to randomly selectSm from M\S; the subproblem stops when the solution tothe master problem provides an ε-approximation solution [6] tothe original problem [see (3)–(6)]. (In our simulation, we takeε = 10−4.) The value ε indicates how far our obtained solutionis away from the optimal solution for the original problem.Reference [6] proved the relationship between the approxima-tion ratio and the number of iterations required. Therefore, itguarantees a high probability that, by adequate iterations, wecan find a solution with an acceptable approximation ratio tothe minimum value. The reason for having an ε-approximation

solution is to have a comparable polynomial time (randomized)algorithm [30].

Until now, we have formulated the basic optimal linkscheduling problem in sensor networks with QoS requirementin Section IV and showed how to use a column-generation-based algorithm to solve the original complex optimizationproblem for a fixed power case. Next, we extend our approachto a more complicated case where variable power is used in thescheduling problem.

VI. OPTIMAL LINK SCHEDULING PROBLEM UNDER

VARIABLE TRANSMIT POWER

Although simultaneous transmissions in each matching canincrease the frequency spatial reuse of the network, the fixedpower allocation, namely, only using the maximum transmitpower P0, may cause strong interference for the other ongoingtransmissions in the neighborhood. Therefore, to further miti-gate the interference and increase the frequency spatial reuse,we introduce a power control scheme into the original optimalscheduling process. The experimental results in Section IXverify that this more flexible scheduling scheme with variabletransmit power leads to better performance.

Like the fixed power case, we first formulate the optimalscheduling problem under variable transmit power as follows:

min{wk,P

(k)i

} S∑k=1

wk (20)

s.t. 0≤P(k)i ≤P0 ∀i∈N ∀k∈{1, . . . , S} (21)

wk∈(0, 1] ∀k∈{1, . . . , S} (22)

αij,{

P(k)i

},{wk}

(u∗

ij

)≥r(ij)

s ∀{i, j}∈E (23)

where

u∗ij =

− log P(ij)err

r(ij)s × D

(ij)max

(24)

αij,{

P(k)i

},{wk}

(u) =S∑

k=1

wk × αij,P

(k)i

(wk × u) (25)

where αij,P

(k)i

(u) is defined in (9). We call the optimal sched-

uler resulting from (20)–(23) the SINR-EC scheduler undervariable power.

If we do not allow interference, i.e., at any time, only onenode is allowed to transmit, then we call the optimal sched-uler resulting from (20)–(23) with SINR in (10) replaced bySNR the NI-TDMA scheduler under variable power. We willuse the NI-TDMA scheduler as a benchmark to evaluate theperformance of our proposed SINR-EC scheduler.

To solve the optimal scheduling problem under variablepower, we again use the column generation method. We explainour idea as follows. The master problem also uses only a subsetof columns indexed by s ∈ {1, . . . , |M|}. The master problemis first initialized with any S ⊂ M that satisfies (5), (6), and


0 ≤ P(k)i ≤ P0 (∀i ∈ N ∀k ∈ {1, . . . , |S|}). Therefore, the

master problem is formulated as follows:

min{wk}

|S|∑k=1

wk (26)

s.t. wk ∈ (0, 1] ∀k ∈ {1, . . . , |S|} (27)

αij,{

P(k)i

},{wk}

(u∗

ij

)≥ r(ij)

s ∀{i, j} ∈ E . (28)

Since this formulation optimizes over a subset S of all feasiblesolutions, the optimal solution to (26)–(28) provides an upperbound for the original scheduling problem. Note that (26) isoptimized over {wk}; the transmit power P

(k)i is determined

by the subproblem, which is described as follows.The subproblem for generating a new column is formulated

as follows: First, randomly select K members from M\S. Kcan take any positive integer not larger than the cardinalityof the set MS. Denoting these K members by Sm(m =1, · · · ,K), i.e., the mth element of the set, for each Sm, thesubproblem is given by

ζm = max{Pi:i∈{i,j}∈Sm}

∑{i,j}∈Sm

α(c1)ij,Pi

(u∗

ij

)/r(ij)

s (29)

s.t. 0 ≤ Pi ≤ P0 ∀i ∈ {i, j} ∈ Sm (30)

where

u∗ij =

− log P(ij)err

r(ij)s × D

(ij)max

(31)

α(c1)ij,Pi

(u) = limt→∞

−1ut

log E

[e−u∫ t

0W log(1+SINRij,m(τ))dτ

]u ≥ 0 (32)

where

SINRij,m(t) =PiGij(t)

ηj +∑

l �=i,j,&{l,j}∈SmPlGlj(t)

. (33)

The maximization in (29) is over {Pi} only, unlike (3). Thisis similar to CDMA power control with a single slot only [28],i.e., choosing the powers that maximize the total capacity (forSm), weighted by {1/r

(ij)s }. Let m∗ = arg maxm ζm. Then,

add the column induced by Sm∗ to S as a new member, andassign the transmit powers{

P(k)i : i ∈ {i, j} ∈ Sm∗

}

= arg max{Pi:i∈{i,j}∈Sm∗}

∑{i,j}∈Sm∗

α(c1)ij,Pi

(u∗

ij

)r(ij)s

. (34)

Since there are exponential numbers of members in M, tohave a polynomial time algorithm, we also need to termi-nate the subproblem when the solution to the master prob-lem provides an ε-approximation solution to the originalproblem.

VII. PERFORMANCE ANALYSIS

In this section, we analyze the performance of our SINR-ECscheduler in terms of admission region of QoS-assured flowsand in terms of throughput gain.

We first examine the performance of our SINR-EC schedulerin the aspect of admission region of QoS-assured one-hopflows. We only consider one-hop flows here since it is mucheasier to analyze the admission region for one-hop flows; ourfuture work will address the admission region of multihopflows. We assume that the system under study has an admissioncontrol module [31] to ensure that the admitted one-hop flowshave their requested QoS satisfied. Assume that there are L QoSclasses. The QoS class l (l = 1, . . . , L) is specified by bit rater(l)s , delay bound D

(l)max, and delay bound violation probability

P(l)err . Assume that for QoS class l (l = 1, . . . , L), the maximum

number of admitted QoS-assured one-hop flows under the NI-TDMA scheme is Nl. Then, the vector [N1, . . . , NL] specifiesa point on the boundary of the admission region or the capacityregion for the NI-TDMA, that is, the NI-TDMA is able to si-multaneously support all these flows, i.e., Nl one-hop flows forQoS class l (l = 1, . . . , L). The flows of all QoS classes sharethe same wireless resource (i.e., the same frequency band), andthere are many Pareto-optimal values for [N1, . . . , NL]. AllPareto-optimal vectors [N1, . . . , NL] specify the boundary ofthe admission region for the NI-TDMA. Compared with theNI-TDMA, the admission region under our SINR-EC scheduleris increased by a factor of �1/

