We consider the following queueing system which arises as a model of a wireless link shared by multiple users. Multiple flows must be served by a "channel" (server). The channel capacity (service rate) changes in time randomly and asynchronously with respect to different flows. In each time slot, a scheduling discipline (rule) picks a flow for service based on the current state of the channel and the queues. We study a scheduling rule, which we call the exponential rule, and prove that this rule is throughput-optimal, i.e., it makes the queues stable if there exists any rule which can do so. In the proof we use the fluid limit technique, along with a separation of time scales argument. Namely, the proof of the desired property of a "conventional" fluid limit involves a study of a different fluid limit arising on a "finer" time scale. In our companion paper [12] it is demonstrated that the exponential rule can be used to provide Quality of Service guarantees over a shared wireless link.

We consider the problem of scheduling CDMA data users on the forward link. The goal is to meet their QoS requirements defined in terms of probabilistic packet delay bounds. The constraint is the limit on the total forward link transmit power. Each user's channel condition is characterized by the forward link power required to achieve a unit data rate. This paper extends the work reported in [1], in which several simplifying assumptions were made, including the assumption that channel conditions are constant in time. In this work, we study a more realistic scenario, in which transmission rates can only be chosen from a discrete finite set, rate scheduling can only be done at discrete scheduling intervals, and, most importantly, the users' channel conditions may vary in time.

It is shown here that stability of the stochastic approximation algorithm is implied by the asymptotic stability of the origin for an associated ODE. This in turn implies convergence of the algorithm. Several specific classes of algorithms are considered as applications. It is found that the results provide (i) a simpler derivation of known results for reinforcement learning algorithms; (ii) a proof for the first time that a class of asynchronous stochastic approximation algorithms are convergent without using any a priori assumption of stability; (iii) a proof for the first time that asynchronous adaptive critic and Q-learning algorithms are convergent for the average cost optimal control problem.

We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient. Running head : Stability and Instability of Queueing Networks Keywords : Multiclass queueing networks, ergodicity, stability, performance analysis. 1 Introduction It has generally been taken for granted in queueing theory that stability of a network is guaranteed so long as the overall traffic intensity is less than unity and in recent years there has been much analysis which supports this belief for special classes of systems, such as single class queueing networks (see Borovkov [2], Sig...

We consider a polling system with a finite number of stations fed by compound Poisson arrival streams of customers asking for service. A server travels through the system and upon arrival at a station the server serves all waiting customers until the queue is empty, where the service time distribution depends on the station. The choice of the station to be visited next as well as the corresponding walking time may depend on the whole current state. Examples are systems with a greedy-type routing mechanism. Under appropriate independence assumptions it is proved that the system is stable if and only if the workload is less than one. POLLING SYSTEM; STABILITY; ERGODICITY OF MARKOV CHAINS; GREEDY SERVER AMS 1991 SUBJECT CLASSIFICATIONS: PRIMARY 60K25, SECONDARY 60J27 # This work was done while the first author held a visiting position at Technical University of Braunschweig. Partial support was provided by the INTAS grant 93--820. 1 1 Introduction Consider a server who visits (polls) ...

We consider a polling system with a finite number of stations fed by compound Poisson arrival streams of customers asking for service. A server travels through the system. Upon arrival at a non-empty station i, say, with x > 0 waiting customers, the server tries to serve there a random number B of customers if the queue length has not reached a random level C < x before the server has completed the B services. The random variable B may also take the value # so that the server has to provide service as long as the queue length has reached size C . The distribution H i,x of the pair (B, C) may depend on i and x while the service time distribution is allowed to depend on i. The station to be visited next is chosen among some neighbors according to a greedy policy. That is to say that the server always tries to walk to the fullest station in his well-defined neighborhood. Under appropriate independence assumptions two conditions are established which are sufficient for stability and su...

We consider a slotted ring that allows simultaneous transmissions of messages by different nodes, known as ring with spatial reuse. To alleviate fairness problems that arise in such networks, policies have been proposed that operate in cycles and guarantee that a certain number of packets, not exceeding a given number called a quota, will be transmitted by every node in every cycle. In this paper, we provide sufficient and necessary stability conditions that implicitly characterize the stability region for such rings. These conditions are derived by extending a technique developed for some networks of queues satisfying a monotonicity property. Our approach to instability is novel and its peculiar property is that it is derived from the instability of a dominant system. Interestingly, the stability region depends on the entire distribution of the message arrival process and the steady-state average cycle lengths or lower dimensional systems, leading to a region with nonlinear boundaries, the exact computation of which is in general intractable. Next, we introduce the notions of essential and absolute stability region. An arrival rate vector belongs to the former region if the system is stable under any arrival distribution with this arrival vector, while it belongs to the latter if there exists some distribution with this rate vector for which the system is stable. Using a linear programming approach, we derive bounds for these stability regions that depend only on conditional average cycle lengths. For the case of two nodes, we provide closed-form expressions for the essential stability region.

This paper surveys such powerful stochastic Lyapunov function methods for general state space Markov processes. Under the hypothesis of '-irreducibility, which is satisfied for a broad class of operations research and diffusion models [23], we obtain criteria for transience; recurrence; positive recurrence; finiteness and convergence of moments; geometric ergodicity; and well behaved solutions to Poisson's equation. 2 Processes and Generators

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated in 1979 by Tsybakov and Mikhailov. Since then several bounds on the stability region have been established, however, the exact stability region is known only for the symmetric system and two users ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating single queue to which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes' criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis...

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