Scheduling in Multi-Channel Wireless Networks

Transcription

1 Scheduing in Muti-Channe Wireess Networks Vartika Bhandari and Nitin H. Vaidya University of Iinois at Urbana-Champaign, USA Abstract. The avaiabiity of mutipe orthogona channes in a wireess network can ead to substantia performance improvement by aeviating contention and interference. However, this aso gives rise to non-trivia channe coordination issues. The situation is exacerbated by variabiity in the achievabe datarates across channes and inks. Thus, scheduing in such networks may require substantia information-exchange and ead to non-negigibe overhead. This provides a strong motivation for the study of scheduing agorithms that can operate with imited information whie sti providing acceptabe worst-case performance guarantees. In this paper, we make an effort in this direction by examining the scheduing impications of mutipe channes and heterogeneity in channe-rates. We estabish ower bounds on the performance of a cass of maxima scheduers. We first demonstrate that when the underying scheduing mechanism is imperfect, the presence of mutipe orthogona channes can hep aeviate the detrimenta impact of the imperfect scheduer, and yied a significanty better efficiency-ratio in a wide range of network topoogies. We then estabish performance bounds for a scheduer that can achieve a good efficiency-ratio in the presence of channes with heterogeneous rates without requiring expicit exchange of queue-information. Our resuts indicate that it may be possibe to achieve a desirabe trade-off between performance and information. 1 Introduction Appropriate scheduing poicies are of utmost importance in achieving good throughput characteristics in a wireess network. The semina work of Tassiuas and Ephremides yieded a throughput-optima scheduer, which can schedue a feasibe traffic fows without resuting in unbounded queues [8]. However, such an optima scheduer is difficut to impement in practice. Hence, various imperfect scheduing strategies that tradeoff throughput for simpicity have been proposed in [5, 9, 10, 7], amongst others. The avaiabiity of mutipe orthogona channes in a wireess network can potentiay ead to substantia performance improvement by aeviating contention and interference. However, this aso gives rise to non-trivia channe coordination issues. The situation is exacerbated by variabiity in the achievabe data-rates across channes and inks. Computing an optima schedue, even in a singe-channe network, is amost aways intractabe, due to the need for goba information, as we as the computationa This research was supported in part by NSF grant CNS , US Army Research Office grant W911NF , and a Vodafone Graduate Feowship. Vartika Bhandari is now with Googe Inc.

2 compexity. However, imperfect scheduers requiring imited oca information can typicay be designed, which provide acceptabe worst-case (and typicay much better average case) performance degradation compared to the optima. In a muti-channe network, the oca information exchange required by even an imperfect scheduer can be quite prohibitive as information may be needed on a per-channe basis. For instance, Lin and Rasoo [4] have described a scheduing agorithm for muti-channe muti-radio wireess networks that requires information about per-channe queues at a interfering inks. This provides a strong motivation for the study of scheduing agorithms that can operate with imited information, whie sti providing acceptabe worst-case performance guarantees. In this paper, we make an effort in this direction, by examining the scheduing impications of mutipe channes, and heterogeneity in channe-rates. We estabish ower bounds on performance of a cass of maxima scheduers, and describe some scheduers that require imited information-exchange between nodes. Some of the bounds presented here improve on bounds deveoped in past work [4]. We begin by anayzing the performance of a centraized greedy maxima scheduer. A ower bound for this scheduer was estabished in [4]. However, in a arge variety of network topoogies, the ower bound can be quite oose. Thus is particuary true for muti-channe networks with singe interface nodes. We estabish an aternative bound that is tighter in a range of topoogies. Our resuts indicate that when the underying scheduing mechanism is imperfect, the presence of mutipe orthogona channes can hep aeviate the impact of the imperfect scheduer, and yied a significanty better efficiency-ratio in a wide range of scenarios.. We then consider the possibiity of achieving efficiency-ratio comparabe to the centraized greedy maxima scheduer using a simper scheduer that works with imited information. We estabish resuts for a cass of maxima scheduers couped with oca queue-oading rues that do not require queue-information from interfering nodes. 2 Preiminaries We consider a muti-hop wireess network. For simpicity, we argey imit our discussion to nodes equipped with a singe haf-dupex radio-interface capabe of tuning to any one avaiabe channe at any given time. A interfaces in the network have identica capabiities, and may switch between the avaiabe channes if desired. Many of the presented resuts can aso be used to obtain resuts for the case when each node is equipped with mutipe interfaces; we briefy discuss this issue. The wireess network is viewed as a directed graph, with each directed ink in the graph representing an avaiabe communication ink. We mode interference using a confict reation between inks. Two inks are said to confict with each other if it is ony feasibe to schedue one of the inks on a certain channe at any given time. The confict reation is assumed to be symmetric. The confict-based interference mode provides a tractabe approximation of reaity whie it does not capture the wireess channe precisey, it is more amenabe to anaysis. Such confict-based interference modes have been used frequenty in the past work (e.g., [11, 4]).

