Delay Efficient Scheduling via Redundant Constraints in Multihop Networks

Transcription

1 Delay Eicient Scheduling via Redundant Constraints in Multihop Networks Longbo Huang, Michael J. Neely Abstract We consider the problem o delay-eicient scheduling in general multihop networks. While the class o max-weight type algorithms are known to be throughput optimal or this problem, they typically incur undesired delay perormance. In this paper, we propose the Delay-Eicient SCheduling algorithm (DESC). DESC is built upon the idea o accelerating queues (AQ), which are virtual queues that quickly propagate the traic arrival inormation along the routing paths. DESC is motivated by the use o redundant constraints to accelerate convergence in the classic optimization context. We show that DESC is throughputoptimal. The delay bound o DESC can be better than previous bounds o the max-weight type algorithms which did not use such traic inormation. We also show that under DESC, the service rates allocated to the lows converge quickly to their target values and the average total network service lag is small. In particular, when there are O() lows and the rate vector is o Θ() distance away rom the boundary o the capacity region, the average total service lag only grows linearly in the network size. Index Terms Queueing, Scheduling, Dynamic Power Allocation, Redundant Constraints, Lyapunov analysis I. INTRODUCTION We consider the problem o routing a set o lows over a general multihop network, which operates in slotted time t = 0,,,... At every time slot, there are random packet arrivals entering the network. These packets eventually need to be routed to their destination. Thereore, at every time slot, the network operator needs to allocate power to the network links and transmit data over the links with the corresponding rates. Depending on the physical coniguration o the network, there will be constraints on how the power and rates can be allocated. The goal o the operator is to ind a joint power allocation, routing and scheduling policy that stabilizes the network and preerably, ensures a low delay o the data. This is a classic routing problem that has been studied in previous work, e.g., [], []. It is known that the class o maxweight type scheduling algorithms are able to stabilize the network whenever the arrival rate vector is within the network capacity region, hence are throughput optimal. For instance, the Longest Connected Queue (LCQ) algorithm [3] and the Longest Connected Group (LCG) algorithm [4] or downlink scheduling problems, and the Dynamic Routing and Power Control (DRPC) and Ehanced-DRPC (EDRPC) algorithms [] [] or general multihop networks are examples o the Longbo Huang (web: longbohu) and Michael J. Neely (web: mjneely) are with the Department o Electrical Engineering, University o Southern Caliornia, Los Angeles, CA 90089, USA. This material is supported in part by one or more o the ollowing: the DARPA IT-MANET program grant W9NF , the NSF grant OCE 05034, the NSF Career grant CCF max-weight type algorithms. However, though being optimal in throughput perormance, the max-weight type algorithms are not guaranteed to yield good delay perormance. Indeed, the known delay-eicient results o the max-weight type algorithms are mainly or single-hop networks. In downlink systems with binary arrival and service variables, LCQ and LCG are shown to be delay-order-optimal, i.e., delay in the network does not grow with the number o users [4] [5]. In a single hop network with arrival rate in a reduced capacity region, the simpler maximal-weight scheduling algorithm is shown to be delay-order-optimal or both i.i.d. and bursty traic [6]. For general multihop networks, a delay bound that is polynomial in the number o nodes is computed in [] or the DRPC algorithm, although the bound is not order-optimal, and [7] derives a lower bound o the network delay. In act, it has been argued that the max-weight type algorithms perorm poorly in delay or multihop networks mainly due to the ollowing two disadvantages [8]: () They maintain a separate queue or each commodity low at each node, and () They may route packets via a long route even i shorter routes are available. Many algorithms have thus been proposed trying to improve the delay perormance o the max-weight type algorithms by alleviating the eect o these two disadvantages. [9] proposes a cluster-based max-weight algorithm to reduce the number o queues maintained at each node. [0] combines the shortest path routing with the maxweight scheduling algorithm. The scheme is able to reduce packet hop counts to the destination but can suer rom large queueing delay. [8] proposes using shadow queues to make scheduling decisions and using a single FIFO queue instead o one queue or each low commodity. Simulation results show that the scheme is able to dramatically reduce delay. [] combines max-weight and dynamic programming and proposes the scheme Opportunistic Routing with Congestion Diversity (ORCD) where packets are routed to the node with the minimum-delay-to-go. In [], the authors also prove that ORCD is throughput optimal. Outside the max-weight context, there has also been work designing and studying delay-eicient algorithms. [3] develops a delay-order-optimal scheduling algorithm or traic in a reduced capacity region. [4] uses a dynamic programming approach to derive delayeicient algorithms or the downlink scheduling problem. [5] and [6] instead use the large deviation approach to study and design algorithms that maximize the queue length decay exponent. However, though the proposed schemes are intuitively delay eicient, it is usually diicult to obtain explicit delay bounds or the non-max-weight type algorithms, and the analytical bounds or the max-weight type schemes are

