Information about Congestion control, routing, and scheduling 2015

Published on September 24, 2015

Author: parryprabhu

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2. QU et al.: WIRELESS NETWORKS WITH INTERFERENCE CANCELATION CAPABILITIES 3109 Clearly, efﬁcient link scheduling together with SIC help in promoting better spatial reuse as well as transmission con- currence, resulting in increased throughput performance. In this paper, we consider a wireless network where nodes are endowed with SIC capabilities, and we study the problem of link scheduling, under the SINR interference model, in the context of a cross-layer network design. We consider a network utility maximization (NUM) problem (similar to [25] and [26]) in a multihop wireless network and decouple the cross-layer optimization into congestion control and routing/scheduling subproblems. The congestion control subproblem can easily be solved at the source node of each ﬂow by only using local information, and following a back pressure framework, the routing/scheduling subproblem is converted into a weight scheduling problem where the weight information can be illustrated as some scale of the queue length at each node. However, as mentioned earlier, previous work has shown that the link scheduling under a binary interference model is an NP-hard problem [2] and under the SINR model as NP- complete [5], both with and without SIC, and polynomial-time approximation algorithms are presented (e.g., [10] and [24]). Scheduling methods such as maximum weight scheduling [11] and greedy maximal scheduling (GMS) [12] have shown to achieve 100% throughput in most practical wireless networks with the second method being more amenable to distributed implementation. In this paper, we will consider the GMS ap- proach for solving the link scheduling problem under the SIC constraints and the physical interference model; now, given the complexity of the scheduling problem in centralized settings, we develop a decentralized method for solving it. Our main contributions are as follows. First, we revise the interference localization method in [19] and show that it can be used to maintain the interference constraints in a network with SIC capabilities. Second, we present a search-based method for determining the minimum interference neighborhood of each link. Our design reveals that the network throughput perfor- mance is mainly dependent on how much local information can be coordinated at each communication link. We show that our decentralized algorithm yields the same maximal schedule obtained by the centralized GMS method. The rest of this paper is organized as follows. In Section II, we brieﬂy survey the work related to cross-layer optimization with and without SIC. Our system model, the interference localization technique, and problem formulation are presented in Section III. Section IV presents the dual decomposition for decoupling the cross-layer design problem as well as the design of our decentralized scheduling method. Finally, numerical results are presented in Section V, and conclusions are drawn in Section VI. II. RELATED WORK Recently, there have been growing interests to exploit inter- ference among adjacent concurrent transmissions to increase the network throughput. Mitran et al. in [13] formulated a cross-layer design optimization to solve the joint problem of routing and scheduling in a multihop wireless network with advanced physical-layer techniques for interference cancela- tion, such as SIC, superposition coding, and dirty-paper coding. The authors formulated the problem of routing and scheduling under the physical interference model as a max–min optimiza- tion problem and developed a column generation method for solving it efﬁciently. The authors have shown that SIC signiﬁ- cantly improves the network performance and, in particular, the max–min per node throughput. Jiang et al. in [18] noted that SIC is a very promising interference exploitation technique for increasing the network throughput due to its ability in enabling multiple concurrent transmissions. Upon developing a cross- layer optimization framework for the routing and scheduling problem, the authors studied the optimal interaction between interference exploitation, through SIC, and interference avoid- ance, through link scheduling. The authors have shown that substantial performance gains can be achieved when both tech- niques are combined. Now, the asymptotic transmission capacity of ad hoc net- works with SIC is studied in [15] and [16]; the former con- sidered that all signals within one hop from transmitters can be successfully decoded, and the latter supposed a more realistic SIC model in their analysis. Sen et al. in [17] studied the extent of throughput gains that is possible under SIC from a MAC- layer perspective. They argued that only little gains could be achieved in restricted scenarios (mainly for upload trafﬁc in wireless local area networks). Furthermore, when transmitters choose their bit rates independently, not much gain can be achieved. However, the authors showed (in a two-transmitter scenario) that one way to maximize the gain is through transmit power level selection such that the feasible bit rate is equal for both transmissions. Lv et al. in [22] proposed a layered protocol model and a layered physical model (to model the interference) and studied the problem of link scheduling to characterize the advantages of SIC. The authors analyzed the capacity of a net- work with SIC and demonstrated the importance of designing SIC-aware scheduling. It was shown that signiﬁcant through- put gains (20%–100%) can be obtained in chain/cell network topologies. The problem of link scheduling with interference cancelation using the SINR interference model is studied in [23] where Yuan et al. assumed multiuser decoding receivers. The authors showed that the optimal scheduling problem with (successive or parallel) interference cancelation is NP-hard and developed compact linear programming (LP) methods for obtaining exact solutions. The authors showed that in the lower SINR regime, interference cancelation yields signiﬁcant im- provements. Similarly, in [24], Goussevskaia and Wattenhofer studied the same problem but developed approximation algo- rithms for solving the scheduling problem in polynomial time. SIC has shown to achieve up to 20% performance gains over networks that do not have interference cancelation capabilities. It should be noted that all of the aforementioned methods solve the link scheduling problem in a centralized manner; given the complexity of the problem, decentralized methods are more practical. In [25], a distributed method that jointly adapts decisions made by different layers is proposed. Chen et al. presented then a dual driven decomposition approach for the original problem, which is further decomposed into two sub- problems (one for congestion control and another for routing/ scheduling), and the three are correlated through Lagrangian