∑Sk=1 wk according to the

following proposition.Proposition 3: Assume that [N1, . . . , NL] specifies a point

on the admission region under the NI-TDMA, and thepercentage of channel use under the NI-TDMA is 100%.Suppose that our SINR-EC scheduler is also able to simul-taneously support all these flows, i.e., Nl one-hop flows forQoS class l (l = 1, . . . , L), and the percentage of channeluse under our SINR-EC scheduler is

∑Sk=1 wk < 1. Then,

[N1 × �1/∑S

k=1 wk , . . . , NL × �1/∑S

k=1 wk ] is within theadmission region under our SINR-EC scheduler, where �x isthe largest integer that is less than or equal to x.

For the proof of Proposition 3, see Appendix C.Next, we consider the performance of our SINR-EC sched-

uler in terms of throughput gain. The following propositionstates that the throughput under our SINR-EC scheduler isincreased by a factor of 1/

∑Sk=1 wk under the same delay

bound and delay bound violation probability.Proposition 4: Assume that [N1, . . . , NL] specifies a point

on the admission region under the NI-TDMA, and the percent-age of channel use under the NI-TDMA is 100%. Supposethat our SINR-EC scheduler is also able to simultaneouslysupport all these flows, i.e., Nl one-hop flows for QoS classl (l = 1, . . . , L), and the percentage of channel use under ourSINR-EC scheduler is

∑Sk=1 wk < 1. If our SINR-EC sched-

uler is allowed to have 100% channel use, then for QoS classl (l = 1, . . . , L), our SINR-EC scheduler can satisfy bit rater(l)s /

∑Sk=1 wk, delay bound D

(l)max, and delay bound violation

probability P(l)err .

For the proof of Proposition 4, see Appendix D.


Fig. 1. Superframe structure of the proposed MAC protocol.

Propositions 3 and 4 are valid for both the fixed powercase and the variable power case. This is because the fixedpower case and the variable power case have the same structurefor time fraction allocation and power allocation, except thatthe variable power case allows powers to be changeable overdifferent slots.

Propositions 3 and 4 show that our SINR-EC schedulerachieves a larger admission region of QoS-assured flows and ahigher throughput than the NI-TDMA. This is due to frequencyspatial reuse and interference mitigation obtained by our SINR-EC scheduler.

VIII. DISTRIBUTED PROTOCOL TO IMPLEMENT OPTIMAL

LINK SCHEDULING

In this section, we present our design of a new MAC protocolto implement the proposed optimal link scheduling. To illustrateit, we consider the case where all transmitting nodes use fixedtransmit power.

First, we use the scheme in the previous works [32], [33] toform clusters of nodes. That is, each cluster will elect a clusterhead. A cluster head is used to coordinate the transmission initi-ation by periodically transmitting a beacon signal so that all theother nodes can set up their networking parameters. The MACis a TDMA-like protocol based on a well-defined superframesimilar to that in IEEE 802.15.3. A superframe consists of abeacon, a contention access period, management channel timeallocations (MCTAs), and channel time allocations (CTAs),as shown in Fig. 1. The MCTAs and CTAs together form acontention-free period. All nodes in a cluster will synchronizeto the cluster head based on the preamble in the beacon fromthe cluster head.

Second, we search for a feasible solution to the optimalscheduling problem. Each cluster head randomly selects onenode from its cluster; all these selected nodes form S1; in slotCTA1, all nodes in S1 can simultaneously transmit at the corre-sponding rate r

(ij)s . SINRij and P

(ij)err will be measured by the

receiver j and piggybacked in the packets from node j to nodei in the next superframe (assuming bidirectional traffic, e.g.,interactive video/audio). SINRij will be used to determine thephysical layer parameters, such as the order of modulation (ifadaptive modulation is used). Denote the estimated P

(ij)err as

P(ij)err . Then, θij(r

(ij)s ) is estimated by θij(r

(ij)s ) = Pr{D(ij) >

0}/E[D(ij)], where E[D(ij)] is the expectation of the delay ofthe head-of-line packet for Link {i, j}, and Pr{D(ij) > 0} isthe probability that D(ij) > 0. Note that a packet that violatesthe delay bound D

(ij)max will be dropped by the transmitter.

Therefore, repeat the same process K − 1 times to obtain setsof nodes S1,S2, . . . ,SK ; the protocol will make sure all Sk aredifferent. Since the problem

maxm∈{1,2,...,K}

∑{i,j}∈Sm

α(c1)ij,Pi

(u∗

ij

)/r(ij)

s (35)

is equivalent to the problem

minm∈{1,2,...,K}

∑{i,j}∈Sm

θij

(r(ij)s

)· D(ij)

max/ log P (ij)err (36)

we can solve the following problem instead of (16)–(19):

m∗ = arg maxm∈{1,2,...,K}

∑{i,j}∈Sm

θij

(r(ij)s

)· D(ij)

max

log P(ij)err

(37)

s.t. P (ij)err ≤ P (ij)

err . (38)

After obtaining m∗, add the column induced by Sm∗ to S as anew member, and assign a slot CTA to all the nodes in Sm∗ .Repeat the same process and add columns to S until (14) and(15) are satisfied.

Third, we determine the length of slot CTAk in the super-frame based on the optimal wk values. For the schedulingscheme under variable transmit power, the process follows thesame way.

IX. SIMULATION RESULTS

A. Simulation Setting

1) Method for Estimating EC-Function αs(u): We simulatethe discrete-time system depicted in Fig. 2. In this system, thedata source generates packets at a constant rate μ. The gene-rated packets are first sent to the (infinite) buffer at the trans-mitter, whose queue length is Q(n), where n refers to thenth sample interval. The head-of-line packet in the queue istransmitted over the fading channel at data rate r(n). The fadingchannel has a random power gain g(n). We use a fluid model,that is, the size of a packet is infinitesimal. In practical systems,the results presented here will have to be modified to accountfor finite packet sizes.

We assume that the transmitter has perfect knowledge of thecurrent channel gains g(n) at each sample interval. Therefore,it can use rate-adaptive transmissions and ideal channel codesto transmit packets without decoding errors. We consider thefollowing two cases.

