Infinitesimal Perturbation Analysis Research Articles

Smoothed perturbation analysis for a class of discrete-event systems

A gradient-estimation procedure for a general class of stochastic discrete-event systems is developed. In contrast to most previous work, the authors focus on performance measures whose realizations are inherently discontinuous (in fact, piecewise constant) functions of the parameter of differentiation. Two broad classes of finite-horizon discontinuous performance measures arising naturally in applications are considered. Because of their discontinuity, these important classes of performance measures are not susceptible to infinitesimal perturbation analysis (IPA). Instead, the authors apply smoothed perturbation analysis, formalizing it and generalizing it in the process. Smoothed perturbation analysis uses conditional expectations to smooth jumps. The resulting gradient estimator involves two factors: the conditional rate at which jumps occur, and the expected effect of a jump. Among the types of performance measures to which the methods can be applied are transient state probabilities, finite-horizon throughputs, distributions on arrival, and expected terminal cost.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Aggregation approximations for sensitivity analysis of multi-class queueing networks

Two methods are presented for estimating performance derivatives from simulation of multi-class queueing networks for sensitivity analysis. The methods use approximate subnetwork aggregation to reduce the problem to a single-class derivative estimation problem with which a modified infinitesimal perturbation analysis algorithm is used. The modified algorithm treats a subnetwork as though it had been aggregated, but is actually applied to the original (non-aggregated) network. The two approaches to aggregation considered represent two extensions of the idea of a Norton equivalent to non-product-form networks, one based on an exact result, the other heuristic. These are of interest beyond the particular application considered here.

Sample path and performance homogeneity of discrete event dynamic systems

In this paper, a criterion for the applicability of the Infinitesimal Perturbation Analysis (IPA) is given. The concepts of sample path and performance homogeneity are discussed in terms of the model of the Generalized Semi-Markov process.

A formal description of discrete event dynamic systems including infinitesimal perturbation analysis

The behaviour of simulation outcomes of discrete event dynamic systems is in focus. This behaviour is characterized by several kinds of performance measures. A description of a wide class of discrete event dynamic systems is arrived at. Within this context a definition of infinitesimal perturbation analysis is given, which is a technique to quantify the dependence of simulation outcomes on changes in continuously varying parameter inputs.

A New Estimator of Sensitivity Measures for Simulations Based on Light Traffic Theory

Consider an open queueing system that is stable for arrival rates, 0 ≤ λ &lt; c. Let f(λ) be some function such as the mean sojourn time of a customer. Normally, light traffic theory is concerned with limits as λ → 0, e.g., f(n)(0), n = 0, 1, …. Here we use light traffic theory to obtain derivatives of f(·) at any value of λ for which the system is stable. The derivatives, f(n)(λ), n = 0, 1, …, are obtained by perturbing a system with arrival rate, λ, with a small auxiliary arrival process. The nth derivative is obtained by looking at sample paths of the original system with k auxiliary customers thrown in, k = 0, 1, …, n. In practice, the only way to obtain f(n)(λ) for λ &gt; 0 is by observing the effect of additional customers on the sample path of a simulation. We make this technique rigorous by showing that a certain function of the sample paths is “admissible.” The analysis suggests a method for analytically bounding the derivatives, and yields a proof that a large class of queueing functions are infinitely differentiable with respect to λ. The calculations are reminiscent of other sensitivity analyses of simulations such as (infinitesimal) perturbation analysis and the likelihood ratio method. The method described here (the “light traffic perturbation method”) shares the rigorous foundation of the likelihood ratio method and numerical evidence suggests that (at least for simple systems) it has convergence properties similar to perturbation analysis. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Infinitesimal Perturbation Analysis: A Tool for Simulation

Perturbation analysis is a technique that expedites the process of performing experiments on discrete-event simulation models. This makes it possible to derive sensitivity estimates from one computer execution of a simulation model. Infinitesimal perturbation analysis (IPA) is one class of algorithms used in perturbation analysis. In this paper, the techniques and algorithms used in simulation to perform infinitesimal perturbation analysis are examined. Each algorithm is discussed in detail, with comments concerning implementation problems and examples with experimental results for serial transfer lines. The results of this paper show that for simple systems, IPA can be easily implemented in a general-purpose simulation language such as SIMAN. Unfortunately, for any given system, parameter or performance measure, the algorithm used to generate the gradient may vary. Additionally, algorithms for more complex classes of problems do not yet exist. This problem hampers the current possibility of incorporating IPA into general-purpose simulation languages.

Infinitesimal Perturbation Analysis: A Tool for Simulation

Perturbation analysis is a technique that expedites the process of performing experiments on discrete-event simulation models. This makes it possible to derive sensitivity estimates from one computer execution of a simulation model. Infinitesimal perturbation analysis (IPA) is one class of algorithms used in perturbation analysis. In this paper, the techniques and algorithms used in simulation to perform infinitesimal perturbation analysis are examined. Each algorithm is discussed in detail, with comments concerning implementation problems and examples with experimental results for serial transfer lines. The results of this paper show that for simple systems, IPA can be easily implemented in a general-purpose simulation language such as SIMAN. Unfortunately, for any given system, parameter or performance measure, the algorithm used to generate the gradient may vary. Additionally, algorithms for more complex classes of problems do not yet exist. This problem hampers the current possibility of incorporating IPA into general-purpose simulation languages.