3. 3110 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 7, JULY 2015 multipliers. A fully distributed algorithm following the decomposition is then presented. In [31] and [32], Joo and Shroff and Joo et al., respectively, revised the distributed scheduling methods in wireless networks and classiﬁed them into different categories. For the graph-based interference model, the authors mainly focused on the problem of schedul- ing and proposed a revision of the GMS algorithm to satisfy the distributed design. For the more practical interference model, Le et al., in [19], investigated the link scheduling problem by considering GMS; to simplify the complex relationship govern- ing the interference, the authors presented a method to localize the interference around each link, thereby each link coordinates its scheduling within a local neighborhood while maintain- ing scheduling feasibility. The authors subsequently devel- oped a decentralized scheduling method, which was shown to yield maximal link scheduling similar to the centralized GMS method. These decentralized methods did not however consider network with interference cancelation capabilities. Note that our work is similar to [25] and [26] in that we try to design an efﬁcient distributed cross-layer method to improve the performance of wireless multihop networks. Our work, however, differs from previous work in that we consider networks with SIC capabilities in our distributed cross-layer design with the SINR interference model. This makes the design of a decentralized method more complicated because of the particularity of the SIC constraints. III. SYSTEM MODEL A. Network Model We consider a network of N nodes and L links; we assume stationary nodes, each equipped with SIC capability. Let dn,n be the Euclidean distance between two nodes n and n , and let Gnn be the channel gain from n to n such that Gnn = d−pl n,n , where pl is the path-loss index. The transmission power at each node is assumed to be ﬁxed and equal to Pw. Let F be the set of ﬂows in the network, where each ﬂow f (f ∈ F) is identiﬁed by a source s(f) and a destination d(f) and a transmission rate yf . B. Interference Model With SIC In the physical interference model (also known as the ad- ditive interference model), in the presence of concurrent trans- missions on neighboring links, one transmission (e.g., on link i) is successful if the SINR at the intended receiver is above a certain threshold β. Then, the physical interference model can be formulated as SINRt(i),r(i) = PwGt(i),r(i) η + ∀ n∈NA−t(i) PwGn,r(i) ≥ β (1) where η is the background noise power. NA is the set of all active nodes in the network, and t(i) and r(i) are the transmitter and the receiver of link i. β is the minimum SINR threshold that must be maintained to support a successful transmission on link i while guaranteeing a tolerable bit error rate. If the SINR requirement is not met, then the received packet cannot be correctly extracted from the received signal. In this paper, we assume β ≥ 1. Now, SIC allows a signal to be correctly decoded in the presence of other concurrent transmissions. Here, the receiver starts decoding the strongest signal from the combined received signal; then, the decoded signal is subtracted, and the process is repeated on the residual signal until the signal of interest is either decoded or no more decoding is possible. This technique therefore allows a signal to be correctly received given that other stronger signals are decoded ﬁrst. Next, we illustrate the SINR constraints in the presence of SIC. Consider two links i and i adjacent to each other. Denote by P1 r(i)(P2 r(i)) the strength of the signal received at destination r(i) from t(i) (respectively, from t(i )) and suppose P2 r(i) > P1 r(i). Here, r(i) will attempt to decode the signal received from t(i ) ﬁrst; this signal can be decoded if SINR2 r(i) = P2 r(i) η + P1 r(i) ≥ β. (2) If the signal of t(i ) is successfully decoded at r(i), then r(i) will subtract it from the combined signal and will attempt to decode the signal arriving from t(i), i.e., SINR1 r(i) = P1 r(i) η ≥ β. (3) The given procedure can be generalized in a straightforward manner to any number of transmissions. C. Link Scheduling With Interference Localization We consider a time-division-multiple-access-based MAC layer where time is divided into slots of equal length, and each time slot has two parts: a schedule and a transmission. The schedule part has several intervals, and each interval is further divided into minislots. We deﬁne the set of links that can be concurrently active in the same time slot (without violating the SINR requirements) as a (feasible) conﬁguration (or conf for short). Then, our objective is to generate a new conﬁguration during the “schedule” period under SIC constraints and trans- mits data during the “transmission” period (each active link will transmit one packet during the “transmission” period). Note that, in a wireless network without SIC capabilities, in [19], Le et al. noted that only those concurrent transmissions within a neighborhood of a particular link may create signiﬁcant cumulative interference at the receiver of this link. Accordingly, they presented an “interference localization” technique that allowed them to decentralize the link scheduling problem. The authors presented a method to determine for each link a neigh- borhood such that interference from active links outside this neighborhood will have negligible impact on received signal at the receiver of this link. Namely, for link l, the maximum interference that can be tolerated is Imax l = PwGt(l),r(l) β − η. (4) Let INl and nINl denote the interference neighborhood and noninterference neighborhood of link l. INl is a circle

