On Scheduling for Minimizing End-to-End Buffer Usage over Multihop Wireless Networks

Transcription

1 On Scheduling for Minimizing End-to-End Buffer Usage over Multihop Wireless Networs V.J. Venataramanan and Xiaojun Lin School of ECE Purdue University Lei Ying Department of ECE Iowa State University Sanjay Shaottai Department of ECE The University of Texas at Austin Abstract While there has been much progress in designing bacpressure based stabilizing algorithms for multihop wireless networs, end-to-end performance e.g., end-to-end buffer usage results have not been as forthcoming. In this paper, we study the end-to-end buffer usage sum of buffer utilization along a flow path over a networ with general topology and with fixed, loopfree routes using a large-deviations approach. We first derive bounds on the best performance that any scheduling algorithm can achieve. Based on the intuition from the bounds, we propose a class of bacpressure-lie scheduling algorithms called αβalgorithms. We show that the parameters α and β can be chosen such that the system under the αβ-algorithm performs arbitrarily closely to the best possible scheduler formally the decay rate function for end-to-end buffer overflow is shown to be arbitrarily close to optimal in the large-buffer regime. We also develop variants which have the same asymptotic optimality property, and also provide good performance in the small-buffer regime. Our results are substantiated using both analysis and simulation. I. INTRODUCTION Scheduling is one of the most critical challenges in multihop wireless networ design due to channel fading and wireless interference. A major breathrough in this area is the bac-pressure algorithm proposed in [], which is throughput optimal, i.e., it can stabilize any traffic flows that can be stabilized by some other algorithms. Significant progress has been made in designing bacpressure based algorithms to imize the networ throughput or networ utility as a function of the throughput [] [6]. While these results provide stabilizing algorithms with throughput-optimality guarantees, they may not have the desired performance in terms of other end-to-end performance metrics such as end-to-end delay. For example, results in [7], [8] have demonstrated that the classic bacpressure algorithm could lead to unnecessarily large delays in multihop networs. Since many emerging applications of wireless networs, such as wireless mesh networs for public safety, wireless sensor networs for unmanned surveillance, and vehicular networs for accident warnings, require delay constrained communication for desired performance, we are interested in scheduling algorithms that not only guarantee the throughput but also have good delay performance. It has been observed in [9], [0] that scheduling algorithms that do not tae queue length information into consideration perform much worse than queue-length-based algorithms. However, the delay analysis for queue-length-based algorithms is challenging because of the dependence of the decision process on the queue length process. Existing results in the literature have focused on order optimality [9], the heavy traffic regimes [] [4] and the large-queue regimes [0], [5] [7], but they only consider single-hop flows. In fact, the coupled arrival/departure processes the departures from the previous hop is the arrivals of current hop of multihop traffic flows have aggravated the difficulty in analyzing the end-to-end delay performance. In this paper, we consider a multihop wireless networ with general topology and with fixed, loop-free routes. We assume that a node maintains a separate queue for each flow passing through it. To simplify the analysis, we use the end-to-end buffer usage to approximate the end-to-end delay, and study the probability that the end-to-end buffer usage exceeds a certain threshold. Let X agg t denote the aggregated queuelength along the route of flow and denote the average rate at which data arrives at the source node of flow. Mathematically, we are interested in characterizing P [ X agg t B ]. We call X agg t the end-to-end buffer usage of flow, which is closely related to the end-to-end delay of flow. For example, assuming that the pacets arrive with a constant rate and all queues are FIFO, then Xagg t is the delay experienced by the pacet of flow that departs the system at time t see [9]. In this case, a scheduling algorithm resulting in a small value of guarantees that the probability that the end-to-end delays are larger than B will also be small. We exploit large-deviations analysis to study this quantity. The main contributions of this paper include: We first derive bounds on the best performance that any scheduling algorithm can achieve. In other words, we obtain a θ 0 such that [ X agg t ] B θ 0 lim inf B log P holds for any scheduling policy. We obtain two fundamental structural properties from this bound: i considering a single multihop flow, the scheduling algorithm should give preference to lins that are closer to the destination; and ii considering multiple

