Distributed Link Scheduling under SINR Model in Multi-hop Wireless Networks

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1 1 Distributed Link Scheduling under SINR Model in Multi-hop Wireless Networks Jin-Ghoo Choi, Member, IEEE, Changhee Joo, Senior Member, IEEE, Junshan Zhang, Fellow, IEEE, and Ness B. Shroff Fellow, IEEE Abstract Link adaptation technologies, such as Adaptie Modulation and Coding (AMC) and Multiple Input Multiple Output (MIMO), are used in adanced wireless communication systems to achiee high spectrum efficiency. Communication performance can be improed significantly by adaptie transmissions based on the quality of receied signals, i.e., the signalto-interference-plus-noise ratio (SINR). Howeer, for multi-hop wireless communications, most link scheduling schemes hae been deeloped under simplified interference models that do not account for accumulatie interference, and cannot fully exploit the recent adances in PHY-layer communication theory. This paper focuses on deeloping link scheduling schemes that can achiee optimal performance under the SINR model. One key idea is to treat an adaptie wireless link as multiple parallel irtual links with different signal quality, building on which we deelop throughput-optimal scheduling schemes using a twostage queueing structure in conjunction with recently deeloped carrier-sensing techniques. Furthermore, we introduce a noel three-way handshake to ensure, in a distributed manner, that all transmitting links satisfy their SINR requirements. We ealuate the proposed schemes through rigorous analysis and simulations. Index Terms Multi-hop wireless networks, link scheduling, CSMA, SINR model. I. INTRODUCTION MULTI-HOP wireless networks hae recently attracted significant attention due to their potential to achiee substantial gains oer their single hop counterparts [1], [2]. Oer the past few years, a cross-layer optimization-based framework has emerged that can be used to maximize the network utility (see [3] and the references therein). The solution to such utility maximization problems inoles soling congestion control, link scheduling, androuting components. The link scheduler determines which link has to be actie at what time and at what power leel. The congestion control adjusts the amount of data injected into the network based on J. Choi is with Yeungnam Uniersity, Gyeongbuk , South Korea ( jchoi@yu.ac.kr). C. Joo is with UNIST, Ulsan , South Korea (corresponding author, cjoo@unist.ac.kr). J. Zhang is with Arizona State Uniersity, Tempe, AZ USA ( junshan.zhang@asu.edu). N. B. Shroff is with the Ohio State Uniersity, Columbus, OH USA ( shroff@ece.osu.edu). Manuscript receied XXXX; reised XXX. This work was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A and NRF ), and in part by the US National Science Foundation under Grants CNS , CNS , CNS , and CNS , and DoD MURI project No. FA and a MURI grant from the Army Research Office W911NF congestion information, which can be obtained by feedback from the receier. The routing determines which path(s) should be taken by packets on a gien flow (or source-destination pair). It has been shown that the queue length information plays an important role in the interactions between these components. Moreoer, under this cross-layer optimization based framework, the scheduling component often turns out to hae the highest complexity, and has thus been the focus of attention. In their seminal work [4], Tassiulas and Ephremides studied joint routing and link scheduling for packet radio networks,assuming no a priori information of input traffic, and deeloped a throughput-optimal solution, so called the back-pressure algorithm. The back-pressure idea has since been applied to arious fields such as input-queued switch [5] and cellular data networks [6]. Also, it has been reformulated as a dynamic power control scheme under time-arying channels [7], and shown to maximize the network utility as a scheduling component [3], [8]. Back-pressure, although throughput-optimal, is a centralized algorithm with high computational complexity, and can be shown to be an NP-hard problem for most interference models [9], [10]. In general, centralized schemes would not work well for multi-node wireless networks due to high complexity for exchanging control messages, and the lack of scalability. Thus, researchers hae expended significant effort in deeloping distributed counterpart of high-performance centralized back-pressure schemes, e.g., [11] [17]. To this end, many studies employ simple combinatorial models for wireless interference, where the interference relationship between wireless links is binary. Notably, the k-hop interference model, under which two links cannot transmit simultaneously if their distance is within k hops [9], has been extensiely studied. The so-called maximal scheduling 1 scheme has been shown to attain a certain fraction of the optimal throughput for certain combinatorial channels [12], [15], and seeral proably efficient distributed random access schemes hae been deeloped in [13], [14]. Recently, it has been shown that throughput optimality can be achieed using traditional Carrier Sense Multiple Access with Collision Aoidance (CSMA/CA) protocols in a new way. In [18], it had been shown that the stationary distribution of link actiations can be deried in a product form. This result has been recently exploited in [19] to deelop fully distributed scheduling schemes to achiee throughput optimality under 1 Also referred to as maximal matching