4. QU et al.: WIRELESS NETWORKS WITH INTERFERENCE CANCELATION CAPABILITIES 3111 centered around the receiver of l and whose radius will be determined later. All links whose transmitters are inside INl will be able to exchange information (therefore coordinate) with the transmitter of link l for scheduling purposes. nINl (complement of INl) contains the set of links whose cumula- tive interference is assumed to be negligible at rl. Le et al. in [19] have shown that given a constant (0 < < 1), for link l to be feasible, the upper bound on the interference coming from active links located in INl should not exceed (1 − )Imax l , and the total interference coming from links located in nINl cannot exceed Imax l . Therefore, when all active links in some set are feasible, we obtain a feasible conﬁguration. Now, to account for the SIC property of the nodes in the network, we ﬁrst revise the interference localization technique presented earlier as follows. For a particular link l, we divide the network into three regions: the strongest signal area, the inter- ference area, and the noninterference area. The strongest signal area (Al) refers to a circular area with radius dt(l),r(l) centered around the receiver of link l. The interference area refers to the ring-like area (INl − Al) with radius from dt(l),r(l) to a certain length λ(l) (λ(l) ≥ dt(l),r(l)) centered around the receiver of link l. The noninterference area denotes the region outside the interference area (nINl). Next, we introduce the deﬁnition of vector −→ λ . Deﬁnition 1: For any feasible link l, λ(l) denotes the lower bound of the radius of the interference neighborhood (INl) such that the total interference coming from links located in the noninterference area does not exceed Imax under SIC constraints. We assume that each link l (l ∈ L) is able to communicate with any link whose transmitter is located in INl, or any link l such that t(l) ∈ INl , to exchange link weight information and coordinate the link scheduling. Deﬁnition 1 implies that there is a relationship between and −→ λ (we will present a procedure to compute −→ λ based on in the following section). Furthermore, can be used to control the potential scheduling overload, and we will verify in Section V that will have a signiﬁcant effect on the achievable network performance. Note that, in any feasible conﬁguration, we assume that all “active” links l (such that t(l ) ∈ Al) have stronger received signals at r(l) than the signal arriving from t(l), and therefore, using SIC, r(l) is capable of successively decoding those signals prior to decoding the signal arriving from t(l). Deﬁnition 2: Given a certain −→ λ , Θ( −→ λ ) is a class of schedul- ing algorithms such that a particular scheduling method will belong to Θ( −→ λ ) if it yields an active schedule satisfying the following constraints: 1) For any active link l ∈ L, the total interference coming from active transmitters located in INl − Al does not exceed (1 − )Imax (l); and 2) let k de- note the link from any active transmitter located in Al to r(l); then, the cumulative interference at r(k) = r(l) coming from active transmitters located in INk − Ak does not exceed (1 − )Imax (k). The second constraint in the given deﬁnition implies that all active transmitters within the neighborhood of r(l) (i.e., in Al) should have their signals decoded at r(l) prior to decoding the signal arriving from t(l). Here, r(l) will attempt to success- fully decode each of these arriving signals (using SIC); this is possible because for each signal, we assume that the cumulative interference from active transmitters located in INk − Ak does not exceed (1 − )Imax (k), which is required for successful decoding of the signal of t(k) at r(l). Combining Deﬁnitions 1 and 2, it can be shown that for any (0 < < 1), there exists a vector −→ λ and a class of scheduling methods Θ( −→ λ ) such that any scheduling algorithm belonging to Θ( −→ λ ) will result in a feasible schedule satisfying the SIC constraints. This is summarized in the following theorem. Theorem 1: Given any 0 < < 1, there exists a vector λ that satisﬁes Deﬁnition 1, and the result of schedule Θ(λ) that satisﬁes Deﬁnition 2 is feasible under SIC constraints. D. Problem Formulation Following the given discussion, we assume that all active links in one conf can simultaneously transmit. Denote by E the set of all possible conﬁgurations/schedules, where each conf is indexed by e. Each conf (e) is represented by a |L|-dimensional rate vector −→r e , where for each link l, re l can be deﬁned as re l = c, if l ∈ e 0, otherwise (5) where c is a constant link transmission rate. We deﬁne the feasible rate region Γ as the convex hull of these rate vectors. We assume through time sharing that all interior points of Γ are attainable. We deﬁne a link-ﬂow matrix v to describe the routing of ﬂows in the network, where each element vlf ∈ v(∀ l ∈ L, f ∈ F) corresponds to the fraction of ﬂow f deliv- ered over link l. We assume a utility function U(yf ) to be twice differentiable, increasing, and strictly concave [25]. Our design target can be summarized in the following utility maximization problem: [OBJECTIVE] max yf ≥0,∀ f∈F f∈F U(yf ). (6) Similar to [26], a feasible routing must satisfy two constraints: the interference and link capacity constraints (7) and the ﬂow balance constraints (8). [CONSTRAINTS] f∈F vlf ∈ Γ ∀ l ∈ L (7) yif + Υ− if ≤ Υ+ if ∀ i ∈ N, f ∈ F (8) where yif (i ∈ N, f ∈ F) denotes the node-ﬂow variable such that yif = yf when i = s(f), and otherwise yif = 0. Υ− if = l∈L:r(l)=i vlf and denotes the fraction of ﬂow f incoming into node i, and Υ+ if = l ∈L:t(l )=i vl f denotes the fraction of ﬂow f outgoing of node i.

