Mary A. Johnson

An investigation of phase-distribution moment-matching algorithms for use in queueing models

Algorithms for matching moments to phase-type distributions are evaluated on the basis of their performance in their intended application, queueing models. The moment-matching algorithms under consideration match two moments to a hyperexponential distribution with balanced means and three moments to a mixture of two Erlang distributions of common order. These algorithms are used to approximate an interarrival-time distribution for a queueing model, and the accuracy of associated performance-measure approximations is then used to evaluate the moment-matching algorithms. Three performance measures are considered, and attention is focussed on the steady-state mean queue length (number in system) of theGI/M/1 queue. Performance-measure approximations are compared to three-moment bounds and performance-measure values arising from hypothetical approximated distributions.

An empirical study of queueing approximations based on phase-type distributions

The GI/PH/1 model (single-server queue with general independent interarrival times and phase-type service times) is used to explore the behavior of phase-type (PH) approximations of interarrival- and service-time distributions. Various PH approximating distributions are considered for five non-PH interarrival-time distributions and two PH service-time distributions. Simple two- and three-moment-matching approximations and, in some cases, approximations obtained by also considering distribution shape are compared. Each approximating distribution is evaluated on the basis of the quality of the corresponding approximations of steady-state measures of the congestion seen by an arriving customer. The traffic intensity and the variability of the distribution (interarrival-or service-time) that is not approximated are shown to have a substantial effect on the accuracy of the queueing approximations. Rules for selecting approximating distributions for queueing applications are suggested

Descriptors of arrival-process burstiness with application to the discrete Markovian arrival process

We use the concept of burstiness to propose new descriptors of arrival processes or, more generally, point or traffic processes. We say that the arrival process is in aburst when its interarrival times are less than or equal to some threshold value; during periods in which interarrival times are greater than the threshold value, the process is in agap. We propose to describe the arrival process in terms of the number of arrivals during a burst (gap) and the duration of a burst (gap). For the case of discrete-time Markovian arrival processes we derive the distribution of the number of arrivals during a burst (gap) and the mean duration of a burst (gap). We present numerical results to illustrate how our descriptors can be used to understand the behavior of an arrival process and the congestion it induces in a queueing system. We also report the results of an experiment which shows a statistical relationship between two of our descriptors and a measure of queueing congestion.

Experimental Evaluation of a Procedure for Estimating Nonhomogeneous Poisson Processes Having Cyclic Behavior

This paper summarizes an experimental evaluation of a procedure for estimating a nonhomogeneous Poisson process having an exponential rate function, where the exponent may include both polynomial and trigonometric components. Maximum likelihood estimates of the unknown continuous parameters of the rate function are obtained numerically, and the degree of the polynomial rate component is determined by a likelihood ratio test. Our evaluation is limited to the case of a known frequency. To examine the small- and large-sample behavior of the estimation procedure for many types of time-dependent arrival processes encountered in previous work, we performed 100 independent replications of twelve selected Poisson processes over two observation intervals. On each replication of each case, we applied the fitting procedure to estimate the parameters of the target process; and we computed the corresponding estimates of the rate and mean-value functions over the observation interval. Evaluation of the fitting procedure was based on plotted tolerance bands for the rate and mean-value functions together with summary statistics for the maximum and average absolute estimation errors in these functions over the observation interval. The experimental results provide substantial evidence of the procedures numerical stability and practical usefulness. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A graphical investigation of error bounds for moment-based queueing approximations

Many approximations of queueing performance measures are based on moment matching. Empirical and theoretical results show that although approximations based on two moments are often accurate, two-moment approximations can be arbitrarily bad and sometimes three-moment approximations are far better. In this paper, we investigate graphically error bounds for two- and three-moment approximations of three performance measures forGI/M/ · type models. Our graphical analysis provides insight into the adequacy of two- and three-moment approximations as a function of standardized moments of the interarrival-time distribution. We also discuss how the behavior of these approximations varies with other model parameters and with the performance measure being approximated.