2 usually not better than those o DRPC [], which will serve as the benchmark algorithm or comparison in this paper. In this paper, we ocus on designing throughput-optimal delay-eicient algorithms using max-weight type approaches. However, we do not try to solve the two disadvantages mentioned above. We instead note that in all the above maxweight type algorithms, only the actual queue sizes are used to build the algorithms. We thus construct the Delay-Eicient SCheduling algorithm (DESC), a max-weight type algorithm, based on the actual network queues and a set o virtual accelerating queues (AQs), which quickly propagate the traic arrival inormation down the routing path. Our approach is motivated by the convergence accelerating techniques in the classic optimization context, where by adding redundant constraints to the original problem, one usually obtain a aster convergence o gradient-type algorithms [7]. As shown in [8], many o the max-weight type algorithms are closely related to solving a discrete version o some rate allocation optimization problem with a dual subgradient method. Thus one major role o the backlogs is to propagate the traic inormation down the routing path. However, i only the actual queue sizes are used to do so, the queues have to build up to create gradients towards the destination, leading to large network delay. In act, such undesired delay perormance can happen even in the simple case o routing a single low over a ixed route using a max-weight type algorithm [8]. DESC instead uses virtual queues to provide traic inormation to intermediate nodes. Under DESC, the convergence o the allocated service rates to their target values is aster than that under the DRPC algorithm. The delay bound we obtain can also be potentially better than those o DRPC. Finally, we note that DESC can be combined with some o the above approaches to also deal with the two aorementioned disadvantages and to urther reduce network delay. Interestingly, we note that the order-optimality proo o the max-weight algorithm or downlink systems in [5] can be viewed as considering all the redundant constraints in some associated optimization problem or the delay analysis. This paper is organized as ollows. In Section II we state our model. We then introduce the notion o an accelerating queue (AQ) and present the DESC algorithm in Section III. Section IV summarizes the perormance results o DESC. The results are then proved in Section V. Section VI considers the case when there is delay in propagating the arrival inormation. Simulation results are in Section VII. For convenience, we summarize all the notations in this paper in Table I. II. NETWORK MODEL We consider a network operator that operates a general multihop network as shown in Fig.. The network is modeled as a graph G = (N, L), where N = {,,..., N} denotes the set o N nodes and L denotes the set o L directed links in the network. Here a link [m, n] where m, n N is in L i there is a communication link rom node m to node n. The network is assumed to operate in slotted time, i.e., t {0,,,...}. The goal o the operator is to support a set o lows going through the network and to achieve good delay perormance. TABLE I TABLE OF NOTATIONS Notation Deinition s, d source and destination nodes o low A number o arrivals or low at time t P, K the routing path o low and its length k (n) the order o node n in path P u (n), l (n) the upstream and downstream nodes o n in P S the aggregate network channel state P the network power allocation vector µ [m,n] the rate allocated to low over link [m, n] at t Q n the queue size o low at node n at t Hn the accelerating queue size o low at node n at t W[m,n] the weight o link [m, n] at time t Dn the messaging passing delay o A or node n at t D upper bound o Dn or all, n, t Lag the approximate network service lag Fig.. &!! # Flows traversing a multihop network. A. The Flow and Routing Model &# &" " % $ '%$()* We assume there are a total o M lows going through the network. We denote the set o lows as F = {,,..., M}. Each low F enters the network rom its source node s and needs to be routed to its destination node d. Let A denote the number o packets o low arriving at its source node at time t. We assume A is i.i.d. every slot and let λ = E { A }. The operator can observe the value o A at every slot. However, the statistics o A may be unknown. We assume there exists some constant A max < such that A A max or all and t. For each pair o nodes s, d, we deine an