5. 3112 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 7, JULY 2015 IV. DISTRIBUTED CROSS-LAYER DESIGN WITH SUCESSIVE INTERFERENCE CANCELLATION A. Dual Decomposition Similar to [25] and [26], we resort to the dual driven Lagrangian decomposition approach to get a distributed so- lution. For completeness, we brieﬂy describe the process of decomposition, and the detailed demonstration can be found in [25]. Consider the dual problem of primal problem (6), i.e., min uif ≥0,∀ i∈N,f∈F D(u) (9) with partial dual function D(u) = max yf ≥0,v∈Γ × ⎧ ⎨ ⎩ f∈F U(yf ) − f∈F i∈N uif yif + Υ− if − Υ+ if ⎫ ⎬ ⎭ (10) where uif is a Lagrangian multiplier, and u = [uif ]i∈N,f∈F . Then, (10) can be further decomposed into the following two subproblems: D1(u) = max yif ≥0 f∈F i∈N (U(yif ) − yif uif ) (11) D2(u) = max v∈Γ f∈F i∈N uif Υ− if − Υ+ if . (12) Here, D1(u) can be solved by (13), which is the standard rate control problem at the source node of each ﬂow, i.e., yf = U −1 (uf ) (13) where uf = uif if node i = s(f). For the routing/scheduling problem D2(u), we have the following identity: D2(u) = max v∈Γ l∈L vlf∗ max f∈F ut(l)f − ur(l)f (14) where f∗ = arg maxf∈F {ut(l)f − ur(l)f }, l ∈ L. Based on (14), the routing/scheduling problem can be solved by the following two-step process. Step 1) For each link l, we can use local informa- tion u to ﬁnd a ﬂow f∗ that satisﬁes f∗ = arg maxf∈F {ut(l)f − ur(l)f }. Let wl = ut(l)f∗ − ur(l)f∗ be the weight of link l. Here, wl can also be interpreted as the scaled queue length at link l with ﬂow f∗ . In each time slot, the links in one conf can be active to send data to the receivers (we assume one packet transmission per active link per time slot). The given algorithm can be interpreted as a back pressure process to solve the routing problem. Step 2) We convert (14) into its reduced form as follows: D2(u) = max v∈Γ l∈L vlf∗ wl. (15) The formulation of (15) can be seen as an ordinary link scheduling problem where each link is associated with its weight wl. However, it is difﬁcult to solve the scheduling prob- lem because the interference relationship under the physical interference model (SINR) is nonconvex and combinatorial. In the sequel, we present a simpliﬁed distributed link scheduling method taking into account the SIC constraints and using the interference localization technique presented earlier. We ﬁrst propose a method to calculate the vector −→ λ under a certain , and then, a distributed scheduling method is proposed under SIC constraints. B. Identifying the Interference Neighborhood Here, we present a binary-search-based method to determine vector −→ λ under a certain value of . Recall that, according to Deﬁnition 1, λ(l) is the lower bound of the radius of the interference neighborhood INl, which guarantees that, under a feasible scheduling method (see Deﬁnition 2), the total interfer- ence coming from links in the noninterference region (nINl) does not exceed Imax . For a link l, denote dmax (l) as the distance from the farthest node in the network to the receiver of link l. The search procedure for link l starts from dmax (l), and we use a bisection search method to reduce the gap between the current search radius and the optimal λ(l). At every level of current radius, we have to determine the maximum inter- ference coming from the active links whose transmitters are located outside the current radius. One simple way to decide the maximum interference is to sum up the received signals from all transmitters outside the current radius; it should be noted that since some of these transmitters may not be active in our conﬁguration, this calculated maximum interference represents an upper bound. An alternative approach is to solve a simple SIC-based scheduling problem on the links whose transmitters are located outside the current radius. We repeat this procedure (of updating the radius of interference neighborhood) for link l until a tolerable performance is attained. This procedure is illustrated in Algorithm 1. Note that Algorithm 1 is a central- ized procedure that needs to be performed only once for a static wireless network. All links in the area outside the current radius are initially stored in a link set Ψ. For current link l, we associate weight attribute wal(l ) = PwGt(l ),r(l), ∀ l ∈ Ψ and 0 otherwise. We also deﬁne a binary variable pi(i ∈ L), which is equal to 1 when link i is active, and otherwise, it is zero. Then, we deﬁne our optimization objective as Maximize : i∈L wal(i)pi. (16) Similarly, we deﬁne another binary variable qt(j)(j ∈ L), which is equal to 1 when node t(j) is an active transmitter, and otherwise, it is zero. Let L+ n be the set of links whose transmitter is node n and L− n be the set of links whose receiver is node n. Therefore, we have qt(j) = i∈L+ t(j) pi (17) i∈L+ n pi ≤ 1. (18)

6. QU et al.: WIRELESS NETWORKS WITH INTERFERENCE CANCELATION CAPABILITIES 3113 In this paper, we only consider the half-duplex mode at each node. Then, our half-duplex constraints can be written as pi + pj ≤ 1 ∀ n ∈ N : i ∈ L− n, j ∈ L+ n . (19) Similar to [18], we use the concept of residual-SINR or r-SINR for short. Due to its interference cancelation capability, a receiver node can sequentially cancel all interfering signals, which are stronger than the one of interest; therefore, one only needs to consider the interference from senders whose signals are weaker than that of the intended one. The r-SINR can be deﬁned as r_SINRt(i),t(i) = PwGt(i),r(i) Gt(k),r(i)≤Gt(i),r(i) k=i,t(i)=t(k) PwGt(k),r(i)qk + η . (20) Indeed, when scheduling variable vp i = 1 (link i is active), this implies that all other stronger received signals from adjacent senders at r(i) have been correctly decoded, and the decoding of the signal of interest at link i is also successful. Namely, if vp i = 1, then the following two constraints should be satisﬁed: r_SINRt(i),r(i) ≥ β, (pi = 1, i ∈ L) (21) r_SINRt(j),r(i) ≥ β, j =i, t(j)=t(i), Gt(j),t(i) ≥Gt(i),r(i), pj = 1, j, i ∈ L) . (22) To convert the SIC constraints into an LP format, we deﬁne a bi- nary variable ρi,t(j) to describe the relationship of pi and qt(j). Let ρi,t(j) = 1 if and only if pi = 1 and qt(j) = 1; otherwise, it is zero. Then, the relationship can be written as follows: pi ≥ ρi,t(j) (23) qt(j) ≥ ρi,t(j) (24) ρi,t(j) ≥ pi + qt(j) − 1. (25) Now, we can use mathematical programming to describe constraints (21) and (22) as PwGt(j),r(i) − Gt(j),r(i)≥Gt(k),r(i) t(k)=t(j) βPwGt(k),r(i)qt(k) − βη ≥ 1 − ρi,t(j) Mi,t(j) Mi,t(j) = PwGt(j),r(i) − Gt(j),r(i)≥Gt(k),r(i) t(k)=t(j) βPwGt(k),r(i) − βη (26) PwGt(i),r(i) − Gt(i),r(i)≥Gt(k),r(i) t(k)=t(i) βPwGt(k),r(i)qk − βη ≥ (1 − pi)Hi Hi =PwGt(i),r(i) − Gt(i),r(i)≥Gt(k),r(i) t(k)=t(i) βPwGt(k),r(i)−βη. (27) Algorithm 1: Determination of Interference Neighbourhood 1 Initialize (0 < < 1); 2 Initialize itrCut; 3 for l : l ∈ L do 4 Initialize curr.decision = 1; 5 Initialize curr.radius = dmax (l); 6 Initialize success.radius = dmax (l); 7 Initialize curr.decision = 1; 8 for i = 1 to itrCut do 9 curr.interval = (dmax (l) − dt(l),r(l))/2i ; 10 curr.radius = curr.radius + (1 − curr. decision) ∗ curr.interval − curr.decision ∗ curr.radius; 11 Generate wal based on curr.radius; 12 Solve optimal objective (16) under constraints (17)–(19), (23)–(27); 13 if (16) > Imax then 14 curr.decision = 0; 15 else 16 success.radius = curr.radius; 17 curr.decision = 1; 18 end 19 λ(l) = success.radius; 20 end 21 end C. Distributed Scheduling Algorithm With SIC Based on Algorithm 1, we can calculate −→ λ under a certain value of . Let Δ1 l be the set of all links k such that t(l) is located in Ak, Δ2 l be the set of all links k such that t(l) is located in INk − Ak, and Δl = Δ1 l Δ2 l . At the beginning of each scheduling period, each link l broadcasts its weight information (wl) to links in Δl and ΔIN l [the set of links whose transmitters are located in the interference neighborhood of link l (i.e., in INl)]. We further divide ΔIN l into two link sets: ΔIN1 l and ΔIN2 l . The former denotes the set of links k such that t(k) is located in Al, and the latter denotes the set of links k such that t(k ) is located in INl − Al. The weight of link l is computed as illustrated in Section IV-A. We assume that each link l(l ∈ L) maintains two local link sets, i.e., the current active link set (currsl for short) and a candidate link set (cansl for short). The former contains links that have been added into a feasible conﬁguration conf in a particular scheduling period, and the latter contains links that are candidate links to be added into conf. At the beginning of each scheduling period, we initialize currsl = ∅ and cansl = {l, Δl ΔIN l }. Each scheduling period consists of several intervals, and in each interval, each link l makes a decision as to whether it should be added into currsl or removed from cansl. Therefore, the purpose of our scheduling method is to generate a new con- ﬁguration that satisﬁes the SIC constraints under the physical interference model such that the sum of the weights of the links in this conﬁguration is the largest possible. Our scheduling method follows the classical GMS method but is implemented in a distributed manner.

7. 3114 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 7, JULY 2015 To make sure that the new conﬁguration, which has been generated, satisﬁes the SIC constraints under the physical in- terference model, each link will execute two main procedures as follows. At the beginning of each scheduling interval, each link l (l ∈ cansl) compares its weight to the weights of links in cansl. If link l has the largest weight among all links in cansl, it will run Algorithm 2 to try to add itself into currsl. The detailed process is illustrated next. Link l ﬁrst broadcasts a REQ message to links in Δl. Any link in Δl will add link l into a local auxiliary link set (auxs for short); now, any link l ∈ {currsl auxsl} will determine whether it remains feasible under SIC constraints upon adding link l to the current schedule as follows. For each link l ∈ {currsl auxsl}, we deﬁne the SIC link set (sicsl for short) as the set of links whose trans- mitters are in ΔIN1 l {currsl auxsl } and receivers are r(l ). According to Theorem 1, any link l ∈ {currsl auxsl} satisﬁes SIC constraints if 1) the total interference (I1) coming from ΔIN2 l is ≤ (1 − )Imax (l ) and 2) all links k ∈ sicsl are feasible (i.e., total interference (I2) coming from ΔIN2 k is ≤ (1 − )Imax (k)). According to the given procedure, if any link l ∈ {currsl auxsl} does not satisfy the SIC constraints, then link l will send an ERROR message to link l indicating that link l cannot be added to the conﬁguration (that is, link l is causing strong interference making the current schedule not feasible). If link l ∈ auxsl does not receive any ERROR message from its neighbors, it adds itself into currsl, removes itself from cansl, auxsl, and broadcasts a SUCCESS message to all its neighbors to update their local link sets (cans, currs, and auxs). Otherwise, it removes itself from cansl, auxsl and broadcasts a REMOVE message to all its neighbors to update their local link sets (cans, auxs). The given process enforces that when adding a new link to a feasible conﬁguration, the current schedule remains feasible under SIC constraints. The main purpose of our second procedure is to remove links in cans (e.g., link l ∈ cansl), which have no chance of being added into currs. After the new conf is generated at each interval, all links l (l ∈ cansl) need to make a decision as to whether they satisfy SIC constraints as follows. For each link l ∈ cansl, we deﬁne another SIC link set (sicsl for short) as the set of links whose transmitters are in ΔIN1 l currsl and receiver is r(l). Similar to the process in the ﬁrst procedure, any link l ∈ cansl does not satisfy SIC constraints if 1) the total interference (I1) coming from ΔIN2 l is > (1 − )Imax (l) or 2) for any link k ∈ sicsl is infeasible (i.e., total interference (I2) coming from ΔIN2 k is > (1 − )Imax (k )). After the given process, if there is a link l in cansl that does not satisfy SIC constraints, then link l will remove itself from cansl and broadcast a REMOVE message to all its neighbors to update their local link sets (cans, currs). The given process makes sure that each link in cans satisﬁes SIC constraints with the current schedule. In our distributed implementation, we set the number of inter- vals per scheduling period to a ﬁxed value. In each scheduling interval, each link will run Algorithms 2 and 3 to generate new feasible schedule/conﬁguration during the scheduling period. Once a schedule is obtained, links that have been selected will transmit in the transmission period one packet each. Algorithm 2: Distributed Scheduling Method With SIC (Link l) 1 Link l broadcast REQ message to all links in Δl; 2 