2 2 combinatorial channel models. The main idea has been generalized to more practical scenarios in a time-slotted system [20], and to utility maximization for multi-channel systems [21]. The combinatorial channel models greatly simplify the interference relationship and make the problem relatiely tractable. Howeer, they do not fully capture some fundamental features of wireless channels. Indeed, the link rate is usually a function of the receied SINR that hinges heaily on the accumulatie interference from all other transmitters. Under simple combinatorial models, this global property of interference is ignored and, thus, a transmitting link considered feasible under those channel models may experience strong interference in practice, and fail to delier data. It has been shown that scheduling schemes that guarantee a certain fraction of the optimal throughput under the k-hop interference model may attain zero-throughput in the worst case under the SINR-based model [22]. In deeloping CSMA scheduling under the SINR interference model, one main difficulty is to ensure the feasibility of a schedule in a distributed manner while maintaining the desired product-form stationary distribution of schedules, whereas this can be easily done under the combinatorial interference models [19] [21]. To this end, the authors in [23] hae deeloped aconseratiecombinatorialmodelthatconformsthesinr requirements, which results in performance degradation while requiring for each link to hae a priori knowledge of the set of links that it can coexist with. In this work, we deelop algorithms that proide feasibility test of a schedule under the SINR interference model, which is the key component of the CSMA scheduling to achiee throughput optimality without any priori knowledge on the network topology. There are also a few works that hae been deeloped non- CSMA scheduling under the SINR channel model. In [24], the greedy maximal scheduling has been modified to operate under SINR channels at the cost of performance degradation. It has been shown that the modified greedy scheme achiees non-diminishing performance. Power control for utility maximization in saturated networks has been deeloped in [25], which, howeer, does not take into account traffic dynamics and has substantial oerhead of control message exchanges since each link requires global channel information. In this paper, we deelop resource allocation schemes in multi-node 2 wireless networks that not only achiee the optimal throughput under SINR channel models, but are also amenable to distributed implementation. The main contributions of our paper are summarized as follows. We model a single physical link as multiple irtual links with different SINR requirements, and are able to formulate the resource allocation as a standard optimization problem. By introducing a two-stage queueing structure, we successfully decompose the oerall problem into two subproblems. By studying the duality problem, we deelop resource allocation schemes that consist of two components: traffic 2 Here, by multi-node wireless systems, we mean wireless systems, where multiple nodes can transmit to multiple destination in a single or multi-hop fashion. Fig. 1. Transmission rate of link l s. SINR. splitter and link scheduler. We proide an optimal traffic splitter, and characterize the property of an optimal link scheduler. We deelop practical solutions by using CSMA techniques that are amenable to distributed implementation. We introduce a noel three-way handshake to satisfy the SINR requirements of each actie link in a distributed manner. Finally, we show through rigorous analysis that the proposed schemes follows the same Marko chain eolution as the optimal solution, and proide simulations to support our ealuations. The rest of this paper is organized as follows. In Section II, we describe the system and channel models. In Section III, we deelop a fully distributed throughput optimal resource allocation scheme using channel sensing. We ealuate our proposed scheme through simulations and make some interesting obserations in Section IV. We conclude our paper in Section V. A. System model II. MODEL DESCRIPTION We model a wireless network by a graph G = (N,L), where N is the set of nodes and L is the set of directed links. We assume the reciprocity of wireless channels for any pair of nodes a and b, i.e.,ifnodea is connected to node b by a link, there always exists a link in the opposite direction from node b to node a. Asessionisastreamofdatainthenetwork from a source node to a destination node. We assume that all sessions traerse one hop only, and there is at most one session per link. This one-hop model is only for ease of exposition and can be extended to multi-hop sessions following the approach in [19]. The system operates in a time-slotted and synchronized fashion. At the beginning of time slot t, databitsareinjected into link l at its source, and wait in the queue until they are sered in a first-come first-sered manner. When the link is actiated, backlogged bits are transmitted and leae the network (since they hae reached the destination). The amount of sered bits is determined by the transmission rate of link l, dependingonthesinr.leta l (t) denote the amount of

3 3 data bits injected into link l in time slot t. Weassumethatthe arrial process {A l (t)} t=0 is stationary and i.i.d. across time and, further, the first and second moments of A l (t) are finite. The aerage arrial rate of data bits at link l is denoted by 1 T 1 λ l := lim T T τ =0 A l(τ). µ l1 r l1 r l2 B. Channel model Let P l denote the transmission power of (the transmitting node of) link l. Weassumethatthetransmissionpoweris fixed. Let G lm represent the channel gain from the transmitter of link l to the receiing node of link m, andletg ll denote the channel gain of link l itself. When the transmitter of link l sends a signal, the corresponding receier attains signal strength of P l G ll,andthereceierofanotherlinkm( l) experiences interference of P l G lm. Let γ l denote the receied SINR at link l. We assume that each link can support V different transmission rates based on the alue of γ l.toelaborate,wegroupthedomain of γ l into V +1 fixed interals using positie constants, Γ 1 < Γ 2 < < Γ V +1 (= ). Atthe-th interal for each {1, 2,...,V}, i.e.,whenγ l [Γ, Γ +1 ),linkl can achiee transmission rate of c l (> 0). Whenγ l [0, Γ 1 ),the transmission rate is 0. Iflinkl transmits data with higher rate transmission mode than c l,thentheprobabilitythereceier can decode the data correctly decreases to an unacceptably low leel. The -th transmission mode with rate c l may correspond to a different AMC scheme and/or a different MIMO antenna configuration [23]. Fig. 1 depicts an example of the rate cure of a link, which is an (unequal) staircase function due to the quantization of the SINR leel. Note that in our definition, the transmission rate monotonically increases, but does not strictly increase with the SINR leel, as shown for =2and 3 in the figure. This piecewise constant rate model is more realistic than continuous rate models such as the Shannon capacity formula, since practical communication systems support only a finite number of transmission modes. Moreoer, by increasing the number of modes V,ourmodel can closely approximate the Shannon capacity formula. We assume heterogeneous link rate cures, i.e., each link has its own cure. We represent each -th transmission mode of (physical) link l by a