1) For the case without link interference, the transmissionrate r(n) is equal to the instantaneous (time-varying)


Fig. 2. Queueing system model used in our simulation.

capacity of the fading channel, i.e.,

r(n) = Bc log2

(1 + g(n) × P0/σ2

n

)(39)

where Bc denotes the channel bandwidth, and the trans-mission power P0 and the noise variance σ2

n are assumedto be constant. The average SNR is fixed in each sim-ulation run, and the average SNR SNRavg = E[g(n) ×P0/σ2]. Since we set E[g(n)] = 1, we have SNRavg =E[g(n) × P0/σ2] = P0/σ2.

2) For the case with link interference, the transmission rater(n) is equal to the instantaneous (time-varying) capacityof the fading channel, i.e.,

r(n) = Bc log2

(1 + g(n) × P0/

(I + σ2

n

))(40)

where I is the variance of the total link interferencepower at receiver j from all the other simultaneoustransmitters k, i.e., I =

∑k P0 · d−ν

kj , where dkj is thedistance between node k and node j, and ν is the path-loss exponent [34]; without loss of generality, we assumethat the total link interference power at receiver j is aGaussian random variable. Note that we use the SINR inthe interference case. The average SINR is also fixed ineach simulation run, and the average SINR SINRavg =E[g(n) × P0/(I+ σ2)]. Since we set E[g(n)]= 1, wehave SINRavg = E[g(n) × P0/(I+ σ2)]= P0/(I+ σ2).

We collect the following measurements from the queueingsystem at the nth sampling epoch (n = 1, 2, . . . , NT ): S(n) isthe indicator of whether a packets is in service (S(n) ∈ {0, 1}),Q(n) is the number of bits in the queue (excluding the packetin service), and τ(n) is the remaining service time of the packetin service (if there is one in service). We calculate the measuredEC function αs(u) by the following procedure:

γ =1

NT

NT∑t=1

S(n) (41)

q =1

NT

NT∑t=1

Q(n) (42)

τs =1

NT

NT∑t=1

τ(n) (43)

θ =γ × μ

μ × τs + q(44)

αs(u) =μ, for u = θ/μ. (45)

In our simulations, the sampling interval δ is set to 1 ms.This is not too far from reality, since third-generation wideband

CDMA systems already incorporate rate adaptation on the orderof 10 ms [35]. Each simulation run is 10 000 s long for all thescenarios to obtain a good estimate by the Monte Carlo method.Since the sampling interval is 1 ms, we have 10 million samplesfor estimation.

2) AR(1) Rayleigh Fading Channel Simulator: Denote h(n)as the voltage gain in the nth sample interval. We generateRayleigh flat-fading voltage gains h(n) by a first-order autore-gressive (AR(1)) model as follows: We first generate h(n) by

h(n) = κ × h(n − 1) + ug(n) (46)

where ug(n) are independent identically distributed complexGaussian variables with zero mean and unity variance perdimension. Then, we normalize h(n) and obtain h(n) by

h(n) = h(n)/

√2

1 − κ2= h(n) ×

√1 − κ2

2. (47)

It is clear that (47) results in E[g(n)] = E[|h(n)|2] = 1. Thecoefficient κ determines the Doppler frequency, i.e., the largerthe κ, the smaller the Doppler frequency. Specifically, thecoefficient κ can be determined by the following procedure:1) Compute the coherence time Tc by [36, p. 165]

Tc ≈ 916πfm

(48)

where the coherence time is defined as the time over which thetime autocorrelation function of the fading process is above 0.5.2) Compute the coefficient κ by1

κ = 0.5δ/Tc . (49)

3) Network Topology and Traffic Model: Next, we describethe network topology and the source traffic demands used inthe simulation. To evaluate the performance of the column-generation-based algorithm for our scheduling problem, weconsider a six-node network and a 20-node network. For thesix-node network, the topology and link QoS requirements areshown in Fig. 3; note that the position of each node in Fig. 3does not correspond to its geographic location, and Fig. 3 onlyshows the connectivity relationship among nodes. More specif-ically, the demand of traffic bit rate vector is [120, 80, 100, 100,110, 90, 90, 95] kb/s, corresponding to links {1, 2}, {2, 3},{3, 4}, {3, 6}, {4, 5}, {5, 3}, {5, 6}, and {6, 1}, respectively.The corresponding maximum delay bound vector is [140, 190,

1The auto-correlation function of the AR(1) process is κm, where m is thenumber of sample intervals. Solving κTc/δ = 0.5 for κ, we obtain (49).


Fig. 3. Topology and traffic load of the six-node network.

TABLE ISIMULATION PARAMETERS

170, 170, 160, 150, 150, 130] ms, and the delay bound violationprobability vector is [3%, 5%, 4%, 4%, 9%, 7%, 6.5%, 6%].These QoS requirements are typical for multimedia streamingapplications.

Denote G as the average channel gain matrix, the elementof which is the average channel gain Gij = E[Gij ], with i asthe row index and j as the column index. Gij is proportionalto d−ν

ij (ν ≥ 2), where dij denotes the transmitter–receiverseparation distance of Link {i, j}. In the simulation, we usethe following G:

G =

⎡⎢⎢⎢⎢⎢⎣

N/A 0.16 0.30 0.70 0.39 1.100.50 N/A 2.30 0.69 1.70 0.230.60 1.40 N/A 0.90 1.06 6.900.10 0.71 3.10 N/A 5.70 0.610.43 2.10 8.00 0.96 N/A 0.840.71 3.60 0.51 1.76 0.27 N/A

⎤⎥⎥⎥⎥⎥⎦× 10−3

where N/A means that a user will not send messages to itself.In our simulation, the power of thermal noise at the receiver isset to 3.34 × 10−9 W, and the maximum transmit power P0 isset to 1 mW, which means that the transmission range coverstens of meters. The bandwidth of each link is 100 kHz.

For the 20-node network used in our simulation, we will notshow the topology and the link QoS requirements here since thedescription is very complicated, e.g., matrix G has 400 entries.

Table I lists the parameters used in our simulations.

B. Simulation Results

In this section, we show simulation results to demonstrate theefficacy of our column-generation-based algorithm for solving

TABLE IIMATCHINGS UNDER FIXED POWER

the optimal link scheduling problem. Our column-generation-based algorithm is implemented in C++ language. The softwareLingo 9.0 with full packages is used as the optimization toolto solve the optimal link scheduling problem. This section isorganized as follows: In Section IX-B1, we evaluate the perfor-mance of our column-generation-based algorithm under fixedpower for the six-node network. Section IX-B2 presents the per-formance results of our column-generation-based algorithm un-der variable power for the six-node network. In Section IX-B3,we evaluate the performance of our column-generation-basedalgorithm under fixed power and variable power for the 20-nodenetwork. In Section IX-B4, we compare on the channel usewith the other two performing approaches from [19] and [27],respectively, as mentioned in Section II, which also stressed thedelay problem.

