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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 8511

Distributed Opportunistic Scheduling WithQoS Constraints for Wireless Networks

With Hybrid LinksWenguang Mao, Xudong Wang, Senior Member, IEEE, and Shanshan Wu

Abstract—Opportunistic scheduling for a wireless network withhybrid links is studied in this paper. Specifically, two link types areconsidered: A link of the first type always has a much lower trans-mission rate than a link of the second type. To avoid starvation inthe first type of links, two link types must be treated differentlyin opportunistic scheduling, and quality of service (QoS) con-straints, such as maximum delay or minimum throughput, mustbe imposed on the first link type. Considering QoS constraints,a distributed opportunistic scheduling scheme is derived basedon the optimal stopping theory. Two scenarios are considered forthe QoS-oriented opportunistic scheduling scheme. In the firstscenario, all links within the same link type follow the same ratedistribution. Thus, QoS constraints are imposed on the entire linktype. In the second scenario, links of the first type follow hetero-geneous rate distributions. Thus, QoS requirements need to beimposed on links with the worst performance. Performance resultsshow that the new opportunistic scheduling scheme outperformsthe existing ones in most scenarios.

Index Terms—Distributed opportunistic scheduling, hybridlinks, optimal stopping theory, quality of service (QoS).

I. INTRODUCTION

I T is common that wireless links in a network have hetero-geneous characteristics such as transmission rates and QoS

requirements. Such links are called hybrid links in this paper.One factor leading to link heterogeneity involves application-specific requirements for different links. For example, someapplications (e.g., control message transmissions in the smartgrid or cognitive radio networks) impose a strong require-ment on the security, and physical layer techniques [1]–[3]are applied to ensure perfect secrecy in corresponding links

Manuscript received June 7, 2015; revised October 5, 2015; acceptedNovember 24, 2015. Date of publication November 30, 2015; date of currentversion October 13, 2016. This work was supported by the National NaturalScience Foundation of China under Grant 61172066. The review of this paperwas coordinated by Prof. C. Assi. (Corresponding author: Xudong Wang.)

W. Mao was with the University of Michigan-Shanghai Jiao Tong UniversityJoint Institute, Shanghai Jiao Tong University, Shanghai 200030, China. Heis now with the Department of Computer Science, The University of Texas atAustin, Austin, TX 78712 USA.

X. Wang is with the University of Michigan-Shanghai Jiao Tong UniversityJoint Institute, Shanghai Jiao Tong University, Shanghai 200030, China (e-mail:[email protected]).

S. Wu was with the University of Michigan-Shanghai Jiao Tong UniversityJoint Institute, Shanghai Jiao Tong University, Shanghai 200030, China. She isnow with the Wireless Networking and Communications Group, The Universityof Texas at Austin, Austin, TX 78701 USA.

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.2015.2504421

(called secure links) [4], [5]. Since perfect secrecy comes atthe cost of degrading channel capacity [6], [7], secure linkshave much lower transmission rates as compared to other links(called regular links). Due to security concern, secure links mayalso demand stringent QoS guarantee. For ease of explanationthroughout this paper, we use secure links and regular links torepresent two link types that follow significantly different ratedistributions.

Packet transmissions in a network with hybrid links canbe conducted in two different approaches: 1) following apure random access medium access control (MAC) protocol;2) based on a scheduling scheme. The former approach issimple and easy to implement, but this may lead to lowthroughput in a network with hybrid links due to the pres-ence of performance anomaly [8], i.e., the wireless mediumis extensively occupied by low-rate transmissions on securelinks. Therefore, the latter approach is necessary to improvethe network throughput. Among existing scheduling schemes,opportunistic scheduling is considered as the most effective toexploit fluctuations in channel conditions to produce significantthroughput gains for the entire network [9]–[11]. The key ideaof opportunistic scheduling is explained as follows: given atransmission opportunity, if a link with the highest transmissionrate is selected, the maximum throughput can be achieved.Unfortunately, these opportunistic scheduling schemes rely onthe existence of the central controller (e.g., the base station incellular networks) and, hence, are hard to implement in ad hocnetworks or wireless mesh networks, where such a centralnode is not readily available. To address this issue, severaldistributed opportunistic scheduling schemes are proposed in[12] and [13], which utilized local information to determinewhether to take transmission opportunities or not. However,the quality of service (QoS) of communication links is nottaken into account in these schemes. A distributed opportunisticscheduling scheme that considers the delay as a QoS metric isdeveloped in [14] based on the scheme in [15]. However, thisscheme is not applicable to hybrid links, as it cannot guaranteeQoS requirements for a specific type of links (e.g., secure links)and, at the same time, maximize the overall throughput. So far,there is a lack of effective distributed opportunistic schedulingto support a network with hybrid links.

To treat hybrid links separately and also support QoS require-ments of a specific type of links, a new distributed opportunisticscheduling scheme is proposed in this paper. It is developedbased on the optimal stopping theory and considering two

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8512 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016

types of links: secure links and regular links. Compared withexisting opportunistic scheduling schemes, the new schedulingscheme is distinct with the following features: 1) The systemoverall throughput is maximized under various QoS constraintsof a specific link type (e.g., secure links); 2) it can be im-plemented as a double-threshold scheduling policy, i.e., onethreshold for each link type, and then, a link determines itstransmission opportunity based on this threshold; 3) the rateheterogeneity among the same type of links is also taken intoaccount to improve QoS of links with low channel quality.Simulations are carried out to evaluate the new opportunisticscheduling scheme. Performance results verify the optimalityof our scheme and demonstrate that QoS of secure links can beeffectively guaranteed under both homogeneous and heteroge-neous rate distribution scenarios. Moreover, results also showthat our scheme outperforms the existing ones in most cases.

The remainder of this paper is organized as follows: Therelated work is summarized in Section II. The system modelfor our opportunistic scheduling is explained in Section III.The QoS-oriented opportunistic scheduling scheme under thescenarios of homogeneous and heterogeneous rate distributionsis derived in Sections IV and V, respectively. Performanceresults are presented in Section VI. Further discussions aboutour scheme are provided in Section VII. This paper is concludedin Section VIII.

II. RELATED WORK

Opportunistic scheduling is an effective way to utilizethe fluctuation of channel conditions to enhance the net-work throughput performance. Several opportunistic schedul-ing schemes (e.g., [9]–[11]) have been proposed for a networkwith a central controller. However, these schemes are notapplicable to wireless mesh/ad hoc networks, where such con-troller does not exist. To solve this problem, many distributedopportunistic scheduling schemes have been developed so far.These schemes exploit local information to determine whetherto take transmission opportunities. In [16], an opportunisticdistributed scheduling is developed, and the resulting systemcapacity is studied based on point process approximation. It isshown that the capacity approaches that of a centralized system,where the best link is always selected to transmit. The focusin [16] is on uplink traffic in a multiple-access channel, whileour paper considers peer-to-peer traffic in an ad hoc network.In [12], a channel-aware ALOHA protocol is developed, wherethe nodes only transmit their packets when their channel gainsare above a given threshold. In addition, other channel-awareALOHA schemes are designed according to decentralizedchannel state information in [17] and [18]. Based on the optimalstopping theory [19], a distributed opportunistic scheduling isderived in [15] to maximize the system overall throughput. In[20], a stopping policy is investigated, when the channel quali-ties for different transmission periods are correlated. In [21], theauthor proposes a distributed opportunistic scheduling schemebased on game theory. In this scheme, an effective mechanismis designed to combat the issue of user selfishness. Althoughthese schemes utilize opportunistic transmissions to improvethe network throughput in a distributed manner, none of them

takes into account the QoS of interested links. Thus, theseschemes are not applicable to scenarios studied in this paper.

Distributed opportunistic scheduling schemes proposed in[13], [14], and [22] are most related to the scheme developedin this paper. In [13] and [22], an opportunistic schedulingscheme is developed to maximize the proportionally fair allo-cation. Based on the control theory, the scheme adapts to thevariation of network load and can dynamically drive the systemto the optimal operation point. This scheme is different fromours in the following points. First, the scheme considers thefairness. Although ensuring fairness is beneficial to improvethe QoS of the links with low transmission rates, it cannotdirectly guarantee a specific QoS requirement on such links(e.g., the delay is less than a specific value or the throughput isgreater than a specific value). In contrast, our scheme directlyfocuses on the QoS. Second, the proportionally fair allocation ismaximized in their scheme instead of the overall throughput ofthe network, as in our scheme. In [14], an opportunistic schedul-ing scheme considering delay QoS is developed based on thescheme proposed in [15]. In this scheme, network centric delayconstraints and individual delay constraints are satisfied basedon different strategies. However, this scheme is not suitable fora network with hybrid links, for two reasons: 1) A network-centric delay constraint cannot guarantee the QoS requirementsof one specific type of links; 2) if the individual delay constraintis applied to each link, the overall throughput is not maximized.

