Peter Purdue

The M/M/1 Queue in a Markovian Environment

This paper discusses the M/M/1 queue embedded in a Markovian environment or, subject to extraneous phase changes. It studies the busy period, equilibrium conditions, and emptiness probabilities by using analytic matric functions, and makes comparisons with the work of Neuts and of Naor and Yechiali.

Stochastic theory of compartments: One and two compartment systems

Previous work on compartmental systems is generalized (i) to allow the particles present at time zero to have a different lifetime distribution than those which arrive after time zero, and (ii) to allow a particle which enters the system at timet to have a lifetime distribution which is a function oft but is otherwise quite general. The one and two compartment models are analyzed under the above conditions and compared to previous results of Thakuret al. (1974), Purdue (1974) and Cardenas and Matis (1974). Finally, some results for the two compartment, reversible system are given. The analysis used is a blend of direct random variable and queueing theoretic techniques.

Stochastic theory of compartments

The stochastic model of a compartment developed by Thakur, Rescigno and Schafer is discussed without using generating functions. The behavior of the mean and variance of the number of particles present as a function of time is also discussed. We also allow both the input and output to be time dependent.

A semi-markov approach to stochastic compartmental models

In the past, various methods using either differential equations or differential-difference equations have been used to analyze stochastic compartmental models. In this paper a semi-Markov process approach is used to provide a framework for analyzing such models. The distribution function of the number of particles in each of the compartments is derived along with the stationary distributions. Various models found in the literature arising from biological and reliability applications are analyzed here using the semi-Markov process technique.

Stochastic theory of compartments: An open, two compartment, reversible system with independent poisson arrivals

This paper discusses two compartment models with interaction allowed between the compartments. The total number of particles in the system at any time is discussed along with the number to the found in each separate compartment. An interesting result is that the number of particles in each of the two compartments areindependent random variables. Some asymptotic results are also given. The paper is a continuation of some earlier work by the author.

The stochastic theory of compartments—A mammillary system

A stochastic mammalillary model with arbitrary holding times for each of the compartments is analyzed here both with and without input. Several results found earlier by other authors for a special case of this model are generalized and some new results are also added. In particular the distribution of the system content is determined at a finite time and also in the steady state. Computable bounds for the probability of exceeding a given threshold in the peripheral compartments are given. The time until the system becomes empty is also discussed.

Variability in a single compartment system: A note on S. R. Bernard's model

In this note we examine a continuous time version of a compartmental model introduced in a discrete time setting by S. R. Bernard. The model allows for more than one particle to leave the system at any time. This introduces additional randomness into the system, over the pure death system and this is reflected in the variance function.

The Single Server Queue in a Markovian Environment

This paper discusses the single server queue wherein the arrival and service rates are subject to random varitions. It is shown how the “Arrival-Service” function is useful in the analysis of such systems. The integral equation of the Busy Period is derived and the equilibrium conditions exhibited.

Non-linear compartmental systems: extensions of S. R. Bernard's urn model

One of the limitations of stochastic, linear compartmental systems is the small degree of variability in the contents of compartments. S. R. Bernards (1981) urn model (S. R. Bernardet al., Bull. math. Biol. 43, 33–45.) which allows for bulk arrivals and departures from a one-compartment system, was suggested as a way of increasing content variability. In this paper, we show how the probability distribution of the contents of an urn model may be simply derived by studying an appropriate set of exchangeable random variables. In addition, we show how further increases in variability may be modeled by allowing the size of arrivals and departures to be random.

An urn model for some evolutionary processes

Abstract Some properties of a particular urn model are developed and applied to problems in language evalution and protein evalution. The connection between these two forms of evolution is also noted.