2 2 flows competing for a single multi-access channel e.g., at the downlin of a single cell in a cellular networ, the scheduling algorithm should give preference to those flows with the largest ratio of aggregate baclog to arrival rate. Based on the structural properties derived from the upper bound, we propose a class of bacpressure-lie scheduling algorithms called αβ-algorithms. Exploiting the large-deviations analysis developed in [9], we show that the parameters α and β can be chosen such that the system under the αβ-algorithm performs arbitrarily closely to the best possible scheduler. Compared to [9], the main contribution of this paper is to design an algorithm that is tailored to minimizing the end-to-end buffer overflow probability in multihop networs. Finally, we develop variants of αβ-algorithms called hybrid αβ-algorithm that have the same asymptotic optimality property, and also provides good performance in the small-buffer regime. Our simulations demonstrate that the hybrid αβ-algorithm performs better than the classic bac-pressure algorithm. II. SYSTEM MODEL A. Traffic Model We study a multihop networ consisting of N nodes, L lins, and K multihop flows. Denote by bl and el the beginning and ending nodes of lin l. The route for each flow is fixed and loop-free. Denote by D the number of nodes through which flow passes, n i the i th node in flow s path, l i the i th lin in flow s path, and K l the set of flows that traverse lin l. Furthermore, let A t denote the amount of data flow injects to source node n at time t. We assume that A t are identically and independently distributed i.i.d. across time slots and E[A t] =. We assume that the average arrival rates are within the capacity region of the networ hence the system is stationary and ergodic and that the arrival processes A t s satisfy a largedeviations principle with rate function L as defined in [9]. B. Channel Model We consider a multihop wireless networ in this paper, where the lins experience interference and fading. We assume that the time is slotted, where the channel state stays constant over each time-slot, and changes at the beginning of each time slot. We let denote the channel state at time-slot t, which is i.i.d. across time slots and taes values from,...,s with probabilities p,...,p S. Now given that = j, Fj l denotes the units of data that can be transmitted over lin l if there is no interference from other lins, and A j denotes the collection of subsets of lins that do not interfere with each other when transmitting simultaneously. An element of A j is called a schedule, which is a length-l binary vector. Consider a schedule a A j, then a l =indicates that lin l transmits under schedule a. Note that if a,...,a l,,...,a L is a possible schedule, then so is a,...,a l, 0,...,a L. Define the sets Êj = {a Fj,...,a LFj L : a A j} and E j = {γ Fj,...,γ LFj L:0 γ i a i for some a A j }. C. Queueing Each node maintains a separate queue for each flow. Let Xi t denote the queue at node i for flow, and let El, Xt denote the units of data of flow transmitted over lin l in time-slot t. We also define E l, Xt = K l El, Xt to be the net amount of data transmitted over lin l. Note that the vector E j, Xt,...,E L j, Xt must belong to the set E j. The queues for flow evolve as follows: X n t + = X n t+a t E l, Xt X n i t + = X n i t+e l i, Xt E l i, Xt for i =2,...,D X n D t = 0 for all time t Xnt = 0 for all other nodes n and all time t Here, we implicitly assume that El, Xt cannot be larger than the available amount of data at the node bl. III. OBJECTIVE, MAIN RESULTS AND INTUITION A. Objective The goal of a scheduling algorithm is to determine El, Xt subject to fading and interference constraints. In this paper, we are interested in desiging a scheduling algorithm that minimizes the following queue-overflow probability: D i= P n i 0 =,...,K >B, 2 where X n i0 denotes the queue-length at the steady state. As we have discussed in the introduction, the quantity in 2 is closely related to the end-to-end delays. Since it is very difficult to precisely characterize the probability of queue overflow 2 for the general networ model that we consider, we use the large-deviations theory to study its asymptotic decay-rate as B. Specifically, define I lim inf J lim sup B log P B log P =,...,K =,...,K D i= X n i 0 >B D i= X n i 0 >B,. The significance of studying these quantities lies in the following approximations: P[M X >B] e JB+oB P[M X >B] e IB+oB, where M X D i= X n i =,...,K, and I and J are upper and lower bounds, respectively, of the asymptotic decay rates.

