On Characterizing the Delay-Performance of Wireless Scheduling Algorithms

Transcription

1 On Characterizing the Delay-Performance of Wireless Scheuling Algorithms Xiaojun Lin Center for Wireless Systems an Applications School of Electrical an Computer Engineering, Purue University West Lafayette, IN 47907, U.S.A. Abstract In this paper we stuy the problem of characterizing the elay performance of wireless scheuling algorithms. In wireless networks operate uner these wireless scheuling algorithms, there often exists a tight coupling between the service-rate process, the system backlog process, the arrival process an the channel variations. Although one can use sample-path large-eviation techniques to form an estimate of the elay-violation probability uner a given offere loa, the formulation leas to a multi-imensional calculus of variations problem that is often very ifficult to solve. In this paper, we evelop a new technique for aressing this complexity issue. Using ieas from the Lyapunov function approach in control theory, our new technique maps the complex multi-imensional calculus of variations problem to a one-imensional calculus of variations problem, an the latter is often much easier to solve. This new technique can potentially be use to stuy the elayperformance of a large class of wireless scheuling algorithms. I. INTRODUCTION A wireless network may be moele as a system of queues with time-varying service rates. The variability in service rates is ue to a number of factors. First, channel faing an mobility can lea to variations in the link capacity even if the transmission power is fixe. Secon, the transmission power can vary over time accoring to the power control policy. Thir, ue to raio interference, it is usually preferable to scheule only a subset of links to be active at each time, an to alternate the subset of activate links over time. All of these factors lea to a variable service rate at each link. When one is to stuy the performance of any system that involves queues, the first question we can ask is whether the system is stable or not. Here, stability means that all queue length (or equivalently, the elay experience by the packets) remains finite. Conversely, we can ask the question that, in orer to maintain stability, what is the largest offere loa that the system can carry. In other wors, what is the capacity region of the system subject to stability. For wireless networks, these questions have le to results on throughput-optimal scheuling algorithms for scheuling wireless resources. (Here we use the term scheuling in the broaer sense, i.e., it can inclue various control mechanisms at the MAC/PHY layer, e.g., link scheuling, power control, an aaptive coing/moulation.) A scheuling algorithm is throughput-optimal if this algorithm can sustain the largest offere loa while keeping the system stable. In other wors, if this algorithm cannot stabilize the system, no other algorithms can. For example, one such throughput-optimal scheuling algorithm is the algorithm propose in the seminal work by Tassiulas an Ephremies in []. This algorithm chooses at each time, among all possible scheules, the one that maximizes the sum of the queue-weighte-rate over all links. This algorithm has been shown to be throughputoptimal, an it has been the basis for many other throughputoptimal scheuling algorithms for both cellular an multihop wireless networks. Once we know about stability, we are then tempte to ask further questions regaring the istribution of queue length (or elay). For example, at a given offere loa, what is the probability that the elay experience by a packet is greater than a given threshol? Or, conversely, what is the largest offere loa that the system can support at a given elay constraint? (In other wors, what is the effective capacity region of the system uner elay constraints?) Clearly, these question are important for applications that require more stringent elay guarantees than just stability. These elay characterization problem for wireless networks can be ifficult to solve. Here we raw a comparison to the elay characterization problem in wireline networks. In wireline networks, even through the exact elay istribution can be ifficult to obtain, there have been a large boy of work, especially those using large-eviation techniques, to obtain sharp estimates of the elay violation probability of a queue. These wireline network results usually assume that the service rate of the queue is fixe (i.e., time-invariant), an the packet arrival process is known. These results allow us to compute the effective banwith of the arrival process from its (known) statistics [2] [7], which can then be use to etermine the traffic carrying capability of the queue at a given elay constraint. In contrast, in wireless networks, the service rate is time-varying. If the service rate process is again known a priori, large-eviation techniques can be use to compute the effective capacity of the service rate process [8], [9], which is a notion similar to the effective banwith of the arrival process. This effective capacity can again be use to etermine the traffic carrying capability of the queue at a given elay constraint. Unfortunately, uner many wireless scheuling algorithms of interest, even the service rate process is unknown a priori. For example, for