for link l in Δl {currsl auxsl} do 3 if link l received REQ message from link l then 4 Link l adds link l into auxsk; 5 Link l calculates cumulative interference I1; 6 if I1 > (1 − )Imax (l )) then 7 Link l broadcasts ERROR message to link l; 8 else 9 Generate sicsl ; 10 for k : k ∈ sicsl do 11 Link k temporarily calculates cumulative interference I2; 12 if I2 > (1 − )Imax (k) then 13 Link l broadcasts ERROR message to link l; 14 end 15 end 16 end 17 end 18 end 19 if Link l receives no ERROR messages then 20 currsl = currsl l; 21 cansl = cansl − l; 22 auxsl = auxsl − l; 23 link l broadcasts a SUCCESS message to all its neigh- bors to update their local link sets(cans, currs, auxs); 24 Goto Algorithm 3; 25 else 26 cansl = cansl − l; 27 link l broadcasts a REMOVE message to all its neigh- bors to update their local link sets(cans, auxs); 28 end Algorithm 3: Distributed Scheduling Method With SIC (Part II) 1 for link k in Δl cansl do 2 if link k received a SUCCESS message from link l then 3 Link k calculates cumulative interference I1; 4 if I1 > (1 − )Imax (k)) then 5 Link k broadcasts a REMOVE message to its neigh- bors to update their local link sets(cans); 6 else 7 Generate sicsk; 8 for i : i ∈ sicsk do 9 Link i calculates cumulative interference I2; 10 if I2 > (1 − )Imax (i) then 11 Link k broadcasts a REMOVE message to its neighbors to update their local link sets(cans); 12 end 13 end 14 end 15 end 16 end

8. QU et al.: WIRELESS NETWORKS WITH INTERFERENCE CANCELATION CAPABILITIES 3115 D. Joint Transport, Routing, and Scheduling With SIC Consider the dual problem (9) and suppose that the function D(u) is not necessarily differentiable. Therefore, (9) can be solved using the subgradient method. Now, it is easy to verify that v+ if (u) − yif (u) + v− if (u) (28) is a subgradient of D(u) at point u. Thus, based on the subgradi- ent method, the update algorithm can be formulated as follows: uif (t+1)= uif (t)+α v+ if (u(t))− yif (u(t))+v− if (u(t)) + (29) where α is a positive step size. Equation (29) achieves optimal- ity when α is set to a sufﬁciently small value [25]. The given dual algorithm (presented in Section IV-A) solves the cross- layer problem through a distributed manner where at the trans- port layer, nodes in the network individually update their prices according to (29) and the source of each ﬂow f individually adjusts its rate (y) according to the local congestion price (u); for solving the routing/scheduling subproblem, we generate a new conf at each time slot by using Algorithms 2 and 3, which work in a distributed manner. We summarize our joint conges- tion control, routing, and scheduling with SIC in Algorithm 4. Algorithm 4: Distributed Cross-Layer Design With SIC 1 Initialize max iteration count as itrmax; 2 Initialize y, v, u; 3 for i = 1 to itrmax do 4 for n ∈ N do 5 node n updates u by calculating (29); 6 end 7 for f ∈ F do 8 Source node of ﬂow f updates y by calculating (13); 9 end 10 for l ∈ L do 11 link l updates wl; 12 end 13 Generate a feasible conf by Algorithms 2 and 3; 14 Data Transmission based on currently obtained schedule; 15 end E. Complexity Analysis The whole procedure includes a centralized process for identifying the interference neighborhood (see Algorithm 1) and a link scheduling process under SIC, which is done in a distributed manner at each iteration (see Algorithms 2 and 3). To analyze the complexity of the ﬁrst part, we convert the inverse LP (ILP) problem into a complete binary tree for the worst case of solving the optimal objective (16). A route that starts from the root of the tree to the leaf is a feasible solution (or schedule). Based on backtracking, it is clear that the time Fig. 1. Eight-node network topology. complexity of a tree traversal search is O(2n ), where n is the number of links in wal. However, in most cases, there is no need for a complete traversal search, owing to the SINR constraints between links. In practice, we can prune some invalid branches (i.e., the branch-and-cut method [34]) to improve the efﬁciency of the search. For the run time complexity of the distributed scheduling with SIC, we omit the communication overload during the scheduling interval and only focus on the worst case compu- tation analysis. As shown in Algorithms 2 and 3, it is easy to verify that the time complexity of links in cansl is O(n) + O(n2 ) = O(n2 ) (i.e., the weight comparison and the order of SIC decoding), where n is the number of feasible neighbors (i.e., active links) for link l. Given that there is no need to com- pare the weight information in links in currsl auxsl sicsl, their run time complexity is O(n2 ). For each link that is not in cansl currsl auxsl sicsl, it keeps silent and therefore no calculation. Hence, the worst-case run time at each link is O(n2 ), which is polynomial. V. NUMERICAL RESULTS Here, we present numerical results to study the performance of the cross-layer design method for solving the problem of joint transport, routing, and scheduling (JTRS) in wireless networks with interference cancelation. We are particularly in- terested in studying the efﬁciency of the distributed scheduling (JTRDS) method with SIC, and we present comparisons with centralized scheduling methods [JTRCS; both Pick & Com- pare (P&C) and GMS]. We also present comparisons of our cross-layer design method with and without SIC to assess the beneﬁts of interference cancelation (JTRDS-SIC and JTRDS, respectively). We use a CPLEX solver to solve the ILP problem in Algorithm 1 to determine the radius of the interference neighborhood for each link. For our evaluation, we consider two random networks (Network 1 and Network 2), with eight nodes (48 links) and 15 nodes (124 links), each randomly distributed in a square region of 100 m × 100 m. The topologies of the networks are shown in Figs. 1 and 2. Under the physical interference model, the transmission power of each node is set to Pw = 0.001 W. We assume the path-loss index pl = 4; the background noise η is set to 10−10 W; the SINR threshold for a successful transmission is β = 1, = 0.05 (unless otherwise speciﬁed); and the transmission capacity of each link is c = 1 (packet/time slot). We assume that there are two ﬂows and four ﬂows in Network 1 and Network 2, respectively.

