Energy Efficient Low Complexity Joint Scheduling and Routing for Wireless Networks

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1 Energy Effiient Low Complexity Joint Sheduling and Routing for Wireless Networks Satya Kumar V, Lava Kumar and Vinod Sharma Department of Eletrial Communiations Engineering, Indian Institute of Siene, Bangalore Abstrat We onsider a multihop wireless network with multiple users. The hannels may experiene fading. The power is onsumed only during transmission and is a general monotonially inreasing funtion of rate. We provide low omplexity algorithms for joint routing, sheduling and power ontrol whih ensure stability of the queues, ertain minimum rates and/or upper bound on the end-to-end, mean delays. Keywords Multihop wireless networks, sheduling, routing, minimizing power, QoS. I. INTRODUCTION A multihop wireless network (MWN) is a olletion of mobile nodes whih are onneted with eah other by wireless media. A MWN has many appliations, mainly in military servies, where there is no pre-existing infrastruture. With emergering eletronis, MWN is not onfined to military and disaster management systems but also widely used in vehileto-vehile ommuniation systems, home networking and peerto-peer networks. Wide-spread use of a MWN is mainly due to the advantages that it offers, suh as, easy deployment, better overage at lower ost (where it is hard to wire) and higher throughput (due to shorter hops). All these benefits our at the ost of routing omplexity, path management, additional delay due to passage of data in multiple hops and limitations in the transmission range due to wireless hannel inherent harateristis suh as fading, path loss, shadowing and interferene. If the transmitting node and the reeiving node are far apart, then the data may need to traverse to the destination in more than one hop. Here, a real hallenge lies in routing and sheduling of the wireless links, whih an beome ompliated due to limited battery at intermediate nodes and half duplex and other interferene onstraints. Furthermore, the appliations arried may need some quality of servie (QoS) onstraints suh as mean end-to-end delay, minimum mean rate guarantee or stability of its queues in the network. Thus, for limited energy MWN systems (suh as sensor networks) *This work was partly supported by a funding from ANRC., minimizing total average power onsumed by the network while providing QoS is an important onsideration. The multihop QoS problem an be solved in either a distributed or a entralized manner. Pioneering work on joint routing, sheduling and power ontrol was provided in [5] whih maximizes a utility funtion under average power onstraint. As this problem is intratable, a heuristi suboptimal algorithm was provided. [] onsidered the problem of ensuring a fair utilization of network resoures by jointly optimizing routing, sheduling and power ontrol and obtained an effiient sub-optimal solution when the nodes may be powered by energy harvesting soures. [8] extended the solution in [5] to a multihop network where different nodes have multiple antennas and presented effiient, fair algorithms. Bak-pressure based algorithms have been used in [7], [9], [0]. These algorithms are entralized, use hannel and queue length information and ensure stability. Upper bounds on mean delays are provided in [0] via Lyapunov approah but these are rather loose and under heavy traffi the mean delays obtained may violate any given mean delay onstraints. A large survey on multihop networks in general, with emphasis on QoS is presented in [6]. For this problem one may onsider using Markov Deision Proesses (MDP). However due to omplex oupled queue dynamis in multihop networks MDP, or approximate MDP tehniques have a large state spae and the omputations beome unrealisti even for small networks [6]. Furthermore, MDP tehniques may not provide any insights in the struture of the optimal poliy and require huge signaling overheads. A. Problem Statement We onsider the problem of joint link sheduling, routing and power ontrol, so as to ensure ertain end-to-end QoS for individual flows. The QoS may be the upper bound on the end-to-end mean delays, minimum rate guarantees or just the stability of the queue. While ensuring the QoS, we would like to minimize the average power onsumed in the network /5/ 205 IFIP 8