acyclic path P between them to be a sequence o nodes P = (n, n,..., n K+ ) such that [n i, n i+ ] L or all i =,..., K, n = s, n K+ = d and n i n j i i j. In a general multihop network, lows can usually be routed to their destination via multiple paths. In this paper, in order to highlight the scheduling component, we assume that every low is routed via a single ixed path P to its destination. The results in this paper can easily be extended to the case where every low is routed over multiple acyclic ixed paths to their destinations. We deine the length o the path P = {s = n,..., n K + = d } or low to be K. We use k (n) to denote node n s order in the path P or all n P, e.g., k (s ) = and k (d ) = K +. For any node n with k (n), we use u (n) to denote its upstream node in the path P ; or any node n with k (n) K, we use l (n) to denote its downstream node in the path P. Note that this routing model is dierent rom the multihop &! &# &"

3 network considered in [], [], where no route inormation is needed. However, this assumption is not very restrictive. In many cases, e.g., the internet, such route inormation can easily be obtained. B. The Transmission Model We assume the channel states o the links are potentially time varying. We use S = (S mn, m, n N ), where S mn is the channel condition between nodes m and n, to denote the aggregate channel state vector o the network. Note that S contains inormation o channels between all pairs o nodes in the network, even pairs that do not have a communication link in between. This is to capture the act that in some cases, though one node can not communicate with another node, their transmissions can still interere with each other. We assume S takes values in a inite state space S. For simplicity, we also assume that S is i.i.d. over slots, but the components o S are allowed to be correlated. The network operator can observe the aggregate channel state inormation S at every time slot, but the distribution o S is not necessarily known. At every time t, ater observing the channel state vector S, the network operator allocates power to each link or data transmission. It does so by choosing a power allocation vector P = (P [m,n], [m, n] L), where P [m,n] denotes the power allocated to link [m, n]. We assume that i S = S S, then P is chosen rom a easible power allocation set associated with S, denoted by P (S). We assume or any S S, P (S) is compact and time invariant. Given the channel state S and the power vector P, the rate over link [m, n] at time t is given by µ [m,n] = Φ [m,n] (S, P ), or some general rate-power unction Φ [m,n] (, ). Now let µ [m,n] be the rate allocated to low over link [m, n] at time t, chosen subject to the ollowing constraint: µ [m,n] µ [m,n]. It is evident that i m u (n), i.e., m is not the upstream node o n in the path P, then µ [m,n] = 0 t. In the ollowing, we assume that there exists some µ max < such that µ [m,n] µ max or all [m, n] L, S S and P P (S). C. Queueing Dynamics and Network Capacity Region Let Q = (Q n), F, n P and t = 0,,,... be the queue backlog vector o the network, in units o packets. We assume the ollowing queueing dynamics or all node n P with k (n) K : Q n(t + ) max[q n µ [n,l (n)], 0] + µ () [u (n),n] In the above equation we assume that when k (n) =, i.e., when node n is the source node o low, µ [u (n),n] = A or all t. The inequality is due to the act that the upstream node may not have enough packets to send. When k (n) = K +, i.e., n = d, we always assume that: Q n = µ = 0 or all t. Throughout this paper, [n,l (n)] Nodes that are not used by any low are assumed to always have zero backlog or all lows. we assume the ollowing notion o queue stability: Q lim sup t t t E { Q n(τ) } <. (),n Deine Λ R M to be the network capacity region, which is the closure o all arrival rate vectors λ = (λ,..., λ M ) T or which under the routing and transmission conigurations there exists a stabilizing control algorithm that ensures (). The ollowing theorem rom [9] gives a useul characterization o the capacity region in our setting and will be useul in the ollowing analysis. Theorem : The capacity region Λ is the set o arrival vectors λ = (λ,..., λ M ) T R M such that there exists a stationary randomized power allocation and scheduling algorithm that allocates power and low rates purely as a unction o S and achieves: E { µ [n,l (n)] } = λ, F, n P : k (n) K, (3) where the expectation is taken with respect to the random channel states and the (possibly) random power allocation and scheduling actions. In the ollowing, we use S-only policies to reer to stationary randomized policies that allocate power and low rates purely as unctions o the aggregate channel state S. D. The Delay Eicient Scheduling Problem Our goal is to ind a joint power allocation and scheduling algorithm that, at every time slot, chooses the right power allocation vector and transmits the right amount o packets or each low, so as to maintain queue stability o the network and to yield good delay perormance. We will reer to this problem as the Delay Eicient Scheduling Problem (DESP). This ramework has been studied by many previous