3 3 X 3 n 3 3 Destination 3 X n Destination X 2 n 2 2 Source X n X n 2 X n 3 Destination 2 3 Destination 2 Fig.. Cellular downlin topology with multiple flows. The optimal algorithm gives preference to serving users with larger value of scaled aggregated queue-length. In this example users and 3 are preferred over user 2. B. Two Structural Principles and The Class of αβ Algorithms In section IV-A, we will derive an upper bound on the best decay rate I. By analyzing this upper bound, we obtain the two basic structural principles: Proposition 2 in Section IV: Considering a tandem networ with a single flow, the optimal algorithm should schedule lins in such a way that lins closer to the destination get preference in service. In other words, all buffering occurs as close to the source node as possible see Figure 2. This result indicates the following structural principle: Principle : give preference to scheduling lins closer to destination. Proposition 3 in Section IV: Consider a cellular downlin topology where there is a single base station serving K cellular users. The optimal algorithm should schedule the user with the largest value of scaled queue baclog i.e. X n t/ see Figure. This result indicates the following structural principle: Principle 2: give preference to serving users with larger value of scaled aggregated queue-length. Based on the observations above, we propose a class of scheduling algorithms parametrized by α and β called αβalgorithms, which is similar to the bac-pressure scheduling algorithm [], but with different ways of defining lin weights. αβ-scheduling algorithm: For each flow, we define V X = D i= X n i α+ α+, and Wl t = V Xt β α β+ Xbl tα Xel tα. The αβ-algorithm assigns a weight W l t to lin l such that W l t = W l t. { K l } At each time slot, the αβ-scheduling algorithm computes an activation vector a A such that the vector e a F,...,a L F L satisfies L l= W l te l = e Ê L W l te l. l= Fig. 2. Tandem topology with single flow. The optimal algorithm gives preference to serving lins near the destination. Hence most of the buffering occurs at source node. If lin a l =, the scheduling algorithm activates lin l and serves flow l witharatemin{f l,x l bl t}, where the flow l satisfies W l l t =W l t. Note that the αβ-algorithm minimizes the drift of the K Lyapunov function = V X β+ β+ in a fluid scaled sense see Proposition 4. When α =and β =,theαβalgorithm is equivalent to the bac-pressure algorithm []. The behavior of the αβ-scheduling algorithm: Consider a flow in the networ, and compare Wl t over each hop l. Consider first the case when all Xnt > 0. Since V Xt β α / β+ is the same for all l, we denote the weight Wl t to be Wl t =κ Xbl tα Xel tα. Note that the weight associated with the last lin is W l D t =κx n D tα. In the scheduling step, the lin with the larger value of Wl t will have the preference to be activated. When α 0, the weight of the last lin W l D t = κx n D tα will dominate the weights, W t, of all other lins since lim α 0 X bl tα X el tα =0, for all lins before the last hop, and lim α 0 X n D tα =. Hence, for α 0, whenever the node n D has pacets to transmit, it will get preference to do so. Similarly, if X n D =0and X n D 2 > 0, then the weight W l D 2 t will dominate all the weights associated with other lins, and node n D 2 will get preference in service. In summary, lins closer to the destination get preference in service, so the αβ-scheduling algorithm satisfies Principle. Now, consider a networ where single-hop flows compete for a single multiaccess channel l e.g., in the downlin of a cell. In this case, the lin weight Wl t simplifies to X n tβ since the baclog at the destination node l