1) Performance Under Fixed Power for the Six-Node Net-work: In this section, we compare the performance of ourSINR-EC scheduler with that of the NI-TDMA under fixedpower for the six-node network.

Table II shows all the feasible matchings2 under fixed powerfor our SINR-EC scheduler. All the links in a matching cansimultaneously transmit while satisfying the QoS requirementof each link, which is specified by bit rate rs, delay boundDmax, and delay bound violation probability Perr; in otherwords, all the links in a matching can simultaneously transmitwhile satisfying (6).

We first run simulations for the NI-TDMA under the settingspecified in Table I and estimate the EC of each channel.Then, we run simulations for our SINR-EC scheduler under thesettings specified by Tables I–III, which achieve the same ECof each channel as that in the NI-TDMA.

From Table III, we know that∑S

k=1 wk = 60%, where wk isnormalized by the 100% channel use of the NI-TDMA. Hence,our SINR-EC scheduler uses 40% less channel resource thanthe NI-TDMA. The saved channel resource can be used toadmit more QoS-assured flows or support a higher throughputfor elastic traffic such as Transfer Control Protocol (TCP)traffic. Compared with the NI-TDMA, the throughput under ourSINR-EC scheduler is increased by a factor of 1/

∑Sk=1 wk =

166.7% under the same delay bound and delay bound violationprobability (see Proposition 4). Our SINR-EC scheduler uses60% total power of all nodes in the case of the NI-TDMA, i.e.,our SINR-EC scheduler uses 40% less total power of all nodes

2A matching or independent edge set in a graph is a set of edges withoutcommon vertices.


TABLE IIISCHEDULE OF OUR SINR-EC SCHEDULER UNDER

FIXED POWER FOR THE SIX-NODE NETWORK

TABLE IVMATCHINGS UNDER VARIABLE POWER

compared with the NI-TDMA. Hence, our SINR-EC scheduleris more energy efficient.

2) Performance Under Variable Power for the Six-NodeNetwork: In this section, we compare our SINR-EC schedulerunder variable power with 1) our SINR-EC scheduler underfixed power and 2) the NI-TDMA under fixed power for thesix-node network.

Table IV shows all the feasible matchings under variablepower for our SINR-EC scheduler. All the links in a match-ing can simultaneously transmit while satisfying the QoS re-quirement of each link, i.e., satisfying (6). It is observed thatvariable power brings about great advantages. First, the numberof feasible matchings becomes much larger. There are totally20 different matchings, i.e., twice of that in the fixed powercase, and even three links could be active simultaneously. Thus,the link scheduling under variable power becomes more flexiblethan that under fixed power. Second, the interference betweenthe simultaneously active links could be greatly mitigated byvariable power allocations so that we can achieve a high degreeof frequency spatial reuse in the whole wireless network.

TABLE VSCHEDULE OF OUR SINR-EC SCHEDULER UNDER

VARIABLE POWER FOR THE SIX-NODE NETWORK

We run simulations for our SINR-EC scheduler under thesettings specified by Tables I, IV, and V, which achieve thesame EC of each channel as that in the NI-TDMA.

From Table V, we know that∑S

k=1wk = 54.66%, wherewk is normalized by the 100% channel use of the NI-TDMA.Hence, our SINR-EC scheduler uses 45.34% less channel re-source than the NI-TDMA. The saved channel resource canbe used to admit more QoS-assured flows or support a higherthroughput for elastic traffic, such as TCP traffic. Comparedwith the NI-TDMA, the throughput under our SINR-EC sched-uler is increased by a factor of 1/

∑Sk=1 wk = 182.9% under the

same delay bound and delay bound violation probability. OurSINR-EC scheduler uses 25.6% total power of all nodes in thecase of the NI-TDMA, i.e., our SINR-EC scheduler uses 74.4%less total power of all nodes compared with the NI-TDMA.

Compared with the results in Section IX-B1, the SINR-ECscheduler under variable power is more power efficient than theSINR-EC scheduler under fixed power; this is because usingfixed maximum power P0 causes a high level of interference,resulting in fewer number of links that can simultaneously beactivated. This becomes evident if we compare Table II withTable IV. As shown in Table II, at most two links can simul-taneously be activated under fixed power, whereas three linkscan simultaneously be activated under variable power, as shownin Table IV. Therefore, the degree of frequency spatial reuseunder variable power is higher than that under fixed power.

3) Performance Under Fixed Power and Variable Powerfor the 20-Node Network: In this section, we study the per-formance of our SINR-EC scheduler under fixed power andvariable power for the 20-node network.

We do not show the topology and the link QoS requirementsfor the 20-node network since the description is very compli-cated. We study the SINR-EC scheduler under four scenarios,i.e., the number of links that have traffic demands are 10, 20,30, and 40, respectively.

We run simulations for our SINR-EC scheduler for bothfixed power and variable power under the settings specified byTable I, which achieves the same EC of each channel as that inthe NI-TDMA.


TABLE VIPERFORMANCE OF OUR SINR-EC SCHEDULER FOR THE 20-NODE

NETWORK: (I) FIXED POWER, (II) VARIABLE POWER

TABLE VIIPOWER PERFORMANCE OF OUR SINR-EC SCHEDULER FOR THE

20-NODE NETWORK: (I) FIXED POWER, (II) VARIABLE POWER

Table VI shows the percentage of channel use, throughputgain, and capacity gain of our SINR-EC scheduler under fixedpower and variable power for four different numbers of linksthat have traffic demands. In Table VI, wk is normalized bythe 100% channel use of the NI-TDMA. The capacity gainof our SINR-EC scheduler over the NI-TDMA is defined by�1/

∑Sk=1 wk = 1, i.e., compared with the NI-TDMA, the

admission region under our SINR-EC scheduler is increased bya factor of �1/

∑Sk=1 wk (see Proposition 3). It is observed that

as the number of traffic links increases, the SINR-EC schedulerunder variable power achieves a faster increase of throughputgain than the SINR-EC scheduler under fixed power. This isbecause under variable power, a larger number of traffic linksprovides an increased opportunity for frequency spatial reuse,whereas under fixed power, a higher level of interference fromsimultaneous transmissions might reduce this opportunity. Thehigher level of interference under fixed power can also beconfirmed in Table VII, where the SINR-EC scheduler underfixed power consumes more power than the SINR-EC schedulerunder variable power.