2 2 B. Our Contribution We present omputationally effiient algorithms, for routing, sheduling and power ontrol for a multihop network whih minimizes total average power onsumed while providing endto-end QoS to individual users. We assume that the transmit rate is a general monotonially inreasing funtion of the power invested at eah node. Although our algorithms are suboptimal in power, these are power-effiient and guarantee endto-end QoS. We also provide onditions for finiteness of stationary queue length moments, rate of onvergene to the stationary distribution and good approximation to mean end-to-end delays for our algorithms. We are not aware of any other work that provides suh results for multihop wireless networks. As against the previous work our algorithms are in losed form or have very low omputational omplexity. Our algorithms do not require queue length information or paket arrival distribution. These algorithms are built on our previous work in [3], [4] and [5]. These algorithms are optimal in the lass of algorithms requiring only hannel gains. In these works it is shown that these algorithms are overall optimal or lose to optimal via omputations on expliit examples. We are not aware of any other work on multihop wireless networks that ensures QoS exept stability, whih again does not minimize power. The paper is organised as follows. In Setion II, we desribe the system model. Setion III results to the single user multihop senario. Setion V extends our results to the multiuser, multihop system. Setion VI develops algorithms to ensure minimum average rate guarantee and/or upper bounds on the end-to-end mean delays to all the users for all the above ases. Setion VII onludes the paper. II. SYSTEM MODEL We onsider a wireless network, modelled as a onneted direted graph G(N, L), where N = {,..., N} and L = {,..., L} represent the set of nodes and the set of direted links respetively. A subset of nodes in the network transmits data to another subset of nodes. Eah soure has a unique destination. We onsider a disrete time slotted system. A stream of pakets that are transmitted from a soure node to its destination node is alled a flow. We denote the set of user flows as F := {, 2,..., M}. A flow arries data pakets, for whih we may either only need to ensure the stability of all the queues in the network or need to guarantee that the flow gets a minimum end-to-end average rate or an upper bound on endto-end average delay. Let A t f be the number of pakets generated by flow f in slot t at its soure and plaed in an infinite buffer queue. We assume that all pakets are of same size. Also, let {A t f, t 0} to independent, identially distributed (iid), independent of streams for the other flows. We assume that all the links in the network are half- duplex. Also, a node an either transmit or reeive data at a given time instant. This assumption is not ritial for our algorithm. Atually we will only assume that the set of links L is divided into independent sets S,..., S d suh that all links within a set an simultaneously transmit with negligible interferene to eah other. However, the links in two different sets will not be allowed to transmit simultaneously. A link an belong to multiple sets. This is a ommonly made assumption in multihop wireless networks [4]. Let Hij t be the hannel gain in a slot t for link (i, j) from node i to node j. We assume that instantaneous hannel gain knowledge is available at node i in the beginning of every time slot and we also assumed that the hannel gain is onstant in a time slot. We also assume that the hannel gain proess {Hij t, t 0} is iid on all links and independent of the link hannel gain proesses of the other links. The hannel gain Hij t takes values on a finite set of values. This an be a good approximation of ontinuous distributions, (e.g., Rayleigh, Riian, Nakagami) if we take the finite set to be large enough. Let Pij t (f) be the power spent by node i to transmit Rij t (f) pakets to node j in time slot t for flow f. Then, often, R t ij(f) = 2 log 2( + G ij P t ij(f)h t ij(f)/σ 2 ij), () where σij 2 is the reeiver noise variane and G ij is a onstant that depends on the modulation and oding used. Our polies will not require fragmentation of pakets. Also, instead of (), Rij t (f) ould as well be a general, nonnegative, monotoni ontinuous, non-dereasing funtion g ij (Pij t (f), Ht ij ) of power Pij t (f) and Ht ij. Let qij t be the queue length at node i for transmission on link (i, j) in the beginning of slot t. Then q t+ ij = (q t ij + A t ij R t ij) +, i, j, t 0, (2) where A t ij is the amount of data arriving at node i in the beginning of slot t and Rij t is the data transmitted on link (i, j). Our objetive is to minimize the total average power onsumed by the system suh that the QoS of different flows is met. Our hallenges lie in sheduling the links, routing the flows and alloating the transmit power on eah link. We transmit the data from a flow on a single path instead of splitting it on multiple paths. This will avoid unneessary delays while splitting and merging (re-ordering) at the destination. We onsider a entralized setup, i.e., system has all the information, inluding hannel gain statistis and average external arrival rates. This will be needed only for developing 9