3 Time is assumed to be sotted with a sot duration of 1 unit time (i.e., we use sot duration as the time unit). In each time sot, the scheduer determines which inks shoud transmit in that time sots, as we as the channe to be used for each such transmission. We now introduce some notation and terminoogy. The network is viewed as a coection of directed inks, where each ink is a pair of nodes that are capabe of direct communication with non-zero rate. L denotes the set of directed inks in the network. C is the set of a avaiabe orthogona channes. Thus, C is the number of avaiabe channes. We say that a scheduer schedues ink-channe pair (,c) if it schedues ink for transmission on channe c. r c denotes the rate achievabe on ink by operating ink on channe c, provided that no conficting ink is aso schedued on channe c. For simpicity, we assume that r c > 0 for a L and c C. 1 The rates r c do not vary with time. We aso define the terms: r max = max L,c C rc, and r min = min L,c C rc. When two conficting inks are schedued simutaneousy on the same channe, both achieve rate 0. β s denotes the sef-skew-ratio, defined as the minimum ratio between rates supportabe over different channes on a singe ink. Therefore, for any two channes c and d, and any ink, we have rd r c β s. Note that 0 < β s 1. β c denotes the cross-skew-ratio, defined as the minimum ratio between rates supportabe over the same channe on different inks. Therefore, for any channe c, and any two inks and : rc r c β c. Note that 0 < β c 1. r c c C Let r = max c C rc. Let σ s = min L r. Note that σ s 1+β s ( C 1). Moreover, in typica scenarios, σ s wi be expected to be much arger than this worst-case bound. σ s is argest when β s = 1, in which case σ s = C. b() and e(), respectivey, denotes the nodes at the two endpoints of a ink. In particuar, ink is directed from node b() to node e(). E(b())and E(e())denote the set of inks incident on nodes b() and e(), respectivey. Thus, the inks in E(b()) and E(e()) share an endpoint with ink. Since we focus on singe-interface nodes, this impies that if ink is schedued in a certain time sot, no other ink in E(b()) or E(e()) can be schedued at the same time. We refer to this as an interface confict. Let A() =E(b()) E(e()). Note that A(). Links in A() are said to be adjacent to ink. Links that have an interface confict with ink are those that beong to E(b()) E(e()) \ {}. Let A max = max A(). I() denotes the set of inks that confict with ink when schedued on the same channe. I() may incude inks that aso have an interface-confict with ink. By convention, is considered incuded in I(). The subset of I() comprising interfering inks that are not adjacent to is denoted by I (), i.e., I () = I() \A(). Let I max = max I (). 1 Though we assume that r c > 0 for a,c, the resuts can be generaized very easiy to hande the case where r c = 0 for some ink-channe pairs.

