IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER Wenguang Mao, Xudong Wang, Senior Member, IEEE, and Shanshan Wu

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1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER Ditributed Opportunitic Scheduling With QoS Contraint for Wirele Network With Hybrid Link Wenguang Mao, Xudong Wang, Senior Member, IEEE, and Shanhan Wu Abtract Opportunitic cheduling for a wirele network with hybrid link i tudied in thi paper. Specifically, two link type are conidered: A link of the firt type alway ha a much lower tranmiion rate than a link of the econd type. To avoid tarvation in the firt type of link, two link type mut be treated differently in opportunitic cheduling, and quality of ervice (QoS) contraint, uch a maximum delay or minimum throughput, mut be impoed on the firt link type. Conidering QoS contraint, a ditributed opportunitic cheduling cheme i derived baed on the optimal topping theory. Two cenario are conidered for the QoS-oriented opportunitic cheduling cheme. In the firt cenario, all link within the ame link type follow the ame rate ditribution. Thu, QoS contraint are impoed on the entire link type. In the econd cenario, link of the firt type follow heterogeneou rate ditribution. Thu, QoS requirement need to be impoed on link with the wort performance. Performance reult how that the new opportunitic cheduling cheme outperform the exiting one in mot cenario. Index Term Ditributed opportunitic cheduling, hybrid link, optimal topping theory, quality of ervice (QoS). I. INTRODUCTION IT i common that wirele link in a network have heterogeneou characteritic uch a tranmiion rate and QoS requirement. Such link are called hybrid link in thi paper. One factor leading to link heterogeneity involve applicationpecific requirement for different link. For example, ome application (e.g., control meage tranmiion in the mart grid or cognitive radio network) impoe a trong requirement on the ecurity, and phyical layer technique [1] [3] are applied to enure perfect ecrecy in correponding link Manucript received June 7, 2015; revied October 5, 2015; accepted November 24, Date of publication November 30, 2015; date of current verion October 13, Thi work wa upported by the National Natural Science Foundation of China under Grant The review of thi paper wa coordinated by Prof. C. Ai. (Correponding author: Xudong Wang.) W. Mao wa with the Univerity of Michigan-Shanghai Jiao Tong Univerity Joint Intitute, Shanghai Jiao Tong Univerity, Shanghai , China. He i now with the Department of Computer Science, The Univerity of Texa at Autin, Autin, TX USA. X. Wang i with the Univerity of Michigan-Shanghai Jiao Tong Univerity Joint Intitute, Shanghai Jiao Tong Univerity, Shanghai , China ( wxudong@ieee.org). S. Wu wa with the Univerity of Michigan-Shanghai Jiao Tong Univerity Joint Intitute, Shanghai Jiao Tong Univerity, Shanghai , China. She i now with the Wirele Networking and Communication Group, The Univerity of Texa at Autin, Autin, TX USA. Color verion of one or more of the figure in thi paper are available online at Digital Object Identifier /TVT (called ecure link) [4], [5]. Since perfect ecrecy come at the cot of degrading channel capacity [6], [7], ecure link have much lower tranmiion rate a compared to other link (called regular link). Due to ecurity concern, ecure link may alo demand tringent QoS guarantee. For eae of explanation throughout thi paper, we ue ecure link and regular link to repreent two link type that follow ignificantly different rate ditribution. Packet tranmiion in a network with hybrid link can be conducted in two different approache: 1) following a pure random acce medium acce control (MAC) protocol; 2) baed on a cheduling cheme. The former approach i imple and eay to implement, but thi may lead to low throughput in a network with hybrid link due to the preence of performance anomaly [8], i.e., the wirele medium i extenively occupied by low-rate tranmiion on ecure link. Therefore, the latter approach i neceary to improve the network throughput. Among exiting cheduling cheme, opportunitic cheduling i conidered a the mot effective to exploit fluctuation in channel condition to produce ignificant throughput gain for the entire network [9] [11]. The key idea of opportunitic cheduling i explained a follow: given a tranmiion opportunity, if a link with the highet tranmiion rate i elected, the maximum throughput can be achieved. Unfortunately, thee opportunitic cheduling cheme rely on the exitence of the central controller (e.g., the bae tation in cellular network) and, hence, are hard to implement in ad hoc network or wirele meh network, where uch a central node i not readily available. To addre thi iue, everal ditributed opportunitic cheduling cheme are propoed in [12] and [13], which utilized local information to determine whether to take tranmiion opportunitie or not. However, the quality of ervice (QoS) of communication link i not taken into account in thee cheme. A ditributed opportunitic cheduling cheme that conider the delay a a QoS metric i developed in [14] baed on the cheme in [15]. However, thi cheme i not applicable to hybrid link, a it cannot guarantee QoS requirement for a pecific type of link (e.g., ecure link) and, at the ame time, maximize the overall throughput. So far, there i a lack of effective ditributed opportunitic cheduling to upport a network with hybrid link. To treat hybrid link eparately and alo upport QoS requirement of a pecific type of link, a new ditributed opportunitic cheduling cheme i propoed in thi paper. It i developed baed on the optimal topping theory and conidering two IEEE. 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2 8512 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 type of link: ecure link and regular link. Compared with exiting opportunitic cheduling cheme, the new cheduling cheme i ditinct with the following feature: 1) The ytem overall throughput i maximized under variou QoS contraint of a pecific link type (e.g., ecure link); 2) it can be implemented a a double-threhold cheduling policy, i.e., one threhold for each link type, and then, a link determine it tranmiion opportunity baed on thi threhold; 3) the rate heterogeneity among the ame type of link i alo taken into account to improve QoS of link with low channel quality. Simulation are carried out to evaluate the new opportunitic cheduling cheme. Performance reult verify the optimality of our cheme and demontrate that QoS of ecure link can be effectively guaranteed under both homogeneou and heterogeneou rate ditribution cenario. Moreover, reult alo how that our cheme outperform the exiting one in mot cae. The remainder of thi paper i organized a follow: The related work i ummarized in Section II. The ytem model for our opportunitic cheduling i explained in Section III. The QoS-oriented opportunitic cheduling cheme under the cenario of homogeneou and heterogeneou rate ditribution i derived in Section IV and V, repectively. Performance reult are preented in Section VI. Further dicuion about our cheme are provided in Section VII. Thi paper i concluded in Section VIII. II. RELATED WORK Opportunitic cheduling i an effective way to utilize the fluctuation of channel condition to enhance the network throughput performance. Several opportunitic cheduling cheme (e.g., [9] [11]) have been propoed for a network with a central controller. However, thee cheme are not applicable to wirele meh/ad hoc network, where uch controller doe not exit. To olve thi problem, many ditributed opportunitic cheduling cheme have been developed o far. Thee cheme exploit local information to determine whether to take tranmiion opportunitie. In [16], an opportunitic ditributed cheduling i developed, and the reulting ytem capacity i tudied baed on point proce approximation. It i hown that the capacity approache that of a centralized ytem, where the bet link i alway elected to tranmit. The focu in [16] i on uplink traffic in a multiple-acce channel, while our paper conider peer-to-peer traffic in an ad hoc network. In [12], a channel-aware ALOHA protocol i developed, where the node only tranmit their packet when their channel gain are above a given threhold. In addition, other channel-aware ALOHA cheme are deigned according to decentralized channel tate information in [17] and [18]. Baed on the optimal topping theory [19], a ditributed opportunitic cheduling i derived in [15] to maximize the ytem overall throughput. In [20], a topping policy i invetigated, when the channel qualitie for different tranmiion period are correlated. In [21], the author propoe a ditributed opportunitic cheduling cheme baed on game theory. In thi cheme, an effective mechanim i deigned to combat the iue of uer elfihne. Although thee cheme utilize opportunitic tranmiion to improve the network throughput in a ditributed manner, none of them take into account the QoS of intereted link. Thu, thee cheme are not applicable to cenario tudied in thi paper. Ditributed opportunitic cheduling cheme propoed in [13], [14], and [22] are mot related to the cheme developed in thi paper. In [13] and [22], an opportunitic cheduling cheme i developed to maximize the proportionally fair allocation. Baed on the control theory, the cheme adapt to the variation of network load and can dynamically drive the ytem to the optimal operation point. Thi cheme i different from our in the following point. Firt, the cheme conider the fairne. Although enuring fairne i beneficial to improve the QoS of the link with low tranmiion rate, it cannot directly guarantee a pecific QoS requirement on uch link (e.g., the delay i le than a pecific value or the throughput i greater than a pecific value). In contrat, our cheme directly focue on the QoS. Second, the proportionally fair allocation i maximized in their cheme intead of the overall throughput of the network, a in our cheme. In [14], an opportunitic cheduling cheme conidering delay QoS i developed baed on the cheme propoed in [15]. In thi cheme, network centric delay contraint and individual delay contraint are atified baed on different trategie. However, thi cheme i not uitable for a network with hybrid link, for two reaon: 1) A networkcentric delay contraint cannot guarantee the QoS requirement of one pecific type of link; 2) if the individual delay contraint i applied to each link, the overall throughput i not maximized. To the bet of our knowledge, the optimal topping theory [19] i firt introduced to derive ditributed opportunitic cheduling cheme in [14] and [15]. The framework of derivation in thi paper i baed on the work in [14] and [15] but make the following nontrivial and important extenion. Firt, in our model, two type of link, which have different rate ditribution function, different QoS requirement, and different tranmiion duration, are conidered [23]. To meet their QoS requirement, two type of link need to be treated differently in the mathematical derivation, which lead to a double-threhold topping policy. Furthermore, the trategie for handling two type of link in thi paper can be eaily extended to upport multiple type of link with variou QoS requirement. Second, in our derivation, we conider both the delay contraint and the throughput contraint, which lead to different derivation to obtain the cheduling cheme. For intance, it i required to deign more complicated profit function for opportunitic cheduling with throughput contraint. Third, in our model, link heterogeneity among the ame type of link i taken into account. In thi cae, we et contraint on a ubet of one type of link, intead of the whole et of link, which complicate the proce of deriving the optimal topping threhold and the maximum expected profit equation. 1 III. SYSTEM MODEL In thi paper, we focu on a ingle-hop ad hoc network where all node can hear each other, following the ame aumption in previou work [13] [15], [21]. Although thi network etup cannot repreent a generic network model, it i actually 1 See Section IV-A.

