A survey of research on the simulation startup problem ... and Power Company, he worked on the development of ... application of time-series analysis techni-.
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A survey of research on the simulation startup problem James R. Wilson and A. Alan B. Pritsker SIMULATION 1978; 31; 55 DOI: 10.1177/003754977803100204 The online version of this article can be found at: http://sim.sagepub.com/cgi/content/abstract/31/2/55
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Keywords:
A survey of research on the simulation
by James R. Wilson and A. Alan B.
a PhD student in the School of Industrial Engineering at Purdue University. He received a BA in mathematics from Rice University and an MS in industrial engineering from Purdue University, with his master’s thesis selected as the best industrial engineering thesis in the United States for 1977. As a research analyst for the Houston Lighting and Power Company, he worked on the development of that company’s corporate model. He has also worked for Pritsker and Associates, Inc., a systems engineering consultant firm. He is a member of SCS, AIIE, TIMS, and ORSA.
A. ALAN B. PRITSKER is a professor of industrial engineering at Purdue University and president of He is the developer of Pritsker and Associates, Inc.
GASP IV, GASP-PL/I, GASP II, GERT, and Q-GERT. Dr. Pritsker was formerly associated with Virginia Polytechnic Institute, Arizona State University, and Battelle Memorial Institute. He holds a BSEE and a MSIE from Columbia University, and a Phd from Ohio He is a registered professional State University. engineer and a member of ACM, AIDS, ORSA, TIMS, AIIE, ASEE, SCS, and PMI. Dr. Pritsker is SIMULATION’s He is a associate editor for combined simulation. Fellow of AIIE, the holder of AIIE’s Distinguished Research Award (1966), the H. B. Maynard Innovative Achievement Award (1978), and Arizona State University’s Faculty Achievement Award (1967).
ABSTRACT To reduce the effects of initial conditions on the results of a simulation experiment, several authors have proposed startup policies that specify how to minimize the warmup period and identify the truncation point beyond which observations are to be recorded and analyzed. Other approaches to this probZem have used results from time-series analysis and queueing theory. This paper surveys research on the simulation startup problem; a companion paper which
follows presents
a
Pritsker
Purdue University Pritsker and Associates, Inc.
startup problem JAMBS R. WILSON is
bias, digital simuation, mean square error, statistics, truncation, variance
West
Lafayette,
Indiana 47907
INTRODUCTION
Simulation models are often used to estimate the performance of steady-state systems. When the behavior of the underlying process is autocorrelated, the initial conditions of the model may produce transient components in the simulated output. Although these transients typically decay geometrically in time, convergence to steady-state conditions can be quite slow.3 This phenomenon often results in substantial and persistent bias in the statistics computed from an observed time series. The startup problem has been recognized for some time. Three approaches to it have evolved: the application of time-series analysis techniques,4,6,7,13 the study of queueing theory models,1,2,10,11 and the development of heuristic rules of thumb.3,5,8,9,12 Although the results derived from time-series analysis and queueing theory are rigorous and precise, they have rather limited applicability. On the other hand, many of the heuristic methods have broader applicability but are ambiguously formulated and have uncertain statistical properties. In general, these heuristic policies specify how to set initial conditions to minimize the warmup period and identify the truncation point beyond which data are not significantly distorted by the initial conditions. Until recently, no attempt had been made to evaluate the performance of these policies. This paper reviews research on the startup problem; a companion paper develops a generalized procedure for evaluating
startup
policies.15
TIP1E-SERIES MODELS
Fishman4 analyzed first-order truncated sample on
a
the effects of initial conditions process. Using the
autoregressive mean
procedure for evaluating startup
when the mean of the truncated sample is to be used to estimate the steady-state mean of the underlying process.