4 4 is 0. The user with the largest value of X n tβ F l β+ will be served. Equivalently, the user with the largest value of X n t F l β will be served. When β is + β very large, the user with the largest value of X n t among the set of users with F l > 0 will have the priority to be served. In other words, the users with larger values of scaled baclogs will get preferences to be served, which satisfies Principle 2. In Section IV, we will exploit large-deviations theory to analyze the performance of the class of αβ-scheduling algorithms, and show that this class of algorithms yield the optimal decay rate as α 0 and β. Letting P αβ represent the stationary probability under the αβ-algorithm, and P π be the stationary probability under any scheduling algorithm π, we will prove the following result: Main Result Proposition 5: Considering the class of αβalgorithms, we have lim lim sup α 0 B logpαβ [ β =,...,K D i= X n i 0 >B] D B logpπ i= [ X n i 0 =,...,K >B], which implies that by choosing α sufficiently small and β sufficiently large, the decay rate of D i= P X n i 0 =,...,K >B for the αβ-algorithm can be made arbitrarily close to the optimal decay rate. IV. ANALYSIS In this section, we will first derive an upper bound on the decay rate I achievable by any scheduling algorithm. Then, by studying the bound under some specific topologies, we will obtain the two structure principles Principles and 2 that lead to the αβ-scheduling algorithm. Finally, we will prove that the class of αβ-algorithms is asymptotically optimal. A. An upper bound on the decay-rate We consider the optimization problem w φ, f defined in 3 below. The quantity f can be interpreted as the long term average rate at which data arrives for flow, can be interpreted as the long term fraction of time-slots for which the channel is in state j, the quantity θl,j can be interpreted as the long term fraction of time that service is given to flow over lin l in channel state j, γ e,j can be interpreted as the long term fraction of time that the schedule e is used when channel state is j, and x n is the average rate of growth of the baclog of flow at node n. w φ, f = min subject to {x n i,θ l,γ e,j} =,...,K i,j x n =[f x n i =[ S { e E j} j= S j= S j= D i= x n i { e Êj} γ e,j θ l,j e l ] + { e Êj} γ e,j θ l i,j e l i { e Êj} γ e,j θ l i,j e l i] + for i =2,...,D γ e,j =, θl,j = { K l } γ e,j 0 for all e, j θl,j 0 for all, l, j. 3 For any given φ and f, the solution to the optimization problem w φ, f is the optimal long term scheduling assignments θl,j and γ e,j such that the long term average rate D of the growth of M X i= i.e. =,...,K x n i is minimized. Interpreting this in another way, w φ, f is a lower bound on the average rate of growth of M X for any algorithm. If there exists an optimal algorithm that can attain this lower bound on the average rate of growth, then it must use the long-term scheduling assignment that corresponds to the solution of w φ, f. We will soon use this optimization problem to obtain basic insights about the behavior of this optimal algorithm. Note that since is in the capacity region, if = p j and f = then the long term average rate of growth of queue baclogs will be zero for any throughput-optimal scheduling algorithm such as the bac-pressure algorithm. Hence it will be zero for the optimal algorithm. i.e. w p, =0. Therefore, to mae M X exceed a large value B, the long term average channel probability must be φ p and/or f. As is typical of large deviations results, this deviation from the norm is associated with a cost. The cost for φ to deviate from p per unit time is given by the relative entropy function H φ p = S j= log φj p j. The cost for f to deviate from per unit time is given by L f K = L f. Consider a timescale of length w φ, f over which the deviation from the norm is given by φ and f. Since w φ, f is the average rate of growth for the optimal algorithm, the system must overflow after the time period w φ, f under all algorithms. Hence the corresponding cost, given by w φ, f [H φ p+l f], provides an upper bound on the minimum cost to overflow for any algorithm. Minimizing this value over φ and f, we then obtain