4 K denotes the maximum number of non-adjacent inks in I () that can be schedued on a given channe simutaneousy if is not schedued on that channe. K ( C ) denotes the maximum number of non-adjacent inks in I () that can be schedued simutaneousy using any of the C channes (without conficts) if is not schedued for transmission. Note that here we excude inks that have an interface confict with. K is the argest vaue of K over a inks, i.e., K = max K. K C is the argest vaue of K ( C ) over a inks, i.e., K C = max hard to see that for singe-interface nodes: K ( C ). Let I max = max I (). It is not K K C min{k C,I max } (1) We remark that the term K as used by us is simiar, but not exacty the same as the term K used in [4]. In [4], K denotes the argest number of inks that may be schedued simutaneousy if some ink is not schedued, incuding inks adjacent to. We excude the adjacent inks in our definition of K. Throughout this text, we wi refer to the quantity defined in [4] as κ instead of K. Let γ be 0 if there are no other inks adjacent to at either endpoint of, 1 if there are other adjacent inks at ony one endpoint, and 2 if there are other adjacent inks at both endpoints. γ is the argest vaue of γ over a inks, i.e., γ = max γ. Load vector: We consider singe-hop traffic, i.e., any traffic that originates at a node is destined for a next-hop node, and is transmitted over the ink between the two nodes. Under this assumption, a the traffic that must traverse a given ink can be treated as a singe fow. The traffic arriva process for ink is denoted by {λ(t)}. The arrivas in each sot t are assumed i.i.d. with average λ. The average oad on the network is denoted by oad vector λ = [λ 1,λ 2,...,λ L ], where λ denotes the arriva rate for the fow on ink. λ may possiby be 0 for some inks. Queues: The packets generated by each fow are first added to a queue maintained at the source node. Depending on the agorithm, there coud be a singe queue for each ink, or a queue for each (ink, channe) pair. Stabiity: The system of queues in the network is said to be stabe if, for a queues Q in the network, the foowing is true [2]: im t sup 1 t t τ=1 E[q(τ)] < where q(τ) denotes the backog in queue Q at time τ (2) Feasibe oad vector: In each time sot, the scheduer used in the network determines which inks shoud transmit and on which channe (reca that each ink is a directed ink, with a transmitter and a receiver). In different time sots, the scheduer may schedue a different set of inks for transmission. A oad vector is said to be feasibe, if there exists a scheduer that can schedue transmissions to achieve stabiity (as defined above), when using that oad vector.

5 Link rate vector: Depending on the schedue chosen in a given sot by the scheduer, each ink wi have a certain transmission rate. For instance, using our notation above, if ink is schedued to transmit on channe c, it wi have rate r c (we assume that, if the scheduer schedues ink on channe c, it does not schedue another conficting ink on that channe). Thus, the schedue chosen for a time-sot yieds a ink rate vector for that time sot. Note that ink rate vector specifies rate of transmission used on each ink in a certain time sot. On the other hand, oad vector specifies the rate at which traffic is generated for each ink. Feasibe rate region: The set of a feasibe oad vectors constitutes the feasibe rate-region of the network, and is denoted by Λ. Throughput-optima scheduer: A throughput-optima scheduer is one that is capabe of maintaining stabe queues for any oad vector λ in the interior of Λ. For simpicity of notation, we use λ Λ in the rest of the text to indicate a oad-vector vector λ ying in the interior of a region Λ. From the work of [8], it is known that a scheduer that maintains a queue for each ink, and then chooses the schedue given by argmax r q r, is throughputoptima for scenarios with singe-hop traffic (q is the backog in ink s queue, and the maximum is taken over a possibe ink rate vectors r ). Note that q is a function of time, and queue-backogs at the start of a time sot are used above for computing the schedue (or ink-rate vector) for that sot. Imperfect scheduer: It is usuay difficut to determine the throughput-optima inkrate aocations, since the probem is typicay computationay intractabe. Hence, there has been significant recent interest in imperfect scheduing poicies that can be impemented efficienty. In [5], cross-ayer rate-contro was studied for an imperfect scheduer that chooses (in each time sot) ink-rate vector s such that q s δ argmax r q r, for some constant δ (0 < δ 1). It was shown [5] that any scheduer with this property can stabiize any oad-vector λ δλ. Note that if a rate vector λ is in Λ, then the rate vector δ λ is in δλ. δλ is aso referred to as the δ-reduced rate-region. If a scheduer can stabiize a λ δλ, its efficiency-ratio is said to be δ. Maxima scheduer: Under our assumed interference mode, a schedue is said to be maxima if (a) no two inks in the schedue confict with each other, and (b) it is not possibe to add any ink to the schedue without creating a confict (either confict due to interference, or an interface-confict). We wi aso utiize the Lyapunov-drift based stabiity criterion from Lemma 2 of [6]. 3 Scheduing in Muti-channe Networks As was discussed previousy, throughput-optima scheduing is often an intractabe probem even in a singe-channe network. However, imperfect scheduers that achieve a fraction of the stabiity-region can potentiay be impemented in a reasonaby efficient manner. Of particuar interest is the cass of imperfect scheduers know as maxima scheduers, which we defined in Section 2. The performance of maxima scheduers