irtual link [l, ], asfirstintroducedin[23].virtuallink [l, ] is said to be actie when link l transmits data using the - th mode. Let x l {0, 1} represent the actiity of irtual link [l, ], i.e.,x l =1if irtual link [l, ] is actie, and x l =0 otherwise. We call x := (x l ) as the irtual link actiation ector. Foragienx, thereceiedsinrofirtuallink[l, ] can be calculated as x l P l G ll γ l (x) = m,w x, (1) mwp m G ml x l P l G ll + η l where η l is the background noise power at the receier of link l. Notethatforasinglelink,onlyoneirtuallinkcanbe actiated at any gien time, and the SINR of actie irtual link [l, ] is equialent to the SINR of physical link l at that time. Hence, if irtual link [l, ] is actie and satisfies the SINR requirement γ l Γ,thetransmissionrateoftheirtuallink is c l,andotherwise,0. Wedenotetheaerage transmission λ l µ lv µ l 2 Fig. 2. Queueing structure of link l. Thesolidarrowindicatesanexample processing path of data bits. 1 T 1 rate of irtual link [l, ] by r l := lim T T t=0 c lx l (t), where x l (t) represents the actiity of [l, ] at time slot t. C. Queueing structure Based on the fact that a physical link accommodates multiple irtual links, we deelop a two-stage queueing structure as shown in Fig. 2. A similar queueing structure has been used for scheduling with multi-channel wireless links [26].All incoming bits are put into the first-stage session queue, and then distributed to V irtual link queues at the second stage, each of which is associated with a irtual link. Data bits arrie at the session queue at rate λ l (bits/slot), and moe from the session queue to the -th irtual link queue at serice rate µ l. Since the data moement between queues are internal, it does not rely on the SINR of the channel. The data bits in the -th irtual link queue will be transmitted to the next node at rate c l when the associated irtual link is actie. This two-stage queueing structure may result in some undesirable effect: a packet should wait until a irtual queue is consumed, and it may experience long delay if the irtual queue that it moes to remains inactie for a long period. We can address the problem by implementing the queueing structure with a single FIFO queue and (1 + V ) counter ariables at each link. Each counter ariable maintains the queue length of the session queue and the irtual queues. When a packet arries at a link, the packet is physically enqueued into the FIFO queue and the counter ariable for the session queue increases. Wheneer a packet needs to moe from the session queue to a irtual link queue, the counter ariable is decreased accordingly for the session queue and the counter corresponding to the irtual link queue is increased by the same amount. Note that the physical packet does not moe and still wait for serice in the FIFO queue during this process. When a irtual link is scheduled to transmit, we transmit the head-of-line packet in the FIFO queue and decrease the counter for the irtual link queue. D. Capacity region We begin with specifying the feasible irtual link actiation ectors. For a irtual link actiation ector x, the associated link actiation ector is defined as ˆx := (ˆx l ),whereˆx l := r lv

4 4 x l, andlinkl is actie if x l =1,andinactieif x l =0.ThereceiedSINRoflinkl can be calculated as γ l (ˆx) = ˆx l P l G ll m ˆx mp m G ml ˆx l P l G ll + η l. (2) Based on the reciprocity of wireless channel, if link l connects node a to node b, therealwaysexistsareerselinkinthe opposite direction from node b to a. Forˆx, wedenoteits reerse link actiation ector as re(ˆx) := (re l (ˆx)), where re l (ˆx) =1if and only if link l is the reerse link of any actie link in ˆx, and0otherwise. Definition 1: Airtuallinkactiationectorx is feasible if 1) x l 1 for all links l, 2) γ l (x) Γ for all actie irtual links [l, ] of x, and 3) γ l (re(ˆx)) Γ ctrl for all actie links l in re(ˆx), where ˆx is the link actiation ector associated with x, andγ ctrl is the minimum SINR requirement for the reerse links. The first condition implies that at most one irtual link can be actiated in a physical link. The second condition requires that all the actie links hae a large enough SINR to support the transmission mode used. The third condition means that the reerse links should satisfy the minimum requirement on SINR, and is needed to guarantee that each receier can feedback control packets to the transmitter through the reerse link. Owing to the tiny size of control packets, small alue of Γ ctrl would be sufficient under the protection of strong channel coding schemes. Without loss of generality, we assume that Γ ctrl Γ 1. The second and third conditions imply that under any feasible actiation ector, control packets can be successfully deliered through both the forward and reerse links. Let F denote the collection of all the feasible irtual link actiation ectors in a network. We consider a link scheduling policy that chooses a irtual link actiation ector x among F and actiates the irtual links [l, ] with x l =1,satisfying the SINR requirements of links. Since the actie irtual link [l, ] has the transmission rate of c l,wedenotetheirtuallink transmission rate ector for x by c x, wherec := (c l ),using the component-wise multiply, i.e., c x =(c l x l ).Then,any achieable ector of aerage irtual link transmission rates r := (r l ) can be obtained by time-interleaing of feasible irtual link actiations, i.e., by a certain combination of c x oer F.Nowwedefinethecapacity region Λ as the set of arrial rate ectors λ := (λ l ) that is no greater than an achieable transmission rate ector component-wise, i.e., Λ := {λ λ l r l l, for some r Co(c x), x F }, (3) where Co( ) represents the conex hull. Clearly, if an arrial rate ector is not in the set Λ,itcannotbeaccommodatedinthe network with any scheduling policy. If a link scheduling policy supports any traffic that is strictly within the capacity region, the scheduler is said to be throughput optimal. Inthiswork, we are interested in deeloping throughput-optimal scheduling schemes that are amenable to distributed implementation. III. LINK SCHEDULING UNDER SINR MODEL We first formulate the problem formally and proide a throughput-optimal solution using the standard optimization techniques. For practical implementation, we deelop a distributed resource allocation scheme with the two-stage queueing structure and a noel three-way handshake, and show that the proposed distributed link scheduling scheme still achiees the optimal throughput. A. Problem formulation Recall