To the best of our knowledge, the optimal stopping the-ory [19] is first introduced to derive distributed opportunisticscheduling schemes in [14] and [15]. The framework of deriva-tions in this paper is based on the work in [14] and [15] butmakes the following nontrivial and important extensions. First,in our model, two types of links, which have different rate dis-tribution functions, different QoS requirements, and differenttransmission durations, are considered [23]. To meet their QoSrequirements, two types of links need to be treated differently inthe mathematical derivations, which leads to a double-thresholdstopping policy. Furthermore, the strategies for handling twotypes of links in this paper can be easily extended to supportmultiple types of links with various QoS requirements. Second,in our derivations, we consider both the delay constraints andthe throughput constraints, which leads to different derivationsto obtain the scheduling scheme. For instance, it is requiredto design more complicated profit functions for opportunisticscheduling with throughput constraints. Third, in our model,link heterogeneity among the same type of links is taken intoaccount. In this case, we set constraints on a subset of one typeof links, instead of the whole set of links, which complicatesthe process of deriving the optimal stopping thresholds and themaximum expected profit equation.1

III. SYSTEM MODEL

In this paper, we focus on a single-hop ad hoc network whereall nodes can hear each other, following the same assump-tion in previous work [13]–[15], [21]. Although this networksetup cannot represent a generic network model, it is actually

1See Section IV-A.

MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8513

Fig. 1. Diagram for channel contention and data transmission.

meaningful in a practical environment. Such a network modelis common in various communication scenarios, e.g., wirelesssensor networks [24], body area networks [25], and device-to-device (D2D) networking in 5G networks [26]. Moreover,a single-hop network can serve as a building block for ageneral multihop network. A common and practical approachto handling packet transmissions in a multihop network is todivide the entire network into several single-hop subnetworksand coordinate transmissions among these subnetworks in ahierarchical way [27]. Within each subnetwork, the schedulingscheme developed in this paper can be adopted. Hence, thestudy on a single-hop network also benefits data transmissionsin a general multihop network.

The carrier sensing is enforced: if a node detects a busymedium, it needs to postpone its own transmission. Moreover,two types of links between nodes in this network areconsidered: 1) secure links for transmitting critical messageswith physical layer security; 2) regular links for other messages.Note that the links considered in this paper are virtual. Thus,a secure link and a regular link can have the same source anddestination nodes. In this case, these two links share a commonphysical link.

When the transmission medium is sensed idle, a node con-tends the channel slot by slot with a fixed access probability(i.e., p-persistent mechanism). To this end, the node transmits apilot packet to its destination node at the beginning of a certaintime slot. If two or more nodes contend the channel in the sametime slot, a collision occurs (denoted by “C” in Fig. 1). If onlyone node sends a pilot packet to the channel, the destination willsuccessfully receive it and reply a confirmation packet. In thiscase, the contention is successful (denoted by “S” in Fig. 1).Note that the confirmation packet is transmitted in the sametime slot with the pilot packet. Thus, the size of a time slot is setlarger than the total length of a pilot packet and a confirmationpacket. If none of the nodes transmits a pilot packet, the channelremains idle (denoted by “I” in Fig. 1).

If a node successfully captures the channel, instead ofproceeding to data packet transmissions directly, it needs todetect current channel quality and follow a decision rule todetermine whether or not a packet can be transmitted: If thecurrent channel quality is low, the node skips the transmissionopportunity to avoid the situation where the wireless mediumis occupied by a low-rate transmission, as shown in Fig. 1.For this purpose, the node needs to know the current channelquality and determine the decision rule. The channel from thesource to the destination is measured by the destination usingthe preamble sequence in the pilot packet, and the results are

carried back by the confirmation packet. The decision rule isderived based on the optimal stopping theory [19], such thatthe network throughput is maximized under the constraints ofQoS requirements. If the opportunity is dropped, the contentionprocess restarts in the next time slot. Otherwise, the node startsits data transmissions. The transmission can last multiple timeslots, as shown in Fig. 1, and we assume that the channelcondition remains unchanged during the transmission process(i.e., block-fading channel). In addition, we assume that thechannel conditions during different transmission periods areindependent, which is the same in the previous work [13], [14].

To clearly present our opportunistic scheduling scheme, sev-eral parameters are defined as follows. As shown in Fig. 1, atime slot has a length of t, and data transmission durationsfor secure links and regular links are assumed constant anddenoted by Ds and Dr, respectively. There are M nodes inthe network. For the sake of clarity, we assume that each nodehas exactly one secure link and one regular link. However,the derivation presented in Sections IV and VI can be directlyapplied to general cases. Given a time slot, nodem contends thechannel for its secure link and regular link with the probability(i.e., the persistent factor) equal to ps,m and pr,m respectively.Furthermore, the probability that the secure link of nodem winsthe channel contention, i.e., Ps,m, is given by

Ps,m = ps,m∏i�=m

(1 − ps,i − pr,i). (1)

Thus, Ps, which is defined as the probability that any securelink successfully contends the channel, can be calculated by∑

Ps,m. Pr,m and Pr are defined for regular links in a similarway. In addition, the transmission rates on the secure link andthe regular link of node m follow the distributions with cumu-lative (probability) density function (CDF) Fs,m(r)(fs,m(r))and Fr,m(r)(fr,m(r)), respectively. As in [13] and [14], thesedistribution functions are assumed known. Moreover, for themathematical tractability, we assume that Fs,m(r) and Fr,m(r)are differentiable, and fs,m(r) and fr,m(r) are greater than zerofor any r > 0. These assumptions are valid for commonly usedrate distribution models [28]. For convenience, let Rs denotethe transmission rate on any secure link. The distribution of Rs,i.e., Fs(r), is given by

Fs(r) =∑m

Ps,m

PsFs,m(r). (2)

For regular links, Rr and Fr(r) are defined similarly.As shown in Fig. 1, if a node successfully contends the

channel at time N and decides to transmit a data packet, thenwe call N a stopping time. Given a stopping time N , TN

denotes the total time for this transmission round, includingcontention period and packet transmission time DN . Here,DN is equal to Ds, if the transmission is on a secure link,and is equal to Dr for the transmission on a regular link. Inaddition, RN is used to denote the transmission rate of thenode. If the transmission in this round is on a secure link, Rs

N

is used to denote the transmission rate (i.e., RN = RsN ). The

time between two successive transmissions on secure links is


Fig. 2. Rate distributions of different links in the homogeneous scenario.

denoted by Ts, which also stands for the delay of secure links.Apparently, the choice of stopping time has influence on Ts.

IV. OPPORTUNISTIC SCHEDULING WITH

HOMOGENEOUS RATE DISTRIBUTIONS

To maximize the system overall throughput under variousQoS constraints for secure links, we develop a new distributedopportunistic scheduling scheme based on the optimal stoppingtheory. First, the homogeneous rate distribution scenario is con-sidered, where the rate distributions of secure links are identicalto each other. In this case, the overall performance of all securelinks can effectively reflect that of each individual link, sinceeach link contributes equally to the overall performance. Thus,here, we consider secure links as a whole and set overall QoSconstraints on all these links. As soon as the overall QoSof secure links is satisfied, the corresponding QoS for eachindividual link is also guaranteed.

In practice, the developed scheduling scheme is useful in twoscenarios: 1) The rate distributions of secure links are closeto each other, as shown in Fig. 2; 2) the rate distributions aredifferent, but only the overall QoS of secure links needs to besatisfied.

A. Opportunistic Scheduling With Throughput Constraint

We define ζ as the set of stopping times as follows:

ζ � {N : N � 1, E[TN ] ≤ ∞} .

In addition, let θs denote the throughput on all secure links andα stand for the minimum throughput requirement. The maxi-mization of overall throughput under the throughput constraintcan be formulated as

maxN∈ζ

E[RNDN ]

E[TN ], subject to θs =

E [RsNDs]

E[Ts]≥ α.

Note that a distributed opportunistic scheduling that maximizesthe overall throughput under QoS constraints is derived in [14]based on the optimal stopping theory, but the solution therein isnot applicable in our scenarios for two reasons. First, there existtwo different types of links in our problem, and the constraintpresented in our formula is applied to one type of links insteadof to all links. This leads to different stopping policies for two

types of links. In addition, in [14], only the delay constraint isconsidered, while in our problem, the throughput constraint isalso studied. Hence, a new derivation is needed.

Let x∗ denote the maximum throughput under the throughputconstraint. According to [19, Th. 6.1], the optimization problemformulated previously is equivalent to

maxN∈ζ

E[RNDN ]− x∗E[TN ]

subject to

αE[Ts]− E [RsNDs] ≤ 0.

The formulated optimization problem is a constrained one. Tosolve this problem, we convert it into an unconstrained onethrough the method of Lagrange multipliers. As proved inAppendix A, the constraint qualification is satisfied in our prob-lem. In this case, the solution for the original problem also max-imizes the converted problem, and the Karush–Kuhn–Tucker(KKT) condition holds [29]. As a result, we can find the optimalsolution for the original problem by solving the convertedproblem. If there are more than one solution for the convertedproblem, we select the one that maximizes the original objectivefunction.