2 a system operate uner the throughput-optimal Tassiulas- Ephremies algorithm of [], or any queue-length base scheuling algorithms, the service rates epen on the queue length, which in turn epen on the arrival process an the channel state, etc. Hence, the statistics of the service rate process is unknown before han. In this case, the elay characterization problem is known to be very ifficult. For these systems, although it is still possible to use samplepath large-eviation techniques to form an estimate of the elay-violation probability [0] [2], such a formulation leas to a multi-imensional calculus of variations problem. Due to the complex coupling between the service rate, the queue length, the arrival process, an the channel state, this multi-imensional calculus of variations problem is very ifficult to solve. Prior successes have been limite to simple systems: either the problem has some restrictive structure (e.g., symmetry among all links) [], or the size of the system is very small (e.g., two links) [0], [2], [3]. In this paper, we propose a new approach to aress this complexity issue. Motivate by the Lyapunov function approach for proving stability of complex systems, we evelop a new technique that maps the complex multi-imensional calculus of variations problem into a one-imensional calculus of variations problem, an the latter is often very easy to solve. The solution to the one-imensional calculus of variations problem will then provie us with an upper boun estimate of the elay violation probability, an consequently, a lower boun estimate of the effective capacity region of the system. For many practical applications, the resulting effective capacity region is useful because the elay constraint is known to be satisfie. We believe that this marriage between sample-path largeeviations an Lyapunov functions can evelop into a powerful theory to characterize the elay performance of wireless systems uner sophisticate scheuling algorithms. We can potentially lower the ifficulty level of the elaycharacterization problem to that of a stability problem. In other wors, for any scheuling algorithm that is provably stable, which usually means that there exists a Lyapunov function, we coul then apply the propose technique to characterize the elay performance. We provie an example of how this approach can be use to solve a more ifficult problem than those stuie in the literature. The rest of the paper is organize as follows. We present the network moel in Section II. We review a formulation of the sample-path large-eviation principle, an ientify the complexity of the associate calculus of variations problem in Section III. Then, in Section IV, we provie a Lyapunov function base approach to aress the complexity issue. In Section V, we provie an example to show how such an approach can be use. Then we conclue. II. THE SYSTEM MODEL We consier the following moel for a wireless system with L links. In orer to moel channel faing, we assume that the system can be in one of S states. We assume a slotte system, an enote the state of the system at time t to be C(t). Further, we assume that the states C(t), t =, 2,... are i.i.., an let p j = P[C(t) = j] enote the probability that the state of the system at time t is j. Let p = [p,..., p S ]. For ease of exposition, in the rest of the paper we also efine Φ j (t) = {C(t)=j} to be the inicator function that the state of the system at time t is j. Let Φ(t) = [Φ (t),..., Φ S (t)]. Clearly, there is a one-to-one mapping between C(t) an Φ(t). Each link correspons to a queue with time-varying service rate. The arrivals at each link i are at a constant rate λ i. The service offere to link i is etermine by the scheuling algorithm, an in general correlates with the service at other links an epens on the system backlog. Let X i (t) enote the backlog at link i at time t, an let X(t) = [X (t),..., X L (t)]. We assume that the service rate offere to link i is a function of the global backlog X(t) an the system state C(t). In particular, let D ij ( X) enote the service offere to link i when the state of the system is j an the global backlog is X. The evolution of the backlog at link i is then given by X i (t+) = [X i (t)+λ i Φ j (t)d ij ( X(t))] +, i =,..., L () where [ ] + enotes the projection to [0, + ). Assume that the system is stationary an ergoic. In this paper, we will focus on stuying the probability that the system backlog excees a certain threshol B. In particular, let ɛ enote our target on the overflow probability, we woul like to ensure that P[ X(0) B] ɛ, (2) where is an appropriately chosen norm, an B is the overflow threshol. Note that the constraint in (2) is equivalent to a constraint on the elay-violation probability when the arrival rates λ i are constant, because the two types of constraints are relate by P[Delay at link i i ] = P[X i (0) λ i i ]. Hence, in the rest of the paper we will often refer to (2) as a elay constraint [9], []. Unfortunately, the problem of calculating the exact probability P[ X(0) B] is often mathematically intractable. In this paper, we are intereste in using large-eviation techniques to compute estimates of this probability. We assume that the following large-eviation result for the backlog process X(t) hols. That is, when B is large, the following limit exists lim B B log P[ X(0) B] = I 0 ( λ), (3) where I 0 ( λ) can be etermine from a sample-path largeeviation principle that we will escribe in Section III. Equation (3) implies that, when B is large, the overflow probability can be approximate as P[ X(0) B] exp( BI 0 ( λ)).