3 MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8513 Fig. 1. Diagram for channel contention and data tranmiion. meaningful in a practical environment. Such a network model i common in variou communication cenario, e.g., wirele enor network [24], body area network [25], and deviceto-device (D2D) networking in 5G network [26]. Moreover, a ingle-hop network can erve a a building block for a general multihop network. A common and practical approach to handling packet tranmiion in a multihop network i to divide the entire network into everal ingle-hop ubnetwork and coordinate tranmiion among thee ubnetwork in a hierarchical way [27]. Within each ubnetwork, the cheduling cheme developed in thi paper can be adopted. Hence, the tudy on a ingle-hop network alo benefit data tranmiion in a general multihop network. The carrier ening i enforced: if a node detect a buy medium, it need to potpone it own tranmiion. Moreover, two type of link between node in thi network are conidered: 1) ecure link for tranmitting critical meage with phyical layer ecurity; 2) regular link for other meage. Note that the link conidered in thi paper are virtual. Thu, a ecure link and a regular link can have the ame ource and detination node. In thi cae, thee two link hare a common phyical link. When the tranmiion medium i ened idle, a node contend the channel lot by lot with a fixed acce probability (i.e., p-peritent mechanim). To thi end, the node tranmit a pilot packet to it detination node at the beginning of a certain time lot. If two or more node contend the channel in the ame time lot, a colliion occur (denoted by C in Fig. 1). If only one node end a pilot packet to the channel, the detination will uccefully receive it and reply a confirmation packet. In thi cae, the contention i ucceful (denoted by S in Fig. 1). Note that the confirmation packet i tranmitted in the ame time lot with the pilot packet. Thu, the ize of a time lot i et larger than the total length of a pilot packet and a confirmation packet. If none of the node tranmit a pilot packet, the channel remain idle (denoted by I in Fig. 1). If a node uccefully capture the channel, intead of proceeding to data packet tranmiion directly, it need to detect current channel quality and follow a deciion rule to determine whether or not a packet can be tranmitted: If the current channel quality i low, the node kip the tranmiion opportunity to avoid the ituation where the wirele medium i occupied by a low-rate tranmiion, a hown in Fig. 1. For thi purpoe, the node need to know the current channel quality and determine the deciion rule. The channel from the ource to the detination i meaured by the detination uing the preamble equence in the pilot packet, and the reult are carried back by the confirmation packet. The deciion rule i derived baed on the optimal topping theory [19], uch that the network throughput i maximized under the contraint of QoS requirement. If the opportunity i dropped, the contention proce retart in the next time lot. Otherwie, the node tart it data tranmiion. The tranmiion can lat multiple time lot, a hown in Fig. 1, and we aume that the channel condition remain unchanged during the tranmiion proce (i.e., block-fading channel). In addition, we aume that the channel condition during different tranmiion period are independent, which i the ame in the previou work [13], [14]. To clearly preent our opportunitic cheduling cheme, everal parameter are defined a follow. A hown in Fig. 1, a time lot ha a length of t, and data tranmiion duration for ecure link and regular link are aumed contant and denoted by D and D r, repectively. There are M node in the network. For the ake of clarity, we aume that each node ha exactly one ecure link and one regular link. However, the derivation preented in Section IV and VI can be directly applied to general cae. Given a time lot, node m contend the channel for it ecure link and regular link with the probability (i.e., the peritent factor) equal to p,m and p r,m repectively. Furthermore, the probability that the ecure link of node m win the channel contention, i.e.,,m,igivenby,m = p,m (1 p,i p r,i ). (1) i m Thu,, which i defined a the probability that any ecure link uccefully contend the channel, can be calculated by P,m. P r,m and P r are defined for regular link in a imilar way. In addition, the tranmiion rate on the ecure link and the regular link of node m follow the ditribution with cumulative (probability) denity function (CDF) F,m (r)(f,m (r)) and F r,m (r)(f r,m (r)), repectively. A in [13] and [14], thee ditribution function are aumed known. Moreover, for the mathematical tractability, we aume that F,m (r) and F r,m (r) are differentiable, and f,m (r) and f r,m (r) are greater than zero for any r>0. Thee aumption are valid for commonly ued rate ditribution model [28]. For convenience, let R denote the tranmiion rate on any ecure link. The ditribution of R, i.e., F (r), igivenby F (r) = m,m F,m (r). (2) For regular link, R r and F r (r) are defined imilarly. A hown in Fig. 1, if a node uccefully contend the channel at time N and decide to tranmit a data packet, then we call N a topping time. Given a topping time N, T N denote the total time for thi tranmiion round, including contention period and packet tranmiion time D N. Here, D N i equal to D, if the tranmiion i on a ecure link, and i equal to D r for the tranmiion on a regular link. In addition, R N i ued to denote the tranmiion rate of the node. If the tranmiion in thi round i on a ecure link, RN i ued to denote the tranmiion rate (i.e., R N = RN ). The time between two ucceive tranmiion on ecure link i

