Walter Böhm

A course on queueing models

QUEUES: BASIC CONCEPTS Introduction Queues: Features and Characteristics Graphical Methods Modelling Scope and Organization MARKOVIAN QUEUES Introduction A Simple Model: Steady-State Behaviour Birth-Death Models: Steady-State Behaviour Erlangian Models: Steady-State Behaviour Transient Behaviour Waiting Time and Littles Formula Busy Periods and Idle Periods 3 Networks of Queues - I Optimization Discussion REGENERATIVE NON-MARKOVIAN QUEUES - I Introduction Markovian Input Models Markovian Service-Time Models Bulk Queues Functional Relations: A Heuristic Approach Busy Periods Discrete-Time Queues Discussion COMPUTATIONAL METHODS - I Introduction Root Finding Methods The State Reduction Method Transient Behaviour: Numerical Approaches Discussion STATISTICAL INFERENCE AND SIMULATION Introduction Statistical Inference Solving Queueing Problems by Simulation A Practical Application Discussion REGENERATIVE NON-MARKOVIAN QUEUES - II Introduction Non-Markovian Queues: Transient Solution Combinatorial Methods Functional Relations Discussion GENERAL QUEUES Introduction Waiting Time and Idle Time: An Analytic Method Bounds for the Average Waiting Time A Heavy Traffic Approximation Diffusion Approximation Waiting Time and Idle Time: A Probabilistic Method Duality Discussion COMPUTATIONAL METHODS - II Introduction The Matrix-Geometric Solution The Block Elimination Method The Fourier Series Method for Inverting Transforms Discussion DISCRETE-TIME QUEUES: TRANSIENT SOLUTIONS Introduction Combinatorial Methods: Lattice Path Approach Recurrence Relations Algebraic Methods Discussion MISCELLANEOUS TOPICS Introduction Priority Queues Queues with Infinite Servers Design and Control of Queues Networks of Queues II Discussion APPENDICES INDEX Each chapter includes exercises and references.

The combinatorics of birth–death processes and applications to queues

In this paper we present a combinatorial technique which allows the derivation of the transition functions of general birth-death processes. This method provides a flexible tool for the transient analysis of Markovian queueing systems with state dependent transition rates, like M/M/c models or systems with balking and reneging.

Transient analysis of queues with heterogeneous arrivals

In this paper we consider a discrete time queueing model where the time axis is divided into time slots of unit length. The model satisfies the following assumptions: (i) an event is either an arrival of typei of batch sizebi, i=1,...,r with probabilityαi or is a depature of a single customer with probabilityγ or zero depending on whether the queue is busy or empty; (ii) no more than one event can occur in a slot, therefore the probability that neither an arrival nor a departure occurs in a slot is 1−γ−⌆iαi or 1−⌆iαi according as the queue is busy or empty; (iii) events in different slots are independent. Using a lattice path representation in higher dimensional space we will derive the time dependent joint distribution of the number of arrivals of various types and the number of completed services. The distribution for the corresponding continuous time model is found by using weak convergence.

Repeat-buying Theory and its Application for Recommender Services

In the context of a virtual university’s information broker we study the consumption patterns for information goods and we investigate if Ehrenberg’s repeat-buying theory which successfully models regularities in a large number of consumer product markets can be applied in electronic markets for information goods, too. First results indicate that Ehrenberg’s repeat-buying theory succeeds in describing the consumption patterns of bundles of complementary information goods reasonably well and that this can be exploited for automatically generating anonymous recommendation services based on such information bundles. An experimental anonymous recommender service has been implemented and is currently evaluated in the Virtual University of the Vienna University of Economics and Business Administration at http://vu.wu-wien.ac.at.

The effect of serial correlation on the in-control average run length of cumulative score charts

Abstract The cumulative sum (CUSUM) technique is well-established in theory and practice of process control. For a variant of the CUSUM technique, the cumulative score chart, we investigate the effect of serial correlation on the in-control average run length (ARL). The Shewhart chart is a special case of the cumulative score chart. Using the fact that the cumulative score statistic is a correlated random walk with a reflecting and an absorbing barrier, we derive an approximate but closed-form expression for the ARL of a control variable that follows a first-order autoregressive process with normally distributed disturbances. We also give an expression for the asymptotic (large in-control ARL) case. Our method of approximation gives ARL values that are in good agreement with Monte Carlo estimates of the true values. For positive serial correlation the ARL decreases with increasing value of the correlation coefficient. For increasing negative serial correlation, the ARL may decrease or increase depending on the choice of the parameters of the chart; parameterizations can be found which are rather insensitive for negative serial correlation. We use our results to give recommendations on how to modify the control chart procedure in the presence of serial correlation.

Transient Analysis of M/M/1 Queues in Discrete Time with General Server Vacations

In this contribution we consider an M/M/1 queueing model with general server vacations. Transient and steady state analysis are carried out in discrete time by combinatorial methods. Using weak convergence of discrete parameter Markov chains we obtain also formulas for the corresponding continuous time queueing model. As a special case we discuss briefly a queueing system with T-policy operating. (authors abstract)

A Kolmogorov-Smirnov Test for r Samples

We consider the problem of testing whether r ≥ 2 samples are drawn from the same continuous distribution F(x). The test statistic we will study in some detail is defined as the maximum of the circular differences of the empirical distribution functions, a generalization of the classical 2-sample Kolmogorov-Smirnov test to r ≥ 2 independent samples. For the case of equal sample sizes we derive the exact null distribution by counting lattice paths confined to stay in the scaled alcove

Simple random walk statistics. Part II: Continuous time results

{\cal A}

On zero-avoiding transition probabilities of an r-node tandem queue: a combinatorial approach

r of the affine Weyl group Ar−1. This is done using a generalization of the classical reflection principle. By a standard diffusion scaling we derive also the asymptotic distribution of the test statistic in terms of a multivariate Dirichlet series. When the sample sizes are not equal the reflection principle no longer works, but we are able to establish a weak convergence result even in this case showing that by a proper rescaling a test statistic based on a linear transformation of the circular differences of the empirical distribution functions has the same asymptotic distribution as the test statistic in the case of equal sample sizes.

Hard and Soft Euclidean Consensus Partitions

In this paper various statistics for randomized random walks and their distributions are presented. The distributional results are derived by means of a limiting procedure applied to the pertaining discrete time process, which has been considered in part I of this work (Katzenbeisser and Panny 1996). This basic approach, originally due to Meisling (1958), seems to offer certain technical advantages, since it avoids the use of Laplace transforms and is even simpler than Fellers randomization technique.