Based on the method of Lagrange multipliers [30], the opti-mization problem is converted to

maxN∈ζ

E[RNDN ]− x∗E[TN ]− λ (αE[Ts]− E [RsNDs]) (3)

where λ is the Lagrange multiplier. By solving this problem,the following proposition can be derived.

Proposition 4.1: The optimal stopping rule for secure linksand regular links is a double-threshold policy. The threshold forsecure links is φs, and that for regular links is φr . If a secure linkwins the channel contention, it does not skip the transmissionopportunity only if Rs ≥ φs; if a regular link successfullycaptures the channel, it takes the transmission opportunity whenRr ≥ φr. The optimal thresholds for secure links and regularlinks are given by {

φs =x∗+λα1+λ

φr = x∗ + λα.(4)

Proof: Let us define the profit for stopping rule N1 (i.e.,the time when a node successfully contends the channel andstarts the transmission) as

RN1DN1

+ λRsN2

Ds − x∗TN1− λαT ′

s

where N2 is the first stopping time for secure links afterN1 (including N1). Thus, if the link that wins the channelcontention at N1 is a secure one, N2 is equal to N1. In addition,T ′s denotes the time from the beginning of the contention period

for stopping rule N1 to the end of the transmission for stoppingrule N2, as shown in Fig. 3. In addition, according to thememoryless characteristic of our system model, we have

E[T ′s] = E[Ts].


Fig. 3. Two cases for stopping rule N1. 1) A secure link wins the channel.2) A regular link wins the channel.

Thus, the maximum expected profit for stopping rules, which isdenoted by L∗, can be expressed as

maxN∈ζ

E[RNDN ]− x∗E[TN ]− λ (αE[Ts]− E [RsNDs]) .

According to [19, Th. 6.1] and KKT conditions, the value ofL∗ is zero. Moreover, it can be observed that the aforemen-tioned expression is identical with the optimization formulationpresented in Section IV-A. Therefore, solving the optimizationproblem in Section IV-A is equivalent to finding the optimalstopping rule that maximizes its expected profit. To maximizethis expectation, the opportunistic scheduling allows the packettransmission only when the system meets the most favorableopportunity: Given a node that successfully contends the chan-nel, if the profit for transmitting immediately is greater thanthe maximum expected profit with skipping the opportunityand waiting for the next stopping time, the node starts thetransmission; otherwise, the opportunity is dropped.

Consider that a node successfully contends the channel forits secure link. If the node takes this transmission opportunity,the profit can be quantified as

RsDs + λRsDs − x∗Ds − λαDs − (x∗ + λα)Tcont

where Tcont is the contention period before this successfulchannel contention. However, if the node skips this opportunity,based on the time-invariant characteristic of the system, themaximum expected profit is given by L∗ − (x∗ + λα)Tcont.Hence, if

(1 + λ)RsDs − x∗Ds − λαDs ≥ L∗

the profit for taking the transmission opportunity is greater thanthat with skipping the opportunity and waiting for the nextstopping time. In this case, the packet on this secure link istransmitted. However, if (1 + λ)RsDs − x∗Ds − λαDs < L∗,transmitting immediately is less favorable than waiting for bet-ter opportunity. In this case, the node drops the transmission andthe channel contention restarts. Therefore, the stopping thresh-old for secure links is given by Rs ≥ ((x∗ + λα)/(1 + λ)) =φs. Similarly, if a regular link succeeds in channel contention,the maximum expected profit with skipping the transmissionopportunity is L∗ − (x∗ + λα)Tcont, while the expected profitfrom taking the transmission opportunity is given by

RrDr+λE [RsNDs]−x∗Dr−λα (Dr+E[Ts])−(x∗+λα)Tcont.

Note that, after the end of current transmission, the expectedwaiting time for the next transmission on secure links isequal to E[Ts], according to the memoryless characteristicof our system model. In addition, based on KKT conditions[30], we have λ(E[Rs

NDs]− αE[Ts]) = 0. Thus, the profitexpression can be simplified as RrDr − x∗Dr − λαDr −(x∗ + λα)Tcont. Therefore, when RrDr−x∗Dr−λαDr≥L∗,the regular packet is transmitted. Otherwise, the transmissionopportunity is skipped. Hence, the transmission threshold forregular links is given by φr = x∗ + λα. The proposition isproved. In addition, the derivation for the optimal stopping ruleswith the delay constraint and double constraints follows thesimilar framework as presented earlier. �

According to Proposition 4.1, it can be shown that φs <x∗ < φr. Thus, packets on regular links can be sent only whenthe current transmission rate is greater than the system expectedthroughput, while messages on secure links are transmittedeven if the transmission rate is less than the throughput value.Such discrimination between two types of links is helpful tofavor the transmissions on secure links and hence provides theQoS on these links.

In Proposition 4.1, the optimal thresholds φs and φr areexpressed in terms of (x∗, λ). Hence, further calculation of φs

and φr requires the knowledge of (x∗, λ). The procedure fordetermining (x∗, λ) is presented as follows.

According to the definition of profit given in the proofof Proposition 4.1, the maximum expected profit L∗ can beexpressed as

L∗=PsE[max(RsDs+λRsDs−x∗Ds−λαDs, L∗)−kt(x∗+λα)]

+ PrE [max (RrDr + λE [RsNDs]− x∗Dr − λαDr

−λαE[Ts], L∗)− kt(x∗ + λα)] (5)

where k is the number of time slots before the first successfulchannel contention. Note that the first expectation in the right-hand side of (5) denotes the maximum expected profit whenthe first successful channel contention is won by a securelink, while the second expectation stands for the maximumexpected profit when a regular link takes the first successfulchannel contention. Since L∗ is zero as explained in the proofof Proposition 4.1, (5) can be simplified as

t(x∗+λα)=DsPs(1+λ)E[(Rs−φs)

+]+DrPrE

[(Rr−φr)

+]

where (·)+ denotes max{·, 0}. In addition, according to KKTconditions, we have λ(E[Rs

NDs]− αE[Ts]) = 0, where⎧⎨⎩E [Rs

NDs] =∫∞φs

rdFs(r)∫∞φs

dFs(r)Ds

E[Ts] =t+Pr(1−Fr(φr))Dr

Ps(1−Fs(φs))+Ds.

(6)

The first expression in (6) is based on the fact that alltransmissions on secure links are conducted with the ratesgreater than φs, while the second expression is derived inAppendix B. Combining the maximum expected profit equa-tion, KKT conditions, and (4), (x∗, λ) is calculated with theLevenberg–Marquardt algorithm (LMA) [31], i.e., a numer-ical method to solve nonlinear equations. Theoretically, theconvergence speed of LMA is similar to that of the widely


known Gauss–Newton method. However, in practice, the imple-mentations of LMA are proved more efficient in most scenarios[31]. MATLAB [32] provides a built-in function that imple-ments the LMA algorithm. We use this function to solve theaforementioned equations to obtain (x∗, λ). Following that, theoptimal threshold pair (φs, φr) can be calculated based on (4).

It is necessary to emphasize that throughput constraint α iseffective only when it falls into a specific range. If α is toosmall, the optimal thresholds derived from previous equationswill be equal to those in the unconstrained case. In this scenario,the throughput constraint is inactive. If α is too large, theconstraint cannot be satisfied, even if skipping all regulartransmissions. To characterize the lower bound and the upperbound for α, we have the following proposition.

Proposition 4.2: The effective range for throughput require-ment α is given by θLs ≤ α ≤ θUs , where θUs is the maximumthroughput of secure links when φr = ∞, and it can be deter-mined by

θUs =Ps

∫∞θUsrdFs(r)

tDs

+ Ps (1 − Fs (θUs )).

In addition, θLs is the throughput of secure links when thethreshold pair is equal to the optimal one (i.e., (φ∗, φ∗))2 forthe unconstrained case, and it is given by

θLs =PsDs

∫∞φ∗ rdFs(r)

t+ PsDs (1 − Fs(φ∗)) + PrDr (1 − Fr(φ∗)).

The detailed proof of this proposition is listed in Appendix C.As the rate distributions of secure links improve, the throughputof secure links with φr = ∞, i.e., θUs , increases due to thehigher link rate for each transmission. In addition, the through-put of secure links with (φs, φr) = (φ∗, φ∗), i.e., θLs , enhancesbecause secure links get more transmission opportunities. Thus,according to the proposition, both the upper bound and thelower bound of the effective range for α will increase withbetter rate distributions of secure links.

