This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...

A framework for joint scheduling and admission control in broadband switching systems is developed according to a principle of separation between these two levels of control. It is shown how an admission control strategy can be tailored to a particular mix of traffic by making use of high-level information from the scheduler. This principle is presented in the context of Asynchronous TimeSharing (ATS), in which explicit guarantees of cell-level and call-level quality of service (QOS) are given to several traffic classes. The separation principle allows the formulation of an optimal admission control policy, which will maximize the expected system utility while maintaining all QOS guarantees. Several heuristic admission control policies are considered, and compared against the optimal policy as a benchmark. The admissible load region is introduced as a means of quantifying the capacity of a switch under the QOS constraints at the cell and call levels. Numerical calculations for a single...

A joint scheduling and admission control algorithm is presented for Asynchronous Time-Sharing (ATS)-based switching nodes carrying real-time traffic. Systems based on ATS guarantee quality of service, at both the levels of cells and calls, for three well-defined traffic classes. A mechanism is outlined by which an admission control strategy can be tailored to a particular mix of traffic by making use of high-level information from the scheduler. This mechanism is based on a principle of separation between scheduling and admission control. A linear programming formulation is used to find the admission control policy which will maximize the expected system utility while maintaining the guaranteed quality of service. The admissible load region is introduced as a means of quantifying the capacity of a switch under the QOS constraints at the cell and call levels. Numerical calculations for a single MAGNET II switching node carrying two classes of real-time traffic are used to illustrate the...

In this paper, we focus on simple exponential approximations for steady-state tail probabilities in G/GI/1 queues based on large-time asymptotics. We relate the large-time asymptotics for the steady-state waiting time, sojourn time and workload. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/GI/1 queues. Numerical examples show that the exponential approximations are remarkably accurate at the 90 th percentile and beyond. Key words: queues; approximations; asymptotics; tail probabilities; sojourn time and workload.

Abstract. This paper gives the first rigorous convergence analysis of analogues of Watkins’s Q-learning algorithm, applied to average cost control of finite-state Markov chains. We discuss two algorithms which may be viewed as stochastic approximation counterparts of two existing algorithms for recursively computing the value function of the average cost problem—the traditional relative value iteration (RVI) algorithm and a recent algorithm of Bertsekas based on the stochastic shortest path (SSP) formulation of the problem. Both synchronous and asynchronous implementations are considered and analyzed using the ODE method. This involves establishing asymptotic stability of associated ODE limits. The SSP algorithm also uses ideas from two-time-scale stochastic approximation. Key words. simulation-based algorithms, Q-learning, controlled Markov chains, average cost control, stochastic approximation, dynamic programming AMS subject classification. 93E20 PII. S0363012999361974 1. Introduction. Q-learning algorithms are simulation-based reinforcement learning algorithms for learning the value function arising in the dynamic programming approach to Markov decision processes. They were first introduced for the discounted

We present a unified model to compute various performance measures when Poissonian and non-Poissonian (renewal) multi-slot traffic streams are offered to a digital link in a (double) loss system. We represent the non-Poissonian arrival process by a matrix-exponential distribution, requiring only that the inter-arrival distribution has a rational Laplace transform. Several distributions are considered as the non-Poissonian traffic. The resulting model uses matrix algebraic techniques only,thus not requiring any complex and/or tedious transform techniques. We also incorporate various control policies in our modeling framework using acceptance functions. Through our computational studies, we conclude that the second and the third moments of the non-Poissonian traffic have significant impact on various performance measures.

We establish a heavy-traffic asymptotic expansion (in powers of one minus the traffic intensity) for the asymptotic decay rates of queue-length and workload tail probabilities in stable infinite-capacity multi-channel queues. The specific model has multiple independent heterogeneous servers, each with i.i.d. service times, that are independent of the arrival process, which is the superposition of independent non-identical renewal processes. Customers are assigned to the first available server in the order of arrival. The heavy-traffic expansion yields relatively simple approximations for the tails of steady-state distributions and higher percentiles, yielding insight into the impact of the first three moments of the defining distributions.

Abstract—In this paper, quality-of-service (QoS) guarantees for multiclass code division multiple access networks are provided by means of cross-layer optimization across the physical and network layers. At the physical layer, the QoS requirements are specified in terms of a target signal-to-interference ratio (SIR) requirement, and optimal target powers are dynamically adjusted according to the current number of users in the system. At the network layer, the QoS requirements are the blocking probabilities and the call connection delays. The network layer guarantees that both physical layer and network layer QoS are met by employing admission control. An optimal admission control policy is proposed based on a semi-Markov decision process formulation. The tradeoff between blocking and delay is discussed for various buffer configurations. The advantage of advanced signal processing receivers is established using a comparative capacity analysis and simulation with the classical scenario in which the system uses matched filter receivers. Index Terms—Admission control, capacity, CDMA, cross-layer, multiuser detection, power control, QoS.

This paper addresses the call admission control problem for the multimedia services that characterize the third generation of wireless networks. In the proposed model each cell has to serve a variety of classes of requests that differ in their traffic parameters, bandwidth requirements and in the priorities while ensuring proper quality of service levels to all of them. A Semi Markov Process is used to model multi-class multimedia systems with heterogeneous traffic behavior, allowing for call transitions among classes. It is shown that the derived optimal policy establishes state-related threshold values for the admission policy of handoff and new calls in the different classes, while minimizing the blocking probabilities of all the classes and prioritizing the handoff requests. It is proven that in restrictive cases the optimal policy has the shape of a Multi-Threshold Priority policy, while in general situations the optimal policy has a more complex shape.

In this paper we study a single-server system with Erlang-r distributed service times and arbitrarily distributed interarrival times. It is shown that the waiting-time distribution can be expressed as a finite sum of exponentials, the exponents of which are the roots of an equation. Under certain conditions for the interarrival-time distribution, this equation can be transformed to r contraction equations, the roots of which can be easily found by successive substitutions. The conditions are satisfied for several practically relevant arrival processes. The resulting numerical procedures are easy to implement and efficient and appear to be remarkably stable, even for extremely high values of r and for values of the traffic load close to 1. Numerical results are presented. Key words. equilibrium distribution, Markov chain, queues AMS subject classifications. 60K25, 90B22 1.