that λ l is the arrial rate of data bits at link l, µ l is the internal serice rate from its session queue to the -th irtual link queue, and r l is the aerage serice rate of the -th irtual link queue. At each time slot, the link scheduler chooses a feasible irtual link actiation ector and seres the chosen irtual links. For feasible irtual link actiation ectors in F, denote the i-th feasible ector by x (i) := (x (i) l ). Suppose that a scheduling policy chooses feasible irtual link actiation ectors following a stationary distribution, and let α i denote the selection probability of the i-th feasible ector. It has been shown in [19] that the scheduling policy achiees the optimal throughput performance if {α i } is the solution to the following optimization problem P: P: maximize α 0, µ 0 i α i log α i subject to λ l µ l, l, µ l r l, l,, α i =1, where α := (α i ) and µ := (µ l ),and0 is a zero ector. The first and second constraints are the stability conditionsfor session queues and for irtual link queues, respectiely. The aerage serice rate r l is a linear function of α and can be written as c l i α ix (i) l.thelastconstraintalongwithα 0 comes from the fact that α is a probability distribution. The objectie of the problem is to maximize the entropy of α. Problem P has a unique optimizer as long as it is feasible because the objectie function is strictly conex and the constraint set is conex. Suppose that an arrial rate ector λ is within the capacity region. According to (3), we can find non-negatie numbers ( α i ) such that λ l ( i α ic l x (i) ) l and i α i =1.Let µ l := i α ic l x (i) l.itiseasytoseethat ( α i ) and ( µ l ) satisfy all the constraints of problem P and the constraint set is not empty. Hence, for any λ Λ, problemp is solable since it is a conex optimization problem. B. Throughput-optimal scheduling We next sole the dual problem of P rather than soling the more challenging primal problem directly. Let q := (q l ) denote the ector of Lagrange multipliers associated with the first constraint, and q := (q l ) represent the ector of Lagrange multipliers associated with the second constraint. From i (4)

5 5 the standard optimization techniques, a partial Lagrangian can be defined as L(α, µ; q, q ) = i α i log α i + l, q l r l + l, µ l (q l q l ) l and the dual problem D of problem P can be formulated as D: minimize sup α 0,µ 0, i αi=1 L(α, µ; q, q ) subject to q 0, q 0. q l λ l, Gien q and q,theobjectiefunctionofproblemd can be decoupled into the following two subproblems, since the last term l q lλ l is a constant. SubD1: SubD2: maximize µ 0 (5) (6) µ l (q l q l ), (7) l, maximize α 0, i αi=1 i α i log α i + l, q l r l. (8) In SubD1, we introduce an additional constraint on the aerage internal serice rates µ l,i.e., µ l C, where C is a constant no smaller than V max l, c l.notethatfrom the second constraint of P, anysolutionwillsatisfythat µ l r l V max r l V max c l C, (9) l, l, indicating that the new constraint does not affect the solution space. We can now rewrite SubD1 as maximize µ 0, µ l (q l q l ). (10) µ l C l The solution to this subproblem can be obtained as follows. First, each link l finds the irtual link such that =argmax (q l q l ), (11) and sets each internal transmission rate µ l as { C, if = µ l = and q l >q l, 0, otherwise. (12) It is noteworthy that problem SubD1 can be soled in a distributed manner by each link with only local information. In SubD2, theoptimalsolutionwillmaximizetheentropy of the probability distribution α for the feasible irtual link actiation ectors, plus the queue weighted rate sum l, q lr l. We consider a stationary link scheduling policy that chooses the i-th member x (i) of feasible set F with probability π i (q )= 1 Z exp ( l, c l q l x (i) l ), (13) where Z is a normalization constant such that i π i(q )= 1. Ithasbeenshownin[19]thataschedulingpolicythat achiees the stationary distribution (13) is an optimal solution to SubD2. Therefore, at each time slot t, wecanuse(12)and(13) to optimally sole SubD1 and SubD2 gien q(t) and q (t), and maximize the alue of the objectie function of problem Algorithm 1 Traffic splitter of link l at time slot t. 1: Find =argmax (Q l (t) Q l (t)). 2: if Q l (t) Q l (t) > 0 then 3: /* Moe min(c, Q l (t)) bits from Q l (t) to Q l (t) */ 4: Q l (t +1)=[Q l (t) C] + ; 5: Q l (t +1)=Q l (t) + min(c, Q l (t)). 6: end if D. Subgradientalgorithmforaconexoptimizationproblem asserts that we can attain the optimal solution of problem D by adjusting q(t) and q (t) as follows: For small enough κ, each Lagrange multiplier is adjusted as [ q l (t +1) = q l (t)+κ ( λ l ) ] µ l, (14) + [ q l (t +1) = q l (t)+κ ( ) ] µ l r l, (15) + where [x] + := max(x, 0). Finally, we confirm that the strong duality holds for problems P and D, whenthearrialrateectorλ is strictly within the capacity region Λ. From(3),therearenon-negatie( α i ) such that λ l = ( i α ic l x (i) ) l and i α i < 1. Leth be aconstantbetween1 and ( i α i) 1.Wesetα i = α i / i α i, and µ l = h i α ic l x (i) l.clearly,theseα and µ satisfy all the inequality constraints of P with strict inequality as well as the equality constraint. Hence the Slater s condition holds, and the strong duality ensures that the optimal alue of problem P equals to the optimal alue of problem D. C. Distributed link scheduling Based on the studies aboe, we next deise practical distributed algorithms that sole the problems SubD1 and SubD2 in a dynamic setting. The solution is diided into two parts: Traffic splitter associated with SubD1 distributes the data bits from the session queue to irtual link queues, and link scheduler associated with SubD2 achiees the stationary distribution (13) for the irtual link actiation ectors. Traffic splitter can be performed at each link in a distributed fashion with only local information; Q l (t) and Q l (t), where Q l (t) denotes the length of session queue at link l and Q l (t) denotes the length of the -th irtual link queue at link l at time slot t. Ateachtimeslott, theinternalsericerate µ l (t) is set to either 0 or constant C of equation (12), which is larger than V max l, c l.thedetailedalgorithmis outlined in Algorithm 1. Basically, the algorithm implements (12). First, each physical link finds the irtual queue with the smallest queue length. Then, if the session queue has a larger queue length than this smallest irtual link queue, it transfers backlogged data bits to the irtual link queue by at most C bits. Otherwise, it does not transfer data during t. To describe our distributed link scheduler, we begin with the time slot structure for scheduling in our time-slotted system. We diide a single slot into a control subslot and a data subslot, as shown in Fig. 3. During the control subslot, which is further diided into RTS, CTS, and REJECT parts, nodes exchange control packets with each other to determine