works, e.g., [], []. It is also well known that the max-weight type algorithms in e.g., [], [9], can be used to stabilize the network whenever the arrival rate vector is in the network capacity region. However, the best known delay bounds o the max-weight algorithms are not order optimal. Indeed, the delay perormance o the max-weight type algorithms can be poor except or scheduling problems in downlink systems and some single hop networks, [5], [6]. III. TOWARDS BETTER DELAY PERFORMANCE In this section, we present the Delay-Eicient SCheduling algorithm (DESC). In the ollowing, we irst introduce the idea o an Accelerating Queue (AQ). We then use these AQs to develop the DESC algorithm. A. Accelerating Queues and Redundant Constraints In this subsection, we irst deine the notion o an accelerating queue (AQ). We then provide an example relating the AQs to redundant constraints in the optimization context. For each low traversing a path P = (n, n,..., n K + ), we create K accelerating queues (AQ) Hn or n {n, n,..., n K } that evolve as ollows: H n(t + ) = max [ H n µ [n,l (n)], 0] (4) +θa + ( θ)µ [u (n),n],

4 where θ (0, ] is a parameter that is chosen independent o the network size and routing conigurations. The choice o θ will be discussed in Section IV-B. Here we similarly deine µ [u (n),n] = A i k (n) =, i.e., n = s, and Hn = µ [n,l (n)] = 0 i k (n) = K +, i.e., n = d, or all and t. These AQs are used to ensure that the instantaneous traic arrival inormation is quickly sent to all the downstream nodes. We emphasize that these AQs are virtual queues and can easily be implemented in sotware. The actual queues in the system will still obey the queueing dynamic in (). Note that (4) requires that the A values be sent to all intermediate nodes instantly, which can easily be done when centralized control is available. We will consider the case when this inormation propagation requires nonzero time in Section VI. We now relate the AQs to redundant constraints in the optimization context. Consider the example shown in Fig., where we try to support the three lows,, 3. For simplicity assume all channels are static, i.e., no channel variation, and the arrivals are constant. At each time, rate needs to be allocated to each link [m, n] L or data transmission via power allocation. Assume that due to physical constraints, e.g., intererence, the obtained rates are restricted to be within some easible rate allocation set X. Our goal is to ind an algorithm to stabilize the network whenever possible. Now let µ i [m,n] be the rate allocated to low i over link [m, n]. It has been shown in [8] that a max-weight type algorithm applied to the above routing problem is closely related to solving the ollowing optimization problem by a dual subgradient method. (P) min s.t. λ µ [,3], µ [,3] µ [3,5], λ µ [,3], µ [,3] µ [3,5], λ 3 µ 3 [,4], (µ i [m,n], i =,, 3, [m, n])t X. Now consider modiying (P) by adding two redundant constraints as ollows: (P) min s.t. λ µ [,3], µ [,3] µ [3,5], θλ + ( θ)µ [,3] µ [3,5], (5) λ µ [,3], µ [,3] µ [3,5], θλ + ( θ)µ [,3] µ [3,5], (6) λ 3 µ 3 [,4], (µ i [m,n], i =,, 3, [m, n])t X. It is easy to see that the set o optimal solutions (possibly empty) o (P) and (P) are always the same. Also, it has been observed in the optimization context, e.g., Page 585 in [7], that adding these redundant constraints usually leads to aster convergence o the variables µ i [m,n] to their target values under gradient type methods, due to the act that these additional constraints eectively reduce the search space o the optimal solution or the optimization methods. Now suppose we try to solve (P) with a dual subgradient method and we assign to the redundant constraints (5) and (6) Lagrange multipliers H 3 and H 3. Then H 3 and H 3 will be updated according to: 3 (t + ) = [ H 3 µ [3,5] ] + + θλ + ( θ)µ [,3], H 3 (t + ) = [ H 3 µ [3,5] ] + + θλ + ( θ)µ [,3], H where [x] + = max[x, 0] and µ [m,n] is obtained by solving some optimization problem. Compare these update rules with (4), we see that the Accelerating Queues correspond exactly to the Lagrange multipliers o the redundant constraints, and hence can be viewed as mimicking the unctionality o the redundant constraints. Thereore, we expect that with the aid o the AQs, the µ values will converge quickly to their [n,l (n)] target values under DESC, and hence the network delay will likely be small. B. The DESC algorithm In this section, we develop the Delay Eicient SCheduling algorithm (DESC) that will be applied to the DESP problem. Delay Eicient SCheduling Algorithm (DESC): Choose a parameter θ (0, ] independent o the network size and routing conigurations. At every time slot t, observe all queue values Q n and Hn, and do: () Link Weight Computing: For all [m, n] L such that there exists a low with m = u (n), ind the low [m,n] such that (ties broken arbitrarily): [m,n] = arg max F:m=u (n) { Q m Q n (7) } +Hm ( θ)hn. Then deine the weight o the link [m, n] to be: [ W[m,n] = max Q [m,n] m Q [m,n] n (8) ] +H [m,n] m ( θ)h [m,n] n, 0. I no such exists, i.e., [m, n] is not used by any low, deine W[m,n] = 0 t. () Power Allocation: Observe the aggregate channel state S, i S = S, choose P P (S) such that: P = arg max P P (S) µ [m,n] W[m,n], (9) [m,n] L where µ [m,n] = Φ [m,n] (S, P ). (3) Scheduling: For each link [m, n] L, allocate the transmission rate as ollows: { µ [m,n] = µ[m,n] i = [m,n] and W [m,n] > 0 0 else. That is, the ull transmission rate over each link [m, n] at time t is allocated to the low [m,n] that achieves the maximum positive weight W[m,n] o the link. I µ [m,n] [m,n] > Q [m,n] m, null bits are transmitted i needed.