4 8514 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 type of link. In addition, in [14], only the delay contraint i conidered, while in our problem, the throughput contraint i alo tudied. Hence, a new derivation i needed. Let x denote the maximum throughput under the throughput contraint. According to [19, Th. 6.1], the optimization problem formulated previouly i equivalent to ubject to max E[R N D N ] x E[T N ] N ζ Fig. 2. Rate ditribution of different link in the homogeneou cenario. denoted by T, which alo tand for the delay of ecure link. Apparently, the choice of topping time ha influence on T. IV. OPPORTUNISTIC SCHEDULING WITH HOMOGENEOUS RATE DISTRIBUTIONS To maximize the ytem overall throughput under variou QoS contraint for ecure link, we develop a new ditributed opportunitic cheduling cheme baed on the optimal topping theory. Firt, the homogeneou rate ditribution cenario i conidered, where the rate ditribution of ecure link are identical to each other. In thi cae, the overall performance of all ecure link can effectively reflect that of each individual link, ince each link contribute equally to the overall performance. Thu, here, we conider ecure link a a whole and et overall QoS contraint on all thee link. A oon a the overall QoS of ecure link i atified, the correponding QoS for each individual link i alo guaranteed. In practice, the developed cheduling cheme i ueful in two cenario: 1) The rate ditribution of ecure link are cloe to each other, a hown in Fig. 2; 2) the rate ditribution are different, but only the overall QoS of ecure link need to be atified. A. Opportunitic Scheduling With Throughput Contraint We define ζ a the et of topping time a follow: ζ {N : N 1,E[T N ] }. In addition, let θ denote the throughput on all ecure link and α tand for the minimum throughput requirement. The maximization of overall throughput under the throughput contraint can be formulated a max N ζ E[R N D N ], ubject to θ = E [R N D ] α. E[T N ] E[T ] Note that a ditributed opportunitic cheduling that maximize the overall throughput under QoS contraint i derived in [14] baed on the optimal topping theory, but the olution therein i not applicable in our cenario for two reaon. Firt, there exit two different type of link in our problem, and the contraint preented in our formula i applied to one type of link intead of to all link. Thi lead to different topping policie for two αe[t ] E [R N D ] 0. The formulated optimization problem i a contrained one. To olve thi problem, we convert it into an uncontrained one through the method of Lagrange multiplier. A proved in Appendix A, the contraint qualification i atified in our problem. In thi cae, the olution for the original problem alo maximize the converted problem, and the Karuh Kuhn Tucker (KKT) condition hold [29]. A a reult, we can find the optimal olution for the original problem by olving the converted problem. If there are more than one olution for the converted problem, we elect the one that maximize the original objective function. Baed on the method of Lagrange multiplier [30], the optimization problem i converted to max E[R N D N ] x E[T N ] λ (αe[t ] E [RN D ]) (3) N ζ where λ i the Lagrange multiplier. By olving thi problem, the following propoition can be derived. Propoition 4.1: The optimal topping rule for ecure link and regular link i a double-threhold policy. The threhold for ecure link i φ, and that for regular link i φ r. If a ecure link win the channel contention, it doe not kip the tranmiion opportunity only if R φ ; if a regular link uccefully capture the channel, it take the tranmiion opportunity when R r φ r. The optimal threhold for ecure link and regular link are given by { φ = x +λα 1+λ φ r = x (4) + λα. Proof: Let u define the profit for topping rule N 1 (i.e., the time when a node uccefully contend the channel and tart the tranmiion) a R N1 D N1 + λr N 2 D x T N1 λαt where N 2 i the firt topping time for ecure link after N 1 (including N 1 ). Thu, if the link that win the channel contention at N 1 i a ecure one, N 2 i equal to N 1. In addition, T denote the time from the beginning of the contention period for topping rule N 1 to the end of the tranmiion for topping rule N 2, a hown in Fig. 3. In addition, according to the memoryle characteritic of our ytem model, we have E[T ]=E[T ].

5 MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8515 Fig. 3. Two cae for topping rule N 1. 1) A ecure link win the channel. 2) A regular link win the channel. Thu, the maximum expected profit for topping rule, which i denoted by L, can be expreed a max E[R N D N ] x E[T N ] λ (αe[t ] E [RN D ]). N ζ According to [19, Th. 6.1] and KKT condition, the value of L i zero. Moreover, it can be oberved that the aforementioned expreion i identical with the optimization formulation preented in Section IV-A. Therefore, olving the optimization problem in Section IV-A i equivalent to finding the optimal topping rule that maximize it expected profit. To maximize thi expectation, the opportunitic cheduling allow the packet tranmiion only when the ytem meet the mot favorable opportunity: Given a node that uccefully contend the channel, if the profit for tranmitting immediately i greater than the maximum expected profit with kipping the opportunity and waiting for the next topping time, the node tart the tranmiion; otherwie, the opportunity i dropped. Conider that a node uccefully contend the channel for it ecure link. If the node take thi tranmiion opportunity, the profit can be quantified a R D + λr D x D λαd (x + λα)t cont where T cont i the contention period before thi ucceful channel contention. However, if the node kip thi opportunity, baed on the time-invariant characteritic of the ytem, the maximum expected profit i given by L (x + λα)t cont. Hence, if (1 + λ)r D x D λαd L the profit for taking the tranmiion opportunity i greater than that with kipping the opportunity and waiting for the next topping time. In thi cae, the packet on thi ecure link i tranmitted. However, if (1 + λ)r D x D λαd <L, tranmitting immediately i le favorable than waiting for better opportunity. In thi cae, the node drop the tranmiion and the channel contention retart. Therefore, the topping threhold for ecure link i given by R ((x + λα)/(1 + λ)) = φ. Similarly, if a regular link ucceed in channel contention, the maximum expected profit with kipping the tranmiion opportunity i L (x + λα)t cont, while the expected profit from taking the tranmiion opportunity i given by R r D r +λe [R N D ] x D r λα (D r +E[T ]) (x +λα)t cont. Note that, after the end of current tranmiion, the expected waiting time for the next tranmiion on ecure link i equal to E[T ], according to the memoryle characteritic of our ytem model. In addition, baed on KKT condition [30], we have λ(e[rn D ] αe[t ]) = 0. Thu, the profit expreion can be implified a R r D r x D r λαd r (x + λα)t cont. Therefore, when R r D r x D r λαd r L, the regular packet i tranmitted. Otherwie, the tranmiion opportunity i kipped. Hence, the tranmiion threhold for regular link i given by φ r = x + λα. The propoition i proved. In addition, the derivation for the optimal topping rule with the delay contraint and double contraint follow the imilar framework a preented earlier. According to Propoition 4.1, it can be hown that φ < x <φ r. Thu, packet on regular link can be ent only when the current tranmiion rate i greater than the ytem expected throughput, while meage on ecure link are tranmitted even if the tranmiion rate i le than the throughput value. Such dicrimination between two type of link i helpful to favor the tranmiion on ecure link and hence provide the QoS on thee link. In Propoition 4.1, the optimal threhold φ and φ r are expreed in term of (x,λ). Hence, further calculation of φ and φ r require the knowledge of (x,λ). The procedure for determining (x,λ) i preented a follow. According to the definition of profit given in the proof of Propoition 4.1, the maximum expected profit L can be expreed a L = E[max(R D +λr D x D λαd,l ) kt(x +λα)] + P r E [max (R r D r + λe [RN D ] x D r λαd r λαe[t ],L ) kt(x + λα)] (5) where k i the number of time lot before the firt ucceful channel contention. Note that the firt expectation in the righthand ide of (5) denote the maximum expected profit when the firt ucceful channel contention i won by a ecure link, while the econd expectation tand for the maximum expected profit when a regular link take the firt ucceful channel contention. Since L i zero a explained in the proof of Propoition 4.1, (5) can be implified a t(x +λα)=d (1+λ)E [ (R φ ) +] +D r P r E [ (R r φ r ) +] where ( ) + denote max{, 0}. In addition, according to KKT condition, we have λ(e[rn D ] αe[t ]) = 0, where E [RN D φ ]= rdf (r) φ df (r) D (6) E[T ]= t+p r(1 F r (φ r ))D r (1 F (φ )) + D. The firt expreion in (6) i baed on the fact that all tranmiion on ecure link are conducted with the rate greater than φ, while the econd expreion i derived in Appendix B. Combining the maximum expected profit equation, KKT condition, and (4), (x,λ) i calculated with the Levenberg Marquardt algorithm (LMA) [31], i.e., a numerical method to olve nonlinear equation. Theoretically, the convergence peed of LMA i imilar to that of the widely