9. 3116 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 7, JULY 2015 Fig. 2. Fifteen-node network topology. Fig. 3. Utility achieved by JTRDS-SIC ( = 0.05) and JTRCS-SIC (P&C). We start by evaluating the performance beneﬁts of JTRDS- SIC in Network 1. There are two ﬂows in the network (Flow 1: node 1 → node 8; Flow 2: node 3 → node 7). We compare our distributed method with P&C, which is a centralized scheduling method and is shown to achieve 100% throughput [19], [20]. In the P&C scheduling method, we randomly generate a maximal schedule under SIC constraints at each time slot (by randomly selecting links to be included in the schedule as long as they satisfy the interference constraints) and compare the current schedule with the schedule generated in the previous time slot; the schedule with larger weight (sum of the link weights) is always selected for data transmission at each time slot. In our distributed method, we set = 0.05. Figs. 3 and 4 show the utility and the congestion price of both methods. Clearly, the ﬁgures show that our distributed method quickly converges to the optimal solution and oscillates around it; however, the centralized P&C has slower convergence time, which is due to the random selection of transmission links to be included in the schedule. To better understand the obtained results, we look at how these two methods route the two ﬂows and the achievable individual ﬂow rate; the results are shown in Tables I and II. We observe that both methods select different routes for the ﬂows and that ﬂow 1 achieves an optimal rate of 0.8519 (using the centralized P&C scheduling method), whereas ﬂow 1 achieves Fig. 4. Congestion prices of JTRDS-SIC ( = 0.05) and JTRCS-SIC (P&C). TABLE I AVERAGE RATES OF FLOWS THROUGH DIFFERENT LINKS WITH JTRCS-SIC (P&C) TABLE II AVERAGE RATES OF FLOWS THROUGH DIFFERENT LINKS WITH JTRDS-SIC TABLE III SOURCE NODE, DESTINATION NODE OF EACH FLOW IN THE NETWORK a ﬂow rate of 0.8332 using the proposed distributed scheduling method and a gap of 2% between the two methods. Flow 2, however, achieves the same ﬂow rate of 1 in both methods. It is to be noted that GMS achieves exactly similar results to our method (results are omitted in the ﬁgures). Next, we consider the larger network (Network 2) with four ﬂows (Flow1–Flow4: Table III). We start by studying the effect of the parameter on the achievable ﬂow rate. We numerically solve our JTRDS-SIC in this 15-node network, and the obtained results are shown in Fig. 5. We observe that when is smaller,

10. QU et al.: WIRELESS NETWORKS WITH INTERFERENCE CANCELATION CAPABILITIES 3117 Fig. 5. Flow rates (under JTRDS-SIC) versus . Fig. 6. Average number of active links per schedule versus . the achievable ﬂow rates are higher, and as increases, the rate decreases. Clearly, when is small, the interference neighbor- hood of a link gets larger (see Fig. 7); therefore, a link l will be able to coordinate its scheduling/transmission with more links in its interference neighborhood, i.e., INl. This implies that, ultimately, each selected schedule may contain, on average, more active links (see Fig. 6), which, in turn, implies better spectrum spatial reuse in the network. However, it should be noted that a larger interference neighborhood may result in higher scheduling overhead to coordinate the selection of the schedule. Now, conversely, a larger implies a smaller inter- ference neighborhood, and as a result, most of the links in the network will be located outside the neighborhood of a particular link, preventing any effective coordination in the selection of the schedule and resulting in lower attainable ﬂow rates. The ﬂow rate continues to decrease until it reaches a mini- mum value (at = 0.7) beyond which it starts to increase. This can be explained as follows. When = 0.7, as we mentioned earlier, the radius of the interference neighborhood is small (see Fig. 7), and thus, fewer links may exist within the (INl − Al) area. Fig. 8 indicates that almost 0 links within that area may be active. However, according to the protocol, the value of the tolerable interference assigned to links within that area Fig. 7. Average radius of neighborhood versus . Fig. 8. Average number of active links in different areas versus . is set to (1 − )Imax ; given that almost no links are active within that area (see Fig. 8), this tolerable interference value is wasted and would have been better off allocated to links outside this interference neighborhood (i.e., nINl), where the other transmission links are located. As increases, the value of (1 − )Imax decreases, and more tolerable interference (i.e., Imax ) is allocated to those links outside the interference neighborhood of link l, and such links will attempt to schedule their transmissions concurrently with link l. Fig. 8 shows that as increases, more links outside the neighborhood are active in the schedule and none of the links within the interference neighborhood are. This explains the behavior of the trafﬁc ﬂow rates shown in Fig. 5 where beyond = 0.7, the rates start to increase. Fig. 6 shows the average number of active links per schedule, and as previously explained, smaller indicates better coordination to construct the schedule and, therefore, more active links per schedule, hence better spatial reuse, and larger yields lower spatial reuse. Finally, we study the beneﬁts of SIC by comparing the performance of JTRDS-SIC with JTRDS, where in the latter, nodes do not have any SIC capabilities. The results (individual ﬂow rates) are shown in Fig. 9, and we use a value of = 0.3. The selection of = 0.3 is motivated by Fig. 7 where we show