5 5 the tightest upper bound given below. Please refer to [9] for a more technical and precise discussion. Define θ 0 = inf { φ, f: w φ, f>0} w φ, f [H φ p+l f] 4 Proposition : For any scheduling policy π, we have lim inf B log Pπ [ =,...,K D i= X n i 0 B] θ 0 Proof: Due to space constraints we do not provide the proof. Please see [20] Proposition shows that the decay rate I of any scheduling algorithm is less than θ 0. B. Insights into the behaviour of optimal algorithm Let us use a heuristic interpretation of the optimization problem w φ, f to obtain insights into the behavior of the optimal algorithm. We will establish the two principles Proposition 2 and Proposition 3 by studying a single flow tandem networ and a multiflow single-hop cellular downlin networ. Tandem topology Figure 2: Consider a tandem topology with a single flow in the networ, i.e., K = and L = D. For simplicity, assume that at most one lin can be activated in a time slot. The optimization problem w φ, f simplifies to w tandem φ, f: w tandem φ, f = min {x n,γ i e,j} subject to D x n =[f x n i =[ S { e E j} j= i= x n i 5 S j= S j= { e Êj} γ e,j e l ] + { e Êj} γ e,j e l i { e Êj} γ e,j e l i] + for i =2,...,D γ e,j = γ e,j 0 for all e, j. Proposition 2: One of the optimal solutions to the optimization problem w tandem φ, f has the following property x n i = 0 for i =2,...,D 6 x n 0 Proof: Since only one lin can be active in a time slot, we have Ê j = {0,...,F l i j,...,0 : i =,...,D } {0,...,0}. To prove this proposition it is sufficient to show that for any feasible assignment x n i,γ e,j, there exists another assignment ˆx n i, ˆγ e,j that satisfies the condition 6 and is such that D i= ˆx n i D i= x n i. i.e. its objective function value 5 is no greater than the original assignment To show this, consider any i 2 such that x n i > 0. We can reduce the value of γ l...,f i and increase j,...,j the value of γ 0,...,0,j, i.e., reducing the fraction of time spent on serving lin l i. This will reduce the value of S j= φ j { e Êj} γ e,je l i and x n i =[ S j= { e Êj} γ e,j e l i S j= { e Êj} γ e,j e l i] +. On the other hand, x n i may increase. If x n i increases, it will increase by at most the same amount by which x n i reduces as long as x n i > 0. We can perform this operation until x n i =0. Hence, in this process, the sum x n i + x n i either stays same or decreases. Applying the above procedure starting from the node n D and woring bacward to node n 2, it is easy to see that there exists an assignment ˆγ e,j resulting in values ˆx n i ; i =,...,D such that D i= ˆx n i D i= x n i and ˆx n i = 0 for i =2,...,D, which leads to the proposition. The significance of Proposition 2 is that it implies the scheduling algorithm should give preference to scheduling lins that are closer to the destination node, and thus significant buffering occurs only at the source node. 2 Cellular topology Figure : Consider a cellular downlin with a single base station and K users. The base station can communicate directly with the users. In this networ, we have L = K and D = 2 for =,...,K and n = n 2 =... = n. Due to wireless interference, we assume that only one user can be served in a timeslot. The optimization problem w φ, f simplifies to w cellular φ, f: w cellular φ, f = min subject to {x n,γ e,j} =,...,K x n i x n =[f { e E j} S γ e,j e l ] + j= { e Êj} for =,...,K γ e,j = γ e,j 0 for all e, j. Since only one lin can be activated in a time slot, we have Ê j = {0,...,F l j,...,0 : =,...,K} {0,...,0}. Proposition 3: One of the optimal solutions to the optimization problem w cellular φ, f has the following property:

6 6 γ...,f l r j,...,j = 0 for any user r such that x r n r r < =,...,K x n. Proof: Assume that the optimal solution to w cellular φ, f is such that there exists some user r such that xr n r < r =,...,K x n, γ l...,f r j,...,j > 0. We will show that it is possible to maintain the value of =,...,K x n while modifying γ e in such a way that γ l...,f r j,...,j =0for any user r such that r xn r < r =,...,K x n. To achieve this, we reduce the value of γ l...,f r and increase the value of γ j,...,j 0,...,0,j. In other words, we reduce the service given to user r, and increase the time that the scheduler spends idling. Due to x this, r n r will increase. This is done till we end up with r either γ l...,f r j,...,j =0and x r n r < r =,...,K x n or γ l...,f r j,...,j 0 and x r n r = r =,...,K x n. Note that the value of =,...,K x n is not affected by this procedure. Therefore, there is an optimal solution that satisfies