5 (4) Queue Update: Update Q n and H n,, n P, according to () and (4), respectively. We note that DESC inherits almost all properties o the previous max-weight type algorithms: it does not require statistical knowledge o the arrival or channels, and it guarantees network stability whenever the arrival rate is inside the network capacity region. DESC is also not very diicult to implement. The link weights can easily be computed locally. The scheduling part can be done node-by-node. The AQs are virtual queues implemented in sotware and can easily be updated by message passing the inormation o A to all the nodes in its path, similar to the assumptions o the internet low models in [0], []. Although DESC requires routing inormation, such inormation is usually not diicult to obtain in many contexts, e.g., the internet. The most complex part is the power allocation computation, which in general can be NPhard. However, it can easily be solved in some special cases, e.g., when transmissions over dierent links do not interere with each other, e.g., internet, or it can be easily approximated at the cost o some network capacity loss [9]. We inally note that DESC is similar to the DRPC algorithm in [], in terms o power allocation and scheduling. Indeed, the major dierence is the use o AQs in DESC. However, we will see that this simple distinction can lead to signiicant dierence in algorithm perormance. IV. DESC: STABILITY AND DELAY PERFORMANCE Now we present the perormance results o DESC and discuss their implications, using the DRPC algorithm in [] as the benchmark algorithm or comparison. The proos will be presented in Section V. Our irst result characterizes the perormance o DESC in terms o queue stability. In particular, we show that whenever the arrival rate vector is within the capacity region, both the actual queues and the AQs are stable. This result shows that the DESC algorithm is throughput optimal. Our second result concerns the dierence between the aggregate packet arrival rates and the aggregate service rates allocated to the lows. This result, as we will see, states that under DESC, the service rates allocated to each low over its path converge quickly to their desired rate values. We now have our irst perormance result o DESC. Theorem : Suppose there exists an ɛ > 0 such that λ + ɛ Λ, then under the DESC algorithm, we have: n P Qn K B K ɛ n P H n n P K B θɛ Hn[θk (n) + ( θ)], K (0) Q s, () θ where is the M-dimensional column vector with every entry equal to, B = max[a max, µ max ] and the notation x is deined: x = lim sup T T T t=0 E { x }, () which denotes the expected time average o a sequence {x, t = 0,,,...}. Here the expectation is taken over the randomness o the arrivals and link states. Note here λ + ɛ Λ means that (λ + ɛ, λ + ɛ,..., λ M + ɛ) T Λ. In other words, we can increase the arrival rates o all lows by the same amount ɛ and the rate vector is still supportable. Thus ɛ can be viewed as measuring how ar is the current rate vector away rom the boundary o the capacity region Λ. Also note that the bound (0) contains the parameters K on the let-hand side, which are the path lengths o the lows. This is due to the ollowing: When analyzing the average total backlog size, one has to compare the drit under DESC with the drit under an S-only strategy. To obtain bounds on the total network backlog, the S-only strategy has to simultaneously ensure that or each low, its average output rate at every node n P (n d ) is larger than its average input rate into that node by some δ n > 0. To guarantee such a stationary randomized policy exists, we need n P δ n ɛ. In our case, all δ n are equal, hence δ n = ɛ/k and K appears on the let-hand side. We note that the bound (0) has the potential to be strictly better than the bounds o the DRPC and EDRPC algorithms in []. Indeed, the corresponding congestion bound under DRPC can be shown to be: n P Qn K K B. (3) ɛ Hence i the second term in (0) is large, then (0) can be strictly better than (3). As we will see in the simulation section, the second term is indeed large and thus the bound in (0) can actually be better than (3) or DRPC. We also note that the bound or the AQs in () is smaller than the bound (3) or the actual queues under DRPC roughly by a actor θk /. Since the AQ sizes measure the convergence speed o the allocated rates under DESC and the actual queue sizes measure the convergence speed o the allocated rates under DRPC, we see rom () and (3) that the allocated rates under DESC converge aster to their target values than under DRPC. To see this better, we deine the ollowing total approximate network service lag unction Lag at time t as: Lag = = [ K A [0, t ] [K t n P µ [n,l (n)] ] [0, t ] A (τ) t µ ], [n,l (n)] (τ) (4) n P where A [0, t ] = t A (τ) and µ [0, t ] = [n,l (n)] t µ (τ) are the total number o arrivals o low [n,l (n)] and the total amount o rate allocated to low over link [n, l (n)] in the time period [0, t ]. Since every arrival rom A needs to be transmitted K times beore reaching the destination, the quantity Lag can be viewed as approximating the total amount o eective unserved data that is currently in the network. Hence any delay eicient algorithm is expected to have a small time average value o