6 8516 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 known Gau Newton method. However, in practice, the implementation of LMA are proved more efficient in mot cenario [31]. MATLAB [32] provide a built-in function that implement the LMA algorithm. We ue thi function to olve the aforementioned equation to obtain (x,λ). Following that, the optimal threhold pair (φ,φ r ) can be calculated baed on (4). It i neceary to emphaize that throughput contraint α i effective only when it fall into a pecific range. If α i too mall, the optimal threhold derived from previou equation will be equal to thoe in the uncontrained cae. In thi cenario, the throughput contraint i inactive. If α i too large, the contraint cannot be atified, even if kipping all regular tranmiion. To characterize the lower bound and the upper bound for α, we have the following propoition. Propoition 4.2: The effective range for throughput requirement α i given by θ L α θu,whereθu i the maximum throughput of ecure link when φ r =, and it can be determined by θ U = θ rdf U (r) t D + (1 F (θ U )). In addition, θ L i the throughput of ecure link when the threhold pair i equal to the optimal one (i.e., (φ,φ )) 2 for the uncontrained cae, and it i given by P θ L D φ rdf = (r) t + D (1 F (φ )) + P r D r (1 F r (φ )). The detailed proof of thi propoition i lited in Appendix C. A the rate ditribution of ecure link improve, the throughput of ecure link with φ r =, i.e.,θ U, increae due to the higher link rate for each tranmiion. In addition, the throughput of ecure link with (φ,φ r )=(φ,φ ), i.e., θ L, enhance becaue ecure link get more tranmiion opportunitie. Thu, according to the propoition, both the upper bound and the lower bound of the effective range for α will increae with better rate ditribution of ecure link. B. Opportunitic Scheduling With Delay Contraint Here, an opportunitic cheduling cheme with delay contraint i tudied. Similar to the network-wide average delay defined in [14], the delay tudied here i impoed on the et of all ecure link, intead of a pecific ecure link. Specifically, the delay impoed on the et of ecure link (denoted by T ) i defined a the time between two ucceive tranmiion on any ecure link, a hown in Fig. 1. In contrat, the delay on a ecure link of node m (denoted by T,m )idefineda the time between two ucceive tranmiion on thi link. In homogeneou cae, if there are n ecure link in total, the average delay on a pecific ecure link i about n time a that on the et of all ecure link, i.e., E[T,m ]=ne[t ]. Baed on thi relationhip, we can et the delay contraint impoed on the et of all ecure link (i.e., T ) according to the delay requirement of individual ecure link. 2 In the uncontrained cenario, the threhold for ecure link and regular link are identical. Let σ tand for the average delay of ecure link and β denote the delay requirement. Thu, we formulate the problem a max N ζ E[R N D N ], ubject to σ = E[T ] β. E[T N ] A dicued in Section IV-A, the previou optimization problem i equivalent to max E[R N D N ] x E[T N ] μ (E[T ] β) N ζ where μ i the Lagrange multiplier. By olving thi problem, the optimal threhold pair can be derived a { φ = x + μ μβ D φ r = x + μ. To further calculate (φ,φ r ), the following equation are needed: t(x + μ) = D E [ (R φ ) +] +P r D r E [ (R r φ r ) +] μ (E[T ] β) =0 where the firt equation i the maximum expected profit equation and the econd one i from KKT condition [30]. The derivation of aforementioned equation and the calculation of the optimal threhold follow the imilar framework preented in Section IV-A. In addition, imilar with throughput contraint, delay requirement β alo ha an effective range, a decribed in the following propoition. Propoition 4.3: The delay contraint β ha a lower effective bound β L and an upper effective bound β U. β L i the minimal poible average delay for ecure link, and it i given by β L = (t/ )+D. β U i the average delay for ecure traffic, when the threhold for ecure link and regular link are et to thee value (i.e., φ for both type of link) in the uncontrained cae, and it can be determined by β U = t + P r (1 F r (φ )) D r (1 F (φ )) + D. Proof: For the lower bound, T include time period t/ for at leat one round of ucceful channel contention and packet tranmiion time D. Hence, the minimum achievable delay requirement i (t/ )+D. For the upper bound, if delay requirement β i greater than β U, then the optimal threhold pair for the uncontrained cae i located in the feaible domain, which indicate that thi optimal olution i alo the one for the contrained problem. In thi cae, the delay contraint i inactive. According to the propoition, the lower bound of the effective range i determined by acce probabilitie of ecure link and unrelated to rate ditribution, while the upper bound decreae a the rate ditribution of ecure link improve.

7 MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8517 C. Opportunitic Scheduling With Throughput and Delay Contraint In ome cenario, both throughput and delay requirement are impoed. In thi cae, the problem for maximizing the overall throughput can be formulated a max N ζ E[R N D N ] E[T N ] ubject to θ = E [R N D ] α and σ = E[T ] β. E[T ] A dicued in Section IV-A, the aforementioned optimization problem i equivalent to max N ζ E[R N D N ] x E[T N ] λ (αe[t ] E [R N D ]) μ (E[T ] β) where λ and μ are the Lagrange multiplier. By olving thi problem, the optimal topping rule i derived. Similar to the cae of a ingle contraint (e.g., throughput or delay), thi rule i alo a double-threhold topping policy, and the optimal threhold are given by {φ = x +λα+μ μβ D 1+λ φ r = x + λα + μ. The calculation of aforementioned threhold require the knowledge of (x,λ,μ), which can be determined with the following equation: t(x + λα + μ) = D (1 + λ)e [ (R φ ) +] + P r D r E [ (R r φ r ) +] (7) { λ (E [RN D ] αe[t ]) = 0 (8) μ (E[T ] β) =0. Equation (7) i the maximum expected profit equation, while (5) and (8) are from KKT condition. Similar to ingle-contraint cenario, there exit an area where both throughput requirement α and delay requirement β are effective. To characterize thi area, we have the following propoition. Propoition 4.4: For a given delay requirement β, the effective range for α i bounded by where [ φ rdf L (r) β (1 F (φ L ))D, φ H rdf (r) β (1 F (φ H )) D ( ) φ L = F 1 1 t+p rd r ( (β D ) ) φ H = F 1 t 1. (β D ) The detailed proof i lited in Appendix D. When α lie on the left ide of the range given in the propoition, the throughput ] Fig. 4. Rate ditribution of different link in heterogeneou cenario. contraint i le tight than the delay contraint. In thi cae, the throughput contraint i inactive. When α i greater than the upper bound of the range, the throughput requirement impoe a tronger contraint on the problem. In thi cae, the delay contraint i inactive. V. O PPORTUNISTIC SCHEDULING WITH HETEROGENEOUS RATE DISTRIBUTIONS Here, the heterogeneou rate ditribution cenario i conidered, where the rate ditribution of a ecure link can be ignificantly different from thoe of other ecure link, a hown in Fig. 4. In thi cenario, the overall performance of ecure link cannot reflect that of each individual link. If we conider all ecure link a a whole a previou ection, the opportunitic cheduling cheme will provide more tranmiion opportunitie to link with good link quality and caue the tarvation of link with poor channel quality, which lead to unacceptable QoS for thee link. To avoid thi ituation, we et QoS contraint for ecure link with the wort link condition, intead of putting QoS requirement on the whole et of ecure link. A. Wort Link Analyi The baic idea behind our cheme for heterogeneou cenario i to et QoS contraint on the ecure link with the wort performance, intead of on the whole et of ecure link. Hence, before impoing QoS contraint, we need to identify uch link. Conidering the ecure link of node m, the average delay of thi link, i.e., E[T,m ], can be expreed a where E[T,m ]= Δ,m (1 F,m (φ )) Δ=t+,i (1 F,i (φ )) D + P r,i (1 F r,i (φ r )) D r. i i The derivation of thi expreion can be found in Appendix B. According to the aforementioned equation, the numerator of delay expreion for all ecure link are exactly the ame. Therefore, the ecure link with the wort delay can be determined, if we can find a link with the minimum denominator (9)