System representations and performance sensitivity estimates of discrete event systems

A recently developed technique, called Infinitesimal Perturbation Analysis (IPA), provides an effective method of estimating the sensitivity of performance of a discrete event system based on one simulation of the system. IPA stimulates the study of the sensitivity estimation. One important property of the IPA sensitivity estimate, which differs from that of the ordinary performance estimate, is that the convergence property of the IPA sensitivity estimate depends on the representation of a system. We propose a formulation of IPA estimates which explicitly indicates this dependency on the system representations. We discuss different aspects of the related issues. In particular, we show that the necessary condition developed by Cao (1987) and Heidelberger et al. (1988) for multiple IPA estimates to simultaneously converge is not necessary for any IPA estimate to converge.

A tutorial on the extended perturbation analysis

In this paper, we explain the simple idea of infinitesimal and extended perturbation analysis, the motivation of the new technique, the difficulties, as well as the advantages and contributions.

Convergence Properties of Infinitesimal Perturbation Analysis Estimates

Infinitesimal Perturbation Analysis (IPA) is a method for computing a sample path derivative with respect to an input parameter in a discrete event simulation. The IPA algorithm is based on the fact that for certain parameters and any realization of a simulation, the change in parameter can be made small enough so that only the times of events get shifted, but their order does not change. This paper considers the convergence properties of the IPA sample path derivatives. In particular, the question of when an IPA estimate converges to the derivative of a steady state performance measure is studied. Necessary and sufficient conditions for this convergence are derived for a class of regenerative processes. Although these conditions are not guaranteed to be satisfied in general, they are satisfied for the mean stationary response time in the M/G/1 queue. A necessary condition for multiple IPA estimates to simultaneously converge to the derivatives of steady state throughputs in a queueing network is determined. The implications of this necessary condition are that, except in special cases, the original IPA algorithm cannot be used to consistently estimate steady state throughput derivatives in queueing networks with multiple types of customers, state-dependent routing or blocking. Numerical studies on IPA convergence properties are also presented.

A Sample Performance Function of Closed Jackson Queueing Networks

A stochastic system such as a queueing network can be specified by system parameters and a set of sequences of random variables that represents the randomness in the system. A “sample performance function” is a measure of system performance as a function of system parameters, for each realization of the random sequences. Although the average of N sample performance functions converges to the expected value of the performance with probability one when N goes to infinity, the average of the derivatives of these N sample performance functions with respect to a parameter may not converge to the derivative of the expected value. In this paper, we study a sample performance function of a closed Jackson queueing network; specifically, the time required by a server to serve a finite number of customers. We show that this sample performance function is a continuous, piecewise linear function of the mean service time. We prove that the average of derivatives of this sample performance function with a given initial state does converge, with probability one, to the derivative of the conditional mean value, given the same initial state. The result shows that the estimate of the derivative of a server's mean throughput in a finite time period with respect to the mean service time obtained by infinitesimal perturbation analysis is strongly consistent.

Infinitesimal perturbation analysis of a birth and death process

Using a birth and death process as an illustrative example, we introduce the notion of alternative representations of stochastic processes and discuss its importance for infinitesimal perturbation analysis derivative estimation. Through a different choice of representation, we are led to an IPA algorithm for a birth and death process better than one discussed by other authors.

Smoothed (conditional) perturbation analysis of discrete event dynamical systems

A new sample path analysis approach based on the smoothing property of conditional expectation for estimating the performance sensitivity of discrete event dynamical systems is proposed. Several examples are presented to show how this approach overcomes a difficulty of the ordinary infinitesimal perturbation analysis. The basic message is that one can get more knowledge about the system performance by observing and analyzing the sample path than by using the conventional simulation approach. It is also pointed out that the classical queueing theory approach for getting the performance sensitivity and the sample path based infinitesimal perturbation analysis approach can be unified in the framework of the new approach, the smoothed (conditional) perturbation analysis.

Infinitesimal perturbation analysis for general discrete event systems

A rigorous extension of the recent perturbation analysis approach to more general discrete event systems is given. First, a general class of systems and performance measures is defined, and some basic reprsentational and linearity properties are derived. Next a sample gradient of performance with respect to a parameter of the system is defined. Then, for certain parameters of such systems, an infinitesimal perturbation analysis algorithm is derived. It is proved that this algorithm gives exact values for the sample gradients of performance with respect to the parameters, by observing only one sample path of the DEDS. (However, the sample gradient may or may not be a good estimate of the gradient of the performance measure; this point is elaborated in the body of the paper.) The computational complexity of this algorithm is bound to be linear in the number of events. These results offer the potential for very efficient calculation of the gradients—a fact that can be used for design/operation of computer systems, communication networks, manufacturing systems, and many other real-world systems, particularly since restrictive assumptions (e.g., exponential distributions) are not required of the system.

Sensitivity of sample values not generated by inversion

The perturbation generation rule of Suri (Ref. 1) is extended to cases where random variables are not generated by inversion. Conditions are given for the use of other methods. Infinitesimal perturbation analysis is shown to be compatible with common random number techniques in cases where finite-difference approximations are not.

Infinitesimal and finite perturbation analysis for queueing networks

The sample-path perturbation analysis technique is extended to include finite (and possibly large) perturbations typically introduced by changes in queue sizes or other parameters. It is shown that there is a natural hierarchy of perturbation analysis which takes care of increasingly large perturbations. Experiments with zero (infinitesimal) and first order (finite) perturbation analysis show that significant accuracy improvement can be obtained with small increase in computational effort.