3 Thus, the problem of estimating the overflow probability is reuce to that of computing the rate I 0 ( λ). Alternatively, using the above approximation, in orer to satisfy the constraint (2), we only nee to ensure that I 0 ( λ) θ log ɛ B. (4) We can then efine the effective capacity region uner the constraint (2) as the set of arrival rates λ such that the above inequality hols. III. THE SAMPLE-PATH LARGE DEVIATION PRINCIPLE In this paper, we will stuy the problem of computing the rate I 0 ( λ) an characterizing the effective capacity region uner the constraint (2). We first escribe how I 0 ( λ) can be etermine from a sample-path large-eviation principle for the backlog process X(t). (Note that establishing such a large-eviation principle is not the main focus of the paper. We refer the reaers to [0] [2] for etails on the technical assumptions uner which such a large-eviation principle hols.) A. Notations We follow the convention in [0], []. For a large enough T, efine the empirical measure process on the time interval [, 0] as s B j (t) = B B(T +t) l=0 {C(l)=j}, for t = k B T, k = 0,..., BT, an by linear interpolation otherwise. Note that, in the above efinition, we have scale both the time an the magnitue. The quantity s B j (t) can be interprete as the sum of the (scale) time in [, t] that the system is at state j. Further, it is easy to check that sb j (t) = t + T for all t [, 0]. Let sb (t) = [s B (t),...s B S (t)]. Further, let φb j (t) = t sb j (t). (Note that the erivative is well efine almost everywhere on [, 0] except when t = k/b T for some integer k.) Let φ B (t) = [φ B (t),..., φ B S (t)]. Note that φb j (t) = for almost all t. Analogously, efine the scale backlog process as, x B i (t) = B X i(b(t + t)), for t = k B T, k = 0,..., BT, an by linear interpolation otherwise. Let x B (t) = [x B (t),..., x B L (t)]. Note that accoring to (), the backlog process x B (t) is relate to the process φ B (t) by x B i (t + /B) xb i (t) /B = λ i D ij ( x B (t)) t+/b t φ B j (s)s, for t = k B T, k = 0,..., BT. (5) Thus, given a particular initial conition x B ( ), Equation (5) efines a mapping f B from the empirical measure process s B (t) to the backlog process x B (t). Further, although we have assume s B (t) to be piecewise linear to begin with, the efinition of the mapping f B can be naturally extene to all absolute continuous functions s B (t). B. The Large-Deviation Principle Let B. We now have a sequence of scale ranom walks s B (t), an they map to a sequence of scale backlog processes x B (t) through the sequence of mappings f B. For any φ 0 an φ j =, efine H( φ p) = φ j log φj p j. The sequence of empirical measure processes s B (t) are known to satisfy a sample-path large eviation principle [4, p76] with large-eviation rate-function Is T ( s( )) given as follows: I T s ( s( )) = H( φ(t) p)t, if s(t) is absolute continuous an component-wise nonecreasing on [, 0], s( ) = 0, an s j(t) = t + T for all t; where φ(t) = t x(t). (Note that φ(t) is well efine almost everywhere on [, 0] since s(t) is absolute continuous on [, 0].) Otherwise, I T s ( s( )) = +. Such a large-eviation principle means that, for any set Γ of trajectories on [, 0] that is a continuity set [4, p5] accoring to the essential supremum norm [4, p76, p352], the probability that the sequence of empirical measure processes s B (t) fall into Γ must satisfy lim B B log P[ sb ( ) Γ] = s( ) Γ IT s ( s( )). (6) The large-eviation rate-function Is T ( ) characterizes how rarely each trajectory is. Note that Is T ( s( )) 0 for all trajectory s( ). The larger the value of Is T ( s( )) is, the further the empirical probability istribution φ(t) eviates from the prior probability istribution p. Hence, the less likely the trajectory s( ) will occur. Equation (6) reflects the wellknown large-eviation philosophy that rare events occur in the most-likely way. Precisely, when B is large, the probability that the empirical measure process s B (t) falls into a set Γ is etermine by the trajectory in Γ that is most likely to occur, i.e., with the smallest Is T ( s( )). Next, assume that the sequence of mappings f B has a limiting mapping f that also maps any absolute continuous empirical measure process s(t) to a backlog processes x(t). Assume that the limiting mapping f is of the form t x i(t) = λ i φ j (t) ij ( x(t)), (7) where φ(t) = t s(t). (Note that this equation may be viewe as the limit of (5) when B, although the function ij ( x(t)) may not be exactly the same as D ij ( x(t)) as we will see in the example in Section V.) Further, assume that the sequence of mappings f B are exponentially equivalent to f [4, p30], an the mapping f is continuous (see [0] for