6 6 Fig. 3. Time slot structure. the set of actie irtual links distributedly. Then during the data subslot, a sender can transmit data to the corresponding receier using the actiated irtual link. We assume the control subslot is substantially small compared to the data subslot and thus throughput loss due to the control subslot is negligible. We also ignore the processing time and the oerhead from the packetization of data bits. Akeystepistochooseafeasibleirtuallinkactiation ector x(t) in a distributed manner such that its distribution follows (13). To this end, we combine two irtual link actiation ectors, i.e., the irtual link actiation ector x(t 1) at the preious slot t 1 and a randomly chosen ector at the beginning of time t, to determine a feasible irtual link actiation ector at time slot t. A challenge here is to ensure that the resultant irtual link actiation ector is feasible under the SINR interference model. We sole the problem by three-way handshake of control packets: Ready-To-Send (RTS), Clear-To-Send (CTS), and REJECT. The oerall procedure for distributed link scheduling can be described as follows. At the beginning of time slot t, eachlink decides if it changes its actiity or not with some probability. Once a link wants to change its actiity state, it randomly chooses a irtual link. The set of the selected irtual links in the network is called as the decision ector. Thentheactie irtual links in x(t 1) 3 and a subset of irtual links in the decision ector participate in feasibility tests by exchanging control packets. As depicted in Fig. 3, the transmitter of each participating irtual link first sends an RTS in the control subslot. The RTS (and also CTS) packet contains link id l and irtuallink id to be actiated. The receier of the RTS returns an CTS in the CTS part, only if the receied SINR is large enough to support the designated irtual link. The CTS packet is also used to test if the SINR of the reerse link is acceptable. If any actie irtual link in x(t 1) fails to meet the required SINR, it broadcasts a REJECT signal in the REJECT part and inalidates the new link actiity state. If no REJECT signal is broadcasted, we obtain new schedule x(t) from the irtual links participated in the feasibility tests, and otherwise, the schedule remains unchanged, i.e., x(t) =x(t 1). Once x(t) is determined, during the following data subslot, the irtual links in x(t) transmit data packets. Detailed descriptions for the selection of the decision ector and the feasibility tests using the three-way handshake are as follows. (Also refer to Algorithms 2 and 3, where Tx(l) and Rx(l) denote the transmitter and the receier of link l, respectiely.) At the beginning of the control subslot, each 3 We say that irtual link [l, ] is in the ector, e.g.,x if the associated indicator function x l is 1. link decides whether it changes the actiity in a probabilistic way. This probability p trial (> 0), calledthetrial probability, can be different for each link. A link l, ifitistochange the actiity, chooses a irtual link at random. The selection probability need not be uniform for all the irtual links in the link. Howeer, each irtual link should hae a non-zero probability, and the probability should be fixed across time slots. We denote the decision ector by m(t) := (m l (t)), where m l (t) {0, 1} specify the selected irtual links, i.e., m l (t) =1if irtual link [l, ] is chosen, and m l (t) =0otherwise. The irtual links in the decision ector can potentially change its actiity from actie to inactie or ice ersa. If no irtual link of link l was actie at time t 1, irtuallink [l, ] in the decision ector will be actie with probability p l (0 <p l < 1) attimet, andremainssilentwithprobability p l := 1 p l.ifairtuallinkoflinkl was actie at time t 1, irtuallink[l, ] in the decision ector checks whether it is the actie irtual link during the preious time slot. If it is, it remains actie with probability p l and will be inactie with probability p l.ifitisnot,theirtuallinkinthedecision ector remains silent and the preiously actie sibling irtual link will be actie in the current slot, where we say two irtual links are siblings if they belong to the same physical link. Now we hae a set of irtual link candidates that can potentially be actie during time slot t, whichconsistofthe irtual links in the decision ector that decides to be actieand the irtual links in x(t 1) that are not in the decision ector. We proceed to test feasibility of these irtual links candidates. We call the candidates that were not actie at t 1 as new applicants. Torepresentnewapplicants,weusetheindicator function n l (t) and its ector n(t) := (n l (t)). Naturally, n l (t) = 1 when irtual link [l, ] is a new applicant, and n l (t) =0otherwise. All the actie irtual links in x(t 1) as well as all the new applicants in n(t) participate in the feasibility test. Note that some in x(t 1) are not candidates, but participate in the test. The inclusion of all actie irtual links of x(t 1) in the test is crucial to ensure the reersibility of Marko chain described in the next section. In the RTS part of the control subslot, each participating irtual link s transmitter sends an RTS, and its corresponding receier measures the SINR from the RTS. Suppose that a irtual link [l, ] transmits an RTS. If the receier passes the test, i.e., if the SINR is large enough to support irtual link [l, ], the receier returns a CTS to the transmitter in the subsequent CTS part. If the receier fails the test, i.e., if the SINR requirement is iolated, the receier takes a different action depending on its actiity during the preious time slot. If irtual link [l, ] was not actie at time t 1 (equialently, if irtual link [l, ] is in n(t)), the receier does nothing 4,andit remains inactie during time t. If irtual link[l, ] was actie at time t 1 (equialently, if irtual link [l, ] is in x(t 1)), the receier will broadcast a REJECT signal 5 in the REJECT part later, and inalidates the new actiation ector. This ends the forward feasibility test by RTS. Note that the aboe feasibility test is conseratie, since not 4 Line 22 in Algorithm 2. 5 Lines 24 and 32 in Algorithm 3.

