N. U. Prabhu

Markov-modulated queueing systems

Markov-modulated queueing systems are those in which the primary arrival and service mechanisms are influenced by changes of phase in a secondary Markov process. This influence may be external or internal, and may represent factors such as changes in environment or service interruptions. An important example of such a model arises in packet switching, where the calls generating packets are identified as customers being served at an infinite server system. In this paper we first survey a number of different models for Markov-modulated queueing systems. We then analyze a model in which the workload process and the secondary process together constitute a Markov compound Poisson process. We derive the properties of the waiting time, idle time and busy period, using techniques based on infinitesimal generators. This model was first investigated by G.J.K. Regterschot and J.H.A. de Smit using Wiener-Hopf techniques, their primary interest being the queue-length and waiting time.

Time-dependent results in storage theory

The probability theory of storage systems formulated by P. A. P. Moran in 1954 has now developed into an active branch of applied probability. An excellent account of the theory, describing results obtained up to 1958 is contained in Morans (1959) monograph, Considerable progress has since been made in several directions-the study ofthe time-dependent behaviour ofstochastic processes underlying Morans original model, modifications of this model, as well as the formulation and solution of new models. The aim of this paper is to give an expository account of these developments; a comprehensive treatment will be found in the authors forthcoming book [Prabhu (1964)].

Stochastic processes : basic theory and its applications

A Review of Probability Distributions and Their Properties Definition and Characteristics of a Stochastic Process Some Important Classes of Stochastic Processes Stationary Processes The Brownian Motion and the Poisson Process, Levy Processes Renewal Processes and Random Walks Martingales in Discrete Time Branching Processes Regenerative Phenomena Markov Chains Tauberian Theorems.

Large sample inference from single server queues

Problems of large sample estimation and tests for the parameters in a single server queue are discussed. The service time and the interarrivai time densities are assumed to belong to (positive) exponential families. The queueing system is observed over a continuous time interval (0,T] whereT is determined by a suitable stopping rule. The limit distributions of the estimates are obtained in a unified setting, and without imposing the ergodicity condition on the queue length process. Generalized linear models, in particular, log-linear models are considered when several independent queues are observed. The mean service times and the mean interarrival times after appropriate transformations are assumed to satisfy a linear model involving unknown parameters of interest, and known covariates. These models enhance the scope and the usefulness of the standard queueing systems.

A storage model with continuous infinitely divisible inputs

This paper presents a general theory of storage for dams subject to a steady release, and with continuous inputs having infinitely divisible distributions. The paper extends some earlier work by Downton, Lindley, Smith and Kendall ((10), see discussion), and gives a detailed account of results previously sketched by the authors (6).

Markov-Modulated Processes and Semiregenerative Phenomena

Foundations of Markov Random Walks and Markov Additive Processes: Theory of Semiregenerative Phenomena Markov Random Walk-Fluctuation Theory and Wiener-Hopf Factorization Further Results for Semiregenerative Phenomena Limits Theorems for Markov Random Walks Markov Renewal and Markov-Additive Processes -- A Review and Some New Results Markov-Additive Processes of Arrivals Application Examples: Markov-Modulated Single-Server Queueing Systems A Storage Model for Data Communications Systems A Markovian Storage Model.

A bibliography of books and survey papers on queueing systems: theory and applications

In response to the Editorial Introduction (Queueing Systems I (1986), 1-4) the author received several suggestions for additional books that deserved mention. Rather than publishing a supplementary note to that article, the author thought it might perhaps be worthwhile to compile a more complete bibliography of books and important papers on queueing theory. The following is such a bibliography. In section 1 we list important books that serve as valuable sources for the work done during the pioneering phase of queueing theory. This list includes the two survey papers by Kendall. Several other papers should perhaps have been included, but the author decided against it. Many of the books listed here are now out of print, and others are out of date. Even so this list provides a historical perspective. Section 2 contains books on elementary queueing theory with emphasis on applications, and in section 3 are listed advanced books and monographs on a wide variety of special topics. Books on computer performance and data communications systems are listed separately in sections 4 and 5, as these two areas have become increasingly important. Problems in teletraffic have motivated much of the earlier research in queueing theory; nevertheless this is an independent topic by itself, so it is covered by section 6, along with other related topics such as road traffic and scheduling. Numerical tables and simulation are included in section 7. In the final section 8 we list books of general interest whose scope goes beyond queueing theory, but are considered to be very important sources on the subject. Many textbooks on probability, statistics and operations research include some material on queueing systems, but they do not merit a place in a bibliography of this type. A few books in languages other than English are included, but this list is far from complete in this respect. The author is grateful to the associate editors for bringing to his attention several important works of reference.

Markov-modulated PH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated EK/G/1 queueing system as a special case.

Markov-modulated Single Server Queueing Systems

We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained. MARKOV-MODULATED QUEUEING SYSTEM; MARKOV RANDOM WALKS; WAITING TIME; IDLE TIME; TIMEREVERSIBILITY AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J20 SECONDARY 60K25

A storage model for data communication systems

We consider a storage model where the input and demand are modulated by an underlying Markov chain. Such models arise in data communication systems. The input is a Markov-compound Poisson process and the demand is a Markov linear process. The demand is satisfied if physically possible. We study the properties of the demand and its inverse, which may be viewed as transformed time clocks. We show that the unsatisfied demand is related to the infimum of the net input and that, under suitable conditions, it is an additive functional of the input process. The study of the storage level is based on a detailed analysis of the busy period, using techniques based on infinitesimal generators. The transform of the busy period is the unique solution of a certain matrix-functional equation. Steady state results are also obtained; these are not obvious generalizations of the results for simple storage models. In particular, a generalization of the Pollaczek-Khinchin formula brings new insight.