4 Fig.. Top: The overflow probability P[ X(0) B] is relate to the most likely path to overflow. Bottom: our new approach maps any multiimensional path x(t) to a one-imensional path V (t). etails of how the continuity of f may be verifie). For any sequence of backlog processes that start from x B ( ) = 0, we can then invoke the contraction principle [4, p3] an obtain a sample-path large-eviation principle for the sequence of backlog processes x B (t) with large-eviation rate-function given by: I T x ( x( )) = s( ): x( )=f( s( )) { } H( φ(t) p)t where φ(t) = t s(t), an the imum is taken over all empirical measure processes s( ) that map to the backlog process x( ) given that x( ) = 0, uner the mapping f. Finally, the event of overflow correspons to x B (0). As B, we have, I 0 ( λ) lim B B log P[ xb (0) ] = {Ix T ( x( )) over all trajectory x( ) that goes from x( ) = 0 for some T > 0 to x(0) = }. (8) The trajectory that attains the imum in (8) is often calle the most likely path to overflow (see Figure ). Clearly, in orer to estimate the overflow probability P[ X(0) B], all we nee is to fin out which trajectory in (8) is the mostlikely path to overflow. C. The Path-Explosion Challenge Unfortunately, the imum in (8) (a calculus of variations problem) is often very ifficult to evaluate, because it is taken over an inite number of multi-imensional paths x(t). To see this, let us efine the local rate-function l( x, y) = {H( φ p) over all φ such that φ j = an y i = λ i φ j ij ( x) for all i}. (9) Then, we have I T x ( x( )) = l( x(t), x (t))t. Note that the local rate-function l( x, y) characterizes how rarely that, given x(t) = x at some time t, x(t) will follow the irection t x(t) = y immeiately after t. Suppose now we enforce a elay constraint in the form of P[ X(0) B] ɛ. Using (8) to approximate this probability, we then nee to ensure that I T x ( x( )) = l( x(t), x (t))t θ log ɛ/b (0) for all sample paths x(t) that go from 0 at some past time to x(0) =. For avance wireless scheuling algorithms like the Tassiulas-Ephremies algorithm [], the complexity of enumerating all such paths soon becomes prohibitive. Prior successes have been limite to simple systems: either the problem has some restrictive structure (e.g., symmetry among all links) [], or the size of the system is very small (e.g., two links) [0], [2], [3]. IV. A NEW APPROACH COMBINING LARGE-DEVIATIONS WITH LYAPUNOV STABILITY In this section, we will evelop a general approach for solving problems (8) an (0). As we iscusse earlier, fining the most likely path to overflow is often a very ifficult problem. In this section, instea of solving the calculus of variations problem on the right-han-sie of (8), we construct a lower boun for it. That is, we will fin a quantity θ 0 such that I T x ( x( )) θ 0 for all trajectory x( ) that goes from x( ) = 0 at some past time to x(0) =. Hence, we obtain an upper boun on the overflow probability. If θ 0 θ, we then obtain a sufficient conition for meeting the constraint (4) on the overflow probability. Consequently, we also obtain a lower boun on the effective capacity region of the system uner the constraint (2). How to fin such a lower boun θ 0? In this section, we evelop a new approach that is motivate by the Lyapunov function approach for proving stability for complex systems [5]. Note that for a complex system like (), it becomes ifficult to even establish stability, i.e., to show that all queues will remain finite. To see this, take the limit B + again for (5). The flui limit of the system is governe by [6] t x i(t) = λ i p j D ij ( x(t)) h i ( x(t)), for all i () or, in vector form t x(t) = h( x(t)). This flui limit ynamics can be viewe as the mean behavior of the system. The original system woul be stable if the solution of the ODE (orinary ifferential equation) in () can be shown to converge to zero from any initial conition [6]. However, solving the above orinary ifferential equation is often very ifficult. Thus, it is usually impossible