6 Lag. The ollowing theorem characterizes the perormance o DESC with respect to the Lag metric. Theorem 3: Suppose there exists an ɛ > 0 such that λ + ɛ Λ, then under the DESC algorithm, we have: Lag K B θ ɛ Q s θ, (5) where the notion x is deined in (). Note that i K = O(N), e.g., M = O(), and ɛ = Θ(), then (5) implies that the average total service lag in the network grows only linearly in the network size. A. Demonstrating Example As a concrete demonstration o Theorems and 3, we look at a simple example shown in Fig., where a low is going through an N + node tandem wireless network, with the source being node and the destination being node N +. Α Fig N- N μ_[,] μ_[,3] A low traversing a tandem. μ_[n, N+] In this case we see that K = N. Suppose we choose θ = in the DESC algorithm, then Theorems and 3 state that under DESC, we have: N Q n N B N nh n, (6) ɛ i= N n= H n NB ɛ n= t N t Lag = N A(τ) N+ Q, (7) n= µ [n,n+] (τ) NB.(8) ɛ Suppose the network parameters are such that λ = Θ() and ɛ = Θ(). Then it is easy to see that the term N n= nh n = Ω(N ). By subtracting out this term, the bound in (6) can likely be small. (7) shows that the time average value o the AQs is O(N), which implies that the rates allocated to the links converge in O(N) time to their target values. (8) shows that the average total network service lag at all the nodes is no more than some Θ(N) constant, whereas the average service lag under DRPC will be Ω(N ) in this case. We will see in the simulation section that under the DESC algorithm, the time average actual queue backlog can indeed be Θ(N) when λ = Θ() and ɛ = Θ(). B. Discussion on the Choice o θ We note that the results in Theorems and 3 hold or any θ (0, ]. Hence the bounds may be optimized by choosing the best θ. Intuitively, using a larger θ, e.g., θ = will lead to a aster convergence o the allocated rates. However, using a θ may also be beneicial in some cases when we want to reduce the impact o the arrival inormation, or example, when the propagated traic inormation may become noisy. V. PERFORMANCE ANALYSIS In this section we analyze the DESC algorithm by using the Lyapunov techniques developed in works [] [] [9]. To start, we irst have the ollowing lemma, which shows that i the current rate vector is strictly inside Λ, then we can ind an S-only policy that oers perturbed rates to all the nodes along the paths or all lows. Lemma : Suppose there exists an ɛ > 0 such that λ+ɛ Λ, then there exists an S-only policy under which: E { µ [n,l (n)] } = λ + δ n, n P : k (n) K (9) or any λ n δ n ɛ and or all F. Proo: By Theorem, we see that i λ + ɛ Λ, then there exists an S-only policy Π that achieves: E { µ [n,l (n)] } = λ + ɛ,, n P : k (n) K. Now we create a new S-only policy Π by modiying Π as ollows. In every time slot, allocate rates to nodes using the policy Π. However, or each n P, in every time slot, transmit packets or low with the corresponding rate with probability (λ +δn)/(λ +ɛ) (this is a valid probability since λ n δn ɛ). It is easy to see that Π is an S-only policy, and that (9) is satisied under Π. We now prove the perormance o DESC. To start, we irst deine the ollowing Lyapunov unction: L(Q, H) = ([Q n] + [H n] ). (0) n P Denote Z = (Q, H), and deine the one-slot conditional Lyapunov drit to be: = E { L(t + ) L Z }, () where we use L as a short-hand notation or L(Q, H). We have the ollowing lemma: Lemma : The drit deined in () satisies: C H { () [n,l (n)] n P ( θ)µ [u (n),n] θa Z } Q { [n,l (n)] µ [u (n),n] Z}. n P where C = K B. Proo: See Appendix A. We are now ready to prove Theorem. We will use the ollowing theorem in [9]. Theorem 4: Let Q be a vector process o queue backlogs that evolves according to some probability law, and let L(Q) be a non-negative unction o Q. I there exists processes and g and positive constants a, b > 0 such that at all time t, we have: then: b lim sup t t t ag b, E { (τ) } a lim sup t t t E { g(τ) }.