Table VII shows the power efficiency of our SINR-EC sched-uler under fixed power and variable power for four differentnumbers of links that have traffic demands. It can be observedthat as the number of traffic links increases, the SINR-ECscheduler saves more power. It is also observed that theSINR-EC scheduler under variable power achieves a higherpower efficiency and a lower level of interference than theSINR-EC scheduler under fixed power.

4) Performance Comparison With the Other Two PerformingSchemes for a 20-Node Network: To evaluate the efficiencyof our SINR-EC scheduler, we compare our scheme with theapproach from Cappanera et al. [19] and the scheme proposedby Djukic and Valaee [27], as mentioned in Section II. Forfairness of comparison, we use the same randomly generated

TABLE VIIICOMPARISON OF THE CHANNEL EFFICIENCY BETWEEN OUR SINR-EC

SCHEDULER AND SCHEMES [19], [29] FOR A 20-NODE NETWORK

20-node network, including the node positions, link numbers,channel conditions, etc., with the same source rate and delaybound requirement.

Table VIII shows the channel use of the three differentschemes, all normalized by the 100% channel use of the NI-TDMA, in a fixed power case. It can be observed that ourSINR-EC scheduler outperforms the other two approaches. Onereason for this is that our scheduler does better for a completelyrandom network and not only for random sink-tree or overlaytree topologies, for example, in scheduling the multiple unicastflows more efficiently.

To summarize, the simulation results have demonstratedthat compared with the NI-TDMA and some other popularapproaches, our SINR-EC scheduler uses less channel resourceand achieves a larger throughput, a larger admission region,and a higher power efficiency. For all the simulations, we haveverified that the QoS requirements (data rate, delay bound, anddelay bound violation probability) of each flow are satisfied.

X. CONCLUSION

In this paper, we have studied the optimal link schedulingproblem in wireless sensor networks. The optimal link sched-uler assigns time slots to different users to minimize channelusage subject to constraints on data rate, delay bound, and delaybound violation probability; we studied the problem underfading channels and an SINR-based interference model. Tothe best of our knowledge, this problem has not been studiedpreviously. We used the EC model to formulate the optimallink scheduling as a mixed-integer optimization problem forboth fixed and variable power cases. Moreover, because themixed-integer optimization problem is NP-hard, we proposeda computationally feasible column-generation-based iterativealgorithm to search for a suboptimal solution to the problem.Finally, to facilitate the implementation of the optimal linkscheduling strategy in practical sensor networks, we designeda distributed MAC protocol. Simulation results show that ourproposed SINR-EC scheduler achieves a larger throughput, alarger admission region, and a higher power efficiency com-pared with the NI-TDMA scheduler.

APPENDIX APROOF OF PROPOSITION 1

Proof: From (1), for Link {i, j}, we have

αij,Pi(u)=− lim

t→∞

1ut

log E

[e−u∫ t

0rij,Pi

(τ)dτ]

∀u≥0

(50)


where rij,Pi(t) is the instantaneous channel capacity of link

{i, j} with transmit power Pi(t), i.e., rij,Pi(t) = W log[1 +

SINRij,Pi(t)], where SINRij,Pi

(t) is defined by (2). Then,for all u > 0, we have

αij,P

(k)i

,wk(u)

(a)=

− limt→∞1t log E

[e−u∫ t

0r

ij,P(k)i

,wk(τ)dτ

]

u

(b)=


[e−u∫ t

0wk×r

ij,P(k)i

(τ)dτ]

u

=

− limt→∞wk

t log E

[e−(wk×u)

∫ t

0r

ij,P(k)i

(τ)dτ]

wk × u= wk × α

ij,P(k)i

(wk × u) (51)

where (a) rij,P

(k)i

,wk(t) is the instantaneous channel capacity

of Link {i, j} with time fraction wk and the correspondingtransmit power P

(k)i , and (b) is due to scaling law in TDMA,

i.e., rij,P

(k)i

,wk(t) = wk × r

ij,P(k)i

(t). �

APPENDIX BPROOF OF PROPOSITION 2

Proof: From (1), for Link {i, j}, we have

αij,Pi(u) = − lim

t→∞

1ut

log E

[e−u∫ t

0rij,Pi

(τ)dτ]

∀ u ≥ 0

(52)

where rij,Pi(t) is the instantaneous channel capacity of link

{i, j} with transmit power Pi(t), i.e., rij,Pi(t) = W log[1 +

SINRij,Pi(t)], where SINRij,Pi

(t) is defined by (2). Then,for all u > 0, the EC function α

ij,{P (k)i

},{wk}(u) for Link {i, j}under the SINR-EC scheduler with time fractions {wk (k =1, . . . , S)} (wk ∈ (0, 1] ∀k) and corresponding transmissionpowers {P (k)

i (k = 1, . . . , S)} is given by

αij,{

P(k)i

},{wk}

(u)

(a)=


[e−u∫ t

0r

ij,{P(k)i },{wk}

(τ)dτ]

u

=


[e−u∫ t

0

∑S

k=1r

ij,P(k)i

,wk(τ)dτ

]

u

(b)=


[e−u∫ t

0

∑S

k=1wk×r

ij,P(k)i

(τ)dτ]

u

=


[e−u∑S

k=1

∫ t

0wk×r

ij,P(k)i

(τ)dτ]

u

=


[∏Sk=1 e

−u∫ t

0wk×r

ij,P(k)i

(τ)dτ]

u

(c)=

− limt→∞1t log

∏Sk=1 E

[e−u∫ t

0wk×r

ij,P(k)i

(τ)dτ]

u

=

− limt→∞1t

∑Sk=1 log E

[e−u∫ t

0wk×r

ij,P(k)i

(τ)dτ]

u

=S∑

k=1

− limt→∞wk

t log E

[e−(wk×u)