Estimation and simulation of nonhomogeneous Poisson processes having multiple periodicities

We develop and evaluate procedures for estimating and simulating nonhomogeneous Poisson processes (NHPPs) having an exponential rate function, where the exponent may include a polynomial component or some trigonometric components or both. Maximum likelihood estimates of the unknown continuous parameters of the rate function are obtained numerically, and the degree of the polynomial rate component is determined by a likelihood ratio test. The experimental performance evaluation for this estimation procedure involves applying the procedure to 100 independent replications of nine selected point processes that possess up to four trigonometric rate components together with a polynomial rate component whose degree ranges from zero to three. On each replication of each process, the fitting procedure is applied to estimate the parameters of the process; and then the corresponding estimates of the rate and mean value functions are computed over the observation interval. Evaluation of the fitting procedure is based on plotted tolerance bands for the rate and mean value functions together with summary statistics for the maximum and average absolute estimation errors in these functions over the observation interval. The experimental results provide substantial evidence of the numerical stability and usefulness of the fitting procedure in simulation applications.

Markov MECO : A simple markovian model for approximating nonrenewal arrival processes

This paper introduces a special case of the Markovian Arrival Process that can easily be used to approximate both the interarrival-time distribution and the autocorrelation function of an arrival process. This model is labeled a Markov MECO (Mixture of Erlangs of Common Order). The proposed interarrival-time approximation matches the first three moments of the interarrival time. The Markov MECO autocorrelation function is geometric and determined by a single parameter, once the interarrival-time distribution is fixed. Closed-form expressions are given for matching either the lag-one autocorrelation or the asymptotic index of dispersion for intervals. By applying the resulting arrival-process approximations to a single-server queue with exponential service times, the approximations are empirically evaluated on the basis of the associated error in the steady-state mean number in the system. In these experiments, the original arrival processes are superpositions of independent renewal processes. The numerical results show that the nonrenewal approximations outperform the corresponding renewal approximation that simply ignores autocorrelation, and in some cases, the improvement is an order of magnitude.

Burstiness descriptors for markov renewal processes and markovian arrival processes

Quantitative descriptors of the burstiness of an arrival process are derived for Markov renewal processes (MRPs) and Markovian arrival processes (MAPs). Our burstiness descriptors are based on simple definitions of a burst and a gap in an arrival process. Briefly, for threshold v, we define a burst to be a (maximal) interval during which all interarrival times are less than or equal to v and a gap to be a (maximal) interval during which all interarrival times are greater than v. Thus, an arrival process alternates between bursts and gaps. For the case of a MRP, we derive the distribution of the number of arrivals in a burst (gap) and the mean of the duration of a burst (gap). These results are then specialized for the case of a MAP. For four example MAP z.repos s, six complementary descriptors are plotted as a function of threshold parameter i. These plots illustrate how our burstiness descriptors can be used to gain insight into the behavior of the arrival process and into the behavior of a queueing sy...

NPPMLE and NPPSIM: Software for estimating and simulating nonhomogeneous Poisson processes having cyclic behavior

We describe portable software for estimating and simulating a nonhomogeneous Poisson process whose rate function is exponential and may include a polynomial or a sinusoidal component. Program NPPMLE computes maximum-likelihood estimates of the rate-function parameters from a series of event epochs. Given the rate-function parameters, program NPPSIM simulates the process via a thinning scheme.

Tchebycheff Systems for Probabilistic Analysis

SYNOPTIC ABSTRACTWe introduce the theory of Tchebycheff systems and highlight theory relevant to probability applications—especially approximation of stochastic models. We restate and interpret definitions and results of Karlin and Studden (1966) and other sources so that their probabilistic significance is readily apparent. We also provide two original theorems and show how they can be used to identify Tchebycheff systems useful in obtaining queueing-approximation error bounds. Several applications are described in sufficient detail to illustrate the usefulness of Tchebycheff systems.