Ward Whitt

Characterizing Superposition Arrival Processes in Packet Multiplexers for Voice and Data

This paper analyzes a model of a multiplexer for packetized voice and data. A major part of the analysis is devoted to characterizing the aggregate packet arrival process resulting from the superposition of separate voice streams. This is done via the index of dispersion for intervals (IDI), which describes the cumulative covariance among successive interarrival times. The IDI seems very promising as a measurement tool to characterize complex arrival processes. This paper also describes the delays experienced by voice and data packets in the multiplexer using relatively simple two-parameter approximations.

The Fourier-series method for inverting transforms of probability distributions

This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdfs) and probability mass functions (pmfs) 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 cdfs can be calculated from generating functions by finite sums without truncation. For other cdfs, 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 purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.

Numerical Inversion of Laplace Transforms of Probability Distributions

We present a simple algorithm for numerically inverting Laplace transforms. The algorithm is designed especially for probability cumulative distribution functions, but it applies to other functions as well. Since it does not seem possible to provide effective methods with simple general error bounds, we simultaneously use two different methods to confirm the accuracy. Both methods are variants of the Fourier-series method. The first, building on Dubner and Abate (Dubner, H., J. Abate. 1968. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. JACM 15 115–123.) and Simon, Stroot, and Weiss (Simon, R. M., M. T. Stroot, G. H. Weiss. 1972. Numerical inversion of Laplace transforms with application to percentage labeled experiments. Comput. Biomed. Res. 6 596–607.), uses the Bromwich integral, the Poisson summation formula and Euler summation; the second, building on Jagerman (Jagerman, D. L. 1978. An inversion technique for the Laplace transform with applications....

Heavy-Traffic Limits for Queues with Many Exponential Servers

Two different kinds of heavy-traffic limit theorems have been proved for s -server queues. The first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below. The second kind, which is often not thought of as heavy traffic, involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed, with the traffic intensities being less than the critical value of one. In each case the sequence of random variables depicting the steady-state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization. However, in an important respect neither procedure adequately represents a typical queueing system in practice because in the (heavy-traffic) limit an arriving customer is either almost certain to be delayed (first procedure) or almost certain not to be delayed (second procedure). Hence, we consider a sequence of ( GI / M / S ) systems in which the traffic intensities converge to one from below, the arrival rates and the numbers of servers go to infinity, but the steady-state probabilities that all servers are busy are held fixed. The limits in this case are hybrids of the limits in the other two cases. Numerical comparisons indicate that the resulting approximation is better than the earlier ones for many-server systems operating at typically encountered loads.

Some Useful Functions for Functional Limit Theorems

Many useful descriptions of stochastic models can be obtained from functional limit theorems invariance principles or weak convergence theorems for probability measures on function spaces. These descriptions typically come from standard functional limit theorems via the continuous mapping theorem. This paper facilitates applications of the continuous mapping theorem by determining when several important functions and sequences of functions preserve convergence. The functions considered are composition, addition, composition plus addition, multiplication, supremum, reflecting barrier, first passage time and time reversal. These functions provide means for proving new functional limit theorems from previous ones. These functions are useful, for example, to establish the stability or continuity of queues and other stochastic models.

Logarithmic asymptotics for steady-state tail probabilities in a single-server queue

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 asymptotics of the form x-1 log P(W > x) -+ -0* as x - o0 for 0* > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Girtner-Ellis condition for the cumulant generating function of the associated partial sums, i.e. n-1 log Eexp (OSn) --+ (0) as n - oo, plus regularity conditions on the decay rate function 0. The asymptotic decay rate 0* is the root of the equation 0(0) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing 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 multiclass queues based on asymptotic decay rates.

Squeezing the Most Out of ATM

Although ATM seems to be the wave of the future, one analysis requires that the utilization of the network be quite low. That analysis is based on asymptotic decay rates of steady-state distributions used to develop a concept of effective bandwidths for connection admission control. The present authors have developed an exact numerical algorithm that shows that the effective-bandwidth approximation can overestimate the target small blocking probabilities by several orders of magnitude when there are many sources that are more bursty than Poisson. The bad news is that the appealing simple connection admission control algorithm using effective bandwidths based solely on tail-probability asymptotic decay rates may actually not be as effective as many have hoped. The good news is that the statistical multiplexing gain on ATM networks may actually be higher than some have feared. For one example, thought to be realistic, the analysis indicates that the network actually can support twice as many sources as predicted by the effective-bandwidth approximation. The authors also show that the effective bandwidth approximation is not always conservative. Specifically, for sources less bursty than Poisson, the asymptotic constant grows exponentially in the number of sources (when they are scaled as above) and the effective-bandwidth approximation can greatly underestimate the target blocking probabilities. Finally, they develop new approximations that work much better than the pure effective-bandwidth approximation.

Deciding which queue to join: Some counterexamples

Consider a queueing system with two or more servers, each with its own queue with infinite capacity. Customers arrive according to some stochastic process e.g., a Poisson process and immediately upon arrival must join one of the queues, thereafter to be served on a first-come first-served basis, with no jockeying or defections allowed. The service times are independent and identically distributed with a known distribution. Moreover, the service times are independent of the arrival process and the customer decisions. The only information about the history of the system available for deciding which queue to join is the number of customers currently waiting and being served at each server. Joining the shortest queue is known to minimize each customers individual expected delay and the long-run average delay per customer when the service-time distribution is exponential or has nondecreasing failure rate. We show that there are service-time distributions for which it is not optimal to always join the shortest queue. We also show that if, in addition, the elapsed service times of customers in service are known, the long-run average delay is not always minimized by customers joining the queue that minimizes their individual expected delays.

Continuity of Generalized Semi-Markov Processes

It is shown that sequences of generalized semi-Markov processes converge in the sense of weak convergence of random functions if associated sequences of defining elements (initial distributions, transition functions and clock time distributions) converge. This continuity or stability is used to obtain information about invariant probability measures. It is shown that there exists an invariant probability measure for any finite-state generalized semi-Markov process in which each clock time distribution has a continuous c.d.f. and a finite mean. For generalized semi-Markov processes with unique invariant probability measures, sequences of invariant probability measures converge when associated sequences of defining elements converge. Hence, properties of invariant measures can be deduced from convenient approximations. For example, insensitivity properties established for special classes of generalized semi-Markov processes by Schassberger (Schassberger, R. 1977. Insensitivity of steady-state distributions of generalized semi-Markov processes, I. Ann. Probab. 5 87--99.), (Schassberger, R. 1978. Insensitivity of steady-state distributions of generalized semi-Markov processes, II. Ann. Probab. 6 85--93.), Konig and Jansen (Konig, D., U. Jansen. 1976. Eine Invarianzeigenschaft Zufalliger Bedienungs-Prozesse mit Positiven Geschwindigkeiten. Math Nachr. 70 321--364.), and Burman (Burman, D. Y. 1981. Insensitivity in queueing systems. Adv. Appl. Probab. To appear.) extend to a larger class of generalized semi-Markov processes.

Multiple channel queues in heavy traffic. II: sequences, networks, and batches

Abstract : Sequences of queueing facilities with r parallel arrival channels and s parallel service channels are studied under the conditions of heavy traffic: the associated sequences of traffic intensities approaching a limit greater than or equal to one. Weak convergence is obtained for sequences of random functions induced in D(0,1) by the basic queueing processes. Sequences of queueing systems in heavy traffic which are networks of the facilities described above are also investigated. Furthermore, customers are allowed to arrive and be served in batches.