B. Opportunistic Scheduling With Delay Constraint

Here, an opportunistic scheduling scheme with delay con-straint is studied. Similar to the network-wide average delaydefined in [14], the delay studied here is imposed on the set ofall secure links, instead of a specific secure link. Specifically,the delay imposed on the set of secure links (denoted by Ts)is defined as the time between two successive transmissionson any secure links, as shown in Fig. 1. In contrast, the delayon a secure link of node m (denoted by Ts,m) is defined asthe time between two successive transmissions on this link.In homogeneous cases, if there are n secure links in total, theaverage delay on a specific secure link is about n times as thaton the set of all secure links, i.e., E[Ts,m] = nE[Ts]. Based onthis relationship, we can set the delay constraint imposed onthe set of all secure links (i.e., Ts) according to the delayrequirement of individual secure link.

2In the unconstrained scenario, the thresholds for secure links and regularlinks are identical.

Let σs stand for the average delay of secure links and β de-note the delay requirement. Thus, we formulate the problem as

maxN∈ζ

E[RNDN ]

E[TN ], subject to σs = E[Ts] ≤ β.

As discussed in Section IV-A, the previous optimization prob-lem is equivalent to

maxN∈ζ

E[RNDN ]− x∗E[TN ]− μ (E[Ts]− β)

where μ is the Lagrange multiplier. By solving this problem,the optimal threshold pair can be derived as

{φs = x∗ + μ− μβ

Ds

φr = x∗ + μ.

To further calculate (φs, φr), the following equations areneeded:

t(x∗ + μ) = PsDsE[(Rs−φs)

+]+PrDrE

[(Rr−φr)

+]

μ (E[Ts]−β) = 0

where the first equation is the maximum expected profit equa-tion and the second one is from KKT conditions [30]. Thederivation of aforementioned equations and the calculation ofthe optimal thresholds follow the similar framework presentedin Section IV-A. In addition, similar with throughput constraint,delay requirement β also has an effective range, as described inthe following proposition.

Proposition 4.3: The delay constraint β has a lower effectivebound βL and an upper effective bound βU . βL is the minimalpossible average delay for secure links, and it is given by βL =(t/Ps) +Ds. βU is the average delay for secure traffic, whenthe thresholds for secure links and regular links are set to thesevalues (i.e., φ∗ for both types of links) in the unconstrained case,and it can be determined by

βU =t+ Pr (1 − Fr(φ

∗))Dr

Ps (1 − Fs(φ∗))+Ds.

Proof: For the lower bound, Ts includes time period t/Ps

for at least one round of successful channel contention andpacket transmission time Ds. Hence, the minimum achievabledelay requirement is (t/Ps) +Ds. For the upper bound, if de-lay requirement β is greater than βU , then the optimal thresholdpair for the unconstrained case is located in the feasible domain,which indicates that this optimal solution is also the one forthe constrained problem. In this case, the delay constraint isinactive. �

According to the proposition, the lower bound of the effectiverange is determined by access probabilities of secure links andunrelated to rate distributions, while the upper bound decreasesas the rate distributions of secure links improve.


C. Opportunistic Scheduling With Throughput andDelay Constraints

In some scenarios, both throughput and delay requirementsare imposed. In this case, the problem for maximizing theoverall throughput can be formulated as

maxN∈ζ

E[RNDN ]

E[TN ]

subject to

θs =E [Rs

NDs]

E[Ts]≥ α and σs = E[Ts] ≤ β.

As discussed in Section IV-A, the aforementioned optimizationproblem is equivalent to

maxN∈ζ

E[RNDN ]− x∗E[TN ]

− λ (αE[Ts]− E [RsNDs])− μ (E[Ts]− β)

where λ and μ are the Lagrange multipliers. By solving thisproblem, the optimal stopping rule is derived. Similar to thecase of a single constraint (e.g., throughput or delay), this ruleis also a double-threshold stopping policy, and the optimalthresholds are given by{

φs =x∗+λα+μ− μβ

Ds

1+λ

φr = x∗ + λα+ μ.

The calculation of aforementioned thresholds requires theknowledge of (x∗, λ, μ), which can be determined with thefollowing equations:

t(x∗ + λα+ μ) = PsDs(1 + λ)E[(Rs − φs)

+]

+ PrDrE[(Rr − φr)

+]

(7){λ (E [Rs

NDs]− αE[Ts]) = 0

μ (E[Ts]− β) = 0.(8)

Equation (7) is the maximum expected profit equation, while(5) and (8) are from KKT conditions.

Similar to single-constraint scenarios, there exists an areawhere both throughput requirement α and delay requirementβ are effective. To characterize this area, we have the followingproposition.

Proposition 4.4: For a given delay requirement β, the effec-tive range for α is bounded by[ ∫∞

φLsrdFs(r)

β (1 − Fs (φLs ))

Ds,

∫∞φHsrdFs(r)

β (1 − Fs (φHs ))

Ds

]

where ⎧⎨⎩φLs = F−1

s

(1 − t+PrDr

Ps(β−Ds)

)φHs = F−1

s

(1 − t

Ps(β−Ds)

).

The detailed proof is listed in Appendix D. When α lies on theleft side of the range given in the proposition, the throughput

Fig. 4. Rate distributions of different links in heterogeneous scenarios.

constraint is less tight than the delay constraint. In this case, thethroughput constraint is inactive. When α is greater than theupper bound of the range, the throughput requirement imposesa stronger constraint on the problem. In this case, the delayconstraint is inactive.

V. OPPORTUNISTIC SCHEDULING WITH HETEROGENEOUS

RATE DISTRIBUTIONS

Here, the heterogeneous rate distribution scenario is con-sidered, where the rate distribution of a secure link can besignificantly different from those of other secure links, asshown in Fig. 4. In this scenario, the overall performance ofsecure links cannot reflect that of each individual link. If weconsider all secure links as a whole as previous sections, theopportunistic scheduling scheme will provide more transmis-sion opportunities to links with good link quality and cause thestarvation of links with poor channel quality, which leads tounacceptable QoS for these links. To avoid this situation, we setQoS constraints for secure links with the worst link conditions,instead of putting QoS requirements on the whole set of securelinks.

A. Worst Link Analysis

The basic idea behind our scheme for heterogeneous sce-narios is to set QoS constraints on the secure links with theworst performance, instead of on the whole set of secure links.Hence, before imposing QoS constraints, we need to identifysuch links. Considering the secure link of node m, the averagedelay of this link, i.e., E[Ts,m], can be expressed as

E[Ts,m] =Δ

Ps,m (1 − Fs,m(φs))(9)

where

Δ= t+∑i

Ps,i (1−Fs,i(φs))Ds+∑i

Pr,i (1−Fr,i(φr))Dr.

The derivation of this expression can be found in Appendix B.According to the aforementioned equation, the numerators ofdelay expressions for all secure links are exactly the same.Therefore, the secure link with the worst delay can be deter-mined, if we can find a link with the minimum denominator


Fig. 5. Value of denominators in delay expressions for secure links.

value in its delay expression. Furthermore, if the worst linkis identified and the delay QoS constraint is set on the link,the delay of any other secure links will also meet this QoSrequirement, since these links have better delay performanceas compared to the worst link.

However, in some scenarios, such a link cannot be deter-mined without the knowledge of the stopping threshold forsecure links. This can be explained with the following example.Consider three secure links with heterogeneous rate distribu-tions. The denominator values in their delay expressions underdifferent thresholds of secure links (i.e., φs) are plotted inFig. 5. If the threshold is equal to a, the denominator of Link 1has the minimum value, and hence, Link 1 is the worst link; ifthe threshold is equal to b, Link 2 is the worst in delay. Namely,with different thresholds φs, the link that experiences the worstdelay performance is different. Therefore, unless we know thethreshold of secure links, we cannot determine the secure linkwith the worst delay performance.

In this case, instead of identifying a unique secure link thatexperiences the worst performance, we consider all links thatpotentially become the worst link in an interested range of φs.We call these links potential-worst links. In the previous exam-ple, both Link 1 and Link 2 can be the worst link under dif-ferent thresholds of secure links. Therefore, Link 1 and Link 2are both potential-worst links. However, Link 3 evidently hasbetter quality than Link 1 and Link 2 and never experiences theworst delay performance. Therefore, Link 3 does not belongto the set of potential-worst links. Generally, potential-worstlinks have relatively poor channel quality. To help these linksto achieve an acceptable performance, we treat these links as agroup and set QoS constraints on the group.

B. Opportunistic Scheduling With the Delay Constraint onPotential-Worst Links

We define ξ as the set that consists of all potential-worstsecure links. Furthermore, the delay on the set of potential-worst links is defined as the time between two successivetransmissions on links belonging to this set. For convenience,Tpw is used to denote this delay.

To avoid the starvation on potential-worst links, we set thedelay requirement on these links as E[Tpw] � γ. Therefore, the

problem of maximizing the overall throughput under the delayconstraint can be formulated as

maxN∈ζ

E[RNDN ]

E[TN ], subject to E[Tpw] ≤ γ

where ζ denotes the set of stopping times. As discussed inSection IV-A, the previous optimization problem is equiv-alent to

maxN∈ζ

E[RNDN ]− x∗E[TN ]− ω (E[Tpw]− γ)

where ω is the Lagrange multiplier. By solving this optimiza-tion problem, we can derive the following proposition.