5 to establish the stability of the system by irectly solving the ODE. To circumvent this ifficulty, we usually fin a Lyapunov function V ( x), such that V ( x) 0, an V ( x) = 0 if an only if x = 0. We then prove stability by showing a negative rift for V ( ), i.e. t V ( x(t)) = ( V x ) T x t δv ( x(t)), (2) where δ is a small positive constant. Thus, if x(t), or equivalently, V ( x(t)), is away from zero, the negative rift will pull them back to zero. The negative rift then provies a sufficient conition for V ( x(t)) 0, which implies that x(t) 0, as t. Here, the key to the Lyapunov function approach is to map the convergence of a multi-imensional path x(t), to the convergence of a oneimensional path V ( x(t)), which is then much easier to show. Since V ( x(t)) 0 provies a sufficient conition for all solutions x(t) of the ODE to go to zero, the Lyapunov function approach allows us to ientify a lower boun on the capacity region of the system (subject to stability). Can we use a similar Lyapunov function approach to characterize the elay performance (an the overlow probability) of wireless scheuling algorithms? Inee, we now show that such an approach is feasible. Without loss of generality, assume that the Lyapunov function V ( ) are chosen such that x = implies V ( x). Accoring to (2), for any w > δv ( x(t)), the trajectory with t V ( x(t)) = w becomes a rare event. Define l V (v, w) = {l( x, y) over all ( x, y) such that ( ) T V V ( x) = v an y = w}. (3) x Let us abuse notation an let V (t) = V ( x(t)). Compare with (9), l V (v, w) becomes the local rate-function for V (t), i.e., it characterizes how rarely that, given V (t) = v at some time t, V (t) will follow the irection t V (t) = w immeiately after t. It is easy to show that Ix T ( x( )) in (0) satisfies I T x ( x( )) = Let θ 0 = { l( x(t), x (t))t l V (V (t), V (t))t. l V (V (t), V (t))t over all trajectory V ( ) that goes from V ( ) = 0 for some T > 0 to V (0) = }. (4) We then obtain a lower boun θ 0 for the calculus of variations problem (8). It is also easy to see that a sufficient conition for all samples paths x(t) to meet the constraint (0) is l V (V (t), V (t))t θ (5) for all one-imension path V (t) that goes from V ( ) = 0 to V (0) =. Again, we have successfully reuce the original multi-imensional calculus of variations problem to a one-imensional problem. The one-imensional calculus of variations problem in (4) an (5) is usually much easier to solve (Fig. ). A Unifie Theory Combining Large-Deviation an Lyapunov Stability: As we have just seen, by combining large-eviations with Lyapunov stability, we have create a powerful theory for characterizing the elay-performance of wireless scheuling algorithms. We can potentially lower the ifficulty level of the elay-characterization problem to that of a stability problem. In other wors, for any scheuling algorithm that is provably stable, which usually means that there exists a Lyapunov function, we coul then apply the above techniques to characterize the elay performance. Since (5) is a sufficient conition to (0), we can obtain an upper boun on the overflow probability, an corresponingly, if a constraint on the overflow proability is impose, we obtain a lower boun on the effective capacity region. The hope of this approach is that, if the function V ( ) is appropriately chosen, we may recover a large fraction of, or even the entire effective capacity region. V. AN EXAMPLE In this section, we apply the methoology of Section IV to the elay-characterization problem in []. The moel of [] is a base-station serving N users (Fig. 2). Packets for user i arrive at a constant rate λ i. Only one user can be scheule for transmission at any time. The faing channel between the base-station an each user is i.i... At each timeslot, a user s channel is ON with probability p, an OFF with probability p. Let F enote the banwith of the system. Hence, if a user s channel is ON an it is scheule for transmission, its service rate is F. The throughput-optimal Tassiulas-Ephremies algorithm [] in this case is the QLB (Queue-Length Base) algorithm, i.e., the base-station shoul scheule the ON user with the longest queue []. The more challenging question is to etermine the effective capacity region of the system, subject to the buffer overflow constraint P[ max X i