Russell C. H. Cheng

Sensitivity of computer simulation experiments to errors in input data

This paper compares two methods of assessing variability in simulation output. The methods make specific allowance for two sources of variation: that caused by uncertainty in estimating unknown input parameters (parameter uncertainty), and that caused by the inclusion of random variation within the simulation model itself (simulation uncertainty). The first method is based on classical statistical differential analysis; we show explicitly that, under general conditions, the two sources contribute separately to the total variation. In the classical approach, certain sensitivity coefficients have to be estimated. The effort needed to do this becomes progressively more expensive, increasing linearly with the number of unknown parameters. Moreover there is an additional difficulty of detecting spurious variation when the number of parameters is large. It is shown that a parametric form of bootstrap sampling provides an alternative method which does not suffer from either problem. For illustration, simulation ...

Generating beta variates with nonintegral shape parameters

A new rejection method is described for generating beta variates. The method is compared with previously published methods both theoretically and through computer timings. It is suggested that the method has advantages in both speed and programming simplicity over previous methods, especially for “difficult” combinations of parameter values.

Confidence Bands for Cumulative Distribution Functions of Continuous Random Variables

Previously suggested methods for constructing confidence bands for cumulative distribution functions have been based on the classical Kolmogorov-Smirnov test for an empirical distribution function. This paper gives a method based on maximum likelihood estimation of the parameters. The method is described for a general continuous distribution. Detailed results are given for a location-scale parameter model, which includes the normal and extreme-value distributions as special cases. Results are also given for the related lognormal and Weibull distributions. The formulas derived for these distributions give a band with exact confidence coefftcient. A chi-squared approximation, which avoids the use of special tables, is also described. An example is used to compare the resulting bands with those obtained by previously published methods.

Two-point methods for assessing variability in simulation output

In simulation experiments, the form of the distribution of input variables is often not known precisely. The simulation output then contains two sources of variation: that caused by uncertainty in estimating unknown parameters, and that caused by the inclusion of random variation within the simulation model itself. Cheng and Holland (1996) have shown how the classical method of statistical differential analysis (often called the δ-method) can be used to assess the degree of variability arising from each source. The disadvantage of the δ-method is that the computational effort needed for this increases linearly with the number of unknown parameters. In this paper it is shown that the method can be modified to assess the combined effect on the response output of variation in all the parameters by making most simulation replications at just twosettings of parameter values, making the method substantially independent of the number of unknown parameters. Thus, for problems where this number is large, such two-...

Optimal pricing policies for perishable products

In many industrial settings, managers face the problem of establishing a pricing policy that maximises the revenue from selling a given inventory of items by a fixed deadline, with the full inventory of items being available for sale from the beginning of the selling period. This problem arises in a variety of industries, including the sale of fashion garments, flight seats, and hotel rooms. We present a family of continuous pricing functions for which the optimal pricing strategy can be explicitly characterised and easily implemented. These pricing functions are the basis for a general pricing methodology which is particularly well suited for application in the context of an increasing role for the Internet as a means to market goods and services.

Calculation of confidence intervals for simulation output

This article is concerned with the calculation of confidence intervals for simulation output that is dependent on two sources of variability. One, referred to as <i>simulation variability</i>, arises from the use of random numbers in the simulation itself; and the other, referred to as <i>parameter variability</i>, arises when the input parameters are unknown and have to be estimated from observed data. Three approaches to the calculation of confidence intervals are presented--the traditional asymptotic normality theory approach, a bootstrap approach and a new method which produces a conservative approximation based on performing just two simulation runs at carefully selected parameter settings. It is demonstrated that the traditional and bootstrap approaches provide similar degrees of accuracy and that whilst the new method may sometimes be very conservative, it can be calculated in a small fraction of the computational time of the exact methods.

Improved Design of Queueing Simulation Experiments with Highly Heteroscedastic Responses

Simulation experiments for analysing the steady-state behaviour of queueing systems over a range of traffic intensities are considered, and a procedure is presented for improving their design. In such simulations the mean and variance of the response output can increase dramatically with traffic intensity; the design has to be able to cope with this complication. A regression metamodel of the likely mean response is used consisting of two factors, namely, a low-degree polynomial and a factor accounting for the exploding mean as the traffic intensity approaches its saturation. The best choice of traffic intensities at which to make simulation runs depends on the variability of the simulation output, and this variability is estimated using analytical heavy traffic results. The optimal numbers of customers simulated at each traffic intensity are built up using a multistage procedure. The asymptotic properties of the procedure are investigated theoretically. The procedure is shown to be robust and to be more efficient than more naive procedures. A result of note is that even when the range of interest includes high traffic intensities, the highest traffic load simulated should remain well away from its upper limit; but the number of customers simulated should be concentrated at the higher traffic intensities used. Empirical results are included for simulations of a single server queue with different priority rules and for a complicated queueing network. These results support the theoretical results, demonstrating that the proposed procedure can increase the accuracy of the estimated metamodel significantly compared with more naive methods.

Selecting input models

Discrete-event simulation almost invariably makes uses of random quantities drawn from given probability distributions to model chance fluctuations. This paper discusses how to choose appropriate distributions. The two main points addressed are (a) the factors which should be considered in selecting input distributions, and (b) how the effect of errors in making an inappropriate or inexact choice will influence the accuracy of final simulation results.

Maximum Likelihood Estimation of Parameters in the Inverse Gaussian Distribution, With Unknown Origin

Maximum likelihood estimation is applied to the three-parameter Inverse Gaussian distribution, which includes an unknown shifted origin parameter. It is well known that for similar distributions in which the origin is unknown, such as the lognormal, gamma, and Weibull distributions, maximum likelihood estimation can break down. In these latter cases, the likelihood function is unbounded and this leads to inconsistent estimators or estimators not asymptotically normal. It is shown that in the case of the Inverse Gaussian distribution this difticulty does not arise. The likelihood remains bounded and maximum likelihood estimation yields a consistent estimator with the usual asymptotic normality properties. A simple iterative method is suggested for the estimation procedure. Numerical examples are given in which the estimates in the Inverse Gaussian model are compared with those of the lognormal and Weibull distributions.

Searching for important factors: sequential bifurcation under uncertainty

The problem of searching for important factors in a simulation model is considered when the simulation output is subject to stochastic variation. Bettonvil and Kleijnen (1996) give a method which they call sequential bifurcation which allows a large number factors to be considered using a relatively small number of simulation runs. They give the method under the assumption that the simulation response contains negligible random error, and show that when the number of important factors is small then the method is effective and efficient. In this paper the method is extended to handle simulations where the response is stochastic and subject to significant error. An attraction of the sequential bifurcation method is its flexibility in exploring the effects of different factors. The approach in this paper is to develop a clear but flexible framework in which the method is used as an exploratory tool. For illustration a numerical example is considered using a simulation metamodel involving 24 factors. The example is quite a testing one because the different factors cover a spread of values of differing importance. The results show that the method is capable of handling such situations.