3 3 the algorithms for routing, sheduling and power ontrol. The atual implementation will require little information. III. OPTIMAL POLICIES: STABILITY In this setion, we onsider joint routing, sheduling and power alloation poliies whih minimize, lim sup n n E (i,j) n t=0 f= suh that the long term average queue length M Pij(f), t (3) E[q ij ] <, (i, j) L. (4) We provide effiient sub-optimal solutions of this problem. For omputational tratability we limit ourselves to poliies whih do not exploit queue length information. We first onsider a single flow ase and later on generalize to the multiuser system. A. Single User, Single Hop We first study the single user, single hop ase and provide optimal polies. We then extend these polies to the single user, multihop ase. Now we keep the notation of Setion II but omit the sub/super sripts f, (i, j). Let E[A] and E[A α ], α be the mean and the αth moment of the number of a arrivals from the soure node in one slot. In the absene of fading, an average power optimal poliy (whih does not onsider the queue length) whih provides the stability of the queue is, { E[A] + δ, w.p. p, R t = (5) E[A] + δ, otherwise, where w.p. p denotes with probability p, δ > 0, is a small positive quantity and p is hosen suh that E[R] = E[A] + δ p + E[A] + δ ( p) = E[A] + ɛ, and ɛ is a small positive quantity. Also, for positive real x, x ( x ) is the smallest (largest) integer ( ) x. Poliy (5) ensures that an integer number of pakets are transmitted in a slot. The poliies obtained by us will be lose to optimal (depending on the small ɛ, δ hosen) but we will ignore this point. Sine (2) is the usual Lindley equation [], with this poliy, the queue length proess {q k } has a unique stationary distribution. Furthermore, if E[A α+ k ] < for α, then E[qk α ] under stationarity is also finite. For (5), the average power onsumption is E[P ] = φ( E[A] + δ )p + φ( E[A] + δ )( p). (6) Now we onsider the hannel with fading. To ensure stability we take E[R] = E[A]+ɛ, where ɛ > 0 is a small onstant. Let h u, u L be the hannel gain with probability of ourrene p l and h u < h u+ for u, u + {,..., L}. Let the power onsumption when a pakets are transmitted and the hannel gain is h be φ(a, h). We assume that φ(a, h) < φ(a +, h) and φ(a, h) > φ(a, h ) for h > h. Optimizing (3) is a disrete optimization problem. A simple iterative algorithm to obtain the optimal solution is as follows. Let R = [R(),..., R(L)] be the rate vetor at some point in the omputation of the algorithm, where R(u) is the number of pakets that are transmitted when the hannel gain is h u. If L u= R(u)pu < E[A] + ɛ then we update the rate vetor by inreasing one of the omponents of the rate vetor by one while the rest of the omponents remain same. The omponent hosen to inrease is arg min Diff(u) = φ(r(u) +, h u ) φ(r(u), h u ). We repeat this till E[R] E[A] + ɛ. Then we stop. At the end, if E[R] = E[A] + ɛ we take that poliy. If E[R] > E[A] + ɛ then we inrease the last rate only with a ertain probability so that the average servie rate E[R] = E[A] + ɛ. Often the optimal solution is even simpler. For example, let R = 2 log 2( + ph σ ) where σ 2 is the noise power. Let the 2 paket size be n bits. If hl h 2 2n, then the optimal solution has the following simple struture. Find an integer m suh that m i= pl+ i p = E[A] E[A] < m+ i= pl+ i. Let p = p m i= p L+ i. Then the optimal poliy is, E[A], if H t = h i, i L + m, R t E[A], if H t = h L m, w.p. ( p ), = (7) E[A], if H t = h L m, w.p. p, E[A], if H t = h i, i L m. The ondition hl h 2 2n is quite a reasonable assumption and is satisfied for most pratial systems. IV. SINGLE USER, MULTIHOP Now we extend the optimal poliy of Setion III to the ase where a single flow traverses multiple hops to reah the destination, where every link in the network may experiene fading. A. Single User, Multihop: No Fading Initially, we onsider the ase, when the network experienes no fading. Let us assume that the route for the flow has been fixed. We will omment on it later. We replae the stability onstraint of all the queues on the route with E[R ij ] E[A] + ɛ. Let S,..., S d be the independent sets of links [8]. We need restrit this set to the links on the route seleted. We shedule the independent sets suh that eah S k is ative for γ k > 0 fration of time and d z= γ k =. The alloation of slots to the different links will be done as follows. A entral authority generates iid random variables Y t in the beginning of eah slot t with probability P [Y t = k] = γ k, k =,..., d. If Y t = k 0