B] ɛ, where X i is the ranom variable i=,...,n that enotes the backlog of user i. The authors of [] assume that all users have the same offere loa, i.e., λ i = λ for all i =, 2,..., N. Uner this assumption, the most likely path to overflow in (8) can be explicitly solve. They then establish the following effective capacity region: [ Nλ min N M N θ log ( p) M +( ( p) M ) exp( F θ M ) ] (6) where θ = log ɛ/b. However, for non-ientical offere loas, it appears very ifficult to follow the solution approach of []. We now use the methoology of Section IV to solve the elay characterization problem when the offere loas λ i are non-ientical. Following the notations in Sections II an III, the set of possible channel states are

6 l( x, y) as l( x, y) = φ j= H( φ p) subject to y i = λ i φ j ij ( x) for all i. Fig. 2. The scheuling problem in cellular networks uner faing channel. S = {(a,..., a N ) a,..., a N = ON or OFF}. The probability that the channel state C(t) at time t is j is given by, p j = p n(j) ( p) N n(j), where n(j) is the number of users with ON channel at state j. When the state is j an the system backlog is x, let I ( x, j) enote the set of those users whose channels are ON an who have the (ientically) largest queue x i among ON users. The evolution of the backlog is then given by (), where the function can be chosen as D ij ( x(t)) = F/ I ( x(t), j) if i I ( x(t), j), an D ij ( x(t)) = 0 otherwise. We can efine f B accoring to (5). As B, the limiting mapping f is given by (7). It is easy to show that the function ij ( ) in (7) must satisfy ij ( x(t)) = 0 if i / I ( x(t), j). Further, if the set I ( x(t), j) contains only one user i, i.e., there is a unique ON user i that has the largest queue at time t, then ij ( x(t)) = F. However, if I ( x(t), j) contains multiple users, the efinition of ij ( x(t)) becomes somewhat involve [0] [2]. Roughly speaking, ij ( x(t)) shoul be efine so that the users in I ( x(t), j) can maintain ientical queues as much as possible. Regarless of the exact form of ij ( ), the following relationship can be shown. For a given trajectory x( ), let I ( x(t)) = {i x i (t) = max k x k (t)} be the set of users with the (ientically) largest queue at time t. (Note that I ( x) is ifferent from I ( x, j) since in the efinition of I ( x) we o not check whether a user is ON or OFF.) Further, let I 2 ( x(t), x (t)) = {i I ( x(t)) t x i(t) = max k I ( x(t)) t x k(t)}. That is, I 2 ( x(t), x (t)) is the set of users that, among those users with the largest queue at time t, also have the largest queue growth rate. In other wors, these set of users will have the largest queue immeiately after time t. Then, immeiately after time t, as long as one user in I 2 ( x(t), x (t) is ON, this group of users collectively must receive the full service rate F. Therefore, using (7), we must have = i I 2( x(t), x (t)) i I 2( x(t), x (t)) t x i(t) λ i F (I 2( x(t), x (t))) φ j. (7) where for any subset A {, 2,...N}, S(A) enotes the set of states j such that some user i A is ON. We can then use (9) an write own the formulation for Let the Lyapunov function be V ( x) = max i x i. Although V ( ) is not ifferentiable at every point, it is sufficient to eal with its one-sie erivative. In the rest of this section, when we use the notation g(t) g(t+s) g(t) t, we will mean lim s 0 s. We thus have V ( x(t)) t = max i I ( x(t)) x i (t), t (Recall that I ( x) = {i x i = max k x k } is the set of users with the ientically largest queue when the system backlog is x.) Thus, using (3), we have, l V (v, w) = subject to l( x, y) max x i = v i max y i = w. i I ( x) Combining the above two optimization problems, we thus have l V (v, w) = φ j= subject to H( φ p) (8) max x i = v i max y i = w i I ( x) y i = λ i φ j ij ( x), for all i. This optimization problem is still quite ifficult to solve. We will be contente to obtain a lower boun for the optimal value. First, recall that I 2 ( x, y) = {i I ( x) y i = max y k}. k I ( x) Let us first compute the imum of (8) for all x, y such that I 2 ( x, y) = M, where M is a given subset of {,..., N}. This sub-optimization can be written as l V,M (v, w) = φ j= subject to H( φ p) (9) x i = v for i M x i v for i / M y i = w for i M y i < w for i I ( x)/m y i = λ i φ j ij ( x) for all i. Note that for all i M = I 2 ( x, y), we have w = λ i φ j ij ( x).