6 1 Identica channes/gains β c β s 1 Fig D visuaization of channe heterogeneity under various assumptions has been studied in much recent work, e.g., [10, 7], with the focus argey on singe-channe wireess networks. The issue of designing a distributed scheduer that approximates a maxima scheduer has been addressed in [3], etc. When there are mutipe channes, but each node has one or few interfaces, an additiona degree of compexity is added in terms of channe seection. In particuar, when the ink-channe rates r c can be different for different inks, and channes c, the scheduing compexity is exacerbated by the fact that it is not enough to assign different channes to interfering inks; for good performance, the channes must be assigned taking achievabe rates into account, i.e., individua channe identities are important. Scheduing in muti-channe muti-radio networks has been examined in [4], which argues that using a simpe maxima scheduer is used in such a network coud possiby ead to arbitrary degradation in efficiency-ratio (assuming arbitrary variabiity in rates) compared to the efficiency-ratio achieved with identica channes. A queue-oading agorithm was been proposed, in conjunction with which, a maxima scheduer can stabiize any vector in ( 1 κ+2) Λ, for arbitrary βc and β s vaues. This rue requires knowedge of of the ength of queues at a interfering inks, which can incur substantia overhead. Whie variabe channe gains are a rea-word characteristic that cannot be ignored in designing effective protocos/agorithms, it is important that the soutions not require extensive information exchange with arge overhead that offsets any performance benefit. In ight of this, it is crucia to consider various points of trade-off between information and performance. In this context, the quantities β s,β c and σ s defined in Section 2 prove to be usefu. The quantities β s and β c can be viewed as two orthogona axes for worst-case channe heterogeneity (Fig. 1). The quantity σ s provides an aggregate (and thus averaged-out) view of heterogeneity aong the β s axis. β s = 1 corresponds to a scenario where a channes have identica characteristics, such as bandwidth, moduation/transmission-rate, noise-eves, etc., and the ink-gain is a function soey of the separation between sender and receiver. β c = 1 corresponds to a scenario where a inks have the same sender-receiver separation, and the same conditions/characteristics for any given channe, but the channes may have different characteristics, e.g., an b channe with a maximum supported data-rate of 11 Mbps, and an a channe with a maximum supported data-rate of 54 Mbps.

7 Vertex representing a ink Channe Interference confict Fig. 2. Exampe of improved bound on efficiency ratio: ink-interference topoogy is a star with a center ink and x radia inks In this paper, we show that in a singe-interface network, a simpe maxima scheduer augmented ( with oca traffic-distribution ) and threshod rues achieves an efficiencyratio at east. The noteworthy features of this resut are: σ s K C +max{1,γ} C 1. This scheduer does not require information about queues at interfering inks. 2. The performance degradation (compared to the scheduer of [4]) when rates are σ variabe, i.e., β s,β c 1, is not arbitrary, and is at worst s C 1+β s( C 1) C 1 C. Thus, even with a purey oca information based queue-oading rue, it is possibe to avoid arbitrary performance degradation even in the worst case. Typicay, the performance woud be much better. ( 3. In many network scenarios, the provabe ower bound of σ s K C +max{1,γ} C ) may actuay be better than κ+2 1. This is particuary ikey to happen in networks with singe-interface nodes, e.g., suppose we have three channes a,b,c with r a = 1,r b = 1,r c = 0.5 for a inks. Then, in the network in Fig. 2 (where the ink-interference graph is a star with x radia vertices, and there are no interface-conficts), K C = 1 x,γ = 0,σ s = 2.5, and we obtain a bound of 0.4x+1.2, whereas the proved ower 1 bound of the scheduer of [4] is x+2. The muti-channe scheduing probem is further compicated if the rates r c are timevarying, i.e., r c = rc (t). However, handing such time-varying rates is beyond the scope of the resuts in this paper, and we address ony the case where rates do not exhibit timevariation. Note that reated prior work on muti-channe scheduing [4] aso addresses ony time-invariant rates. 4 Summary of Resuts For muti-channe wireess networks with singe-interface nodes, we present ower bounds on the efficiency-ratio of a cass of maxima scheduers (incuding both centraized and distributed scheduers), which indicate that the worst-case efficiency-ratio can be higher when there are mutipe channes (as compared to the singe-channe case). More specificay, we show that: The number of inks schedued by any maxima scheduer are within at east a δ fraction of the maximum number of inks activated by any feasibe schedue, where: { } C δ = max K C + max{1,γ} C, 1 max{1,k + γ}