Proposition 5.1: The optimal stopping rule that maximizesthe overall throughput under the delay constraint on potential-worst links is a double-threshold policy, and the optimal thresh-old pair is given by{

φs = x∗ + ω −∑

i∈ξ Ps,i

Ps

γωDs

φr = x∗ + ω.

In addition, the value of (x∗, ω) can be determined with thefollowing equations:⎧⎪⎨⎪⎩t(x∗ + ω) = PsDsE [(Rs − φs)

+] + PrDrE [(Rr − φr)+]

+(Ps−

∑i∈ξ Ps,i)(

∑i∈ξ Ps,i)γω

PsFΔ

ω (E[Tpw]− γ) = 0

where FΔ denotes the difference between the value of therate distribution function of non-potential-worst secure linksat φs (i.e., Fs,nw(φs)) and that of potential-worst links (i.e.,Fs,pw(φs)). The proof of this proposition can be found inAppendix E.

C. Effectiveness of the Proposed Scheme

The proposed scheme can effectively guarantee that the delayon the set of potential-worst links is less than a specified valueγ. We assume that there are np links in this set. If these linkshave comparable performance for a given threshold pair derivedfrom the aforementioned equations, then the delay of each linkis approximately equal to npγ. Since the worst link belongs tothe set of potential-worst links, the delay of all other securelinks outside the set cannot be longer than npγ. Thus, in thiscase, the delay performance of each secure link is guaranteedto be not worse than a specific level (i.e., npγ).

If potential-worst links do not have comparable performancegiven the derived threshold pair, the delay of the worst link canbe longer than npγ. In this case, the proposed scheme cannotguarantee the delay performance of each secure link. However,by setting constraints on the whole set of potential-worst links,the scheme is still beneficial to avoid severe starvation in low-quality secure links.

Another possible solution to handling heterogeneous scenar-ios is to separate the worst links from the secure links, treatthem as different link types, and derive a certain threshold foreach of them. This solution can achieve better performance


TABLE INORMALIZED SNR FOR SECURE LINKS AND REGULAR

LINKS OF DIFFERENT NODES

than the proposed scheme here, since there are more degreesof freedom to select thresholds for various links. However,it increases the number of link types, which leads to highcomplexity in scheduling. Nevertheless, even after separatingthe worst links from each other, heterogeneity still exists amongother secure links. As a result, more types of links need tobe considered, which further increases the complexity of thescheduling scheme and becomes impractical.

VI. PERFORMANCE EVALUATION

Here, the new scheduling scheme is evaluated by several ex-periments. In our simulation, we consider a network consistingof five nodes,3 and each of them maintains its own regular linkand secure link to other nodes. The transmission rate on a linkis assumed to be equal to the channel capacity given by

R = log(1 + ρ|H |2

)nats/s/Hz (10)

where H denotes the random channel gain that follows acomplex Gaussian distribution with the variance equal to 1,and ρ is the normalized SNR for the link. In the simulationsfor homogeneous scenarios, all links belonging to the sametype are set with identical normalized SNR, as given in Table I,while in heterogeneous cases, the links of Node 1 and Node 2evidently have worse link quality, as shown in Table I. Inaddition, the transmission duration for regular links is equalto 30 time slots, while that for secure links varies in differentexperiments and its default value is 30 time slots. Moreover, toreflect the traffic load of the network, channel occupancy ratiois introduced and defined as

P = 1 −∏m

(1 − ps,m − pr,m).

Without being explicitly specified, ps,m and pr,m, for any nodem, are equal to 0.1 in our experiments. In this case, the channeloccupancy ratio is 0.672.

Based on aforementioned parameters, our scheme is eval-uated with MATLAB programs. Given a specific setting(including thresholds for different types of links, channel ac-cess probabilities for nodes, transmission duration, etc.), ournetwork simulation program runs for 107 time slots, and perfor-mance results, such as overall throughput, throughput on securelinks, and delay of secure links, are recorded. Note that the per-formance variation introduced by channel condition/mediumaccess randomness is negligible with the specified running time(i.e., 107 time slots).

3Since our method is insensitive to the number of nodes in the network, asimulation with five nodes is enough to demonstrate the network performance.

Fig. 6. Optimal throughput under various QoS constraints. (a) Homogeneousscenario 1. (b) Homogeneous scenario 2. (c) Heterogeneous scenario.

A. Homogeneous Scenarios

To verify the optimality of threshold pairs derived inSection IV, we compare the throughput performance of ourscheme (denoted by QDOS) with the optimal throughput ob-tained by the enumeration method (denoted by Enum.) undervarious QoS constraints. In the enumeration method, we searchover all possible threshold combinations (φs, φr), and ournetwork simulation program runs for each combination. Basedon simulation results, threshold combinations that do not guar-antee the QoS constraints are removed. Among the remainingones, the combination leading to the maximum overall through-put is selected, and the corresponding maximum value isrecorded.

The comparison results between QDOS and Enum. undervarious QoS constraints are shown in Fig. 6(a) and (b). It canbe observed that the throughput of QDOS is always equal tothe optimal value obtained through enumerating all possiblethreshold pairs. These results demonstrate the optimality of ourscheme.

Performance results of our scheduling scheme with differentQoS constraints are shown in Fig. 7. For comparison, per-formance of another two access schemes is provided: 1) thepure random access scheme, namely, the scheme with bothφs and φr equal to zero; and 2) the distributed opportunisticscheme (DOS) proposed in [15]. In Fig. 7(a), we know that


Fig. 7. Throughput and delay with different schemes under homogeneous scenarios. (a) Overall throughput. (b) Throughput of secure links. (c) Delay ofsecure links.

TABLE IITHROUGHPUT AND DELAY WITH DOUBLE CONSTRAINTS UNDER

HOMOGENEOUS SCENARIOS

the overall throughput of our scheme is significantly higherthan that of the random access scheme. In addition, when thechannel occupancy ratio is greater than 0.2, the throughputperformance loss of our scheme, as compared to the DOSscheme, is within 14%. This result indicates that the overallthroughput of our scheme is not significantly compromised.Fig. 7(b) and (c) shows the QoS of secure links under differentschemes. The results indicate that there is no QoS guaranteeon secure links in the DOS scheme. In contrast, our schemewith the throughput constraint can effectively guarantee thethroughput performance of secure links, while our schemewith the delay constraint successfully controls the delay onsecure links to an acceptable level. Moreover, note that thethroughput requirement α = 0.4 is not satisfied in the lowchannel occupancy ratio range, when only the delay constraintis imposed on our scheme. In addition, the delay requirementβ = 75 is violated in the high channel occupancy range, if ourscheme only sets the throughput constraint. Therefore, if boththroughput QoS and delay QoS are required, our schedulingscheme with double constraints needs to be applied. In Table II,the overall throughput (θtotal) and QoS of secure links for ourscheme with double constraints are summarized. The resultsshow that both delay (σs) and throughput (θs) requirementsof secure links are satisfied at any channel occupancy ratio (P ).This confirms the effectiveness of our scheme.

In addition, our scheme is compared to the approach pro-posed in [13] (ADOS) in two cases. In the first case, thetransmission duration for secure links is equal to that for regularones, i.e., Ds = 30. In this case, the delay QoS requirement forsecure links is set to 75 time slots as before. In the second case,the transmission duration for secure links is equal to five time

Fig. 8. Performance comparison between QDOS and ADOS under homoge-neous scenarios. (a) Case 1. (b) Case 2.

slots. This case is also a typical one: in many scenarios, mes-sages requiring high-level security (e.g., bank/game accountinformation) usually have much smaller sizes than regular ones(e.g., peer-to-peer (P2P) streams) [33]. In the second case, thedelay QoS requirement for secure links is set to 25 time slots.In addition, since the ADOS scheme cannot directly guaranteea specific QoS requirement for secure links, as discussed inSection II, we impose a weight coefficient for secure links in theobjective function of [13], and we keep tuning this coefficientuntil the delay QoS constraint is satisfied.

The performance results under two schemes, including theoverall throughout (marked as THO) and the throughput ofsecure links (marked as THS), are shown in Fig. 8. In thefirst case, the performance gain of QDOS over ADOS is about2%. In the second case where Dr/Ds becomes larger, the gainincreases to 5%. This can be explained as follows. Accordingto [13, eq. (12)], when Dr/Ds increases, the ADOS schemeprovides more channel access opportunities to secure links toguarantee the fairness. More of a preference on secure links has


Fig. 9. Throughput and delay with different schemes in the heterogeneous scenario. (a) Overall throughput. (b) Delay of secure links. (c) Throughput ofsecure links.

a negative impact on the overall throughput and leads to largerperformance gap as compared to QDOS.