7 7 TABLE I TYPES OF CONTROL PACKETS AND SIGNAL THAT CAN BE GENERATED BY LINK l. (YES) IMPLIES THAT THE CONTROL PACKET OR SIGNAL WILL BE GENERATED CONDITIONALLY, BASED ON THE PASS OR FAIL OF THE FEASIBILITY TESTS. RTS CTS REJECT s.t. [l, ] x(t 1), [l, ] / n(t) Yes (Yes) (Yes) s.t. [l, ] / x(t 1), [l, ] n(t) Yes (Yes) No s.t. [l, ] / x(t 1), [l, ] / n(t) No No No all the senders of RTS will transmit data during the data subslot (een when there are no REJECT signals in the REJECT part): Some irtual links in n(t) may fail the feasibility test and stay inactie during the data subslot, and some irtual links that were actie in x(t 1) will oluntarily gie up being actie and turn off their radio signal. This implies the receiers of actie links may achiee higher SINR than that in the feasibility test. In the CTS part, the reerse feasibility test, which is similar to the forward feasibility test of the RTS, proceeds and the transmitter checks whether it receies a SINR higher than Γ ctrl and successfully decodes the CTS. If the transmitter passes the test and receies no REJECT signal before the data subslot, it will transmit data bits during the data subslot. If the transmitter fails the test, depending on its actiity during the preious time slot, the transmitter remains silent 6 (if n l (t) =1), or broadcasts a REJECT signal 7 in the following REJECT part (if x l (t 1) = 1). This reerse feasibility test is also conseratie for the same reason as in the forward feasibility test. In the REJECT part, the transmitter and receier of links in x(t 1) may send REJECT signals if the SINR requirement is iolated. If a REJECT signal is broadcasted, all the links will set their actiities to the alue of the preious time slot, i.e., x l (t) =x l (t 1) for all l,. Inthissense,theREJECT signal inalidates the new irtual link actiation ector. Since the REJECT signal does not necessarily carry any specific information, it would suffice to receie a signal with high enough strength compared to the background noise leel. Similar ideas can be found in a different context of wireless local area networks [27] and wireless mesh networks [28]. In a small network, we can assume that all nodes perceie a signal from any node. In a larger network, we may need to employ a flooding method to relay the signal oer the network, ineitably with additional control oerheads. Howeer, since the signal does not carry data, multiple nodes can transmit the REJECT signals at the same time, which enables quick propagation of the signal throughout the network. In this work, we assume that the REJECT signal is quickly relayed to all nodes in the network within the period of the REJECT part. Detailed implementation of the signal flooding is beyond the scope of the paper. Finally, in the data subslot, irtual link [l, ] transmits data bits at rate c l if x l (t) =1.TableIsummarizesthetypes of control packets that can be generated by a link during the 6 Line 25 in Algorithm 2. 7 Lines 27 and 32 in Algorithm 3. control subslot, based on its membership in x(t 1) and n(t). Airtuallinkcannotbelongtobothx(t 1) and n(t) at the same time. Note that i) if a receier sends a CTS packet, it will not broadcast a REJECT signal, ii) it is possible for both the transmitter and receier of a link to broadcast REJECT signals simultaneously, and iii) new applicants are not allowed to generate REJECT signals, which is the key difference between Algorithms 2 and 3. Note that the flooding-like control (i.e., REJECT signal) incurs additional oerhead. Howeer, a certain leel of global information is required for scheduling under the SINR-based interference model, since een a small change of the transmission power may cause a constraint iolation of a far-away transmission. In this work, we reduce the amount of control oerhead for link actiity coordination under the SINR model to a minimal leel. Exchanging control messages often incurs significant oerhead due to inclusion of addresses, delimiters, checksum, etc, in each message. We aoid such large oerhead by using a simple signal, which carries only a single bit information. Thus it can be easily implemented without much complexity, een without decoding, e.g., by using energy detection, and can propagate through the network quickly by simply repeating. All the other complex operations hae been designed to operate in a distributed manner. D. Throughput optimality of distributed link scheduler In this section, we show that the proposed link scheduler indeed achiees optimal throughput. Specifically, we show that the scheduler yields the same steady-state probability distribution as (13), which implies an optimal solution to SubD2. Werepresentthesetofactieirtuallinksinairtual link actiation ector as the corresponding capital letter. For example, we denote the decision set associated with decision ector m =(m l ) by M := { [l, ] m l =1}. We begin with some properties of the stochastic process {x(t)} t=0. Lemma 1: {x(t)} t=0 is a Discrete-Time Marko Chain (DTMC). Proof: The inputs of Algorithms 2 and 3 are the preious irtual link actiation ector x(t 1), trialprobability(p trial ), actiation probabilities (p l ),andthefixedprobabilitiesfor constructing the decision ector. Since the current state does not depend on the history before time t 1, thediscretetime process {x(t)} t=0 is a Marko Chain. The following two lemmas assert that the Marko Chain walks through all and only the feasible irtual link actiation ectors of the network. Lemma 2: With a feasible initial irtual link actiation ector, e.g., x(0) = 0, theirtuallinkactiationectorx(t) under our proposed scheduler is feasible, i.e., x(t) F,for all t 0. Proof: Anewsetofirtuallinksispermittedtobeactie at the same time only if it passes both the forward and reerse feasibility tests. The tests guarantee that each irtual link of the new actiation ector satisfies the SINR requirements. Moreoer, the proposed algorithm allows for only a single irtual link in a physical link to be actie in a time slot.

