Consider a queue which serves traffic from a number of distinct sources and which is required to deliver a performance guarantee, expressed in terms of the mean delay or the probability the delay exceeds a threshold. For various simple models we show that an effective bandwidth can be associated with each source, and that the queue can deliver its performance guarantee by limiting the sources served so that their effective bandwidths sum to less than the capacity of the queue. Keywords: large deviations, M/G/1 queue, circuit-switched network, connection acceptance control. 1.

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...

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have small-tail asymptotics of the form x - 1 logP(W > x) - q * as x for q * > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Ga .. rtnerEllis condition for the cumulant generating function of the associated partial sums, i.e., n - 1 log Ee qS n y(q) as n , plus regularity conditions on the decay rate function y. The asymptotic decay rate q * is the root of the equation y(q) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general nondecreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multi-class queues based on asymptotic decay rates.

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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.

Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative real-time delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the last customer to enter service (LES), (ii) the delay experienced so far by the customer at the head of the line (HOL), and (iii) the delay experienced by the customer to have arrived most recently among those who have already completed service (RCS). We compare these delay-history estimators to the estimator based on the queue length (QL), which requires knowledge of the mean interval between successive service completions in addition to the queue length. We characterize performance by the mean squared error (MSE). We do analysis and conduct simulations for the standard GI/M/s multi-server queueing model, emphasizing the case of large s. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation to its mean value serves as a refined estimator. For all three candidate delay estimators, the MSE relative to the square of the mean is asymptotically negligible in the many-server and classical heavy-traffic limiting regimes.

This paper is concerned with the explicit evaluation of the steady--state distribution of the waiting time W in a many--server queue with (possibly heterogeneous) servers each having a phase--type service time distribution. Consider first the most classical case GI=PH het =c of renewal arrivals (we later generalize to MAP=PH het =c, i.e. a Markovian arrival process as in Neuts [15]). Denote the interarrival distribution by H and the service time distribution of server i by F i . It is assumed that any F i is phase--type, say with representation (fi i ; S i ) where fi i is a m i --dimensional row vector and S i is a m i \Theta m i --matrix. 2 S. Asmussen, J.R. Mller / Calculation of the waiting time That is, the density is \Gammafi i e S i x S i e (throughout, e denotes the column vector of ones with the appropriate dimension depending on the context). In the homogeneous case F 1 = : : : = F c = F , m i = m, we write GI=PH=c. The service time is FCFS in the sense that the customers form a single line in the order of arrival and joins the next server to become available. In the homogeneous case, this suffices for a complete model description, whereas in the heterogenous case, we also need to describe the rule for the allocation of a server to a customer arriving in a not--all--busy period, that is, a period in which some or all servers are idle. We will consider two situations, one where any of s ! c idle servers is selected with equal probability 1=s, and one where the servers have priorities in the sense that the customer selects server i before server j when i ! j (other Markovian rules can easily be treated with small modifications of the analysis given for these two cases). The waiting time of a customer is the time from his arrival until service starts; it can b...

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Research w1der contract No. Nonr-855(09) for research in probability and statistics at the University of