∫ t

0r

ij,P(k)i

(τ)dτ]

wk × u

=S∑

k=1

wk × αij,P

(k)i

(wk × u) (53)

where (a) rij,{P (k)

i},{wk}(τ) is the instantaneous channel ca-

pacity of Link {i, j} under the SINR-EC scheduler with timefractions {wk (k = 1, . . . , S)} (wk ∈ (0, 1] ∀k) and corre-sponding transmission powers {P (k)

i (k = 1, . . . , S)}, (b) isdue to scaling law in TDMA, i.e., r

ij,P(k)i

,wk(t) = wk ×

rij,P

(k)i

(t), and (c) is because the channel gains in different slots

are independent of each other. �

APPENDIX CPROOF OF PROPOSITION 3

Proof: We first prove the case where∑S

k=1 wk = 0.5,i.e., 1/

∑Sk=1 wk = 2. Assume that an existing one-hop flow

of QoS class l over Link {i, j} under our SINR-EC scheduleris assigned with time fractions {wk (k = 1, . . . , S)} (wk ∈(0, 1] ∀k) and corresponding transmission powers {P (k)

i (k =1, . . . , S)}. Since

∑Sk=1 wk = 0.5, we can admit an additional

flow of QoS class l over Link {i, j} under our SINR-EC sched-uler with time fractions {wk (k = S + 1, . . . , 2S)} (wk ∈(0, 1] ∀k) and corresponding transmission powers {P (k)

i (k =S + 1, . . . , 2S)}, where wk = wk−S (k = S + 1, . . . , 2S), andP

(k)i = P

(k−S)i (k = S + 1, . . . , 2S); hence,

∑2Sk=S+1 wk =

0.5, and the total channel use under our SINR-EC scheduler is∑2Sk=1 wk = 1. In other words, for each existing one-hop flow

of QoS class l over Link {i, j}, we can admit an additional flowof QoS class l over Link {i, j} with the same time fractionsand transmission powers. Then, the additional Nl flows of QoSclass l (l = 1, . . . , L) under our SINR-EC scheduler have theirrequested QoS satisfied since they are allocated with the sametime fractions and transmission powers, and the interferencepatterns are exactly the same as the existing Nl flows of QoSclass l (l = 1, . . . , L). Hence, [2N1, . . . , 2NL] is within theadmission region under our SINR-EC scheduler.

Now, we prove the case where �1/∑S

k=1 wk > 2.Assume that an existing one-hop flow of QoS class l overLink {i, j} under our SINR-EC scheduler is assigned withtime fractions {wk (k = 1, . . . , S)} (wk ∈ (0, 1] ∀k) and


corresponding transmission powers {P (k)i (k = 1, . . . , S)}.

We can admit additional �1/∑S

k=1 wk − 1 flows of QoSclass l over Link {i, j} under our SINR-EC scheduler.The first additional flow is assigned with time fractions{wk (k = S + 1, . . . , 2S)} (wk ∈ (0, 1] ∀k) and correspondingtransmission powers {P (k)

i (k = S + 1, . . . , 2S)}, where

wk = wk−S (k = S + 1, . . . , 2S), and P(k)i = P

(k−S)i (k =

S + 1, . . . , 2S). The second additional flow is assigned withtime fractions {wk (k = 2S + 1, . . . , 3S)} (wk ∈ (0, 1] ∀k)and corresponding transmission powers {P (k)

i (k =2S + 1, . . . , 3S)}, where wk = wk−2S (k = 2S + 1, . . . , 3S),and P

(k)i = P

(k−2S)i (k = 2S + 1, . . . , 3S). Repeat the

same assignments for other additional flows. In otherwords, for each existing one-hop flow of QoS class l overLink {i, j}, we can admit additional �1/

∑Sk=1 wk − 1

flows of QoS class l over Link {i, j} with the same timefractions and transmission powers as the existing one. Then,the additional Nl(�1/

∑Sk=1 wk − 1) flows of QoS class

l (l = 1, . . . , L) under our SINR-EC scheduler have theirrequested QoS satisfied since they are allocated with the sametime fractions and transmission powers, and the interferencepatterns are exactly the same as the existing Nl flows ofQoS class l (l = 1, . . . , L). The total channel use under ourSINR-EC scheduler is (

∑Sk=1 wk) × (�1/

∑Sk=1 wk ) ≤ 1.

Hence, [N1 × �1/∑S

k=1 wk , . . . , NL × �1/∑S

k=1 wk ]is within the admission region under our SINR-ECscheduler. �

APPENDIX DPROOF OF PROPOSITION 4

Proof: Without loss of generality, we examine the flowof an arbitrary QoS class l ∈ {1, . . . , L} and assume thatthe flow is over a certain link {i, j} ∈ E . From the for-mulation of our scheduling problem, i.e., (3)–(6), we knowthat, when the percentage of channel use

∑Sk=1 wk < 1,

our SINR-EC scheduler guarantees αij,{P (k)

i},{wk}(u

∗ij) ≥ r

(l)s

in (6), where u∗ij = − log P

(l)err/r

(l)s × D

(l)max. Since the QoS

class l that we examine is arbitrary, our SINR-EC schedulerguarantees

αij,{

P(k)i

},{wk}

(u∗

ij

)≥ r(l)

s ∀l ∈ {1, . . . , L} (54)

which means that, when the percentage of channel use∑Sk=1 wk < 1, our scheduler can satisfy bit rate r

(l)s , delay

bound D(l)max, and delay bound violation probability P

(l)err for

all QoS classes, i.e., l = 1, . . . , L.Now, we derive the EC for the case where the percentage

of channel use under our SINR-EC scheduler is 100%. As-sume that an existing one-hop flow of QoS class l over Link{i, j} under our SINR-EC scheduler [see (3)–(6)] is assignedwith time fractions {wk (k = 1, . . . , S)} (wk ∈ (0, 1] ∀k) andcorresponding transmission powers {P (k)

i (k = 1, . . . , S)}.Then, 100% channel use under our SINR-EC scheduler canbe achieved by time fractions {wk/

∑Sk=1 wk (k = 1, . . . , S)}

(wk ∈ (0, 1] ∀k) and corresponding transmission powers

{P (k)i (k = 1, . . . , S)}. Then, the EC of Link {i, j} under

100% channel use is

αij,{

P(k)i

},{wk/

∑S

k=1wk}

(u∗

ij

S∑k=1

wk

)

(a)=

S∑k=1

wk∑Sk=1 wk

× αij,P

(k)i

(wk∑S

k=1 wk

× u∗ij

S∑k=1

wk

)

=1∑S

k=1 wk

S∑k=1

wk × αij,P

(k)i

(wk × u∗

ij

)(b)=

1∑Sk=1 wk

αij,{

P(k)i

},{wk}

(u∗

ij

)(c)

≥ r(l)s /

S∑k=1

wk ∀l ∈ {1, . . . , L} (55)

where (a) is due to (12) in Proposition 2, (b) is alsodue to (12) in Proposition 2, and (c) is due to (54).Since α

ij,{P (k)i

},{wk/∑S

k=1wk}

(u∗ij

∑Sk=1wk) ≥ r

(l)s /

∑Sk=1wk

and u∗ij

∑Sk=1wk = −log P

(l)err/((r(l)

s /∑S

k=1wk) × D(l)max), our

SINR-EC scheduler with 100% channel use can satisfy bit rater(l)s /

∑Sk=1 wk, delay bound D

(l)max, and delay bound violation

probability P(l)err for QoS class l (l = 1, . . . , L). �

REFERENCES

[1] P. Gupta and P. Kumar, “The capacity of wireless networks,” IEEE Trans.Inf. Theory, vol. 46, no. 2, pp. 388–404, Mar. 2000.