7 Proo: (Theorem ) Rearranging the terms in (), we obtain: C + H { ne θa Z } (3) n P + [ Q s + ( θ)hs ] E { A Z }, n P :k (n) K E { µ [n,l (n)] [ Q n Q l (n) +H n ( θ)h l (n) ] Z }. Now compare (3) and the DESC algorithm and recall that µ [m,n] = Φ [m,n] (S, P ), we see that the DESC algorithm chooses the power allocation vector and allocates transmission rates to lows every time slot to minimize the right-hand side (RHS) o the drit expression (3). Because the RHS o () and (3) are equivalent, the drit value satisies the ollowing: C H { (4) [n,l (n)] n P ( θ)µ [u (n),n] θa Z } Q { n P [n,l (n)] µ [u (n),n] Z}, where µ corresponds to the rate allocated to Flow [n,l (n)] on link [n, l (n)] at time t by any other alternative algorithms. Now since λ+ɛ Λ, by Lemma, we see that there exists an S-only policy that chooses the power allocation vector P and allocates transmission rates µ purely as [n,l (n)] a unction o the aggregate channel state S, and yields: E { µ [n,l (n)] Z} = λ + k (n)ɛ K, (5) or all F and n P : k (n) K. Thus: E { µ [n,l (n)] } = λ + k (n)ɛ K. (6) Now plug this alternative algorithm into (4), we have: C Q n ɛ (7) K n P Hn [θk (n) + ( θ)]ɛ, K n P which by Theorem 4 implies: [ Q ] n + H n[θk (n) + ( θ)] K K n P Rearranging terms, we have: Qn K B K ɛ n P n P C ɛ = K B. ɛ Hn[θk (n) + ( θ)]. K (8) (9) This proves (0). Now similar as above, but plug in (4) another alternative S-only policy that yields or all F: E { µ [n,l (n)] Z} = λ + ɛ, n P : k (n) K. (30) Such an algorithm exists by Lemma. We then obtain: C Q s ɛ Hnθɛ (3) n P H s ( θ)ɛ. Using the act that H s ( θ)ɛ 0, this implies that: Q s θ + H n K B θɛ n P (3) Proving the theorem. Now we prove Theorem 3: Proo: (Theorem 3) For a low F, let its path be P = {n, n,..., n K +}. From (4), it is easy to show that or all t, we have [9]: which implies: t t Hn A (τ) µ [n (τ),,n ] t µ [n (τ) t,n ] A (τ) Hn. (33) Repeating the above or n, we have: H n t [θa (τ) + ( θ)µ [n (τ)] t,n ] µ [n (τ),n 3] t t A (τ) µ [n (τ) (,n 3] θ)h n, where the second inequality ollows rom (33). Hence: t t Hn + ( θ)hn A (τ) µ [n (τ).,n 3] More generally, we have or all i =,..., K that: i t t ( θ) i j Hn j A (τ) µ [n (τ). i,n i+] j= Summing up all i =,,..., K, we have: K i= j= i ( θ) i j Hn j K [ t i= t A (τ) µ [n (τ) i,n i+] ]. However: K i ( θ) i j Hn j = i= j= K Hn i [ K i ( θ) j] i= K Hn θ i, i= j=0

8 which implies: K [ t i= t ] A (τ) µ [n (τ) i,n i+] K Hn θ i. i= Summing this over all F and using () in Theorem proves Theorem 3. VI. DESC UNDER DELAYED ARRIVAL INFORMATION Here we consider the case when the time required to propagate the arrival inormation A is nonzero. Such a case can happen, or instance, when there is no central controller and thus message passing is required to propagate the A values. Let D n be the delay (number o slots) to propagate the A inormation rom s to node n P at time t. We assume there exists a constant D < such that D n D or all, n, t. Note that in this case, we can no longer use (4) to update the AQs due to the message passing delay. Instead, we modiy the DESC algorithm to use the delayed traic inormation. Speciically, we create a set o AQs using the delayed traic inormation A (t D) as ollows: For all 0 t < D, let H n = 0 and or all t D, we update the AQs according to: H n(t + ) = max [ H n µ [n,l (n)], 0] (34) +θa (t D) + ( θ)µ [u (n),n]. We then deine the ollowing Delayed-DESC algorithm to perorm power allocation and scheduling. Delayed-DESC: Choose a parameter θ (0, ] independent o the network size and routing conigurations. At every time slot, observe all queue values Q n(t D) and Hn, and do: ) Link Weight Computing: For all [m, n] L such that there exists a low with m = u (n), ind the low [m,n] [m,n] = such that (ties broken arbitrarily): arg max F:m=u (n) { Q m(t D) Q n(t D) (35) } +Hm ( θ)hn. Then deine the weight o the link [m, n] to be: [ W[m,n] = max Q [m,n] m (t D) Q [m,n] n (t D) (36) ] +H [m,n] m ( θ)h [m,n] n, 0. I no such exists, i.e., [m, n] is not used by any low, deine W[m,n] = 0 t. ) Power Allocation and Scheduling are the same as DESC except that the weights are now given by (36). 3) Queue Update: Update Q n and Hn or all n, according to () and (34), respectively. Speciically, Delayed-DESC is the same as DESC except that at every time slot t, it uses (Q(t D), H) as the queue backlog vector to perorm power and rate allocation, and the AQ values are updated according (34) instead o (4). The perormance o Delayed-DESC is summarized in the ollowing theorem. Theorem 5: Suppose there exists an ɛ > 0 such that λ + ɛ Λ, then under the Delayed-DESC algorithm with the parameter D, we have: Qn K ( + D)B K ɛ n P n P Hn[θk (n) + ( θ)ɛ], K H n K ( + D)B θɛ n P Lag K ( + D)B θ ɛ Q s, θ Q s θ, where x represents the expected time average o the sequence {x} t=0 and is deined in (). Note that since we typically only need a ew bits to represent the A values, and we can pass these AQ inormation at a much aster time scale, i.e., not necessarily once per slot, the D value will typically be very small. In this case, Theorem 5 states that Delayed-DESC perorms nearly as well as DESC. Proo: (Theorem 5) Let Ẑ = (Q(t D), H) be the delayed queue state and deine the drit to be: E { L(t + ) L Ẑ}, where i t D < 0 we deine Q n(t D) = 0. Now using Lemma, we see that: C H { (37) [n,l (n)] n P n P E ( θ)µ [u (n),n] θa (t D) Ẑ} { Q n [ µ [n,l (n)] µ [u (n),n] ] Ẑ}, where C = K B. Denote the RHS o (37) as R. Using the act that or all and n P : Q n(t D) DB Q n Q n(t D) + DB, we have: Q [ n µ [n,l (n)] µ [u (n),n] ] n P DB n P + Q [ n(t D) µ [n,l (n)] µ [u (n),n] ], n P where B = max[a max, µ max ]. Plug this into (37), we get: R C + K DB (38) Q { n(t D)E µ [n,l (n)] µ Ẑ} [u (n),n] n P H { [n,l (n)] n P ( θ)µ [u (n),n] θa (t D) Ẑ}.