8 8518 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 problem of maximizing the overall throughput under the delay contraint can be formulated a max N ζ E[R N D N ], ubject to E[T pw ] γ E[T N ] where ζ denote the et of topping time. A dicued in Section IV-A, the previou optimization problem i equivalent to max N ζ E[R N D N ] x E[T N ] ω (E[T pw ] γ) Fig. 5. Value of denominator in delay expreion for ecure link. value in it delay expreion. Furthermore, if the wort link i identified and the delay QoS contraint i et on the link, the delay of any other ecure link will alo meet thi QoS requirement, ince thee link have better delay performance a compared to the wort link. However, in ome cenario, uch a link cannot be determined without the knowledge of the topping threhold for ecure link. Thi can be explained with the following example. Conider three ecure link with heterogeneou rate ditribution. The denominator value in their delay expreion under different threhold of ecure link (i.e., φ ) are plotted in Fig. 5. If the threhold i equal to a, the denominator of Link 1 ha the minimum value, and hence, Link 1 i the wort link; if the threhold i equal to b, Link 2 i the wort in delay. Namely, with different threhold φ, the link that experience the wort delay performance i different. Therefore, unle we know the threhold of ecure link, we cannot determine the ecure link with the wort delay performance. In thi cae, intead of identifying a unique ecure link that experience the wort performance, we conider all link that potentially become the wort link in an intereted range of φ. We call thee link potential-wort link. In the previou example, both Link 1 and Link 2 can be the wort link under different threhold of ecure link. Therefore, Link 1 and Link 2 are both potential-wort link. However, Link 3 evidently ha better quality than Link 1 and Link 2 and never experience the wort delay performance. Therefore, Link 3 doe not belong to the et of potential-wort link. Generally, potential-wort link have relatively poor channel quality. To help thee link to achieve an acceptable performance, we treat thee link a a group and et QoS contraint on the group. B. Opportunitic Scheduling With the Delay Contraint on Potential-Wort Link We define ξ a the et that conit of all potential-wort ecure link. Furthermore, the delay on the et of potentialwort link i defined a the time between two ucceive tranmiion on link belonging to thi et. For convenience, T pw i ued to denote thi delay. To avoid the tarvation on potential-wort link, we et the delay requirement on thee link a E[T pw ] γ. Therefore, the where ω i the Lagrange multiplier. By olving thi optimization problem, we can derive the following propoition. Propoition 5.1: The optimal topping rule that maximize the overall throughput under the delay contraint on potentialwort link i a double-threhold policy, and the optimal threhold pair i given by { φ = x i ξ + ω,i γω D φ r = x + ω. In addition, the value of (x,ω) can be determined with the following equation: t(x + ω) = D E [(R φ ) + ]+P r D r E [(R r φ r ) + ] + ( i ξ,i)( i ξ,i)γω P F Δ ω (E[T pw ] γ) =0 where F Δ denote the difference between the value of the rate ditribution function of non-potential-wort ecure link at φ (i.e., F,nw (φ )) and that of potential-wort link (i.e., F,pw (φ )). The proof of thi propoition can be found in Appendix E. C. Effectivene of the Propoed Scheme The propoed cheme can effectively guarantee that the delay on the et of potential-wort link i le than a pecified value γ. We aume that there are n p link in thi et. If thee link have comparable performance for a given threhold pair derived from the aforementioned equation, then the delay of each link i approximately equal to n p γ. Since the wort link belong to the et of potential-wort link, the delay of all other ecure link outide the et cannot be longer than n p γ. Thu, in thi cae, the delay performance of each ecure link i guaranteed to be not wore than a pecific level (i.e., n p γ). If potential-wort link do not have comparable performance given the derived threhold pair, the delay of the wort link can be longer than n p γ. In thi cae, the propoed cheme cannot guarantee the delay performance of each ecure link. However, by etting contraint on the whole et of potential-wort link, the cheme i till beneficial to avoid evere tarvation in lowquality ecure link. Another poible olution to handling heterogeneou cenario i to eparate the wort link from the ecure link, treat them a different link type, and derive a certain threhold for each of them. Thi olution can achieve better performance

9 MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8519 TABLE I NORMALIZED SNR FOR SECURE LINKS AND REGULAR LINKS OF DIFFERENT NODES than the propoed cheme here, ince there are more degree of freedom to elect threhold for variou link. However, it increae the number of link type, which lead to high complexity in cheduling. Neverthele, even after eparating the wort link from each other, heterogeneity till exit among other ecure link. A a reult, more type of link need to be conidered, which further increae the complexity of the cheduling cheme and become impractical. VI. PERFORMANCE EVALUATION Here, the new cheduling cheme i evaluated by everal experiment. In our imulation, we conider a network coniting of five node, 3 and each of them maintain it own regular link and ecure link to other node. The tranmiion rate on a link i aumed to be equal to the channel capacity given by R =log ( 1 + ρ H 2) nat//hz (10) where H denote the random channel gain that follow a complex Gauian ditribution with the variance equal to 1, and ρ i the normalized SNR for the link. In the imulation for homogeneou cenario, all link belonging to the ame type are et with identical normalized SNR, a given in Table I, while in heterogeneou cae, the link of Node 1 and Node 2 evidently have wore link quality, a hown in Table I. In addition, the tranmiion duration for regular link i equal to 30 time lot, while that for ecure link varie in different experiment and it default value i 30 time lot. Moreover, to reflect the traffic load of the network, channel occupancy ratio i introduced and defined a P = 1 (1 p,m p r,m ). m Without being explicitly pecified, p,m and p r,m, for any node m, are equal to 0.1 in our experiment. In thi cae, the channel occupancy ratio i Baed on aforementioned parameter, our cheme i evaluated with MATLAB program. Given a pecific etting (including threhold for different type of link, channel acce probabilitie for node, tranmiion duration, etc.), our network imulation program run for 10 7 time lot, and performance reult, uch a overall throughput, throughput on ecure link, and delay of ecure link, are recorded. Note that the performance variation introduced by channel condition/medium acce randomne i negligible with the pecified running time (i.e., 10 7 time lot). 3 Since our method i inenitive to the number of node in the network, a imulation with five node i enough to demontrate the network performance. Fig. 6. Optimal throughput under variou QoS contraint. (a) Homogeneou cenario 1. (b) Homogeneou cenario 2. (c) Heterogeneou cenario. A. Homogeneou Scenario To verify the optimality of threhold pair derived in Section IV, we compare the throughput performance of our cheme (denoted by QDOS) with the optimal throughput obtained by the enumeration method (denoted by Enum.) under variou QoS contraint. In the enumeration method, we earch over all poible threhold combination (φ,φ r ), and our network imulation program run for each combination. Baed on imulation reult, threhold combination that do not guarantee the QoS contraint are removed. Among the remaining one, the combination leading to the maximum overall throughput i elected, and the correponding maximum value i recorded. The comparion reult between QDOS and Enum. under variou QoS contraint are hown in Fig. 6(a) and (b). It can be oberved that the throughput of QDOS i alway equal to the optimal value obtained through enumerating all poible threhold pair. Thee reult demontrate the optimality of our cheme. Performance reult of our cheduling cheme with different QoS contraint are hown in Fig. 7. For comparion, performance of another two acce cheme i provided: 1) the pure random acce cheme, namely, the cheme with both φ and φ r equal to zero; and 2) the ditributed opportunitic cheme (DOS) propoed in [15]. In Fig. 7(a), we know that

10 8520 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 Fig. 7. Throughput and delay with different cheme under homogeneou cenario. (a) Overall throughput. (b) Throughput of ecure link. (c) Delay of ecure link. TABLE II THROUGHPUT AND DELAY WITH DOUBLE CONSTRAINTS UNDER HOMOGENEOUS SCENARIOS the overall throughput of our cheme i ignificantly higher than that of the random acce cheme. In addition, when the channel occupancy ratio i greater than 0.2, the throughput performance lo of our cheme, a compared to the DOS cheme, i within 14%. Thi reult indicate that the overall throughput of our cheme i not ignificantly compromied. Fig. 7(b) and (c) how the QoS of ecure link under different cheme. The reult indicate that there i no QoS guarantee on ecure link in the DOS cheme. In contrat, our cheme with the throughput contraint can effectively guarantee the throughput performance of ecure link, while our cheme with the delay contraint uccefully control the delay on ecure link to an acceptable level. Moreover, note that the throughput requirement α = 0.4 i not atified in the low channel occupancy ratio range, when only the delay contraint i impoed on our cheme. In addition, the delay requirement β = 75 i violated in the high channel occupancy range, if our cheme only et the throughput contraint. Therefore, if both throughput QoS and delay QoS are required, our cheduling cheme with double contraint need to be applied. In Table II, the overall throughput (θ total ) and QoS of ecure link for our cheme with double contraint are ummarized. The reult how that both delay (σ ) and throughput (θ ) requirement of ecure link are atified at any channel occupancy ratio (P ). Thi confirm the effectivene of our cheme. In addition, our cheme i compared to the approach propoed in [13] (ADOS) in two cae. In the firt cae, the tranmiion duration for ecure link i equal to that for regular one, i.e., D = 30. In thi cae, the delay QoS requirement for ecure link i et to 75 time lot a before. In the econd cae, the tranmiion duration for ecure link i equal to five time Fig. 8. Performance comparion between QDOS and ADOS under homogeneou cenario. (a) Cae 1. (b) Cae 2. lot. Thi cae i alo a typical one: in many cenario, meage requiring high-level ecurity (e.g., bank/game account information) uually have much maller ize than regular one (e.g., peer-to-peer (P2P) tream) [33]. In the econd cae, the delay QoS requirement for ecure link i et to 25 time lot. In addition, ince the ADOS cheme cannot directly guarantee a pecific QoS requirement for ecure link, a dicued in Section II, we impoe a weight coefficient for ecure link in the objective function of [13], and we keep tuning thi coefficient until the delay QoS contraint i atified. The performance reult under two cheme, including the overall throughout (marked a THO) and the throughput of ecure link (marked a THS), are hown in Fig. 8. In the firt cae, the performance gain of QDOS over ADOS i about 2%. In the econd cae where D r /D become larger, the gain increae to 5%. Thi can be explained a follow. According to [13, eq. (12)], when D r /D increae, the ADOS cheme provide more channel acce opportunitie to ecure link to guarantee the fairne. More of a preference on ecure link ha