8 A centraized greedy maxima (CGM) scheduer achieves an efficiency-ratio which is at east σ max{ s } This constitutes an improvement over the ower K C +max{1,γ} C, 1 max{1,k+γ} bound for the CGM scheduer proved in [4]. Since K C min{k C,I max } κ C, this new bound on efficiency-ratio can often be substantiay tighter. We show that any maxima scheduer, in conjunction with a simpe oca queueoading rue, and a threshod-based ink-participation rue, achieves an efficiency- ( ratio of at east σ s K C +max{1,γ} C ). This scheduer is of significant interest as it does not require information about queues at a interfering inks. Due to space constraints, proofs are omitted. Pease see [1] for the proofs. Note that the text beow makes the natura assumption that two inks that confict with each other (due to interference or interface-confict) are not schedued in the same timesot by any scheduer discussed in the rest of this paper. 5 Maxima Scheduers We begin by presenting a resut about the cardinaity of the set of inks schedued by any maxima scheduer. Theorem 1. Let S opt denote the set of inks schedued by a scheduer that seeks to maximize the number of inks schedued for transmission, and ets max denote the set of inks activated by any maxima scheduer. Then the foowing is true: { S max max C K C + max{1,γ} C, 1 max{1,k + γ} The proof is omitted due to ack of space. Pease see [1]. } S opt (3) 6 Centraized Greedy Maxima Scheduer A centraized greedy maxima (CGM) scheduer operates in the manner described beow. In each timesot: 1. Cacuate ink weights w c = q r c for a inks and channes c. 2. Sort the ink-channe pairs (,c) in non-increasing order of w c. 3. Add the first ink-channe pair in the sorted ist (i.e., the one with highest weight) to the schedue for the timesot, and remove from the ist a ink-channe pairs that are no onger feasibe (due to either interface or interference conficts). 4. Repeat step 3 unti the ist is exhausted (i.e., no more inks can be added to the schedue). In [4], it was shown that this centraized greedy maxima (CGM) scheduer can achieve an approximation-ratio which is at east ( 1 κ+2) in a muti-channe muti-radio network, where κ is the maximum number of inks conficting with a ink that may

9 possiby be schedued concurrenty when is not schedued. This bound hods for arbitrary vaues of β s and β c, and variabe number of interfaces per node. However, this bound can be quite oose in muti-channe wireess networks where each device has one or few interfaces. In this section, we prove an improved bound on the efficiency-ratio achievabe with the CGM scheduer for singe-interface nodes. We aso briefy discuss how it can be used to obtain a bound for muti-interface nodes. Theorem 2. Let S opt denote the set of inks activated by an optima scheduer that chooses a set of ink-channe pairs (,c) for transmission such that w c is maximized. Let c () denote the channe assigned to ink S opt by this optima scheduer. Let S g denote the set of inks activated by the centraized greedy maxima (CGM) scheduer, and et c g () denote the channe assigned to a ink S g. Then: w cg() S g w c () S opt { max σ s K C + max{1,γ} C, 1 max{1,k + γ} } (4) The proof is omitted due to ack of space. Pease see [1]. Theorem 2 eads to the foowing resut: Theorem 3. The centraized greedy maxima (CGM) scheduer can stabiize the δ- reduced rate-region, where: { } σ s δ = max K C + max{1,γ} C, 1 max{1,k + γ} Proof. We earier discussed a resut from [5] that any scheduer, which chooses rateaocation s such that q s δ argmax q r, can stabiize the δ-reduced rate-region. Using Theorem 2 and this resut, we obtain the above resut. We remark that the above bound is independent of β c. 6.1 Mutipe Interfaces per Node We now describe how the resut can be extended to networks where each node may have more than one interface. Given the origina network node-graph G = (V, E), construct the foowing transformed graph G = (V,E ): For each node v V, if v has m v interfaces, create m v nodes v 1,v 2,...v mv in V. For each edge (u,v) E, where u,v have m u,m v interfaces respectivey, create edges (u i,v j ),1 i m u,1 j m v, and set q (ui,v j ) = q (u,v). Set the achievabe channe rate appropriatey for each edge in E and each channe. For exampe, assuming that the channe-rate is soey a function of u,v and c, then: for each channe c, set r c (u i,v j ) = rc (u,v).