the r x property that γ l...,f r n =0for any user r with r j,...,j < r =,...,K x n. The significance of proposition 3 is that it tells us that the optimal value of w cellular φ, f is achieved by serving the users with the largest average rate of growth of end-to-end baclog scaled by the average arrival rate. In other words, the optimal scheduling algorithm should give preference to those users with the largest ratio of end-to-end baclog to arrival rate. C. Asymptotic optimality of the class of αβ-algorithms Based on the two principles above, we propose the class of αβ scheduling algorithms as described in Section III-B. We will use the Lyapunov-function-based large deviations approach developed in [9] to show that this class of algorithms is asymptotically optimal as α 0 and β. Next, we first show that the αβ algorithm is large deviations decay rate optimal for minimizing P[V X0 >B], where V is a Lyapunov function defined to be: V X K = V X β+ β+. 7 = We can show that the αβ-algorithm minimizes the drift of this Lyapunov function in a fluid-sample-path sense see the proof of Proposition 4 in [20]. Since lim α 0,β V X D i= = X n i =,...,K, the function V X can be viewed as an approximation of our objective function M X with the parameters α and β controlling the degree of approximation. Letting P αβ represent the stationary probability under the αβ-algorithm and P π be the stationary probability under any scheduling algorithm π, we first have the following proposition. Proposition 4: The quantity lim B logpαβ [V X0 >B] exists and for any scheduling policy π, lim B logpαβ [V X0 >B] B logpπ [V X0 >B]. Proof: It follows from our prior wor [9] that a scheduling algorithm minimizing the drift of a Lyapunov function will imize the decay rate of the probability that the Lyapunov function value exceeds a large threshold. Please refer to [20] for details. Now we proceed to show that the class of αβ-algorithms is asymptotically optimal in terms of the large deviations decay rate for the stationary probability P[M X0 >B]. Proposition 5: Considering the αβ-scheduling algorithm, we have lim lim sup α 0 β B logpαβ [ B logpπ [ =,...,K =,...,K D i= X n i 0 >B] D i= X n i 0 >B]. Proof: Using Proposition 4 and standard inequalities, it can be shown that K β+ N α α+ lim sup B logpαβ [ =,...,K D i= X n i 0 >B] D B logpπ i= [ X n i 0 =,...,K >B] from which the result follows. Please refer to [20] for details. V. PRACTICAL SCHEDULING ALGORITHMS WITH GOOD DELAY PERFORMANCE So far, we have seen that by choosing α sufficiently small and β sufficiently large, we can mae the αβ-algorithm have a large deviations decay rate that is arbitrarily close to the optimal decay rate Proposition 5. Note that large deviations behaviour deals with the regime of large buffer lengths. In this section, we will discuss how to design a hybrid algorithm that has the same large deviations decay rate performance of an αβ-algorithm while at the same time having a better performance in the regime of small buffer lengths. To achieve this goal, the hybrid algorithm emulates the behaviour of βalgorithm for small queue baclogs and the behaviour of αβalgorithm for large queue baclogs. Consider a three-node, two-lin tandem networ with one flow as shown in Fig. 3. Since there is only one flow, we will

7 7 Source n X n2 Fig. 3. n 2 l l 2 Three node tandem topology Serve l2 γ F l X n Serve l Destination Decision boundary +, F l2 F l2 Fig. 4. State space plot for β-algorithm. γ is a parameter used to define the decision boundary. simplify some of the notation by not specifying the flow e.g. we will use X ni instead of X n i. Due to interference, only one lin can be served. The scheduler must decide whether to serve l or l2. Consider the αβ-algorithm. The state space i.e. a plot of X n vs. X n2 is divided into two regions by the line specified by X n α X n2 α F l = X n2 α F l2 see Fig. 5. If the state i.e. the ordered pair X n,x n2 falls in the region above the line, lin l2 will be served. If the state falls in the region below the line, l will be served. In either case, as a consequence of being served, the state will move towards