11 MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8521 Fig. 9. Throughput and delay with different cheme in the heterogeneou cenario. (a) Overall throughput. (b) Delay of ecure link. (c) Throughput of ecure link. a negative impact on the overall throughput and lead to larger performance gap a compared to QDOS. B. Heterogeneou Scenario The cheme derived for heterogeneou cenario are evaluated here. A et of link with different normalized SNR i ued in the evaluation, and their SNR value are given in Table I. Baed on thee SNR, the CDF of tranmiion rate on thee link are determined, according to (10). Following the definition of potential-wort link in Section V-A, we identify thee link (i.e., ecure link of Node 1 and Node 2 in Table I) and impoe QoS contraint on them. To verify the optimality of the QDOS cheme derived for heterogeneou cenario, we compare QDOS with Enum., a dicued in Section VI-A. In thi experiment, the delay contraint for potential-wort link varie from 100 time lot to 1200 time lot. The comparion reult are hown in Fig. 6(c). It can be oberved that the throughput of QDOS under variou delay QoS requirement i alway equal to the optimal throughput obtained from the enumeration method. Thi confirm that the derived threhold pair for heterogeneou cenario are optimal. To tudy the performance of QDOS for heterogeneou cenario, we compare it with variou cheme. Firt, we evaluate the performance difference between homogeneou and heterogeneou QDOS. To evaluate the performance of QDOS for heterogeneou cenario, a et of link with different normalized SNR i given in Table I. In thi cenario, we require that the delay QoS of each ecure link i controlled to the level of 600 time lot or le. If the QDOS deigned for homogeneou cae i applied, all ecure link are equally treated, and the previou QoS requirement on each individual ecure link lead to a delay contraint equal to 120 on the whole et of ecure link (there are five ecure link in total). With the QDOS deigned for heterogeneou cenario, the delay contraint i only impoed on potential-wort ecure link, and the previou QoS requirement convert to a delay contraint equal to 300 on the potential-wort link et (there are two ecure link in thi et). The imulation reult under two QDOS cheme (QDOS for homogeneou cenario and QDOS for heterogeneou cenario) are given in Fig. 9. For comparion, the performance of DOS propoed in [15] i alo provided. The overall throughput and the throughput of ecure link are plotted in Fig. 9(a). It can be found that the overall throughput under QDOS for heterogeneou cenario degrade 17.8% and 13.1%, a compared to that under DOS and that under QDOS for homogeneou cenario, repectively. The drop in the overall throughput i due to the fact that more tranmiion opportunitie are provided to ecure link (particularly potential-wort one) in QDOS for heterogeneou cenario, which i reflected by the throughput of ecure link under three cheme, a hown in Fig. 9(a). QoS performance for each ecure link under two QDOS cheme i hown in Fig. 9(b) and (c). With the QDOS deigned for homogeneou cenario, the delay QoS of potential-wort ecure link (ecure link of Node 1 and Node 2) everely violate the requirement et previouly, and the throughput performance of thee link indicate evere tarvation. With the QDOS deigned for heterogeneou cenario, both delay and throughput performance of potential-wort ecure link are ignificantly improved, and the delay QoS of each ecure link meet the performance requirement, which demontrate the neceity and effectivene of conidering heterogeneou cenario in QDOS. The QDOS cheme for heterogeneou cenario i alo compared to ADOS in two cae. In the firt cae, the tranmiion duration for ecure link (D ) i 30 time lot, which i equal to that for regular link (D r ). In thi cae, the delay requirement i given by 300 time lot. In the econd cae, D r remain to be 30 time lot, while D i et to five time lot. Thi etting i common in real communication cenario: ecure link are uually ued to tranmit the mot critical meage uch a control frame, which i horter than regular data frame. In thi cae, the delay contraint i et to 50 time lot to uit the need of ecure link. Similar to the experiment in homogeneou cenario, the objective function in [13] i modified by adding a weight coefficient for ecure link, and the coefficient i tuned until the delay QoS requirement i atified. The comparion reult are hown in Fig. 10. In the firt cae, the overall throughput of QDOS i 12% lower than that of ADOS, while in the econd cae, our cheme outperform ADOS by 16%. The reult indicate that, when the ratio between D r and D varie, the relative performance gain of QDOS over ADOS alo change. Thi can be explained by

12 8522 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 Fig. 10. Performance comparion between QDOS and ADOS under heterogeneou cenario. (a) Cae 1. (b) Cae 2. conidering two factor. On the one hand, for the ake of fairne, ADOS provide more channel acce opportunitie to low-quality link under heterogeneou cenario, which caue the degradation of the overall throughput. On the other hand, in the ADOS cheme, each node ha individual threhold for determining whether to take tranmiion opportunitie. Compared to unified threhold for all node, thi cheme i beneficial to improve the overall throughput under heterogeneou cenario. When the firt factor dominate a in the econd cae (larger D r /D ), our cheme i better than ADOS. If the econd factor dominate a in the firt cae (maller D r /D ), our cheme doe not outperform ADOS. The comparion reult indicate that our cheme for heterogeneou cenario can be further improved if each node ha individual threhold a ADOS. Thi i ubject to future reearch. VII. DISCUSSION A. Beyond Two Type of Link In previou ection, two type of link, i.e., ecure link and regular link, are conidered. Our cheme can be extended to upport multiple type of link with variou QoS requirement. The cheme upporting multiple type of link follow the ame framework a developed in Section IV, and the key tep for deriving optimal threhold for different type of link are ummarized a follow: 1) Convert the objective function (i.e., the overall throughput) baed on the optimal topping theory [19] and the method of Lagrange multiplier[30]a(3) insection IV-A. Let {λ i } (i [1,n]) denote the Lagrange multiplier for QoS contraint, where n i the total number of thee contraint. 2) Define the profit function a that in Section IV-A according to QoS requirement. The expectation of the defined profit mut have the ame expreion a the converted objective function. 3) Derive the optimal threhold with repect to {λ i } and x and the maximum expected profit equation following the ame approache a thoe in Section IV-A and Appendix E. 4) Determine {λ i } and x baed on KKT condition (n equation) and the maximum expected profit equation. Finally, the optimal threhold can be calculated. A the number of link type grow, the number of QoS contraint on variou type of link linearly increae. To derive and calculate the optimal threhold (i.e., the third tep and the fourth tep) with n QoS contraint, 2 n cae need to be conidered ince each contraint can be active or not. Thu, the complexity of determining optimal threhold exponentially increae a the number of link type grow. Such growing complexity limit the number of link type that can be conidered in our cheme. The cheme propoed in Section V provide a remedy to thi iue. With thi cheme, we can provide QoS to a group of link, even if there are heterogeneou rate ditribution among the group, which can be conidered a the union of everal link et with homogeneou link ditribution. In thi way, we decreae the number of link type conidered in the cheduling cheme, which reduce the complexity of the cheme. B. Information Exchange for QDOS In our cheme, each node need rate ditribution and channel acce probabilitie of other node to calculate the optimal threhold. A mentioned previouly, we aume that thee parameter are known by each node, a in previou work [13], [14]. However, in reality, rate ditribution and channel acce probability of each node are collected by the node itelf, and thee need to be exchanged among different node. For thi purpoe, a tartup phae can be introduced. In thi phae, node exchange their channel acce probabilitie and rate ditribution information with other. After thi phae, each node can calculate the optimal threhold and initiate data tranmiion following the cheduling cheme propoed in thi paper. If a node detect the variation of it parameter after the tartup phae, it notifie the change to other node by piggybacking the new parameter in it data tranmiion. Other node keep overhearing data tranmiion in the medium, extract the new parameter, and update the optimal threhold. Generally, when thee parameter vary lowly (i.e., large channel coherence time), the performance penalty introduced by the aforementioned information exchange cheme i minimal, ince the overhead i negligible a compared to a large amount of data tranmiion. However, if thee parameter vary quickly (i.e., mall coherence time), it i difficult to catch up the variation of thee parameter through information exchange. How to make a cheduling cheme (including our cheme and the related one [13], [14]) perform well under uch cenario i an intereting but challenging problem, which demand future reearch. VIII. CONCLUSION Opportunitic cheduling conidering QoS contraint for hybrid link of a wirele network ha been tudied in thi