policies
as
an
estimator of the
steady-state
mean
Px, he showed
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LIST OF SYMBOLS
that bias
deleting the first d observations reduces the and increases the variance VeX E(X of
,- ~ )X
the statistic
n,d*
examples considered, square
error
of the
n,)
In the
specific
numerical
truncation increased the
sample J
mean
according
This last equation suggests that after m/Rp observations, the initial transients have disappeared; consequently the truncation point should be
mean
to
~B
a finite-order autoregressive process, Fishman expressed m and Ro directly in terms of the autoregressive parameters. To apply these results to a simulation-generated time series, Fishman developed an algorithm to identify and estimate an appropriate autoregressive representation. The corresponding
For
Fishman concluded that the bias reduction produced by truncation must be carefully weighed against the increased variance caused by the loss of information.
estimates of m and d are then used to construct fidence intervals for the steady-state mean
and Sussmanl3 extended Fishman’s work on first-order autoregressive processes. They considered the more general problem of selecting both a truncation point and a replication count that minimizes the mean square error of the overall sample mean, subject to a budget constraint on the number of observations. Numerical examples reinforced Fishman’s conclusion that truncation is not always desirable. Furthermore, it was found that the optimal number of replications is not always one but is relatively insensitive to the budget constraint.
the distribution of waiting time single-server queueing system (GI/G/1/°°) having an &dquo;empty and idle&dquo; initial condition and first-come, first-served queue discipline. Here Xt is the waiting time of the tth customer, and the initial condiIf the steady-state coeffition implies that X, 0. cient of variation of the waiting time is at least
Turnquist
Fishman6
also examined more general time-series models. For many processes, the variance of the sample mean may be approximated by
QUEUEING
MODELS
Blomqvistl analyzed
in
a
=
one,
i.e., if
MSE(Xn dlXl
showed that the truncation point d=
is minimized size n is sufficiently large. Furthermore, condition (5) was shown to hold in the special case of exponential arrival or service processes, provided that the traffic intensity is less than one.
Blomqvist at
0’when
Madanskyll that the sample size n is sufficiently large. is the steady-state autocovariance of lag u, and hence Rp is the steady-state variance of the process.) Equating this estimate to the analogous result based on a sample of Q independent observations, Fishman concluded that
provided (Here Ru
0)
=
the
sample
examined the process Xn number in the for the single-server queueing system (M/M/1/oo). Considering Xn rather than Xn &dquo;d as an estimator of the steady-state mean, Madansky showed 0 minimizes that the initial condition Xo for n sufficiently large, and that increased run length is more effective than replication in reducing 0). Unfortunately, the is really the statistic of interest, sample mean and the extension of Madansky’s methods to study is not obvious.
system
=
at time n
=
MSE(XnIXO)
xnd
Thus the number of autocorrelated observations &dquo;equivalent&dquo; to one independent observation is given by
con-
ux’
MSE(Xn~X~
=
A~5E’(Z ,J~J Cheng2
studied the effects of initial conditions on the tradeoff between run length and replication count
56 Downloaded from http://sim.sagepub.com at NORTH CAROLINA STATE UNIV on June 12, 2009
in the simulation of finite-state Markov processes. When the initial state probability distribution differs from the equilibrium distribution, Cheng showed that both the bias and the mean square error of the overall untruncated sample mean are minimized by making a single run. The effects of initial conditions on the variance of the untruncated sample mean is more complicated; Cheng explored these effects in detail for the two-state process.
points Elog(n), logfs(n)}], sample size n at which the graph approximately linear with slope -2. Plot the
d
TR6
=
[Gordon9l
Make replications to estimate the value E(Xn) and set d sample size n at which the bias of is less than e. The grand mean X(n,k) versus n will appear to stabilize at some sample size i.e.,
expected
The somewhat limited scope of the theoretical results has prompted various authors to propose rules of thumb for setting the initial condition Xo and the truncation point d in a simulation experiment. Some of the most frequently cited rules are summarized below; others are discussed in Wilson.14 Initial condition rules IC1
Start the system
IC2
Set
Xo
as
Xo
as
as
close to the
[Conway3]
{Xt : t TR2
TR3
nor =
steady-state
mean
as
is,
set
d
=
0.
Set d so that Xd+1 is neither the minimum of the remaining set
d+1,
...,
n}.
Set d = n when {sgn(Xt - Xn) : 1 :5 t 5 n} contains k runs - i.e., when the time 1
