Marcel F. Neuts

Efficient Algorithmic Solutions to Exponential Tandem Queues with Blocking.

Stable queuing systems consisting of two groups of servers, having exponential service times, placed in tandem and separated by a finite buffer, are shown to have a steady-state probability vector of matrix-geometric form. The queue is stable as long as the Poisson arrival rate does not exceed a critical value, which depends in a complicated manner on the service rates, the numbers of servers in each group, the size of the intermediate buffer and the unblocking rule followed when system becomes blocked. The critical input rate is determined in a unified manner.For stable queues, it is shown how the stationary probability vector and other important features of the queue may be computed. The essential step in the algorithm is the evaluation of the unique positive solution of a quadratic matrix equation.

On the use of phase type distributions in reliability modelling of systems with two components

SummaryAssuming that the time-to-failure and repair time distributions areof phase type, a variety of reliability models with a small number of components may be studied in terms of finite-state Markov processes. Although the state spaces of these processes are typically large, their infinitesimal generators are highly structured. By utilizing the formalism of PH-distributions, it is possible to construct efficient algorithms to evaluate a large number of quantities of interest. Some new properties of PH-distributions are also established.ZusammenfassungFür Verteilungen der Lebensdauer und der Reparaturdauer vom Phasentyp lassen sich bei kleiner Komponentenanzahl eine Reihe von Zuverlässigkeitsmodellen mit Hilfe von Markoff-Prozessen mit endlichem Zustandsraum untersuchen. Zwar sind die Zustandsräume dieser Prozesse groß, doch sind die Matrizen der Übergangsraten stark strukturiert. Durch Anwendung des Formalismus der Verteilungen vom Phasentyp lassen sich effiziente Algorithmen zur Berechnung einer großen Anzahl interessierender Größen entwickeln. Daneben werden einige neue Eigenschaften der Verteilungen vom Phasentyp dargestellt.

Asymptotic Behavior of the Stationary Distributions in the GI/PH/c Queue with Heterogeneous Servers.

SummaryThis paper deals with the stablec-server queue with renewal input. The service time distributions may be different for the various servers. They are however all probability distributions of phase type. It is shown that the stationary distribution of the queue length at arrivals has an exact geometric tail of rate η, 0<η<1. It is further shown that the stationary waiting time distribution at arrivals has an exact exponential tail of decay parameter ξ>0. The quantities η and ξ may be evaluated together by an elementary algorithm. For both distributions, the multiplicative constants which arise in the asymptotic forms may be fully characterized. These constants are however difficult to compute in general.

Matrix-analytic methods in queuing theory☆

Abstract This review describes the development during the past decade of a number of matrix-analytic methods for the study of a variety of stochastic models, primarily queues but also certain models for dams and inventories. This work originated in the search for algorithmic methods and has led to results that are well-suited for computer implementation. It has also required a reexamination of the theoretical approaches to these stochastic models, which may now be analyzed by purely probabilistic methods rather than by techniques from complex analysis. The author also argues the case for an algorithmic optic on the problems of applied probability and sketches the main steps in the construction of computer codes for the evaluation of stationary distributions of interest. The approach is built on certain nonlinear matrix equations that arise naturally in the study of structured Markov chains which are generalizations of the chains embedded in the classical GI/M/1 and M/G/1 queues.

A single server queue with platooned arrivals and phase type services

Abstract A semi-Markovian point process which qualitatively models platooned arrivals is introduced. This process is used as the input to a single server queue in which the service times are independent and have a common distribution of phase type. It is shown that this queue has an embedded Markov chain of a particular block-partitioned type, whose invariant probability vector in the stable case is of matrix-geometric form. Detailed algorithms for the computation of the steady-state features of the queue are obtained and a representative numerical example is discussed.

Algorithmic methods for multi-server queues with group arrivals and exponential services

Abstract We discuss algorithms for the computation of the steady-state features of the c -server queue with exponential service times and bounded group arrivals. While our methods are valid for general interarrival time distributions, we treat in detail the simplifications obtained by using a distribution of phase type. Oueue length densities at times prior to arrivals and at arbitrary times are obtained in a modified matrix-geometric form. Their means and variances are found in computationally tractable forms. We also present algorithmic methods for the waiting time and the virtual waiting time distributions of a customer in a group and obtain the means and variances of these distributions in tractable forms. Numerical examples show that the effects of changing various parameters of the queuing model may so be examined at a small computational cost.

The abscissa of convergence of the laplace-stieltjes transform of a ph-distribution

If a probability distribution of phase type has an irreducible representation (α,T), the abscissa of convergence of its Laplace-Stieltjes transform is shown to be the eigenvalue of maximum real part of the matrix T.

The superposition of two PH-renewal processes

A large variety of stochastic models may be viewed as systems of two interacting renewal processes. Unless at least one of these is a Poisson process, the analytic properties of the model are difficult to study and are often intractable. It is clear that the greater analytic tractability of such models as the M/G/1 and GI/M/1 queues is due to the memory-less property of the exponential distribution. This tractability is gained, however, at the price of a severe distributional assumption. A useful compromise is found in the introduction of the probability distributions of phase type (PH-distributions). For these, a formalism of matrix manipulations has been developed which extends the elementary properties of the exponential distribution, yet is general and versatile enough to cover the probability distributions needed in a large variety of modelling problems.

On the coefficient of variation of mixtures of probability distributions

We obtain tight upper and lower bounds on the coefficient of variation of mixtures of m probability distributions on [0,∞). The lower bound is explicit; the upper bound is computed by an elementary algorithm. In some particular cases, the upper bound may also be obtained in an analytically explicit form.