13 MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8523 paper. Given different cenario of rate ditribution, two QoS cheduling cheme were derived baed on the optimal topping theory. Thee cheme balance throughput and QoS guarantee of hybrid wirele link. Performance reult howed that the QoS of a pecific link type could be guaranteed without ignificantly compromiing the overall throughput of hybrid link. Although thi paper take ecure link and regular link a an example of hybrid link, the QoS-oriented opportunitic cheduling cheme derived in thi paper are completely applicable to other cenario of hybrid link. A indicated by imulation reult, adopting individual threhold for each link of a node i beneficial to improve the overall throughput in heterogeneou cenario. How to extend thi paper to upport individual threhold for each node i ubject to future reearch. APPENDIX A CONSTRAINT QUALIFICATION To check whether the contraint qualification hold for our optimization problem, we firt define g 1 = E [R N D ] and g 2 = E[T ]. The value of g 1 and g 2 are determined by the deciion rule for keeping or dropping the tranmiion opportunitie. In our paper, thi rule i characterized by the two threhold φ (for ecure link) and φ r (for regular link). If the current tranmiion rate upported by the link that capture the channel i higher than the correponding threhold, the tranmiion opportunity i taken. Otherwie, the opportunity i dropped. Thu, we have g 1 = ( g1 φ g 1 φ r ) and g 2 = ( g2 φ g 2 φ r ). According to (6), it can be hown that g 1 = f (φ ) φ rdf (r) φ f (φ )(1 F (φ )) φ (1 F (φ )) 2 > f (φ ) φ φ df (r) φ f (φ )(1 F (φ )) (1 F (φ )) 2 = 0. In addition, we have ( g 1 / φ r )=0. Similarly, we can derive that g 2 φ = (t + P r (1 F r (φ r )) D r ) f (φ ) (1 F (φ )) 2 g 2 φ r = P rf r (φ r )D r (1 F (φ )). It can be oberved that g 2 / φ i alway greater than zero, while g 2 / φ r i alway le than zero. Furthermore, the contraint in our problem can be expreed a G 1 = E [R N D ] αe[t ]=g 1 αg 2 G 2 = E[T ] β = g 2 β. Thu, the gradient of G1 andg2aregivenby G 1 = g 1 α g 2, and G 2 = g 2. To verify linear independence contraint qualification, we need to invetigate whether the gradient of active contraint are linearly independent. If only the contraint G 2 i active (in the optimization problem in Section IV-B or Section IV-C), we only need to verify that G 2 0. Note that G 2 = g 2 and g 2 / φ r i alway le than zero. Hence, G 2 i not equal to 0 for any (φ,φ r ). If only the contraint G 1 i active (in the optimization problem in Section IV-A or Section IV-C), we only need to verify that G 1 0. Since( g 1 / φ r )=0and G 1 = g 1 α g 2, we can derive that ( G 1 / φ r )= α( g 2 / φ r ). Becaue throughput requirement α i greater than zero and g 2 / φ r i alway le than zero, G 1 / φ r i alway greater than zero. Hence, G 1 i not equal to 0 for any (φ,φ r ). When both G 1 and G 2 are active (in the optimization problem in Section IV-C), aume that there exit a pair (φ,φ r ), uch that G 1 and G 2 are linearly dependent. Then, we have G 1 = c G 2,wherec i a contant. Following that, it can be hown that g 1 α g 2 = c g 2. If o, we have g 1 = (α + c) g 2. Since ( g 1 / φ r )=0 while ( g 2 / φ r ) < 0, we can conclude that (α + c) i equal to zero, which further lead to g 1 = 0. However, we have hown that ( g 1 / φ ) i alway greater than 0. Thi i a contradiction. Hence, for any pair (φ,φ r ), G 1 and G 2 are linearly independent. The aforementioned derivation how that, for our optimization problem in Section IV-A C, the linear independence contraint qualification hold with repect to any threhold pair (φ,φ r ). APPENDIX B DERIVATION OF THE DELAY EXPRESSION To derive the expreion of the average delay for the ecure link of node m, i.e., E[T,m ],wedefinep 1 a the probability that, given a time lot, the ecure link uccefully contend the channel and take the tranmiion opportunity following the topping rule (ee Fig. 3). Thu, p 1 can be expreed a p 1 =,m (1 F,m (φ )). In addition, p 2 i defined a the probability that other ecure link or regular link win the channel contention in a given time lot and take the tranmiion opportunity following the topping rule. Hence, it can be hown that p 2 =,i (1 F,i (φ )) + P r,i (1 F r,i (φ r )). i m i For convenience, we ue p 2, and p 2,r to denote the firt term and the econd term in the right-hand ide of the aforementioned equation, repectively. Furthermore, the probability p 3, which i defined a the probability that no tranmiion will tart at the end of a given time lot, can be expreed a p 3 = 1 p 1 p 2. Conidering the firt time lot after a tranmiion on the ecure link of node m, if thi link win the channel contention again

14 8524 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 and the current tranmiion rate exceed the threhold for ecure link, another tranmiion will tart on thi link. In thi cae, the delay i given by T,m = t + D. If a tranmiion tart on other link at the end of the lot (we ue C 2 to denote thi cae), according to the memoryle characteritic of our ytem model, the expected delay of the ecure link of node m under thi cae can be denoted a E[T,m C 2 ]=t + D 0 + E[T,m ] where D 0 denote the expected tranmiion time after the end of the given time lot under C 2 and can be expreed a D 0 = p 2, D + p 2,r D r. p 2 p 2 If no tranmiion tart at the end of the lot (we ue C 3 to denote thi cae), the expected delay i given by Therefore, it can be hown that E[T,m C 3 ]=t + E[T,m ]. E[T,m ]=p 1 (t+d )+p 2 (t+d 0 +E[T,m ])+p 3 (t+e[t,m ]). Hence, we have where E[T,m ]= t + p 1D + p 2 D 0 p 1 = Δ=t+ i,i (1 F,i (φ )) D + i Δ,m (1 F,m (φ )) P r,i (1 F r,i (φ r )) D r. The average delay on the et of ecure link can be derived in a imilar way. APPENDIX C PROOF OF PROPOSITION 4.2 The overall throughput of ecure link i the ummation of throughput of each ecure link. Hence, we have θ = M m=1 θ,m = M m=1 E[R,m D ] E[T,m ] where θ,m denote the throughput on the ecure link of node m, R,m tand for the rate for the tranmiion on thi link, and T,m indicate the time between two ucceive tranmiion on the link. Furthermore, it can be hown that θ = M m=1 φ rdf,m(r) 1 F,m (φ ) D t+ i,i(1 F,i (φ ))D + i P r,i(1 F r,i (φ r ))D r,m (1 F,m (φ )) The previou equation can be implified a θ = D φ rdf (r) t + D (1 F (φ )) + P r D r (1 F r (φ r )). (11). For the lower bound, if the throughput requirement α i le than θ L,i.e., D φ rdf α< (r) t + D (1 F (φ )) + P r D r (1 F r (φ )) the optimal threhold pair (φ,φ ) for the uncontrained cae i located in the feaible domain, which indicate that thi pair i alo the optimal olution for the contrained problem. In thi cae, the throughput contraint i inactive. For the upper bound, let θ U denote the maximum poible throughput on ecure link. Apparently, the throughput requirement α cannot be greater than θ U. To determine θ U,weetφ r to approach and derive the optimal threhold for ecure link that maximize the ytem overall throughput following the imilar framework preented in Section IV-A. It can be hown that φ i equal to the maximum overall throughput x.inaddition, ince the threhold for regular link approache infinity, there i no throughput on regular link, and the overall throughput i equal to the throughput of ecure link. Thu, we have θ U = x = φ. Therefore, baed on (11), the maximum throughput of ecure link can be determined by P θ U θ rdf = U (r) t D + (1 F (θ U )). The propoition i proved. APPENDIX D PROOF OF PROPOSITION 4.4 The average delay for ecure link i given by β = E[T ]= t + P r (1 F r (φ r )) D r (1 F (φ )) Thu, we have the following inequalitie: t (1 F (φ )) + D β It follow that t (β D ) (1 F (φ )) + D. t + P r D r (1 F (φ )) + D. t + P rd r (β D ). Since F ( ) i a rate ditribution function, it value increae with φ. Thu, it can be hown that ( F 1 1 t + P ) ( ) rd r φ F 1 t 1. (β D ) (β D ) For convenience, the lower bound and the upper bound in the aforementioned inequalitie are denoted by φ L and φh, repectively. Furthermore, we conider E[RN D ], which i given by E [RN D φ rdf (r) ]= (1 F (φ )) D.