4 4 then slot t is assigned to the independent set S k. If in a slot a link has hannel gain Hij t and the link is ative in that slot then it will transmit Rij t pakets from its queue in that slot, whih is a funtion of γ k s and Hij t. Define for link (i, j), = d k:(i,j)ɛs k γ k. To get the optimal power, we solve the optimization problem, subjet to, min φ ij ( E[A] + ɛ ) (i,j) d k= γ k =, 0 < γ k <, k, where φ ij (a) is the power needed to transmit a pakets on link (i, j). If φ ij are onvex, then it is a onvex optimization problem. The orresponding optimal rates are R t ij = { E[A]+δ w.p. p ij, E[A]+δ otherwise, where p ij is suh that E[R] = E[A] + ɛ = E[A]+δ p ij + E[A]+δ ( p ij ). The total average power onsumed by the network on the given route, an be made lose to E[P ] = (i,j) φ ij ( E[A] (8) )p ij + φ ij ( E[A] )( p ij ). (9) Thus, we should selet the route whih minimizes (9). But this involves searhing in an exponential number of routes. One sub-optimal solution is to give eah link (i, j) the ost φ ij ( E[A]+ɛ d ) and use Dijkstra s (say) algorithm to obtain the optimal route. In partiular, this will be a good solution for (7), espeially at large E[A]. One an further improve over it by using greedy iterative algorithms over the ost funtion (9). In suh algorithms, a link onsuming one of the highest powers from the urrently onsidered route is seleted (say link (i, j)) and removed from the network. Then the least ost path from the soure to destination is omputed again via Dijkstra s algorithm. If this provides lower (9) then it is kept otherwise we go bak to the previous path. Again the proess is repeated. B. Single User, Multihop: With Fading Next we onsider the ase when the hannels experiene fading. Let P [Hij t = hu ij ] = pu ij and βu ij be the probability that link (i, j) will transmit pakets when it is allowed to transmit in hannel state h u ij. Then, for a fixed route, to minimize power, we optimize over βij u and γ j. Over the links on the route, min φ(, h u ij)p u ijβij u (0) (i,j) u subjet to, p u ijβij u E[A] + ɛ, () u 0 βij u, βij u, (2) d k= γ k =, 0 < γ k <, k. (3) This is a non-onvex optimization problem. However, if we fix γ k s, it beomes a Linear Program (LP) in βij u. We iterate greedily over γ k s and solve LP s to obtain a better solution. On the other hand, if βij u are (approximately) known (e.g.(7)), then it is an LP in γ k s. Considering the example (7), a good route an be seleted by obtaining the least ost algorithm via Dijkstra algorithm [2], by taking the link ost for link (i, j) as E[ H ij ]. Let A t ij be the number of pakets arriving to queue i for link (i, j) in slot t. The sequene {Rij t } provided by the above algorithm is iid for eah (i, j). However {A t ij, t 0} is not neessarily iid if i is not the soure node. In the following we study the proess {qij t, t 0} for the above queueing system. Let q t = (q t (l), l =,..., L ) be the queue length proess along the given route. Also let P t (l) be the distribution of ql t and P t = (P t (l), l =,..., L ). Theorem. For the above poliy {q t } proess has a unique stationary distribution π. Starting from any initial onditions, q t onverges in total variation to π. Also, if E[A α+ ] < then E π [q α (l)] < for l =,..., L. Furthermore, P t π β β2 α where 0 < β 2 <, β is a positive onstant and. denotes the total variation norm. Proof: If i is the soure node and link (i, j) is the first link on the path, then A t ij = At is an iid sequene and we have ensured that E[Rij t ] E[A] + ɛ. Thus from the usual results on GI/GI/ queues ([]), qij t = qt () has a unique stationary distribution and q t () onverges in total variation to the stationary distribution. Also if E[A α+ ] < for an α, then E[(qij t )α ] < under stationarity. For the seond queue q t (2) on the route, let the input proess be {A t (2)} and the transmission rate proess be {R t (2)}, whih is given by the optimal poliy. From the first queue result, we also have that {A t (2)}, whih is output proess of the first queue, also has a stationary distribution. Furthermore, although it is not iid, it is a regenerative proess with regeneration epohs the time slots when the first queue is empty. Let its regeneration length be τ(). Under the above assumptions we also have E[(τ()) α ] < if E[A α ] < for α []. Sine {R t (2)} is iid, {(A t (2), R t (2))} is also regenerative with the same regeneration epohs. Also, τ() is