8 8 Algorithm 2 Link scheduling for preiously inactie link l. 1: 1) At the beginning of control subslot: 2: Decide if link l will try to change its actiity with probability p trial. 3: if Link l is to change the actiity then 4: Select {1, 2,..., V } at random. 5: x l (t) =1with probability p l ; 6: x l (t) =0with probability p l =1 p l. 7: x l (t) =x l (t 1) for. 8: else 9: x l (t) =x l (t 1) for all s. 10: end if 11: 12: 2) In RTS part of control subslot: 13: if x l (t) =1then 14: Tx(l) sends RTS(l, ). 15: Rx(l) measures SINR l from RTS(l, ). 16: end if 17: 18: 3) In CTS part of control subslot: 19: if SINR l Γ then 20: Rx(l) sends CTS(l, ). 21: else 22: x l (t) =0. 23: end if 24: if Tx(l) fails to decode CTS(l, ) then 25: x l (t) =0. 26: end if 27: 28: 4) In REJECT part of control subslot: 29: if Tx(l) detects REJECT then 30: x l (t) =x l (t 1) for all s. 31: end if 32: 33: 5) In data subslot: 34: if x l (t) =1then 35: Tx(l) transmits data bits at the rate of c l. 36: end if Therefore, the selected irtual link actiation ector should be feasible if it passes the feasibility tests. If the new set is rejected, then the actiation ector at time t should be also feasible since x(t) =x(t 1). Therefore,x(t) is feasible for all t 0 by induction. Lemma 3: In {x(t)} t=0, astatex can reach any x F with positie probability in a finite number of steps. Proof: We show that by the proposed scheduling, the null state 0 can reach an arbitrary feasible link actiation ector with some positie probability in a single step, and ice ersa, where the null state implies that all links in the network are inactie. 1) Suppose that the network is currently in the null state. For any x F,ifthedecisionsetcoincideswithX and all the irtual links in the decision set determine to be actie, then they will pass the feasibility tests and clearly, x is accepted as the next irtual link actiation ector. Algorithm 3 Link scheduling for preiously actie link l. 1: 1) At the beginning of control subslot: 2: Decide if link l will try to change its actiity with probability p trial. 3: if Link l is to change the actiity then 4: Select {1, 2,..., V } at random. 5: if x l (t 1) = 1 then 6: x l (t) =1with probability p l ; 7: x l (t) =0with probability p l =1 p l. 8: else 9: x l (t) =0. 10: end if 11: x l (t) =x l (t 1) for. 12: else 13: x l (t) =x l (t 1) for all s. 14: end if 15: 16: 2) In RTS part of control subslot: 17: Tx(l) sends RTS(l, ). 18: Rx(l) measures SINR l from RTS(l, ). 19: 20: 3) In CTS part of control subslot: 21: if SINR l Γ then 22: Rx(l) sends CTS(l, ). 23: else 24: Rx(l) constructs and holds REJECT. 25: end if 26: if Tx(l) fails to decode CTS(l, ) then 27: Tx(l) constructs and holds REJECT. 28: end if 29: 30: 4) In REJECT part of control subslot: 31: if Tx(l) or Rx(l) is holding REJECT then 32: REJECT is broadcasted. 33: end if 34: if Tx(l) detects REJECT then 35: x l (t) =x l (t 1) for all s. 36: end if 37: 38: 5) In data subslot: 39: if x l (t) =1then 40: Tx(l) transmits data bits at the rate of c l. 41: end if 2) Conersely, suppose that x is the current irtual link actiation ector. If the decision set coincides with X and all the irtual links in the decision set intend to turn off, the null state will become the irtual link actiation ector of the next slot. From 1) and 2), astatex can reach any x F possibly ia the null state with a positie probability in a finite number of steps. We next show that the corresponding Marko Chain has the same distribution as (13) under our proposed scheduler. Since the Marko Chain is aperiodic and irreducible from Lemma 3, it has a unique stationary distribution. Focusing on the transition of two irtual link actiation ectors, we

9 9 characterize the probability distribution that satisfies detailed balance conditions. Then the reersibility of the Marko Chain indicates that it is the stationary distribution. We consider a state transition at time slot t, andomitthe parameter t (or t 1) ifthereisnoconfusion.ifanyreject signals are transmitted during the control subslot of time slot t, then there is no state transition. Henceforth, we consider only the case when new irtual link actiation ector is accepted. Suppose that at time slot t, theirtuallinkactiationector changes from x to x.letx and X denote the set of all the actie irtual links in x and x,respectiely.letm denote the decision set chosen in the process of the transition. Note that for irtual link [l, ] X, allitssiblingirtuallinksare blocked from being actie at time slot t under our scheduling scheme, i.e., [l, w] / X for w. (Seelines3 14 in Algorithm 3.) We denote the blocked irtual links by B([l, ]), and extend the notation by letting B(A) denote the set of irtual links blocked by the set of irtual links A. Further, N represents the set of new applicants for actiation, i.e., irtual links that were not actie in the preious time slot but participate in the feasibility tests in this time slot. Let subset D N denote the set of the irtual links discouraged from being actiated in the process of the feasibility tests, i.e., the new applicants that could not meet the SINR requirement. In this setting, the irtual links in the network can be classified as shown in Table II, based on their action in the time slot. Each irtual link belongs to either the decision set M or its complement M c.theirtuallinksinm can be further subdiided as follows. We also proide a diagram in Fig. 4 to help readers understand the table, where the set M is represented by grey areas and the set N is by slanted areas, respectiely. M X\X : Virtual links that were actie at t 1 and are inactie at t. TheselinkswillexchangeRTS/CTS in the control subslot but do not transmit data in the data subslot. Under our scheduling algorithms, for a irtual link to change its state from actie to inactie, it should be included in the decision set M. Hence, X \ X M, andthusm X\X = X \ X. M X X :Virtuallinksthatwereactieatt 1 and remain actie at t. TheselinkswillexchangeRTS/CTSanddo transmit data bits in the data subslot. They can be also written as M X X = M (X X ). M X \X: Virtuallinksthatwereinactieatt 1 and are actie at t. Theselinksarenewapplicants,participatein