7 Summing over all i I 2 ( x, y), an using (7), we have, M w = λ i F φ j, i M (M) where M enotes the carinality of M. (Recall also that S(M) is the set of states j such that some user in M is ON.) We can then relax the constraints of (9) as: lv,m (v, w) = φ j= subject to H( φ p) (20) M w = λ i F i M (M) φ j. This subproblem is solve in [], an the solution is given by lv,m (v, w) = u log where u = u ( p) +( u) log u M ( p). M i M λi M w F lv (v, w) =. Let lv,m (v, w). (2) M {,2,...,N} Since (20) is a relaxation of (9), we have, l V (v, w) = l V,M(v, w) lv,m (v, w) = l V (v, w). M M Therefore, in orer to ensure that l( x(t), x (t))t θ for all trajectory x( ) that goes from x( ) = 0 at some past time to x(0) =, it is sufficient to ensure that θ { lv (V (t), V (t))t over all trajectory V ( ) that goes from V ( ) = 0 for some T > 0 to V (0) = }. (22) Note that l V (v, w) in (2) oes not epen on v. Therefore, the trajectory V ( ) that attains the imum in (22) is in fact very easy to solve [7, p520], an the imum is equal to w 0 lv (v, w)/w. Therefore, using the efinition of l V (, ) an l V,M (, ), it is then sufficient to ensure that where θ w 0 = M w 0 = M w l V (v, w) w l V,M (v, w) 0 u i M λ i F M i M λ i uf D M (u p) D M (u p) = u u log ( p) + ( u) log u M ( p). M Note that the above conition is equivalent to θ 0 u i M λ i F M i M λ i uf D M (u p) (23) for all M {, 2,..., N}. For a fixe M, the conition (23) is shown in [] to be equivalent to λ i M [ log ( p) M θ i M +( ( p) M ) exp( F θ ] M ). (24) Thus, we obtain a lower boun on the effective capacity region as { λ Inequality (24) hols for all M {, 2,..., N}}. (25) Remark: Note that (25) reuces to (6) when all λ i are equal. Thus, we not only reprouce a lower boun on the effective capacity region for the case with ientical offere loas (which is the same as the effective capacity region foun in []), but also solve the more general problem with non-ientical offer loas. VI. CONCLUSIONS In this paper we stuy the problem of characterizing the elay performance of complex wireless scheuling algorithms. Our main contribution is to evelop a new technique for aressing the complexity issue of the calculus of variations problem involve in the sample-path large eviation approach. Our new technique combines samplepath large eviations with Lyapunov stability, which may evelop into a powerful approach to stuy a large class of scheuling algorithms. We also illustrate the potential of such an approach through an example. REFERENCES [] L. Tassiulas an A. Ephremies, Stability Properties of Constraine Queueing Systems an Scheuling Policies for Maximum Throughput in Multihop Raio Networks, IEEE Transactions on Automatic Control, vol. 37, no. 2, pp , December 992. [2] F. P. Kelly, Effective Banwith in Multiclass Queues, Queueing Systems, vol. 9, pp. 5 6, 99. [3] A. I. Elwali an D. Mitra, Effective Banwith of General Markovian Traffic Sources an Amission Control of High Spee Networks, IEEE/ACM Transactions on Networking, vol., no. 3, pp , June 993. [4] G. Kesiis, J. Walran, an C.-S. Chang, Effective Banwith for Multiclass Markov Flui an other ATM Sources, IEEE/ACM Transactions on Networking, vol., no. 4, pp , Aug [5] D. D. Botvich an N. G. Duffiel, Large Deviations, the Shape of the Loss Curve, an Economies of Scale in Large Multiplexers, Queueing Systems, vol. 20, pp , 995. [6] N. G. Duffiel an N. O Connell, Large eviations an overflow probabilities for the general single server queue, with application, Math. Proc. Cambr. Phil. Soc., vol. 8, pp , 995. [7] C. Courcoubetis an R. Weber, Buffer Overflow Asymptotics for a Buffer Hanling Many Traffic Sources, Journal of Applie Probability, vol. 33, pp , 996. [8] D. Wu an R. Negi, Effective Capacity: A Wireless Link Moel for Support of Quality of Service, IEEE Transactions on Wireless Communications, vol. 2, no. 4, pp , July [9] A. Eryilmaz an R. Srikant, Scheuling with Quality of Service Constraints over Rayleigh Faing Channels, in Proceeings of the IEEE Conference on Decision an Control, [0] S. Shakkottai, Effective Capacity an QoS for Wireless Scheuling, available at shakkott/pub.html, [] L. Ying, R. Srikant, A. Eryilmaz, an G. E. Dulleru, A Large Deviations Analysis of Scheuling in Wireless Networks, in Workshop on Rare Events in Communication Networks, EURANDOM, February 2005.

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