Percy H. Brill

An Algorithm to Compute the Waiting Time Distribution for the M/G/1 Queue

In many modern applications of queueing theory, the classical assumption of exponentially decaying service distributions does not apply. In particular, Internet and insurance risk problems may involve heavy-tailed distributions. A difficulty with heavy-tailed distributions is that they may not have closed-form, analytic Laplace transforms. This makes numerical methods, which use the Laplace transform, challenging. In this paper, we develop a method for approximating Laplace transforms. Using the approximation, we give algorithms to compute the steady state probability distribution of the waiting time of an M/G/1 queue to a desired accuracy. We give several numerical examples, and we validate the approximation with known results where possible or with simulations otherwise. We also give convergence proofs for the methods.

AN EMBEDDED LEVEL CROSSING TECHNIQUE FOR DAMS AND QUEUES

The new concept of embedded level crossings is combined with the old principle of stationary set balance to produce an alternative approach for obtaining the steady-state distribution of the level in a dam with general release rule. The method yields the steady state distribution of the customer waiting time in the GI/G/1 queue as a special case. Results for a dam in which the instantaneous release rate is proportional to the level, and for the M/G/1, GI/M/1, E,/M/1 and DIM/1 queues are derived using the new technique.

Internet-Type Queues with Power-Tailed Interarrival Times and Computational Methods for Their Analysis

Internet traffic flows have often been characterized as having power-tailed (long-tailed, fat-tailed, heavy-tailed) packet interarrival times or service requirements. In this work, we focus on the development of complete and computationally efficient steady-state solutions of queues with power-tailed interarrival times when the service times are assumed exponential. The classical method for obtaining the steady-state probabilities and delay-time distributions for the G/M/1 (G/M/ c) queue requires solving a root-finding problem involving the Laplace-Stieltjes transform of the interarrival-time distribution function. Then the exponential service assumption is combined with the derived geometric arrival-point probabilities to get both the limiting general-time state and delay distributions. However, in situations where there is a power tail, the interarrival transform is typically quite complicated and never analytically tractable. In addition, it is possible that there is only a degenerate steady-state system-size probability distribution. Thus, an alternative approach to obtaining a steady-state solution is typically needed when power-tailed interarrivals are present. We exploit the exponentiality of the steady-state delay distributions for the G/M/1 and G/M/ c queues, using level-crossings and a transform approximation method, to develop an alternative root-finding problem when there are power-tailed interarrival times. Extensive computational results are given.

An M/G/1 retrial queue with balking and retrials during service

We consider an M/G/1 retrial queue with general retrial times. Customers may balk or renege at particular times. The server is subject to breakdown (with repairs). While the server is being repaired, the customer in service can either remain in the service position or leave and return, while maintaining its rights to the server. We find a stability condition for this system. In the steady state, the joint distribution of the server state and queue length is obtained, leading to useful measures of the system, such as the probability of an empty system, the mean number of customers in the retrial queue and the expected retrial time.

Measurement of adaptivity and flexibility in production systems

Abstract This paper introduces concepts of adaptivity which quantify how machine flexibility changes as the environment shifts. The definition of adaptivity utilizes flexibility measures based on task sets, weights of importance, and machine performance. Measures of adaptivity are given from one situation to another, over a time interval, and instantaneously in time. Examples are presented.

A level crossing quantile estimation method

We introduce a nonparametric quantile estimation method by applying a level crossing empirical function which will be defined in this paper, and also introduce a computational method for the new estimator. A comparison of the new quantile estimation method with the usual kernel quantile estimation method based on the classical empirical distribution function is included. Computational results show that the new method is more efficient than the usual method in many cases.

Analysis of net inventory in continuous review models with random lead time

Abstract This paper derives the steady state distribution of net inventory in a continuous review inventory system under an (s, S) policy in which the demand process is Poisson, ordering decisions are based on net inventory, and lead times are random. The paper assumes that a single supplier satisfies the systems orders and may supply other similar systems as well. It develops useful approximations for the stationary distribution of net inventory, and also relates our results to previous work which utilizes ordering policies based on inventory position. The analysis of the model applies level crossing theory.

Analytical Distribution of Waiting Time in the M/{iD}/1 Queue

We give an analytical formula for the steady-state distribution of queue-wait in the M/G/1 queue, where the service time for each customer is a positive integer multiple of a constant D > 0. We call this an M/{iD}/1 queue. We give numerical algorithms to calculate the distribution. In addition, in the case that the service distribution is sparse, we give revised algorithms that can compute the distribution more quickly.

Using quantile estimates in simulating internet queues with Pareto service times

It is readily apparent how important the Internet is to modern life. The exponential growth in its use requires good tools for analyzing congestion. Much has been written recently asserting that classical queueing models assuming Poisson arrivals or exponential service cannot be used for the accurate study of congestion in major portions of the Internet. Internet traffic data indicate that heavy-tailed distributions (e.g., Pareto) serve as better models in many situations for packet service lengths. But these distributions may not possess closed-form analytic Laplace transforms; hence, much standard queueing theory cannot be used. Simulating such queues becomes essential; however, previous research pointed out difficulties in obtaining the usual moment performance measures such as mean wait in queue. We investigate the use of quantile estimates of waiting times (e.g., median instead of mean), which appear to be considerably more efficient when service times are Pareto.

An Algorithm for Fitting Heavy-Tailed Distributions via Generalized Hyperexponentials

In this paper, we propose an algorithm to fit heavy-tailed (HT) distribution functions by generalized hyperexponential (GH) distribution functions. A discussion of the steps, usage, and accuracy of the GH algorithm is given. Several examples in this paper show that the proposed method can be applied to fit HT distributions with a completely monotone probability density function (pdf) very well, like the Pareto distribution and the Weibull distribution with the shape parameter less than one, as well as HT distributions whose pdf is not completely monotone, like the lognormal distribution. In addition, we provide an example that shows that the proposed method can be applied to density estimation of real data presenting a heavy tail.