15 MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8525 Since E[RN D ] i a function increaing with φ, it can be hown that φ rdf L (r) (1 F (φ L )) D E [RN D φ rdf ] H (r) (1 F (φ H ))D. Moreover, baed on E[R N D ] αe[t ]=0andE[T ]=β, we have E [R N D ]=αβ. Hence, it can be hown that φ rdf L (r) φ rdf β (1 F (φ L ))D α H (r) β (1 F (φ H )) D. Thi prove Propoition 4.4. APPENDIX E PROOF OF PROPOSITION 5.1 We can define the profit of topping rule N a R N D N x T N ωt pw + γω where T pw denote the time from the beginning of the contention period for topping rule N to the end of the next tranmiion on potential-wort link. Furthermore, L i defined a the maximum expected profit for topping rule and ha the identical expreion a the optimization formulation preented in Section V-B. Similar to the derivation in Section IV-A, to olve the optimization problem in Section V-B, we only need to find the optimal topping rule that maximize the expected topping profit. In addition, note that L i equal to zero. If a link from the et ξ take the channel uccefully, the profit for tranmitting a packet on thi link can be expreed a R,pw D x D ωd + γω (x + ω)t cont (12) where R,pw i the tranmiion rate on thi link and follow the ditribution 1 F,pw (r) = m ξ P,m F,m (r).,m m ξ If a ecure link that i not a potential-wort one win the channel contention, the profit for tranmitting immediately i given by R,nw D x D ωd ωe[t pw ]+γω (x + ω)t cont where R,nw denote the tranmiion rate on the link that uccefully take the channel and follow the ditribution 1 F,nw (r) = m ξ P,m F,m (r).,m m ξ According to the KKT condition [30], ω(e[t pw ] γ) =0. The aforementioned profit expreion can be implified a R,nw D x D ωd (x + ω)t cont. A a reult, the average profit for tranmitting immediately on the ecure link that win the channel contention can be denoted a i ξ R D,i [x D +ωd γω+(x +ω)t cont ] i ξ,i [x D +ωd +(x +ω)t cont ] = R D x i ξ D ωd +,i γω (x +ω)t cont where R follow the ditribution i ξ,i F,pw (r)+ P i ξ,i F,nw (r) =F (r). Thu, if thi profit i greater than the maximum expected profit with kipping thi tranmiion and waiting for the next topping time, i.e., R D x i ξ D ωd +,i γω L then the packet i tranmitted. Hence, the threhold for ecure link i given by φ = x i ξ + ω,i γω. D The regular cae can be derived imilarly a that in Section IV-A. If R r D r x D r ωd r ωe[t pw ]+γω L namely, R r x + ω, the regular packet i tranmitted on the link that win the channel contention. In ummary, the optimal threhold pair i given by { φ = x i ξ + ω,i γω D (13) φ r = x + ω. To calculate the optimal threhold pair, the maximum expected profit equation i derived a follow. Conidering the firt ucceful channel contention, if a potential-wort ecure link win the channel contention, it tart it tranmiion only when R,pw φ. In thi cae, the profit i given by R,pw D x D ωd + γω (x + ω)kt where k i the number of time lot before the firt ucceful channel contention. Note that the aforementioned equation i identical with (12), when T cont i equal to kt. On the other hand, if R,pw <φ, the tranmiion opportunity i kipped, and then, the maximum expected profit i L (x + ω)kt. Combining the previou two cae, if a potential-wort link win the channel contention, the maximum profit following the optimal topping rule can be expreed a (R,pw φ ) + i ξ D +γω,i u(r,pw φ ) (x +ω)kt where ( ) + denote max{, 0}, andu( ) i the tep function.

16 8526 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 10, OCTOBER 2016 Similarly, if a ecure link that i not potential-wort win the contention, the maximum profit i given by (R,nw φ ) + i ξ D γω,i u(r,nw φ ) (x +ω)kt. If a regular link contend the channel uccefully, the maximum profit can be derived a (R r φ r ) + D r (x + ω)kt. Therefore, we have L = [,i E (R,pw φ ) + D + γω i ξ,i u i ξ ] (R,pw φ ) (x + ω)kt + [,i E (R,nw φ ) + i ξ D γω,i u i ξ ] (R,nw φ ) (x + ω)kt + P r E [ (R r φ r ) + D r (x + ω)kt ]. Rearranging the aforementioned equation, we get (x + ω)t = D E [ (R φ ) +] +P r D r E [ (R r φ r ) +] + ( i ξ,i)( i ξ,i)γω (F,nw (φ ) F,pw (φ )). (14) In addition, baed on the KKT condition, we have where T pw can be expreed a ω (E[T pw ] γ) =0 (15) T pw = t + P r (1 F r (φ r )) D r + (1 F (φ )) D (. i ξ,i) P (1 F,pw (φ )) Combining (13) (15), (x,ω) can be olved with LMA [31]. Following that, the optimal threhold pair can be calculated according to (13). REFERENCES [1] S. 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17 MAO et al.: DISTRIBUTED OPPORTUNISTIC SCHEDULING WITH QoS CONSTRAINTS FOR WIRELESS NETWORKS 8527 Wenguang Mao received the B.S. degree in 2011 and the M.S. degree in 2014 from the Shanghai Jiao Tong Univerity, Shanghai, China, where he wa a Reearch Aitant with the Wirele and Networking (WANG) Laboratory. He i currently working toward the Ph.D. degree with the Department of Computer Science, The Univerity of Texa at Autin, Autin, TX, USA. Hi current reearch interet include mobile computing and wirele networking. Xudong Wang (SM 08) received the Ph.D. degree in electrical and computer engineering from Georgia Intitute of Technology, Atlanta, GA, USA, in Augut He i currently with the Univerity of Michigan- Shanghai Jiao Tong Univerity (UM-SJTU) Joint Intitute, Shanghai Jiao Tong Univerity, Shanghai, China. He i a Ditinguihed Profeor (Shanghai Oriental Scholar) and i the Director of the Wirele and Networking (WANG) Laboratory. He i alo an Affiliate Faculty Member with the Department of Electrical Engineering, Univerity of Wahington, Seattle, WA, USA. Since Augut 2003, he ha been a Senior Reearch Engineer, Senior Network Architect, and R&D Manager with everal companie. He ha been actively involved in R&D, technology tranfer, and commercialization of variou wirele networking technologie. Hi reearch interet include wirele communication network, mart grid, and cyberphyical ytem. He hold everal patent on wirele networking technologie, and mot of hi invention have been uccefully tranferred to product. Dr. Wang erve a an Editor for the IEEE TRANSACTIONS ON MOBILE COMPUTING, the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, and Ad Hoc Network by Elevier. He wa the Demo Cochair of the ACM International Sympoium on Mobile Ad Hoc Networking and Computing (ACM MOBIHOC) in 2006, a Technical Program Cochair of the Wirele Internet Conference (WICON) in 2007, and a General Cochair of WICON in He ha been a technical committee member of many international conference. Shanhan Wu received the B.S. degree in 2011 and the M.S. degree in 2014 from the Shanghai Jiao Tong Univerity, Shanghai, China, where he wa a Reearch Aitant with the Wirele and Networking (WANG) Laboratory. She i currently working toward the Ph.D. degree with the Wirele Networking and Communication Group, The Univerity of Texa at Autin, Autin, TX, USA, working on large-cale machine learning algorithm and highdimenional tatitic. Her current reearch interet include machine learning algorithm with large-cale or treaming data input, ubmodular function, convex optimization, and deign and analyi of calable algorithm in ditributed or multicore ytem.