the decision boundary. For β-algorithm, we obtain the state space shown in Fig. 4. For αβ-algorithm with small α, we obtain the state space X n2 Serve l2 γ + /α, /α F l F l2 F l2 X n Serve l Decision boundary Fig. 5. State space plot for αβ-algorithm. γ is a parameter used to define the decision boundary. X n2 Serve l2 Fig. 6. X n B n,b n 2 Serve l State space plot for hybrid algorithm. Decision boundary shown in Fig. 5. In the case of small α, because the decision line moves toward the x-axis, the state of the system tends to squeeze out towards the right. Hence, for the αβ-algorithm with small α, the state of the system tends to stay further away from the origin in comparison to the β-algorithm. This leads to larger values of M X=X n + X n2 / Note that this is not in contradiction to our theoretical result since our theoretical result is a result on the asymptotic rate of decay at large buffer levels. To overcome this problem, we can construct a hybrid policy which behaves lie β-algorithm for small buffer lengths and then switches to αβ-algorithm when the buffer lengths are larger. The state space for this hybrid policy is shown in Fig. 6. The decision boundary for the hybrid policy is composed of the decision boundaries shown in Fig. 4 and 5 with the point Bn,B n 2 being the point of concatenation. The details of the hybrid algorithm are described next. A. Hybrid algorithms The hybrid algorithm uses the following function to assign weight to the lins. W l t = { K l } V Xt β α [ZX bl t ZX el t], where for any flow and i th node in the path of flow, { X ZX n i = n i if X n i <B n i [X n i B n i α + B n i ] otherwise. B n i are threshold values which are constants. The motivation behind the function Z is as follows. For now, let us assume that we now the values of B n i. Since we want the hybrid to emulate β-algorithm for small values of baclog, we want ZX n i = X n i when X n i is less than a certain threshold B n i. Further, since we want to emulate αβ-algorithm with small value of α when the baclog is large, we want ZX n i = X n i α when X n i > B n i. However, a problem with these two choices is that the function ZX n i is not continuous at B n i. Hence we modify the value of ZX n i for X n i >B n i to [X n i B n i α + B n i ]. Now we loo at one way of choosing the values B n i. Consider again the single flow tandem networ of Fig. 3. From the definition of the hybrid policy, we see that when X n >Bn and X n2 >Bn 2, the decision boundary is determined by the line X n Bn α + Bn X n2 B n 2 α B n 2 = F l X n2 B n 2 α + B n 2 F l2. With some algebra, it can be shown that the line can be

8 8 expressed as γ F l + B F l2 n α + B n, 8 γ F l2 B n 2 α + B n 2 using a parameter γ. Since we want the decision boundary of the hybrid policy to loo lie figure 6, the line 8 should be of the form ˆγ + F l F l2 α, F l2 α +K,K 2,where ˆγ is a parameter and K and K 2 are constants. This provides us with the following solution for Bn,B n 2. Bn,B l n 2=κ/F l2 l2 +/F, /F where κ is some constant. Generalizing this idea to a single flow tandem networ with D nodes, we obtain B n,b n 2,...,B n D D = κ i= F l i D, i=2 F l i,..., F l D Note that the threshold values Bn i depend on the channel state. We impose an additional constraint that the sum of the thresholds should be constant, i.e. for any channel state, D i= Bn i = B B. This gives us κ = D D. q= p=q F l p Extending this idea to our general system model, we obtain the following expression for B n i B n i = B D q= D p=i F l p D p=q F l p where B is a user-defined parameter. Since the values B n i are constants, the hybrid algorithm has the same large deviations behaviour as the αβ-algorithm. Formally speaing, we have the following result. Proposition 6: Let P αβ hyb denote the stationary probability for the hybrid policy with parameters α and β. Then, lim lim sup α 0 B logpαβ hyb [ β B logpπ [ =,...,K =,...,K D i= X n i 0 >B] D i= X n i 0 >B]. Proof: Due to space constraints, we only highlight the main ideas of the proof. Similar to the proof of Proposition 4, we can show that the hybrid policy with parameters α, β minimizes the drift of the