William G. Marchal

Characterizations of generalized hyperexponential distribution functions

This paper examines in detail the class of generalized hyperexponential (GH) probability distribution functions. The family is compared to and contrasted with similar popular classes of distributions used in stochastic modeling. Each of these families arises from a desire to preserve the computationally attractive feature of “memorylessness” possessed by the exponential probability distribution while extending the representations to a broader class in order to approximate an arbitrary probability distribution function. Thus the simple structure and attractive properties of the GH probability distribution functions are presented with a view toward facilitating the mathematical operations which frequently occur in practice.

State dependence in M / G /1 server-vacation models

This paper examines a generalization of the exhaustive and one-at-a-time-discipline M/G/1 server vacation models. This alternative model is viewed as a state-dependent (nonvacation) M/G/1 queue in which the original service times are extended to include a (possibly zero length) state-dependent vacation after each service. Such a vacation policy permits greater flexibility in modeling real problems, and does, in fact, subsume most prior M/G/1 approaches. This device reveals a fundamental decomposition somewhat like that previously established for the classical vacation disciplines. In addition, necessary and sufficient conditions for system ergodicity are established for the state-dependent vacation policy, and some comments are offered on computations.

Robustness of Rootfinding in Single-Server Queueing Models

There has been frequent controversy over the years regarding the use of numerical rootfinding for the solution of queueing problems. It has been said that such problems quite often present computational difficulties. However, it turns out that rootfinding in queueing is so well structured that problems rarely occur. There are fundamental properties possessed by the well-known queueing models that eliminate classical rootfinding problems. Most importantly, we show that distinctness of roots is common within simply determined regions in the complex plane and provide conditions under which the characteristic equations for the G/EK/1 and EK/G/1 models have easily found, distinct roots. Furthermore, we show that the characteristic equation for the more general G/GEK/1 model has a collection of real and complex roots which are effectively distinct and located in clearly defined regions of the complex domain. Extensive computational results are given to support our contentions. INFORMS Journal on Computing, ISSN...

Distribution Estimation Using Laplace Transforms

We propose two related methods for deriving probability distribution estimates using approximate rational Laplace transform representations. Whatever method is used, the result is a Coxian estimate for an arbitrary distribution form or plain sample data, with the algebra of the Coxian often simplifying to a generalized hyperexponentia l or phase-type. The transform (or, alternatively, the moment-generating function) is used to facilitate the computations and leads to an attractive algorithm. For method one, the first 2N - 1 derivatives of the transform are matched with those of an approximate rational function; for the second method, a like number of values of the transform are matched with those of the approximation. The numerical process in both cases begins with an empirical Laplace transform or truncation of the actual transform, and then requires only the solution of a relatively small system of linear equations, followed by root finding for a low-degree polynomial. Besides the computationally attractive features of the overall procedure, it addresses the question of the number of terms, or the order, involved in a generalized hyperexponential, phase-type, or Coxian distribution, a problem not adequately treated by existing methods. Coxian distributions are commonly used in the modeling of single-stage and network queueing problems, inventory theory, and reliability analyses. They are particularly handy in the development of large-scale model approximations.

A note on generalized hyperexponential distributions

We update earlier results on the relationship of generalized hyperex-ponential (GH) distributions to other types of CDFs in the Coxian family, particularly the phase types. Most specifically, we utilize properties of these Coxian-type distributions to offer new approaches to determining when a given rational transform corresponds to a GH distribution.

Technical Note—Some Simpler Bounds on the Mean Queuing Time

This note presents a general lower bound for the mean queuing time in a stationary GI/G/c queue. It also discusses the application of the bound to an M/G/c system and the general areas of usefulness.