9 We can similarly see that the power and rate allocation o Delayed-DESC minimizes the RHS o (38). Hence the above inequality holds i we plug in any alternative power and rate allocation policy. Thus under Delayed-DESC, we have: C + K DB (39) Q { n(t D)E µ [n,l µ Ẑ} (n)] [u (n),n] n P H { [n,l (n)] n P ( θ)µ [u (n),n] θa (t D) Ẑ}, where µ is the rate allocated to low over link [n,l (n)] [n, l (n)] by any alternative policy. We can now carry out a similar argument as in the proo o Theorem and prove the theorem. The details are omitted or brevity. VII. SIMULATION Here we provide simulation results o DESC. The network topology and low coniguration are shown in Fig. 3. We assume the channel conditions are independent and each link [m, n] is i.i.d. every slot being ON with probability 0.8 and OF F with probability 0.. When the channel is ON, we can allocate one unit o power and transmit two packets; otherwise we can send zero packets. We urther assume that all links can be activated without aecting others. However, a node can only transmit over one link at a time, though it can simultaneously receive packets rom multiple nodes. Each low i is an independent Bernoulli process with A i = with probability λ i / and A i = 0 else. The rate vector is given by λ = (λ, λ, λ 3, λ 4 ) T = (0.8, 0.4, 0., 0.6) T. We simulate the system with (h = η, υ = η ), where η, h and υ are parameters in Fig. 3. Note that in this case, N = 3η + 7. The η value is chosen to be {0, 50, 00, 00, 500}. We use θ = 0.5. that n P H n n P Q n,. By equation () o Theorem, this implies that : Q n H n K B. θɛ n P n P Since we also have K = O(N) and ɛ = Θ() in this example, we see that indeed n P Q n = O(N) in this case. This suggests that DESC can potentially be a way to achieve delay-order-optimal scheduling in general multihop networks. Fig. 5 shows that the total average rates allocated to Flow and over their paths, i.e., tk n i P µ [n i,l (n i)] [0, t ] and tk n j P µ [n j,l (n [0, t ] with µ k j)] [0, t ] = [n i,l k (n i)] t µ k [n i,l k (n i)] (τ), converge quickly to above the actual average arrival rates i.e., t A [0, t ]. Whereas the corresponding rates under DRPC converge very slowly rom below to the actual average arrival rate. These plots suggest that the poor delay perormance o many max-weight type algorithms in multihop networks can be due to slow convergence o the corresponding service rates "=0.5 H H Q Q ! H4 H3 Q4 Q ! Fig. 4. Q i and H i, i =,, 3, 4, are the average total actual and AQ backlog sizes o low i, respectively. 0.8 s s 3 3+h η-k η- η d Avg. Rate Avg. Rate to Flow by DPRC Flow Avg. Arrival Rate Avg. Rate to Flow by DESC 3 s3 d3 4 s4 d Time x ν d Fig. 3. A Network with 4 Flows. η is s path length, h measures the path overlap length o and, and υ is the vertical path length o. Avg. Rate Avg. Rate to Flow by DRPC Flow Avg. Arrival Rate Avg. Rate to Flow by DESC Fig. 4 and Fig. 5 show the simulation results. Fig. 4 shows that under DESC, the average total backlogs o Flow and scale only linearly in N. This is in contrast to the example provided in [8], which shows that the average backlog grows quadratically with N under the usual max-weight scheduling policy. We also see that the average total backlog o Flow 3 and 4 remains roughly the same. This is intuitive, as their path lengths do not grow with N. Interestingly, we observe Fig. 5. UP: the average rate allocated to Flow (η = 00); DOWN: the average rate allocated to Flow (η = 00). Time VIII. CONCLUSION In this paper, we consider the problem o delay-eicient scheduling in general multihop networks. We develop a maxweight type Delay Eicient SCheduling Algorithm (DESC). x 0 4

10 We show that DESC is throughput optimal and derive a queueing bound which can potentially be better than previous congestion bounds on max-weight type algorithms. We also show that under DESC, the time required or the allocated rates to converge to their target values scales only linearly in the network size. This contrasts with the usual max-weight algorithms, which typically require a time that is at least quadratic in the network size. APPENDIX A PROOF OF LEMMA Proo: Deine B = max[a max, µ max ] and denote [x] + = max[x, 0]. From the queueing dynamic () we have or all node n P with k (n) K that: [Q n(t + )] = ( [Q n µ [n,l (n)] ]+ + µ [u (n),n] ) = ( [Q n µ [n,l (n)] ]+) + [µ [u (n),n] ] +([Q n µ [n,l (n)] ]+ )µ [u (n),n] [Q n µ [n,l (n)] ] + B + Q nµ [u (n),n]. The inequality holds since or any x R, we have ([x] + ) x, and ([Q n µ [n,l (n)] ]+ )µ [u (n),n] Q nµ. We also use in the above equation that [u (n),n] i k (n) =, i.e., node n is the source node o low, then µ [u (n),n] = A or all t. Thus by expanding the term [Q n µ [n,l (n)] ], we have or all n P with k (n) K that: [Q n(t + )] [Q n] + B Q n [ µ [n,l (n)] µ [u (n),n] ]. Note that i k (n) = K +, i.e., n = d, we have Q n = 0 or all t, and so [Q n(t + )] [Q n] = 0, t. Summing the above over all n,, and multiply by, we have: [Q n(t + )] [Q n] (40) n P n P Y B Q [ n µ [n,l (n)] µ [u (n),n] ], n P where Y = K. Repeat the above argument on the term n P [Hn(t + )] n P [Hn], we obtain: [H n(t + )] [H n] (4) n P n P Y B Hn[µ [n,l (n)] ( θ)µ [u (n),n] n P θa ]. Now adding (40) to (4), taking expectations conditioning on Z = (Q, H), and letting C = Y B = K B proves the lemma. REFERENCES [] L. Tassiulas and A. Ephremides. Stability properties o constrained queueing systems and scheduling policies or maximum throughput in multihop radio networks. 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