5 5 aperiodi. Then, from [6], if τ() E A t (2) < E [τ()] E [ R t (2) ], (4) t= the proess {A t, q t (), q t (2)} is also regenerative with regeneration length τ(2) with finite mean. Sine at regeneration epohs, the first queue is empty and τ() is a stopping time, τ() τ() E A t (2) = E A t = E [τ()] E [A] t= t= and hene (4) is satisfied if E[A] < E[R t (2)], whih has been ensured by our algorithm. Also, from [6], if for α, α+ α+ τ() τ() E A t (2) = E A t <, (5) t= t= then E [(q(2)) α ] <. But, (5) holds when E [ A α+] <. Under the same onditions, from [6] we also get E[τ(2) α+ ] < (see also [6]). This argument extends diretly to the queues q t (l), l 3. Thus we get that E π [q t (l) α ] < for l =,..., L and the regeneration length τ of the {q t } proess satisfies E[τ α+ ] <. Then the rate of onvergene P t π β β2 α follows from the results on regenerative proesses [6]. It is useful to ompute the mean queue length E π [q()], under stationarity. Sine {A t, t 0} is iid and {R t ()} are iid, E[q()] an be diretly approximated from the approximations for GI/GI/ queues ([2], [3]) where ρ = E[q] ρde[a](c2 A + C2 R ). (6) 2( ρ) E[A] E[R()], C2 A = V ar[a] E[A] 2, C2 R = V ar(r()) E[R()] 2 (7) exp[ d = 2( ρ) 3ρ ( CR 2 )2 CR 2 + ], if C C2 R 2 <, A exp[ ( ρ)) (C2 R ) CR 2 + ], if C 4C2 R 2. A (8) For E π [q(l)], l >, {A t (l), t 0} is not iid. However A t (l) R t (l ) for all t 0. Thus, if we onsider a disrete queue with {R t (l )} arrival proess and {R t (l), l 0} as the servie proess, assuming stability, using approximation (6) provides an approximate upper bound. We will see from simulations that this an be a good upper bound. This is espeially so sine minimizing power ensures that we have some what heavy trafii situation at eah queue. Thus, we will use these bounds in Setion VI to obtain poliies whih guarantee mean end-to-end delays for individual flows. L 5,7 L 7,9 L 9,0 L 0,2 total ˆq q ˆP P TABLE I. C. Example SINGLE USER: AVERAGE QUEUE LENGTHS AND AVERAGE POWERS For simulations, we onsider a multihop network of 20 nodes. Node i and node j are onneted, if i j 2, and i j; otherwise the hannel gains are so bad that we ignore those hannels. If i > j then link (i, j) takes hannel gains from the set i j, and if i < j then the link (i, j) takes hannel gain from the set 2+j i, where, ={, 2, 3, 4, 5, 6 }, 2 = {, 3.03, 5.79, 9.8, 3.3, 7.58 }, 3 = {, 2.46, 4.7, 6.06, 8., 0.27 }, 4 = {, 3.48, 7.22, 2.2, 8., 25.5 }. Channel gains from set i hold with the probabilities P = { 0.2, 0., 0.5, 0.25, 0.2, 0. }, P 2 = { 0.5, 0., 0.25, 0.2, 0., 0.2 }, P 3 = { 0., 0.2, 0.3, 0.2, 0., 0. }, P 4 = { 0.2, 0.2, 0., 0., 0.2, 0.2 } respetively. The power required for given rate r and hannel gain h is 22r h for all hannels. We onsider the ost of link (i, j) as E[ H ij ]. By using Dijkstra algorithm, we find the shortest ost path from soure node 5 to destination node 2, whih is We take the independent sets by half-duplex onstraints. The two independent sets are {(5, 7), (9, 0)} and {(7, 9), (0, 2)}. The input proess generated at soure node 5 has Binomial distribution with mean 3. We have provided servie rate E[R] = 3.2 on all links. We take γ = γ 2 = 2 and minimize the total average power on all the links in the network by substituting in (0)-(3) and solving the LP. Also, we have simulated the system for 0 6 slots. In Table we provide the results, where ˆq and ˆP are the theoretial average queue lengths from (6) and the orresponding powers, while q and P are the respetive simulated values. We observe that (6) provides an approximation for the first queue but an upper bound for the other queues. These approximations/bounds are generally quite good. Also, the theortial power is an upper bound on the atual value sine these provide servie rates whih are a little higher than the atual rate E[A]. We have also solved the full nononvex optimization problem (0)-(3) for loal minima starting with 5 different initial onditions. The total simulated average power onsumption in the network for the best loal minima is 298 and the theoretial value is The total simulated mean end-toend queue lengths of the network via simulation is 62 and via the approximation (6) is 76. We see that solving the muh simpler LP for γ = γ 2 = 2 provides a reasonably good solution for this example. 2

