On Taylor series expansions for waiting times in tandem queues: an algorithm for calculating the coefficients and an investigation of the approximation error

doi:10.1016/S0166-5316(99)00043-7

Performance Evaluation

Volume 38, Issues 3–4, December 1999, Pages 153-173

https://doi.org/10.1016/S0166-5316(99)00043-7 Get rights and content

Abstract

Recently, a Taylor series expansion was developed for expected stationary waiting times in open (max,+)-linear stochastic systems with Poisson input process; these systems cover various instances of queueing networks.

As an application, we present an algorithm for calculating the coefficients for infinite capacity tandem queueing networks with discrete service-time distributions. The algorithm works quite efficiently if the random vector of the service times of all servers is concentrated at a small number of atoms. We investigate the relative error of the Taylor approximation by simulation; in many cases, it follows very well a simple expression which holds exactly for independent, exponentially distributed servers.

Introduction

Efficient tools for the analysis of discrete event systems (DES) are required in many applications. As analytical methods fail in most cases, approximation schemes for computing performance measures have been developed. Here we deal with a Taylor series expansion, which is derived in [2] for the expectation of certain stationary waiting times in open (max,+)-linear stochastic systems with Poisson input. In [4], [5], it is generalized to more complex stationary characteristics such as higher moments, Laplace transforms, distribution functions, tail probabilities, etc., and also to transient performance measures. Various instances of queueing networks and manufacturing models can be represented as (max,+)-linear systems (e.g. acyclic or cyclic fork-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking, synchronized queueing networks, Kanban systems, Job–Shop systems); these correspond to the so-called “event graphs” in the theory of Petri nets. Service times need not be exponential; moreover, they may be dependent. The well aquainted tools of the Petri net formalism can be used for modeling these systems, and once a Petri net model is established, the (max,+)-linear system equations can be generated automatically.

These features make the proposed expansion technique a promising tool. However, the expansion coefficients are stated in terms of expectations of certain transformed random variables, where the transformations include maximization and summation. The number of variables increases with the size of the system and with the order of the coefficients. The evaluation of these expectations involves complicated high-dimensional integration; in [5] an algorithm is presented for this purpose that makes use of a computer algebra system like Maple or Mathematica. In some cases, this approach leads to simple closed expressions for the coefficients.

To study the applicability of the approach, we describe an algorithm for calculating the coefficients of the expansion given in [2] for an infinite capacity tandem queueing network with discrete service-time distributions. In a simulation study, we investigate the accuracy of the approximation, depending on the number of coefficients which have been computed. For independent service times, it seems that the relative error can be approximated very well by a simple expression which holds exactly for exponentially distributed service times.

The coefficients can be computed quite efficiently for networks where the random vector of service times is concentrated at a small number of atoms. A network of this type represents, for example, a manufacturing system, where jobs out of a small number of different types arrive in random order.

Section snippets

Taylor series expansions for (max, +)-linear systems

In [2], a Taylor series expansion is given for (max,+)-linear systems, i.e. for systems which are linear in the so-called (max,+)-algebra. This algebra is based on the two operations (⊕,⊗), which are defined in terms of conventional operations for $x,y∈ R$ by $x⊕y≔ max {x,y}$ and $x⊗y≔x+y.$

The neutral element with respect to ⊕ is ε≔−∞; as in conventional algebra, ⊗ has priority over ⊕ in all arithmetic expressions. The (max,+)-algebra is presented in detail in [1].

Let A and B be matrices of size p×q and q×r

Tandem queues

Consider a network of β single-server FIFO queues with infinite capacity in tandem (Fig. 1).

Here T_n is the arrival epoch of the nth customer in the network and σ_nⁱ is the nth service time in the ith queue. If X_nⁱ represents the time of the beginning of the nth service in queue i, then it is shown in [2] that the vector X_n≔(X_n¹,…,X_n^β)^T satisfies a β-dimensional recurrence relation (1) with the following matrices:

•
A_n is of size β×β, the entries (A_n)_i,j are given by Eq. (8): $(A_{n})_{i,j} ≔ ε, if i<j, ∑ k=j i−1 σ_{}$

Preliminaries

Let ${σ_{n}}_{n∈Z}$ be a sequence of independent replications of σ. For calculating a_kⁱ,1≤i≤β, we need the random vectors σ₀,σ₋₁,…,σ_−k. These have the same joint distribution as σ₀,σ₁,…,σ_k (here the time order can be reversed), this follows from the independence of the vectors σ_n. The coefficients a_kⁱ can be calculated by (7) in terms of σ₀,σ₁,…,σ_k if the definition of A_n and D_k is modified in the following way: $(A_{n})_{i,j} ≔ ε, if i<j, ∑ k=j i−1 σ_{n−1}^{k} +σ_{n}^{j}, if i≥j,$ $D_{k} ≔ ⊗ n=1 k A_{n} ⊗B_{k} for k≥1.$ We will use this modification

A simple expression

Here we shall investigate the error of the approximation (6) to the steady state waiting times in a tandem queue. For the M/M/1 results used in this section, the reader is referred to [6] or [11]. Consider the following assumption:

Assumption 3

At each server, the customers arrive according to a Poisson process with intensity λ.

Assumption 3 is valid if the service times of different servers are independent and all service times are exponentially distributed or if there are two independent servers and the

Conclusions

We consider stationary waiting times in infinite capacity tandem queues with Poisson input. For discrete service time distributions, a numerical procedure for calculating the coefficients of a Taylor series expansion for average waiting times is proposed. It makes use of a general approximation technique for open (max,+)-linear stochastic systems which has been developed in [2]. For the truncation error, depending on the order m of the approximation, simple guidelines can be given: For

Acknowledgements

We thank Volker Schmidt for drawing our attention to the subject, and Sven Hasenfuss and Volker Schmidt for extremely valuable suggestions and discussions.

We are also grateful to two referees for helpful comments which have considerably improved the organization of the paper, and to Astrid Ruck and Heiko Gerlach for their assistance at the simulation studies.

Wilfried Seidel received his Ph.D. degree in mathematics from the Ludwig-Maximilians-University of Munich in 1983 and his Habilitation in statistics from the University of the Federal Armed Forces Hamburg in 1992. From 1983 until 1992 he was research assistant and from 1992 until 1997 he was Acting Chair of Statistics and Operations Research at several German universities. Since January 1998 he has been Prof. for Mathematical Methods of Economic Sciences at the University of the Federal Armed

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His research interests include stochastic Petri nets and computational statistics, in particular analysis of mixture distributions, experimental design and gamma-minimax tests. Prof. Seidel is a member of the Institute of Mathematical Statistics, the German Statistical Society and the GOR (Gesellschaft für Operations Research).

Kai von Kocemba is currently an IT consultant at SYSECA, Germany, and a graduate student at the Institute for Mathematical Stochastics, University of Hamburg. His research interests are Discrete Event Dynamic Systems (DEDS), especially manufacturing queueing networks.

Klaus Mitreiter was born in Hamburg, Germany in 1969. He is currently a graduate student of Computer Science at the University of Hamburg; his main research interests are Petri nets, in particular object-oriented nets, scenario nets and reference nets.

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