B. Heterogeneous Scenarios

The schemes derived for heterogeneous scenarios are evalu-ated here. A set of links with different normalized SNRs is usedin the evaluations, and their SNR values are given in Table I.Based on these SNRs, the CDFs of transmission rates on theselinks are determined, according to (10). Following the definitionof potential-worst links in Section V-A, we identify these links(i.e., secure links of Node 1 and Node 2 in Table I) and imposeQoS constraints on them.

To verify the optimality of the QDOS scheme derived for het-erogeneous scenarios, we compare QDOS with Enum., as dis-cussed in Section VI-A. In this experiment, the delay constraintfor potential-worst links varies from 100 time slots to 1200 timeslots. The comparison results are shown in Fig. 6(c). It canbe observed that the throughput of QDOS under various delayQoS requirements is always equal to the optimal throughputobtained from the enumeration method. This confirms that thederived threshold pairs for heterogeneous scenarios are optimal.

To study the performance of QDOS for heterogeneous sce-narios, we compare it with various schemes. First, we evaluatethe performance difference between homogeneous and hetero-geneous QDOS. To evaluate the performance of QDOS forheterogeneous scenarios, a set of links with different normal-ized SNRs is given in Table I. In this scenario, we require thatthe delay QoS of each secure link is controlled to the level of600 time slots or less. If the QDOS designed for homogeneouscases is applied, all secure links are equally treated, and theprevious QoS requirement on each individual secure link leadsto a delay constraint equal to 120 on the whole set of securelinks (there are five secure links in total). With the QDOS de-signed for heterogeneous scenarios, the delay constraint is onlyimposed on potential-worst secure links, and the previous QoSrequirement converts to a delay constraint equal to 300 on thepotential-worst link set (there are two secure links in this set).

The simulation results under two QDOS schemes (QDOS forhomogeneous scenarios and QDOS for heterogeneous scenar-ios) are given in Fig. 9. For comparison, the performance of

DOS proposed in [15] is also provided. The overall throughputand the throughput of secure links are plotted in Fig. 9(a). Itcan be found that the overall throughput under QDOS for het-erogeneous scenarios degrades 17.8% and 13.1%, as comparedto that under DOS and that under QDOS for homogeneous sce-narios, respectively. The drop in the overall throughput is dueto the fact that more transmission opportunities are providedto secure links (particularly potential-worst ones) in QDOS forheterogeneous scenarios, which is reflected by the throughputof secure links under three schemes, as shown in Fig. 9(a). QoSperformance for each secure link under two QDOS schemes isshown in Fig. 9(b) and (c). With the QDOS designed for ho-mogeneous scenarios, the delay QoS of potential-worst securelinks (secure links of Node 1 and Node 2) severely violatesthe requirement set previously, and the throughput performanceof these links indicates severe starvation. With the QDOS de-signed for heterogeneous scenarios, both delay and throughputperformance of potential-worst secure links are significantlyimproved, and the delay QoS of each secure link meets the per-formance requirement, which demonstrates the necessity andeffectiveness of considering heterogeneous scenarios in QDOS.

The QDOS scheme for heterogeneous scenarios is also com-pared to ADOS in two cases. In the first case, the transmissionduration for secure links (Ds) is 30 time slots, which is equal tothat for regular links (Dr). In this case, the delay requirementis given by 300 time slots. In the second case, Dr remains tobe 30 time slots, while Ds is set to five time slots. This settingis common in real communication scenarios: secure links areusually used to transmit the most critical messages such ascontrol frames, which is shorter than regular data frames. Inthis case, the delay constraint is set to 50 time slots to suit theneed of secure links. Similar to the experiment in homogeneousscenarios, the objective function in [13] is modified by adding aweight coefficient for secure links, and the coefficient is tuneduntil the delay QoS requirement is satisfied.

The comparison results are shown in Fig. 10. In the firstcase, the overall throughput of QDOS is 12% lower than thatof ADOS, while in the second case, our scheme outperformsADOS by 16%. The results indicate that, when the ratio be-tween Dr and Ds varies, the relative performance gain ofQDOS over ADOS also changes. This can be explained by


Fig. 10. Performance comparison between QDOS and ADOS under heteroge-neous scenarios. (a) Case 1. (b) Case 2.

considering two factors. On the one hand, for the sake offairness, ADOS provides more channel access opportunities tolow-quality links under heterogeneous scenarios, which causesthe degradation of the overall throughput. On the other hand,in the ADOS scheme, each node has individual thresholds fordetermining whether to take transmission opportunities. Com-pared to unified thresholds for all nodes, this scheme is ben-eficial to improve the overall throughput under heterogeneousscenarios. When the first factor dominates as in the second case(larger Dr/Ds), our scheme is better than ADOS. If the secondfactor dominates as in the first case (smaller Dr/Ds), ourscheme does not outperform ADOS. The comparison resultsindicate that our scheme for heterogeneous scenarios can befurther improved if each node has individual thresholds asADOS. This is subject to future research.

VII. DISCUSSION

A. Beyond Two Types of Links

In previous sections, two types of links, i.e., secure links andregular links, are considered. Our scheme can be extended tosupport multiple types of links with various QoS requirements.The scheme supporting multiple types of links follows thesame framework as developed in Section IV, and the key stepsfor deriving optimal thresholds for different types of links aresummarized as follows:

1) Convert the objective function (i.e., the overall through-put) based on the optimal stopping theory [19] and themethod of Lagrange multipliers[30]as(3) inSection IV-A.Let {λi} (i ∈ [1, n]) denote the Lagrange multipliers forQoS constraints, where n is the total number of theseconstraints.

2) Define the profit function as that in Section IV-A accord-ing to QoS requirements. The expectation of the definedprofit must have the same expression as the convertedobjective function.

3) Derive the optimal thresholds with respect to {λi} andx∗ and the maximum expected profit equation follow-ing the same approaches as those in Section IV-Aand Appendix E.

4) Determine {λi} and x∗ based on KKT conditions(n equations) and the maximum expected profit equation.Finally, the optimal thresholds can be calculated.

As the number of link types grows, the number of QoSconstraints on various types of links linearly increases. Toderive and calculate the optimal thresholds (i.e., the third stepand the fourth step) with n QoS constraints, 2n cases need to beconsidered since each constraint can be active or not. Thus, thecomplexity of determining optimal thresholds exponentially in-creases as the number of link types grows. Such growing com-plexity limits the number of link types that can be consideredin our scheme. The scheme proposed in Section V provides aremedy to this issue. With this scheme, we can provide QoS to agroup of links, even if there are heterogeneous rate distributionsamong the group, which can be considered as the union of sev-eral link sets with homogeneous link distributions. In this way,we decrease the number of link types considered in the schedul-ing scheme, which reduces the complexity of the scheme.

B. Information Exchange for QDOS

In our scheme, each node needs rate distributions and chan-nel access probabilities of other nodes to calculate the optimalthreshold. As mentioned previously, we assume that these pa-rameters are known by each node, as in previous work [13],[14]. However, in reality, rate distribution and channel accessprobability of each node are collected by the node itself, andthese need to be exchanged among different nodes. For thispurpose, a startup phase can be introduced. In this phase, nodesexchange their channel access probabilities and rate distribu-tion information with others. After this phase, each node cancalculate the optimal thresholds and initiate data transmissionsfollowing the scheduling scheme proposed in this paper. If anode detects the variation of its parameters after the startupphase, it notifies the change to other nodes by piggybackingthe new parameters in its data transmission.

Other nodes keep overhearing data transmissions in themedium, extract the new parameters, and update the optimalthreshold. Generally, when these parameters vary slowly (i.e.,large channel coherence time), the performance penalty intro-duced by the aforementioned information exchange scheme isminimal, since the overhead is negligible as compared to a largeamount of data transmissions. However, if these parametersvary quickly (i.e., small coherence time), it is difficult tocatch up the variation of these parameters through informationexchange. How to make a scheduling scheme (including ourscheme and the related ones [13], [14]) perform well undersuch scenarios is an interesting but challenging problem, whichdemands future research.

VIII. CONCLUSION

Opportunistic scheduling considering QoS constraints forhybrid links of a wireless network has been studied in this


paper. Given different scenarios of rate distributions, two QoSscheduling schemes were derived based on the optimal stoppingtheory. These schemes balance throughput and QoS guaran-tee of hybrid wireless links. Performance results showed thatthe QoS of a specific link type could be guaranteed withoutsignificantly compromising the overall throughput of hybridlinks. Although this paper takes secure links and regular linksas an example of hybrid links, the QoS-oriented opportunisticscheduling schemes derived in this paper are completely ap-plicable to other scenarios of hybrid links.

As indicated by simulation results, adopting individualthresholds for each link of a node is beneficial to improve theoverall throughput in heterogeneous scenarios. How to extendthis paper to support individual thresholds for each node issubject to future research.

APPENDIX ACONSTRAINT QUALIFICATION

To check whether the constraint qualification holds for ouroptimization problems, we first define

g1 = E [RsNDs] and g2 = E[Ts].