6 6 We also used iterative greedy algorithm to improve the route. Link (5, 7) was removed from the network. Then the least ost path obtained was This path was used with γ = γ 2 = 2 and the optimization LP redued the total power by 30. However removing link (9, 0) and getting a new lowest ost link from 5 to 2 led to inrease in the total power onsumed. V. MULTIUSER, MULTIHOP In this setion, we onsider the ase where multiple souredestination pairs exist in the multihop network. Our objetive is to stabilize all the queues in the network. The hannels may experiene fading. Let the average arrival rate of flow f be E[A f ]. We onsider the poliies whih minimize (3) subjet to (4). Let the routes for the M users be fixed. For stability of the queues we require E[R ij ] f E[A f ] + ɛ, for eah link (i, j) where the summation is over the flows passing through the link (i, j). The notation remains as in Setion II. Also, as before γ k is the probability with whih independent set S k will transmit in a slot and is the probability with whih link (i, j) will be allowed to transmit. Also, let βij u be the probability that node i will transmit pakets on link (i, j) when its hannel gain is h u ij and it is allowed to transmit. We onsider the optimization problem, subjet to, u min (i,j) u φ(, h u ij)p u ijβ u ij (9) p u ijβij u E[A] + ɛ, (i, j), (20) 0 βij u, (i, j),, u, (2) βij u, (i, j), u, (22) d γ k =, γ k > 0, k. (23) k= The objetive funtion and the onstraints are non-onvex but given all the γ k s it beomes an LP for whih we an easily get the optimal solution. Then we iterate greedily to improve over to better γ k s. Similarly, if all βij u are known, then it is an LP in γ k s and this takes are of the important example of (7). For given fixed routes, for eah of the soure-destination pairs, the above poliy provides us with a network of feedforward queues. For eah link (i, j) its queue gets servied at rate Rij t, whih forms an iid sequene. Using results from [6], Theorem extends to this setup also. Let q t = (qij t, (i, j) L) and P t = (Pij t, (i, j) L) the orresponding (i, j) distributions. Theorem 2. For the given poliy, {q t } has a unique stationary distribution π. Starting from any initial distribution, P t onverges in total variation [ to its stationary distribution π. Also, if for an α, E (A f ) α+] for eah f then E π [(q ij ) α ] < for eah (i, j). Furthermore, P t π β 3 β α 4 for some 0 < β 4 < and β 3 a positive onstant. Approximations for mean queue lengths E π [q ij ] are obtained as in the last setion where we need to onsider the statistis of the input proess A t ij at link (i, j). To get the upper bound system, this proess is replaed by the iid input (k,i) L Rt k,i. A. Example We onsider the network example of Setion IV with 3 users. We onsider Binomial arrivals with mean for the first user, Poisson arrivals with mean.05 for the seond user and a disrete distribution with values {0,, 2, 3} taken with probabilities {0.35, 0.35, 0.2, 0.} for the third user. The objetive is to provide stability of all queues. Servie rate at eah link is taken 0. more than the respetive mean values. We simulated the system for 0 6 slots. We used Dijkstra s algorithm to get the least ost paths. The route for the first user is 3 9 7, for the seond user is and for the third user is In this example, we get 4 independent sets S = {(5,7), (0,2), (,9), (5,7)}, S 2 = {(7,9), (3,), (7,9), (0,2)}, S 3 = {(9,0), (5,7), (3,), (5,7)}, S 4 = {(9,7), (0,2), (3,), (7,9)}. We solved the optimization problem (9)-(23) by taking γ i = 4, for all i. We have tabulated the simulated and theoretial average queue lengths and average powers in Tables II, III, IV for the three flows. In Tables II and III, links (3, ) and (0, 2) appear in three of the four independent sets. Thus, their are 3 4, resulting in substantially lower power onsumption on these links. We also solved the optimization problem (9) fully for loal optima, starting with 5 different initial onditions. For the best loal optimum, the total simulated and theoretial powers onsumed by the three flows is and The end-to-end simulated mean delays for the three flows are 38.7, 64.82, 4.42 and the respetive theoretial end-toend mean delays are 45.42, , From Tables II- IV the total simulated and theoretial powers onsumed by the three flows is 34.24, , 3.0 and 39.37, , 3.38 The end-to-end simulated mean delays for the three flows are58., 57.28, 24.2 and respetive theoretial values are 6.40, 85.43, Again we see that the muh simpler 3