[2] X. Huang and B. Bensaou, “On max-min fairness and scheduling inwireless ad-hoc networks: Analytical framework and implementation,” inProc. ACM MobiHoc, 2001, pp. 221–231.

[3] X. Lin, N. Shroff, and R. Srikant, “A tutorial on cross-layer optimiza-tion in wireless networks,” IEEE J. Sel. Areas Commun., vol. 24, no. 8,pp. 1452–1463, Aug. 2006.

[4] H. Luo and V. Bharghavan, “A new model for packet scheduling in multi-hop wireless networks,” in Proc. 6th Annu. Int. Conf. Mobile Comput.Netw., 2000, pp. 76–86.

[5] S. Borbash and A. Ephremides, “Wireless link scheduling with powercontrol and SINR constraints,” IEEE Trans. Inf. Theory, vol. 52, no. 11,pp. 5106–5111, Nov. 2006.

[6] K. Jansen and L. Porkolab, “On preemptive resource constrained schedul-ing: Polynomial-time approximation schemes,” SIAM J. Discr. Math.,vol. 20, no. 3, pp. 545–563, 2007.

[7] Y. Hou, Y. Shi, and H. Sherali, “Optimal spectrum sharing for multi-hop software defined radio networks,” in Proc. IEEE INFOCOM, 2007,pp. 1–9.

[8] Y. Shi and Y. Hou, “Optimal power control for programmable radionetworks,” in Proc. IEEE INFOCOM, 2007, pp. 1694–1702.

[9] J. W. S. Kompella and A. Ephremides, “A cross-layer approach to optimalwireless link scheduling with SINR constraints,” in Proc. IEEE MILCOM,Orlando, FL, 2007, pp. 1–7.

[10] D. Wu and R. Negi, “Effective capacity: A wireless link model for supportof quality of service,” IEEE Trans. Wireless Commun., vol. 2, no. 4,pp. 630–643, Jul. 2003.

[11] B. Hajek and G. Sasaki, “Link scheduling in polynomial time,” IEEETrans. Inf. Theory, vol. 34, pt. 1, no. 5, pp. 910–917, Sep. 1988.

[12] S. Ramanathan and E. Lloyd, “Scheduling algorithms for multihop radionetworks,” IEEE/ACM Trans. Netw., vol. 1, no. 2, pp. 166–177, Apr. 1993.

[13] I. Chlamtac and S. Kutten, “A spatial reuse TDMA/FDMA for mobilemulti-hop radio networks,” in Proc. IEEE INFOCOM, 1985, vol. 1,pp. 389–394.

[14] T. ElBatt and A. Ephremides, “Joint scheduling and power control forwireless ad hoc networks,” IEEE Trans. Wireless Commun., vol. 3, no. 1,pp. 74–85, Jan. 2004.

[15] K. Kar, X. Luo, and S. Sarkar, “Delay guarantees for throughput-optimal wireless link scheduling,” in Proc. IEEE INFOCOM, 2009,pp. 2331–2339.


[16] Y. Wang, W. Wang, X. Li, W. Song, and I. Li, “Interference-aware jointrouting and TDMA link scheduling for static wireless networks (HTML),”IEEE Trans. Parallel Distrib. Syst., vol. 19, no. 12, pp. 1709–1726,Dec. 2008.

[17] L. Badia, A. Erta, L. Lenzini, and M. Zorzi, “A general interference-aware framework for joint routing and link scheduling in wireless meshnetworks,” IEEE Netw., vol. 22, no. 1, pp. 32–38, Jan./Feb. 2008.

[18] C. Hong and A. Pang, “3-approximation algorithm for joint routingand link scheduling in wireless relay networks,” IEEE Trans. WirelessCommun., vol. 8, no. 2, pp. 856–861, Feb. 2009.

[19] P. Cappanera, L. Lenzini, A. Lori, G. Stea, and G. Vaglini, “Link schedul-ing with end-to-end delay constraints in wireless mesh networks,” in Proc.IEEE Int. Symp. WoWMoM, 2009, pp. 1–9.

[20] Z. Fang and B. Bensaou, “Fair bandwidth sharing algorithms based ongame theory frameworks for wireless ad-hoc networks,” in Proc. 23rdAnnu. Joint Conf. INFOCOM, 2004, vol. 2, pp. 1284–1295.

[21] L. Tassiulas and S. Sarkar, “Maxmin fair scheduling in wireless net-works,” in Proc. IEEE INFOCOM, 2002, pp. 763–772.

[22] Y. Hou, Y. Shi, and H. Sherali, “Rate allocation in wireless sensor net-works with network lifetime requirement,” in Proc. 5th ACM Int. Symp.Mobile Ad Hoc Netw. Comput., 2004, pp. 67–77.

[23] K. Wang, C. Chiasserini, J. Proakis, and R. Rao, “Distributed fair schedul-ing and power control in wireless ad hoc networks,” in Proc. IEEEGLOBECOM, 2004, vol. 6, p. 3556.

[24] A. Behzad and I. Rubin, “Optimum integrated link scheduling and powercontrol for multihop wireless networks,” IEEE Trans. Veh. Technol.,vol. 56, no. 1, pp. 194–205, Jan. 2007.

[25] J. Tang, G. Xue, C. Chandler, and W. Zhang, “Link scheduling with powercontrol for throughput enhancement in multihop wireless networks,”IEEE Trans. Veh. Technol., vol. 55, no. 3, pp. 733–742, May 2006.

[26] Y. Wu, Y. Zhang, and Z. Niu, “Non-preemptive constrained linkscheduling in wireless mesh networks,” in Proc. IEEE GLOBECOM,2008, pp. 1–6.

[27] P. Djukic and S. Valaee, “Delay aware link scheduling for multi-hopTDMA wireless networks,” IEEE/ACM Trans. Netw., vol. 17, no. 3,pp. 870–883, Jun. 2009.