10 The transformed graph G comprises ony singe-interface inks, and thus Theorem 2 appies to it. Moreover, it is not hard to see that a schedue that maximizes q r in G aso maximizes q r in G. Thus, the efficiency-ratio from Theorem 2 for network graph G yieds an efficiency-ratio for the performance of the CGM scheduer in the muti-interface network. We briefy touch upon how one woud expect the ratio to vary as the number of interfaces at each node increases. Note that the efficiency-ratio depends on β s, C,K C,γ. Of these β s and C are aways the same for both G and G. γ is aso aways the same for any G derived from a given node-graph G, as it depends ony on the number of other node-inks incident on either endpoint of a node-ink in G (which is a property of the node topoogy, and not the number of interfaces each node has). However, K C might potentiay increase in G as there are many more non-adjacent interfering inks when each interface is viewed as a distinct node. Thus, for a given number of channes C, one woud expect the provabe efficiency-ratio to initiay decrease as we add more interfaces, and then become static. Whie this may initiay seem counter-intuitive, this is expained by the observation that mutipe orthogona channes yieded a better efficiency-ratio in the singe-interface case since there was more spectra resource, but imited hardware (interfaces) to utiize it. Thus, the additiona channes coud be effectivey used to aeviate the impact of sub-optima scheduing. When the hardware is commensurate with the number of channes, the situation (compared to an optima scheduer) increasingy starts to resembe a singe-channe singe-interface network. 6.2 Specia Case: C Interfaces per Node Let us consider the specia case where each node in the network has C interfaces, and achievabe rate on a ink between nodes u,v and a channes c C is soey a function of u,v and c (and not of the interfaces used). In this case, it is possibe to obtain a simper transformation. Given the origina network node-graph G = (V, E), construct C copies of this graph, viz., G 1,G 2,...,G C, and view each node in each graph as having a singeinterface, and each network as having access to a singe channe. Then each network graph G i can be viewed in isoation, and the throughput obtained in the origina graph is the sum of the throughputs in each graph. ( From ) Theorem 2, in each graph we can 1 max{1,k+γ} show that the CGM scheduer is within ( overa network, the CGM scheduer is within 1 max{1,k+γ} of the optima. Thus, even in the ) of the optima. 7 A Rate-Proportiona Maxima Muti-Channe (RPMMC) Scheduer In this section, we describe a scheduer where a ink does not require any information about queue-engths at interfering inks. The set of a inks in denoted by L. The arriva process for ink is i.i.d. over a time-sots t, and is denoted by {λ (t)}, with E[λ (t)] = λ. We make no assumption about independence of arriva processes for two inks, k. However, we consider ony