7 7 optimization LP with γ i = 4, i =,..., 4 provides a solution whih is very lose to the best loal optimum obtained. From L 3, L,9 L 9,7 total ˆq q ˆP P TABLE II. MULTIUSER : FLOW L 5,7 L 7,9 L 9,0 L 0,2 total ˆq q ˆP P TABLE III. MULTIUSER: FLOW 2 L 5,7 L 7,9 total ˆq q ˆP P TABLE IV. MULTIUSER: FLOW 3 Table II-IV, we see that node 9 transmits on 4 links, resulting in 4 independent sets. This is aused by traffi from soure nodes 5,3 passing through node 9. Consequently all the links passing through node 9 onsuming high power. Thus we removed link (7,9) and reomputed least ost route for soure node 5. This resulted in new path Consequently, the routes beame disjoint, resulting in drasti redution in power. The total theoretial powers on the 2 routes is 30.27,5.04. VI. RATE GUARANTEES, MINIMUM DELAY CONSTRAINTS In this setion, we first onsider flows whih always have data to transmit, e.g., TCP onnetions arrying long files. The QoS requirement is to provide ertain minimum rate to eah flow. Subjet to this, we want to minimize the total network average power (3). If we replae ɛ and δ by zero in Setions III, IV, V, we get optimal polies that guarantee minimum rates whih also minimize the total average power onsumption in the system. Stability of intermediate queues may still be of onern. This is beause, even with TCP onnetions, we do not want the queueing delays to tend to infinity. Thus, we an make ɛ, δ = 0 at the soure node but keep them positive at all intermediate nodes. Now we onsider flows whih require an upper bound on mean end-to-end delay. This an be a first level QoS for real time traffi and an also be useful for data traffi. First onsider a single flow with iid input {A t f, t 0}. Suppose its route has been fixed. We desire that its mean end-to-end delay be D. By Little s law, it provides an upper bound E[A f ]D on the mean end-to-end queue length on the fixed route (assuming all other deterministi proessing, propagation delays et. have been taken into aount). We split this mean end-to-end queue length bound on the given route into individual mean queue length bounds q,..., q L L suh that q i = E[A f ]D (whih we an optimize over). i= Now we solve the optimization problem (0) subjeted to ()- (3) and the additional onstraints orresponding to the E[q(l)] upper bounds (6) orresponding to q,..., q L. The resulting problem is non-onvex and hene one obtains loal minima (sine it is a smooth optimization problem) using standard algorithms. Starting from random initial onditions and taking the best loal minimum an provide a good solution. The above sheme an be made to provide an upper bound on end-to-end mean delays on the intermediate nodes for TCP flows also (thus providing minimum rate guarantee as well as an upper bound on the end-to-end mean delay in the network). The extension to multiuser ase is obvious. In fat we an onsider the system where some flows have only stability requirements, some minimum rate guarantees and some endto-end mean delay requirement in the system and formulate a single optimization problem). We an redue the total power requirement if we give priority to the delay sensitive traffi on any given link over the other traffi. In that ase the mean delay requirement gets relaxed ompared to the ase when it is not given priority. A. Example We demonstrate the above algorithm on the network of 20 nodes with the speifiations in Setion IV-C. There are three users. The first user requires only the stability of its queue, seond user requires its end-to-end mean delay 30 slots and the third user requires the minimum rate guarantee of.05 pakets/slot. The respetive routes are ; and We have 3 independent sets, S = {(7,9), (9,7)}, S 2 = {(5,7), (9,0), (7,6)} and S 3 = {(6,7), (9,), (9,7) }. The arrival distribution for first and the seond user are Poisson and Binomial with means,.5. While the third user always has data to transmit. We split the end-to-end mean of 45 into three parts 0, 20, 5 along the route (optimization over this split should also be done, but is being ignored here; we only want to show that any split is aheivable by our algorithm). Stability for user is attained by serving it at more than the arrival rate at eah link. We took γ i = 3 for i =, 2, 3 and solved the optimization problem. The mean queue lengths for user 2 are provided in Table V. The average powers on the orresponding links are also provided. We see that the simulation end-to-end mean queue lengths satisfy the upper bound speified. From simulations, we saw that user 3 gets the rate.05 pakets/se and all the queues are stable. 4