11. 3118 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 7, JULY 2015 Fig. 9. Achievable ﬂow rates (JTRDS-SIC versus JTRS ( = 0.3)). that both methods have close average radius for the interference neighborhood. Fig. 9 shows that ﬂows achieve much higher rates in a network with SIC capabilities (almost twice the rate is achieved by most of the ﬂows). To better understand, we observe from Fig. 6 that the JTRDS-SIC method (when = 0.3) has a much better schedule length than that of the JTRDS method; indeed, the schedule length (i.e., number of active links per selected schedule) of JTRDS-SIC is almost twice that of JTRDS. This shows that SIC is effectively managing the interference in the network and promoting transmission con- currence, leading to better achievable ﬂow rates in the network. VI. CONCLUSION In this paper, we have studied the beneﬁts of SIC in improv- ing the performance of wireless networks. We considered solv- ing an NUM problem in the context of cross-layer optimization of the joint congestion control, routing, and scheduling problem under the SINR interference model. Through dual decompo- sition, we divided our design problem into a congestion con- trol subproblem and a routing/scheduling subproblem. Given the complexity of the scheduling subproblem, we presented a decentralized method for solving the link scheduling prob- lem. Our decentralized method beneﬁts from the interference localization concept to help neighboring links coordinate their transmissions, taking into account SIC constraints, and without causing sufﬁcient interference that may corrupt ongoing sched- uled transmissions. Our approach to solving the joint design problem is completely decentralized. We have numerically solved the cross-layer optimization, and we have shown that our distributed resource allocation method achieves very close results to centralized methods (e.g., our achieved results are below 2% from the centralized P&C scheduling method, which achieves 100% throughput performance, and we obtain similar results to the centralized GMS). We also studied the beneﬁts of SIC, and we have shown that the ﬂows in the network may achieve up to twice the achievable rates in a network with- out SIC. We have shown that networks with SIC capabilities promote better transmission concurrence and, therefore, better spectrum reuse. REFERENCES [1] T.-S. Kim, H. Lim, and J. C. Hou, “Understanding and improving the spa- tial reuse in multihop wireless networks,” IEEE Trans. Mobile Comput., vol. 7, no. 10, pp. 1200–1212, Oct. 2008. [2] G. Sharma, R. R. Mazumdar, and N. B. Shroff, “On the complexity of scheduling in wireless networks,” in Proc. 12th Annu. Int. Conf. Mobile Comput. 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12. QU et al.: WIRELESS NETWORKS WITH INTERFERENCE CANCELATION CAPABILITIES 3119 [25] L. Chen, S. H. Low, M. Chiang, and J. C. Doyle, “Cross-layer congestion control, routing and scheduling design in ad hoc wireless networks,” in Proc. IEEE INFOCOM, 2006, pp. 1–13. [26] Z. Feng and Y. Yang, “Joint transport, routing and spectrum sharing optimization for wireless networks with frequency-agile radios,” in Proc. IEEE INFOCOM, 2009, pp. 1665–1673. [27] Z. Shao, M. Chen, S. Avestimehr, and S. R. Li, “Cross-layer optimization for wireless networks with deterministic channel models,” IEEE Trans. Inf. Theory, vol. 57, no. 9, pp. 5840–5862, Sep. 2011. [28] M. Chiang, S. H. Low, A. R. Calderbank, and J. C. Doyle, “Layering as optimization decomposition: A mathematical theory of network architec- tures,” Proc. IEEE, vol. 95, no. 1, pp. 255–312, Jan. 2007. [29] M. F. Uddin, H. M. K. AlAzemi, and C. Assi, “Optimal ﬂexible spectrum access in wireless networks with software deﬁned radios,” IEEE Trans. Wireless Commun., vol. 10, no. 1, pp. 314–324, Jan. 2011. [30] M. F. Uddin and C. Assi, “Joint routing and scheduling in WMNs with variable-width spectrum allocation,” IEEE Trans. Mobile Comput., vol. 12, no. 11, pp. 2178–2192, Nov. 2013. [31] C. Joo and N. B. Shroff, “Local greedy approximation for scheduling in multihop wireless networks,” IEEE Trans. Mobile Comput., vol. 11, no. 3, pp. 414–426, Mar. 2012. [32] C. Joo, X. Lin, J. Ryu, and N. B. Shroff, “Distributed greedy approxi- mation to maximum weighted independent set for scheduling with fading channels,” in Proc. MobiHoc, 2013, pp. 89–98. [33] M. Dinitz, “Distributed algorithms for approximating wireless network capacity,” in Proc. IEEE INFOCOM, 2010, pp. 1–9. [34] J. E. Mitchell, “Branch-and-cut algorithms for combinatorial optimiza- tion problems,” in Handbook of Applied Operations Research. London, U.K.: Oxford Univ. Press, 2000. Long Qu received the B.S. degree in communica- tion engineering from Zhengzhou University, Henan, China, in 2010. He is currently working toward the Ph.D. degree in communication and information systems with Ningbo University, Zhejiang, China. From December 2012 to December 2013, he was a visiting Ph.D. student with Concordia University, Montreal, QC, Canada. His current research interests include cross-layer design in wireless communica- tion systems and wireless network optimization. Jiaming He received the M.S. and Ph.D. degrees from Zhejiang University, Hangzhou, China, in 1993 and 1996, respectively. He is currently a Professor with Ningbo Univer- sity, Zhejiang, China. His research interests include broadband wireless communication systems. Chadi Assi (SM’08) received the B.Eng. degree from the Lebanese University, Beirut, Lebanon, in 1997 and the Ph.D. degree from the City University of New York (CUNY), New York, NY, USA, in 2003. He is currently a Full Professor with the Concor- dia Institute for Information Systems Engineering, Concordia University, Montreal, QC, Canada. Be- fore joining Concordia University, he was a Visiting Researcher with Nokia Research Center, Boston, MA, USA, where he worked on quality of service in passive optical access networks. His main research interests include networks and network design and optimization. His current research interests include network design and optimization, network modeling, and network reliability. Dr. Assi is on the Editorial Board of the IEEE COMMUNICATIONS SUR- VEYS AND TUTORIALS, IEEE TRANSACTIONS ON COMMUNICATIONS, and IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He was a recipient of the prestigious Mina Rees Dissertation Award from CUNY in August 2002 for his research on wavelength-division multiplexing optical networks.

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