Lyapunov function V X in a fluid-sample-path sense. This is because the behavior of an algorithm for fluid-sample-paths is determined by its behavior when queue lengths are large. Since the hybrid algorithm behaves lie the αβ-algorithm when queue lengths are large, Fig System topology for simulation 7 Destination 8 9 Destination 2 Destination 3 & 4 0 it has the same behavior as the αβ-algorithm as far as fluidsample paths are concerned. Then, by using the arguments used in the proof of Proposition 5, we obtain K β+ N α α+ lim sup B logpαβ hyb [ B logpπ [ and hence the result. =,...,K =,...,K D i= X n i 0 >B] D i= X n i 0 >B] VI. SIMULATION For simulations, we consider the networ shown in Fig. 7. The networ consists of 0 nodes and 9 lins. Due to fading, the capacity of each lin taes the value 0 or 0 with a probability of 0.5. The fluctuations of the capacity are i.i.d over time and across lins. In each time slot, due to interference only one lin may be active. There are 4 flows traversing the networ and data arrives at each flow according to a Bernoulli process. In each time slot, unit of data arrives with a probability of 0.52 and no data arrives otherwise. The arrival process is i.i.d across flows. Flow passes through nodes, 3, 5, 6, 7, 8. Flow 2 passes through nodes 2, 3, 5, 6, 7, 9. Flow 3 passes through nodes 2, 3, 5, 6, 7, 0 and flow 4 passes through nodes 4, 5, 6, 7, 0. The quantity we are interested in is the stationary probability P[M X0 >B] where M X is given by 0.52 {X + X3 + X5 + X6 + X7 + X8, X2 2 + X3 2 + X5 2 + X6 2 + X7 2 + X9 2, X2 2 + X3 2 + X5 2 + X6 2 + X7 2 + X0, 2 X4 4 + X5 4 + X6 4 + X7 4 + X0} 4 In Fig.8 we present plots of P[M X0 >B] vs. B for the αβ-algorithm with four different choices for the pair of parameters α and β. We see that as expected from theory, α =, β =0algorithm has a better performance than the α =, β =algorithm. However, as we decrease α, we see that α =0.5, β =0does worse than the α =, β =0algorithm but still marginally better than α =, β =algorithm. Further, α =0.3, β =0does much worse although at much larger values of B not shown, the decay rate will be better. In Fig.9 we compare the performance of the α =, β = algorithm, α =, β =0algorithm, hybrid algorithm with

9 9 PMX>B α= β= α= β=0 α=0.3 β=0 α=0.5 β= B Fig. 8. Plot of P[M X0 >B] vs. B for αβ-algorithm for various values of α and β. PMX>B α= β= α= β=0 α=0.3 β=0 B * =50 hybrid α=0.5 β=0 B * =50 hybrid B Fig. 9. Plot of P[M X0 >B] vs. B for αβ-algorithm and hybrid algorithm for various values of α and β. α =0.5, β =0, B =50and hybrid algorithm with α =0.3, β =0, B =50. The hybrid algorithms perform much better than either the α =, β =or the α =, β =0algorithm. Also, both hybrid algorithms have similar performance which suggests that there is no advantage to decrease α further. Note that the α =, β =algorithm is the same as the well nown bac-pressure scheduling algorithm []. Hence, our simulation results show that the hybrid algorithm for the case of small α and large β and the αβ-algorithm for the case of large β perform much better than bacpressure algorithm. VII. CONCLUSION Using a large deviations framewor, we obtain insights into the design of optimal algorithms for minimizing the end-toend buffer usage in a multiflow multihop wireless networ. We propose a class of algorithms called αβ-algorithms and variants called hybrid αβ-algorithm that can be made to perform arbitrarily close to optimal in a large deviations sense by reducing α and increasing β. Through simulations, we show that the class of hybrid algorithms has good performance in the small buffer regime as well. Our result is based on a very general system model and hence can provide insight in a wide range of scheduling scenarios. Acnowledgement: This wor was partially supported by DTRA grant HDTRA , NSF grants CNS , CNS , CNS , CNS , and the DARPA ITMANET program. REFERENCES [] L. Tassiulas and A. Ephremides, Dynamic Server Allocation to Parallel Queues with Randomly Varying Connectivity, IEEE Transactions on Information Theory, vol. 39, no. 2, pp , March 993. 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