The values of g1 and g2 are determined by the decision rulefor keeping or dropping the transmission opportunities. In ourpaper, this rule is characterized by the two thresholds φs (for se-cure links) and φr (for regular links). If the current transmissionrate supported by the link that captures the channel is higherthan the corresponding threshold, the transmission opportunityis taken. Otherwise, the opportunity is dropped. Thus, we have

∇g1 =

(∂g1∂φs∂g1∂φr

)and ∇g2 =

(∂g2∂φs∂g2∂φr

).

According to (6), it can be shown that

∂g1∂φs

=fs(φs)

∫∞φs

rdFs(r)− φsfs(φs) (1 − Fs(φs))

(1 − Fs(φs))2

>fs(φs)

∫∞φs

φsdFs(r) − φsfs(φs) (1 − Fs(φs))

(1 − Fs(φs))2 = 0.

In addition, we have (∂g1/∂φr) = 0. Similarly, we can derivethat

∂g2∂φs

=(t+ Pr (1 − Fr(φr))Dr) fs(φs)

Ps (1 − Fs(φs))2

∂g2∂φr

=−Prfr(φr)Dr

Ps (1 − Fs(φs)).

It can be observed that ∂g2/∂φs is always greater than zero,while ∂g2/∂φr is always less than zero. Furthermore, theconstraints in our problem can be expressed as

G1 = E [RsNDs]− αE[Ts] = g1 − αg2

G2 = E[Ts]− β = g2 − β.

Thus, the gradients of G1 and G2 are given by

∇G1 = ∇g1 − α∇g2, and ∇G2 = ∇g2.

To verify linear independence constraint qualification, we needto investigate whether the gradients of active constraints arelinearly independent. If only the constraint G2 is active (in theoptimization problems in Section IV-B or Section IV-C), weonly need to verify that ∇G2 �= 0. Note that ∇G2 = ∇g2 and∂g2/∂φr is always less than zero. Hence, ∇G2 is not equalto 0 for any (φs, φr).

If only the constraint G1 is active (in the optimization prob-lems in Section IV-A or Section IV-C), we only need to verifythat ∇G1 �=0. Since (∂g1/∂φr)=0 and ∇G1=∇g1−α∇g2,we can derive that (∂G1/∂φr) = −α(∂g2/∂φr). Becausethroughput requirement α is greater than zero and ∂g2/∂φr isalways less than zero, ∂G1/∂φr is always greater than zero.Hence, ∇G1 is not equal to 0 for any (φs, φr).

When both G1 and G2 are active (in the optimization prob-lem in Section IV-C), assume that there exists a pair (φs, φr),such that ∇G1 and ∇G2 are linearly dependent. Then, we have∇G1 = c∇G2, where c is a constant. Following that, it canbe shown that ∇g1 − α∇g2 = c∇g2. If so, we have ∇g1 =(α+ c)∇g2. Since (∂g1/∂φr) = 0 while (∂g2/∂φr) < 0,we can conclude that (α+ c) is equal to zero, which furtherleads to ∇g1 = 0. However, we have shown that (∂g1/∂φs)is always greater than 0. This is a contradiction. Hence, forany pair (φs, φr), ∇G1 and ∇G2 are linearly independent.The aforementioned derivation shows that, for our optimizationproblems in Section IV-A–C, the linear independence constraintqualification holds with respect to any threshold pair (φs, φr).

APPENDIX BDERIVATION OF THE DELAY EXPRESSION

To derive the expression of the average delay for the securelink of node m, i.e., E[Ts,m], we define p1 as the probabilitythat, given a time slot, the secure link successfully contends thechannel and takes the transmission opportunity following thestopping rule (see Fig. 3). Thus, p1 can be expressed as

p1 = Ps,m (1 − Fs,m(φs)) .

In addition, p2 is defined as the probability that other securelinks or regular links win the channel contention in a giventime slot and take the transmission opportunity following thestopping rule. Hence, it can be shown that

p2 =∑i�=m

Ps,i (1 − Fs,i(φs)) +∑i

Pr,i (1 − Fr,i(φr)) .

For convenience, we use p2,s and p2,r to denote the first termand the second term in the right-hand side of the aforemen-tioned equation, respectively. Furthermore, the probability p3,which is defined as the probability that no transmission willstart at the end of a given time slot, can be expressed as

p3 = 1 − p1 − p2.

Considering the first time slot after a transmission on the securelink of node m, if this link wins the channel contention again


and the current transmission rate exceeds the threshold forsecure links, another transmission will start on this link. In thiscase, the delay is given by

Ts,m = t+Ds.

If a transmission starts on other links at the end of the slot(we use C2 to denote this case), according to the memorylesscharacteristic of our system model, the expected delay of thesecure link of node m under this case can be denoted as

E[Ts,m|C2] = t+D0 + E[Ts,m]

where D0 denotes the expected transmission time after the endof the given time slot under C2 and can be expressed as

D0 =p2,sp2

Ds +p2,rp2

Dr.

If no transmission starts at the end of the slot (we use C3 todenote this case), the expected delay is given by

E[Ts,m|C3] = t+ E[Ts,m].

Therefore, it can be shown that

E[Ts,m]=p1(t+Ds)+p2(t+D0+E[Ts,m])+p3(t+E[Ts,m]) .

Hence, we have

E[Ts,m] =t+ p1Ds + p2D0

p1=

Δ

Ps,m (1 − Fs,m(φs))

where

Δ= t+∑i

Ps,i (1−Fs,i(φs))Ds+∑i

Pr,i (1−Fr,i(φr))Dr.

The average delay on the set of secure links can be derived in asimilar way.

APPENDIX CPROOF OF PROPOSITION 4.2

The overall throughput of secure links is the summation ofthroughput of each secure link. Hence, we have

θs =M∑

m=1

θs,m =M∑

m=1

E[Rs,mDs]

E[Ts,m]

where θs,m denotes the throughput on the secure link of nodem, Rs,m stands for the rate for the transmission on this link, andTs,m indicates the time between two successive transmissionson the link. Furthermore, it can be shown that

θs =

M∑m=1

∫∞φs

rdFs,m(r)

1−Fs,m(φs)Ds

t+∑

i Ps,i(1−Fs,i(φs))Ds+∑

i Pr,i(1−Fr,i(φr))Dr

Ps,m(1−Fs,m(φs))

.

The previous equation can be simplified as

θs =PsDs

∫∞φs

rdFs(r)

t+ PsDs (1 − Fs(φs)) + PrDr (1 − Fr(φr)). (11)

For the lower bound, if the throughput requirement α is lessthan θLs , i.e.,

α <PsDs

∫∞φ∗ rdFs(r)

t+ PsDs (1 − Fs(φ∗)) + PrDr (1 − Fr(φ∗))

the optimal threshold pair (φ∗, φ∗) for the unconstrained caseis located in the feasible domain, which indicates that this pairis also the optimal solution for the constrained problem. In thiscase, the throughput constraint is inactive.

For the upper bound, let θUs denote the maximum possiblethroughput on secure links. Apparently, the throughput require-ment α cannot be greater than θUs . To determine θUs , we set φr

to approach ∞ and derive the optimal threshold for secure linksthat maximizes the system overall throughput following thesimilar framework presented in Section IV-A. It can be shownthat φs is equal to the maximum overall throughput x∗. In ad-dition, since the threshold for regular links approaches infinity,there is no throughput on regular links, and the overall through-put is equal to the throughput of secure links. Thus, we have

θUs = x∗ = φs.

Therefore, based on (11), the maximum throughput of securelinks can be determined by

θUs =Ps

∫∞θUsrdFs(r)

tDs

+ Ps (1 − Fs(θUs )).

The proposition is proved.

APPENDIX DPROOF OF PROPOSITION 4.4

The average delay for secure links is given by

β = E[Ts] =t+ Pr (1 − Fr(φr))Dr

Ps (1 − Fs(φs))+Ds.

Thus, we have the following inequalities:

t

Ps (1 − Fs(φs))+Ds ≤ β ≤ t+ PrDr

Ps (1 − Fs(φs))+Ds.

It follows that

t

Ps(β −Ds)≤ (1 − Fs(φs)) ≤

t+ PrDr

Ps(β −Ds).

Since Fs(·) is a rate distribution function, its value increaseswith φs. Thus, it can be shown that

F−1s

(1 − t+ PrDr

Ps(β −Ds)

)≤ φs ≤ F−1

s

(1 − t

Ps(β −Ds)

).

For convenience, the lower bound and the upper bound in theaforementioned inequalities are denoted by φL

s and φHs , respec-

tively. Furthermore, we consider E[RsNDs], which is given by

E [RsNDs] =

∫∞φs

rdFs(r)

(1 − Fs(φs))Ds.


Since E[RsNDs] is a function increasing with φs, it can be

shown that∫∞φLsrdFs(r)

(1 − Fs (φLs ))

Ds ≤ E [RsNDs] ≤

∫∞φHsrdFs(r)

(1 − Fs (φHs ))

Ds.