[28] R. Yates and C. Huang, “Integrated power control and base station assign-ment,” IEEE Trans. Veh. Technol., vol. 44, no. 3, pp. 638–644, Aug. 1995.

[29] I. Hicks, J. Warren, D. Warrier, and W. Wilhelm, “A branch-and-priceapproach for the maximum weight independent set problem,” Network,vol. 46, no. 4, pp. 198–209, Dec. 2005.

[30] R. Motwani and P. Raghavan, “Randomized algorithms,” ACM Comput.Surv., vol. 28, no. 1, pp. 33–37, 1996.

[31] D. Wu, “QoS provisioning in wireless networks,” Wireless Commun.Mobile Comput., vol. 5, no. 8, pp. 957–970, Dec. 2005.

[32] X. Du, D. Wu, W. Liu, and Y. Fang, “Multiclass routing and mediumaccess control for heterogeneous mobile ad hoc networks,” IEEE Trans.Veh. Technol., vol. 55, no. 1, pp. 270–277, Jan. 2006.

[33] X. Du and D. Wu, “Adaptive cell relay routing protocol for mobile adhoc networks,” IEEE Trans. Veh. Technol., vol. 55, no. 1, pp. 278–285,Jan. 2006.

[34] T. Rappaport, Wireless Communications: Principles and Practice.Upper Saddle River, NJ: Prentice-Hall, 2001.

[35] H. Holma and A. Toskala, WCDMA for UMTS: Radio Access for ThirdGeneration Mobile Communications. New York: Wiley, 2000.

Qing Wang received the B.S. degree in informationscience and engineering from Shandong University,Jinan, China, in 2006. He is currently working to-ward the Ph.D. degree with the Department Elec-tronic Engineering, Tsinghua University, Beijing,China.

From 2008 to 2009, he was a Research Scholarwith the Department Electrical Engineering, Uni-versity of Florida, Gainesville. His research interestis in the general area of information systems andnetworks. Specifically, he has been working on wire-

less sensor and ad hoc networks, network information theory and cross layerdesign, etc.

Mr. Wang received the Best Paper Award at the 2010 International WirelessCommunications and Mobile Computing Conference. He serves as a Reviewerfor the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS, and several internationalconferences.

Dapeng Oliver Wu (S’98–M’04–SM’06) receivedthe B.E. degree in electrical engineering fromHuazhong University of Science and Technology,Wuhan, China, in 1990, the M.E. degree in electricalengineering from Beijing University of Posts andTelecommunications, Beijing, China, in 1997, andthe Ph.D. degree in electrical and computer engineer-ing from Carnegie Mellon University, Pittsburgh, PA,in 2003.

Since 2003, he has been with the faculty of theDepartment of Electrical and Computer Engineering,

University of Florida, Gainesville, where he is currently an Associate Professor.His research interests are in the areas of networking, communications, signalprocessing, computer vision, and machine learning.

Dr. Wu received the University of Florida Research Foundation ProfessorshipAward in 2009, the Air Force Office of Scientific Research Young InvestigatorProgram (YIP) Award in 2009, the Office of Naval Research YIP Award in2008, the National Science Foundation CAREER award in 2007, the IEEECircuits and Systems for Video Technology Transactions Best Paper Award forYear 2001, and the Best Paper Award at the International Conference on Qualityof Service in Heterogeneous Wired/Wireless Networks in 2006. He currentlyserves as an Associate Editor for the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS, the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS

FOR VIDEO TECHNOLOGY, the Journal of Visual Communication and ImageRepresentation, and the International Journal of Ad Hoc and UbiquitousComputing. He was the founding Editor-in-Chief of the Journal of Advancesin Multimedia between 2006 and 2008 and an Associate Editor for the IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY between 2004 and 2007. Hehas also been a Guest Editor for the IEEE JOURNAL ON SELECTED AREAS IN

COMMUNICATIONS, Special Issue on Cross-layer Optimized Wireless Multi-media Communications. He will serve as Technical Program Committee (TPC)Chair for IEEE INFOCOM 2012 and has served as TPC Chair for the 2008IEEE International Conference on Communications, the Signal Processingfor Communications Symposium, and as a member of executive committeeand/or technical program committee of over 50 conferences. He has served asChair for the Award Committee and Chair of Mobile and Wireless MultimediaInterest Group, Technical Committee on Multimedia Communications, IEEECommunications Society.

Pingyi Fan (SM’09) received the B.S. degree fromHebei University, Tianjin, China, in 1985, the M.S.degree from Nankai University, Tianjin, China, in1990, and the Ph.D. degree from Tsinghua Univer-sity, Beijing, China, in 1994.

From August 1997 to March 1998, he was aResearch Associate with the Hong Kong Universityof Science and Technology, Kowloon, Hong Kong.From May 1998 to October 1999, he was a ResearchFellow with the University of Delaware, Newark. InMarch 2005, he was a Visiting Professor with NICT,

Japan. From June 2005 to July 2005 and August 2006 to September 2010,he was with the Hong Kong University of Science and Technology. He waspromoted to Full Professor in 2002 and is currently a Professor with theDepartment of Electronic Engineering, Tsinghua University. His main researchinterests include beyond-third-generation technology in wireless communica-tions such as multiple-input–multiple-output, orthogonal frequency divisionmultiplexing, multicarrier code-division multiple access, space–time coding,low-density parity check design, network coding, network information theory,cross layer design, etc.

Dr. Fan is an overseas member of the Institute of Electrical, Information, andCommunication Engineers. He has attended and/or organized many interna-tional conferences, including as Technical Program Committee (TPC) Cochairof the 2010 IEEE International Conference on Wireless Communications, Net-working, and Information Security and TPC member of the IEEE InternationalConference on Communications, Globecom, Wireless Communications andNetworking Conference, Vehicular Technology Conference, etc. He has servedas an Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS,the Interscience International Journal of Ad Hoc and Ubiquitous Computing,and the Wiley Journal of Wireless Communication and Mobile Computing. Heis also a Reviewer for more than 14 international journals, including ten IEEEjournals and three EURASIP journals. He has received some academic awards,including the 2008 IEEE Wireless Communications and Networking Best PaperAward, the 2010 Association for Computing Machinery International WirelessCommunications and Mobile Computing Conference Best Paper Award, andIEEE Communication Society Excellent Editor Award for IEEE TRANSAC-TIONS ON WIRELESS COMMUNICATIONS in 2009.

Documents

Delay-Constrained Optimal Link Scheduling in Wireless Sensor Networks