8 8 Required ˆq Simulated q Sim. Power Th. Power 0, 20, 5.9, 4.23, 0.5 2, 2600, 47 56, 56000, 83 7, 30, , 24.35, 6. 57, 5450, , 6790, 32 5, 35, , 28.89, , 4547, , 5485, , 40, , 27.79, , 4353, , 577, , 37.5, , 27.7, , 4463, , 5352, 60 TABLE V. MULTIUSER WITH QOS: FLOW 2 The flows and 2 are passing via the ommon link (7, 9). If, we give priority to the real time data of user 2 over user on link (7, 9), the simulated power at link (7, 9) redues to while the theoretial power is The power onsumed at the other links stays same. We observe that link (7, 9) onsumes muh more power than the other two links. Thus, we inrease the mean queue length requirement of link (7, 9) to 30 and that of link (6, 7) and (9, ) to 7 and 8 respetively. This redued the power of link (7, 9) substantially. Next we further inreased the mean queue length required at link (7, 9) to 35 leading to further redution in power at (7, 9) and also of the total power. Further inrease in mean queue length at (7, 9) to 37.5 led to an inrease in the total power. These details are provided in Table V. One an do similar optimization of splitting the end-to-end mean queue length for the ase when the dealy sensitive traffi is given priority. VII. CONCLUSIONS We have onsidered the problem of joint routing, sheduling and power ontrol in a multihop network. Our main objetive is to minimize the total average power while providing end-toend QoS to all users. QoS an be the stability of the queues, a minimum rate guarantee per or an upper bound on the end-toend mean delay for a flow. Power is onsumed only during the transmission of data. We have onsidered the ase when the power is a general non-dereasing funtion of transmission rate. Our (sub)optimal polies are easy to implement and omputationally very effiient. [6] Y. Cui V. K. N. Lau R. W. H. Huang S. Zhang, A Survey on Delay- Aware Resoure Control for Wireless Systems Large Deviation Theory, Stohasti Lyapunov Drift and Distributed Stohasti Learning, arxiv: v [s.pf], Ot. 20. [7] L. Georgiadis, M. J. Neely, and L. Tassiulas, Resoure alloation and ross-layer ontrol in wireless networks, Foundations and Trends in Networking, Vol., No., pp. 44, [8] V. Harish, M. Rahul, V. Sharma, Joint Routing Sheduling and Power ontrol for Multihop MIMO Networks, National Conferene on Communiations (NCC), India, Feb [9] L. Huang and M. J. Neely, Delay redution via lagrange multipliers in stohasti network optimization, IEEE Transations on Automati Control, pp , Apr. 20. [0] L. Huang, S. Moeller, M. J. Neely, and B. Krishnamahari, Lifobakpressure ahieves near optimal utility-delay tradeoff, WiOpt, pp , May 20. [] V. Joseph, V. Sharma and U. Mukherji, Joint power ontrol, sheduling and routing for multihop energy harvesting sensor networks, Proeedings of the 4th ACM workshop on Performane monitoring et., PM2HW2N, pp , [2] W. Kramer, M. Langenbah-Beltz, Approximate formulae for the delay in the queueing systems GI/GI/, Eighth International Telemetri Congress, Melbourne, pp , 976. [3] V. Satya Kumar, V. Sharma, Low Complexity Power and Sheduling Poliies for Wireless Networks, Information Theory and Appliations Workshop (ITA), Sandiego, 204. [4] V. Satya Kumar, V. Sharma, Effiient Low Complexity Power Alloation Poliies for Wireless Communiation Systems Guaranteeing QoS, tenth International Conferene on Signal Proessing and Communiations (SPCOM 204), Indian Institute of Siene, July 204. [5] V. Satya Kumar, V. Sharma, Joint Routing, Sheduling and Power Control Providing QoS for Wireless Multihop Networks, National Conferene on Communiations (NCC), India, Feb [6] V. Sharma, Some limit theorems for regenerative queues, Queueing Systems, Vol. 30, pp , 998. REFERENCES [] S. Asmussen, Applied probability and queues, Seond edition, Springer- Verlag, [2] D. P. Bertsekas and R. G. Gallager, Data Networks, Prentie Hall, Seond edition, 992. [3] J. Boxma, V. Sharma and D. K. Prasad, Performae Analysis of a FLuid Queue with Random Servie Rate in Disrete Time, Springer- Verlang, Berlin Heidelberg, pp , [4] L. Bui, A. Eryilmaz, Member, R. Srikant, and X. Wu, Asynhronous Congestion Control in Multi-Hop Wireless Networks With Maximal Mathing-Based Sheduling, IEEE/ACM Transations on networking, Vol. 6, No. 4, Aug [5] M. Cao, V. Raghunathan, S. Hanly, V. Sharma and P. R. Kumar, Power ontrol and Transmission sheduling for network utility maximization in wireless networks, Proeedings of the 46th IEEE Conferene on Deision and Control, New Orleans, LA, USA, De