RTS/CTS exchange, and transmit data in the data subslot. From X \ X M, wehaethatm X \X = X \ X. M B(X X ):VirtuallinksinM that are blocked to be actie at t due to some irtual link in X X.Theycan be written as M B(X X ) = M B(X X ). M D :Virtuallinksthatfailthefeasibilitytests.Allthe irtual links in this set are new applicant since the state transition is assumed to occur. They may fail either the forward or reerse feasibility test, hence they can exchange either RTS only or RTS/CTS depending on cases. From D N M, wehaem D = D. M others :TherestirtuallinksinM \((X X ) B(X X ) D) that oluntarily gie up trying to be actie with Fig. 4. An illustration for irtual link classification. probability p l.theydonotparticipateinthefeasibility tests, and thus do not contribute to the state transition. For a irtual link in M c,ifthelinkwasactieatt 1, it will participate in the RTS/ CTS exchange and remain actie at t. Wedenotetheseirtuallinksby(M c ) X,whichequals to M c X. SinceX \ X does not share any members with M c,i.e.,m c (X \ X )=, wecanrewriteitas(m c ) X = M c (X X ).Ontheotherhand,ifairtuallinkinM c was not actie at t 1, itdoesnotexchangethecontrolpackets and remains silent. Gien X and X,adecisionsetthattriggersatransition from X to X may not be unique. Let M(X, X ) denote the collection of all such decision sets that can make the transition. Also, for a decision set M M(X, X ),therecanbemultiple sets of new applicants that incur the transition from X to X.SinceN consists of two disjoint sets X \ X and D, it suffices to determine N using D, X, andx.hence,we describe the transition from X to X using D instead of N. Let D(X, X ; M) denote the collection of all possible D s that make the transition from X to X gien M. Now, we calculate the transition probability from X to X as follows. Let β(m) denote the probability of a decision set M M(X, X ) being chosen randomly. Each irtual link in M X\X switches to the inactie state oluntarily with probability p l and each irtual link in M X X stays actie with probability p l.amongm, irtuallinksinb(x X ) are blocked (with probability 1) anddoesnotaffectthestate transition probability. Lastly, each of the rest irtual links in M \(X B(X X )) becomes a new applicant with probability p l or oluntarily remains inactie with probability p l.allthe new applicants try to be actiated in the current slot. Among them, irtual links in M X \X pass the feasibility tests, while the others in D D(X, X ; M) fail the tests and are not allowed to be actie. We hae denoted the irtual links that oluntarily gie up by M others.weemphasizethedependency of M others on D by M others (D). Naturally,irtuallinksof M c hae the same actiity states as the preious time slot with probability 1. Hence,thestatetransitionprobabilityfromx to x can be obtained as (16) in Fig. 5. The following lemma and corollary lead to the main result

10 10 TABLE II CLASSIFICATION OF VIRTUAL LINKS. Sets of irtual links Control packets Data transmission Equialent form M X\X RTS/CTS X \ X M X X RTS/CTS Yes M (X X ) M M X \X RTS/CTS Yes X \ X M B(X X ) M B(X X ) M D RTS or RTS/CTS D M others M \ ((X X ) B(X X ) D) M c (M c ) X RTS/CTS Yes M c (X X ) (M c ) others M c \ (X X ) P (x, x )= = P (x, x) = = M M(X,X ) M M(X,X ) M M(X,X) M M(X,X ) β(m) β(m) β(m) β(m) [l,] M X\X [l,] X\X p l [l,] X \X p l [l,] X\X p l p l [l,] M X X [l,] M (X X ) [l,] M (X X) [l,] M (X X ) p l p l p l p l [l,] M X \X p l [l,] X \X p l [l,] X\X p l [l,] X \X p l D D(X,X ;M) D D(X,X ;M) D D(X,X;M) D D(X,X ;M) ( [l,] D ( [l,] D ( [l,] D ( [l,] D p l p l p l p l [l,] M others (D) [l,] M others (D) [l,] M others (D) [l,] M others (D) ) p l ) p l. ) p l ) p l. (16) (17) Fig. 5. Forward and reerse state transition probabilities. of this section. Lemma 4: M(X, X )=M(X,X). Proof: Suppose that state X changes to X with decision schedule M M(X, X ).TheirtuallinksofM can be classified as follows. Virtual links in M X X stays actie. Virtual links in M X\X becomes in actie. Virtual links in M X \X becomes actie. Virtual links in M B(X X ) remains inactie since blocked. The set of the rest irtual links is denoted by M rest. We now focus on M rest.amongtheseirtuallinks,wedenote the set of irtual links that oluntarily become inactie by M others, and the others should be inactie by failing the feasibility test, in order to make the state transition from X to X,anddenotedbyM D,i.e.,M rest = M others M D.Note that the irtual links participate in the feasibility test under our algorithm include X, X \ X, andm D = M rest \ M others, and all the irtual links in M D fail the feasibility test. Let us consider the reerse transition, i.e., from X to X. Suppose that we hae the same decision schedule M = M and the same set M others = M others of irtual links becomes inactie oluntarily. Since X X M \ M others,witha certain probability: Virtual links in M X X stays actie. Virtual links in M X \X becomes in actie. Virtual links in M X\X becomes actie. Virtual links in M B(X X) remains inactie since blocked. The set of the rest irtual links is denoted by M rest. In this case, we hae M rest = M rest because M = M and M others = M others. Also,thesetoftheirtuallinks participate in the feasibility test would be X (X \ X ) (M rest \M others ),whichisidenticaltothesetofirtuallinks participated in the feasibility test for transition from X to X. Hence, due to the identical interference relationship, all the irtual links in M rest \ M others = M rest \ M others should fail the feasibility test. This completes the transition from X to X. The proof of Lemma 4 immediately implies that in both transitions, the set of the irtual links that fail the feasibility test is also identical, which leads to the following corollary. Corollary 1: D(X, X ; M) =D(X,X; M). We now can obtain the following proposition. Proposition 1: The probability distribution π(x; q )= 1 p l, (18) Z p l [l,] X where Z is the normalization factor, is the stationary state distribution of the Marko Chain {x(t)} t=0.further,bysetting p l = exp(c lq l ) 1+exp(c l q l ),weobtainthestateprobability π(x; q )= 1 Z exp ( ) c l q l x l. (19) l,

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