Moreover, based on E[RsNDs]− αE[Ts] = 0 and E[Ts] = β,

we have

E [RsNDs] = αβ.

Hence, it can be shown that∫∞φLsrdFs(r)

β (1 − Fs (φLs ))

Ds ≤ α ≤∫∞φHsrdFs(r)

β (1 − Fs (φHs ))

Ds.

This proves Proposition 4.4.

APPENDIX EPROOF OF PROPOSITION 5.1

We can define the profit of stopping rule N as

RNDN − x∗TN − ωT ′pw + γω

where T ′pw denotes the time from the beginning of the con-

tention period for stopping rule N to the end of the next trans-mission on potential-worst links. Furthermore, L∗ is defined asthe maximum expected profit for stopping rules and has theidentical expression as the optimization formulation presentedin Section V-B. Similar to the derivations in Section IV-A, tosolve the optimization problem in Section V-B, we only needto find the optimal stopping rule that maximizes the expectedstopping profit. In addition, note that L∗ is equal to zero.

If a link from the set ξ takes the channel successfully, theprofit for transmitting a packet on this link can be expressed as

Rs,pwDs − x∗Ds − ωDs + γω − (x∗ + ω)Tcont (12)

where Rs,pw is the transmission rate on this link and followsthe distribution

Fs,pw(r) =1∑

m∈ξ Ps,m

∑m∈ξ

Ps,mFs,m(r).

If a secure link that is not a potential-worst one wins the channelcontention, the profit for transmitting immediately is given by

Rs,nwDs − x∗Ds − ωDs − ωE[Tpw] + γω − (x∗ + ω)Tcont

where Rs,nw denotes the transmission rate on the link thatsuccessfully takes the channel and follows the distribution

Fs,nw(r) =1∑

m �∈ξ Ps,m

∑m �∈ξ

Ps,mFs,m(r).

According to the KKT conditions [30], ω(E[Tpw]− γ) = 0.The aforementioned profit expression can be simplified as

Rs,nwDs − x∗Ds − ωDs − (x∗ + ω)Tcont.

As a result, the average profit for transmitting immediatelyon the secure link that wins the channel contention can bedenoted as

RsDs −∑

i∈ξ Ps,i

Ps[x∗Ds+ωDs−γω+(x∗+ω)Tcont]

−Ps−

∑i∈ξ Ps,i

Ps[x∗Ds+ωDs+(x∗+ω)Tcont]

= RsDs−x∗Ds−ωDs+

∑i∈ξ Ps,i

Psγω − (x∗+ω)Tcont

where Rs follows the distribution∑i∈ξ Ps,i

PsFs,pw(r) +

Ps −∑

i∈ξ Ps,i

PsFs,nw(r) = Fs(r).

Thus, if this profit is greater than the maximum expectedprofit with skipping this transmission and waiting for the nextstopping time, i.e.,

RsDs − x∗Ds − ωDs +

∑i∈ξ Ps,i

Psγω ≥ L∗

then the packet is transmitted. Hence, the threshold for securelinks is given by

φs = x∗ + ω −∑

i∈ξ Ps,i

Ps

γω

Ds.

The regular case can be derived similarly as that inSection IV-A. If

RrDr − x∗Dr − ωDr − ωE[Tpw] + γω ≥ L∗

namely, Rr ≥ x∗ + ω, the regular packet is transmitted on thelink that wins the channel contention. In summary, the optimalthreshold pair is given by{

φs = x∗ + ω −∑

i∈ξ Ps,i

Ps

γωDs

φr = x∗ + ω.(13)

To calculate the optimal threshold pair, the maximum ex-pected profit equation is derived as follows. Considering thefirst successful channel contention, if a potential-worst securelink wins the channel contention, it starts its transmission onlywhen Rs,pw ≥ φs. In this case, the profit is given by

Rs,pwDs − x∗Ds − ωDs + γω − (x∗ + ω)kt

where k is the number of time slots before the first successfulchannel contention. Note that the aforementioned equation isidentical with (12), when Tcont is equal to kt. On the otherhand, if Rs,pw < φs, the transmission opportunity is skipped,and then, the maximum expected profit is L∗ − (x∗ + ω)kt.Combining the previous two cases, if a potential-worst linkwins the channel contention, the maximum profit following theoptimal stopping rule can be expressed as

(Rs,pw−φs)+Ds+γω

∑i�∈ξ Ps,i

Psu(Rs,pw−φs)−(x∗+ω)kt

where (·)+ denotes max{·, 0}, and u(·) is the step function.


Similarly, if a secure link that is not potential-worst wins thecontention, the maximum profit is given by

(Rs,nw−φs)+Ds−γω

∑i∈ξ Ps,i

Psu(Rs,nw−φs)−(x∗+ω)kt.

If a regular link contends the channel successfully, the max-imum profit can be derived as (Rr − φr)

+Dr − (x∗ + ω)kt.Therefore, we have

L∗ =

⎛⎝∑

i∈ξPs,i

⎞⎠E [

(Rs,pw−φs)+Ds + γω

Ps−∑

i∈ξ Ps,i

Psu

× (Rs,pw−φs)− (x∗ + ω)kt

]

+

⎛⎝Ps−

∑i∈ξ

Ps,i

⎞⎠E[

(Rs,nw−φs)+Ds−γω

∑i∈ξ Ps,i

Psu

× (Rs,nw−φs)− (x∗ + ω)kt

]+ PrE

[(Rr − φr)

+Dr − (x∗ + ω)kt].

Rearranging the aforementioned equation, we get

(x∗ + ω)t = PsDsE[(Rs−φs)

+]+PrDrE

[(Rr−φr)

+]

+(Ps −

∑i∈ξ Ps,i)(

∑i∈ξ Ps,i)γω

Ps

× (Fs,nw(φs)−Fs,pw(φs)) . (14)

In addition, based on the KKT conditions, we have

ω (E[Tpw]− γ) = 0 (15)

where Tpw can be expressed as

Tpw =t+ Pr (1 − Fr(φr))Dr + Ps (1 − Fs(φs))Ds(∑

i∈ξ Ps,i

)(1 − Fs,pw(φs))

.

Combining (13)–(15), (x∗, ω) can be solved with LMA [31].Following that, the optimal threshold pair can be calculatedaccording to (13).

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http://www.math.ucla.edu/ tom/Stopping/contents.html

http://www.math.ucla.edu/~tom/Stopping/contents.html

http://www.math.ucla.edu/~tom/Stopping/contents.html


Wenguang Mao received the B.S. degree in 2011and the M.S. degree in 2014 from the ShanghaiJiao Tong University, Shanghai, China, where hewas a Research Assistant with the Wireless andNetworking (WANG) Laboratory. He is currentlyworking toward the Ph.D. degree with the Depart-ment of Computer Science, The University of Texasat Austin, Austin, TX, USA.

His current research interests include mobilecomputing and wireless networking.

Xudong Wang (SM’08) received the Ph.D. degreein electrical and computer engineering from GeorgiaInstitute of Technology, Atlanta, GA, USA, inAugust 2003.

He is currently with the University of Michigan-Shanghai Jiao Tong University (UM-SJTU) JointInstitute, Shanghai Jiao Tong University, Shanghai,China. He is a Distinguished Professor (ShanghaiOriental Scholar) and is the Director of the Wirelessand Networking (WANG) Laboratory. He is also anAffiliate Faculty Member with the Department of

Electrical Engineering, University of Washington, Seattle, WA, USA. SinceAugust 2003, he has been a Senior Research Engineer, Senior Network Archi-tect, and R&D Manager with several companies. He has been actively involvedin R&D, technology transfer, and commercialization of various wireless net-working technologies. His research interests include wireless communicationnetworks, smart grid, and cyberphysical systems. He holds several patentson wireless networking technologies, and most of his inventions have beensuccessfully transferred to products.

Dr. Wang serves as an Editor for the IEEE TRANSACTIONS ON MOBILECOMPUTING, the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY,and Ad Hoc Networks by Elsevier. He was the Demo Cochair of the ACMInternational Symposium on Mobile Ad Hoc Networking and Computing(ACM MOBIHOC) in 2006, a Technical Program Cochair of the WirelessInternet Conference (WICON) in 2007, and a General Cochair of WICONin 2008. He has been a technical committee member of many internationalconferences.

Shanshan Wu received the B.S. degree in 2011 andthe M.S. degree in 2014 from the Shanghai JiaoTong University, Shanghai, China, where she was aResearch Assistant with the Wireless and Network-ing (WANG) Laboratory. She is currently workingtoward the Ph.D. degree with the Wireless Network-ing and Communications Group, The University ofTexas at Austin, Austin, TX, USA, working onlarge-scale machine learning algorithms and high-dimensional statistics.

Her current research interests include machinelearning algorithms with large-scale or streaming data input, submodularfunctions, convex optimization, and design and analysis of scalable algorithmsin distributed or multicore systems.

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