11 the cass of arriva processes for which E[λ (t)λ k (t)] is bounded, i.e., E[λ (t)λ k (t)] η for a L,k L, where η is a suitabe constant. Consider the foowing scheduer: Rate-Proportiona Maxima Muti-Channe (RPMMC) Scheduer Each ink maintains a queue for each channe. The ength of the queue for ink and channe c at time t is denoted by q c (t). In time-sot t: ony those ink-channe pairs with q c (t) rc participate, and the scheduer computes a maxima schedue from amongst the participating inks. The new arrivas during this sot, i.e., λ (t) are assigned to channe-queues in proportion to the rates, i.e., λ c (t) = λ (t)r c r b b C Theorem 4. The RPMMC scheduer stabiizes the queues in the network for any oadvector within the δ-reduced rate-region, where: δ = σ s K C + max{1,γ} C The proof is omitted due to space constraints. Pease see [1]. Coroary 1 The efficiency-ratio of the RPMMC scheduer is aways at east: ( )( σs 1 ) C K + max{1,γ} Proof. The proof foows from Theorem 4 and (1). 8 Discussion The intuition behind the RPMMC scheduer is simpe: by spitting the traffic across channes in proportion to the channe-rates, each ink sees the average of a channerates as its effective rate. This heps avoid worst-case scenarios where the ink may end up being repeatedy schedued on a channe that yieds poor rate on that ink. The agorithm is made attractive by the fact that no information about queues at interfering inks is required. Furthermore we showed that the efficiency-ratio of the RPMMC scheduer is aways at east ( σs C )( 1 K+max{1,γ} ). Note that 1+β s ( C 1) σ s C. Thus, the efficiency ratio of this agorithm does not degrade indefinitey as β s becomes smaer. Moreover, in many practica settings, one can expect σ s to be Θ( C ) and the performance woud be much better compared to the worst-case of σ s = 1+β s ( C 1). 9 Future Directions The RPMMC scheduer provides motivation for further study of scheduers that work with imited information. The scheduer of Lin-Rasoo [4] and the RPMMC scheduer represent two extremes of a range of possibiities, since the former uses information

12 from a interfering inks, whie the atter uses no such information. Evidenty, using more information can potentiay aow for a better provabe efficiency-ratio. However, the nature of the trade-off curve between these two extremities is not cear. For instance, an interesting question to ponder is the foowing: If interference extends up to M hops, but each ink ony has information upto x < M hops, what provabe bounds can be obtained? This woud hep quantify the extent of performance improvement achievabe by increasing the information-exchange, and provide insights about suitabe operating points for protoco design, since contro overhead can be a concern in rea-word network scenarios. References 1. V. Bhandari. Performance of wireess networks subject to constraints and faiures. Ph.D. Thesis, UIUC, L. Georgiadis, M. J. Neey, and L. Tassiuas. Resource aocation and cross-ayer contro in wireess networks. Found. Trends Netw., 1(1):1 144, C. Joo and N. B. Shroff. Performance of random access scheduing schemes in muti-hop wireess networks. In Proceedings of IEEE INFOCOM, pages 19 27, X. Lin and S. Rasoo. A Distributed Joint Channe-Assignment, Scheduing and Routing Agorithm for Muti-Channe Ad-hoc Wireess Networks. In Proceedings of IEEE INFOCOM, pages , May X. Lin and N. B. Shroff. The impact of imperfect scheduing on cross-ayer rate contro in wireess networks. In Proceedings of IEEE INFOCOM, pages , M. J. Neey, E. Modiano, and C. E. Rohrs. Dynamic power aocation and routing for time varying wireess networks. In Proceedings of IEEE INFOCOM, G. Sharma, R. R. Mazumdar, and N. B. Shroff. On the compexity of scheduing in wireess networks. In MobiCom 06: Proceedings of the 12th annua internationa conference on Mobie computing and networking, pages ACM, L. Tassiuas and A. Ephremides. Stabiity properties of constrained queueing systems and scheduing poicies for maximum throughput in mutihop radio networks. IEEE Transactions on Automatic Contro, 37(12): , Dec X. Wu and R. Srikant. Scheduing efficiency of distributed greedy scheduing agorithms in wireess networks. In Proceedings of IEEE INFOCOM, X. Wu, R. Srikant, and J. R. Perkins. Queue-ength stabiity of maxima greedy schedues in wireess networks. In Workshop on Information Theory and Appications, X. Wu, R. Srikant, and J. R. Perkins. Scheduing efficiency of distributed greedy scheduing agorithms in wireess networks. IEEE Trans. Mob. Comput., 6(6): , 2007.