In this paper, we discuss the factors affecting the initialization bias in discrete event simulation. Specifically, we assume that the time average is used to find the equilibrium expectation of a certain variable , say the number in a queueing network, and we would like to minimize the mean squared error (MSE) between the time average of and its equilibrium expectation. To do this, a warm-up period is often used during which no data is collected, and we want to find the length of this period such that the MSE is minimal. We show that if starting in what Tocher calls a “typical condition”, warm-ups tend to be redundant. This result is strengthened by theoretical arguments and numerical experiments. If starting in a typical state is inconvenient, warm-up periods should be used, and methods to find optimal warm-up periods are discussed. The numerical methods used for our experiments do not rely on Monte Carlo simulation. Instead, we determine the MSE of the time average by the randomization method and other deterministic methods.
PasupathyRSchmeiserB. The initial transient in steady state point estimation: context, a biography, the MSE criterium, and the MSER statistic. In: JohanssonBJainSMontaya-TorresJHuganJYücesanE (eds), Proceedings of the 2010 Winter Simulation Conference. Piscataway, NJ: The Institute of Electrical and Electronics Engineers, Inc., 2010, pp. 184–197.
3.
PawlikowskiK. Steady-state simulation of queueing processes: a survey of problems and solutions. ACM Comput Surveys1990; 22(2): 123–170.
4.
McClamonMA. Detection of Steady State in Discrete Event Systems: An Analysis of Heuristics. School of Engineering and Applied Science, University of Viginia. Charlottesville, Virginia, 1990.
5.
FranklinWWWhiteKP. Stationarity tests and MSER-5: exploring the intuition behind mean-squared-error-reduction in detecting and correcting initialization bias. In: MasonSJHillRRMönchLRoseOJeffersonTFowlerJW (eds), Proceedings of the 2008 Winter Simulation Conference. Piscataway, NJ: The Institute of Electrical and Electronics Engineers, Inc., 2008, pp. 541–546.
6.
SánchezPSWhiteKP. Interval estimation using replications and MSER truncation. In: JainSCreaseyRRHimmelspachJWhiteKPFuM (eds), Proceedings of the 2011 Winter Simulation Conference. Piscataway, NJ: The Institute of Electrical and Electronics Engineers, Inc., 2011, pp. 488–494.
7.
WhiteKP. An effective truncation heuristic for bias reduction in simulation output. Simulation1997; 69(6): 323–334.
8.
WhiteKPRobinsonS. The problem of the initial transient (again), or why MSER works. J Simulation2010; 4: 268–272.
9.
MokashiACTejadaJJYousefiSXuTWilsonJRTafazzoliA. Performance comparison of MSER-5 and N-Skart on the simulation start-up problem. In: JahanssonBJainSMontoya-TorresJHuganJY (eds), Proceeding of 2010 Simulation Conference, pp. 971–982.
10.
TafazzoliASteigerNMWilsonJR. N-Skart: a nonsequential skewness and autoregression-adjusted batch-means procedure for simulation analysis. IEEE Trans Automat Control2011; 56(2): 254–264.
11.
MadanskyA. Optimal conditions for a simulation problem. Operat Res1976; 24: 572–577.
12.
GrassmannWK. Warm-up periods in simulation can be detrimental. Probab Eng Informat Sci2008; 22: 415–429.
13.
WilsonJRPriskerAAB. Evaluation of startup policies in simulation experiments. Simulation1978; 31: 79–88.
GrossDMillerDR. The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operat Res1984; 32: 343–361.
19.
CiardoG. SMART, a Stochastic Model Checking Analyser for Reliability and Timing. Available at: http://www.cs.urs.edu/~ciardo/SMART (accessed 8 March 2011).
20.
SandersWH. Möbius, Model-Based Environment for Validation of System Reliability, Availability, Security and Performance. Available at: https://www.mobius.illinois.edu/papers.html (accessed 8 March 2011).
21.
GrassmannWK. Means and variances of time averages in Markovian environments. Eur J Operat Res1987; 31: 132–139.
22.
ReynoldsJF. The covariance structure of queues and related processes–a survey of recent work. Adv Appl Prob1975; 7: 383–415.
23.
HordiijkAIglehartDLSchassbergerR. Discrete methods for simulating continuous time Markov chains. Adv Appl Prob1976; 13: 772–788.
24.
ReynoldsJF. Some theorems on the covariance structure of Markov chains. J Appl Prob1972; 9: 214–218.
25.
GrassmannWK. Means and variances in Markov reward systems. In: MeyerCDPlemmonsRJ (eds), Linear Algebra, Markov Chains and Queueing Models (The IMA Volumes in Mathematics and Its Applications, vol. 48). New York: Springer-Verlag, 1993, pp. 193–204.
LawAMKeltonWD. Simulation Modelling and Analysis. 3rd edn.New York: McGraw-Hill, 2000.
28.
RobinsonS. Simulation. The Practice of Model Develpment and Use. Chichester: John Wiley & Sons, 2004.
29.
TocherKD. The Art of Simulation. London: English University Press, 1963.
30.
ParsenE. Stochastic Processes. San Francisco, CA: Holden Day, Inc., 1962.
31.
AsmussenSGlynnPW. Stochastic Simulation: Algorithms and Analysis (Stochastic Modelling and Applied Probability, vol. 57). New York: Springer-Verlag, 2007.
32.
GrassmannWK. Initial bias and estimation error in discrete event simulation. In: HighlandHGChaoYWMadrigalO (eds), Proceedings of the 1982 Winter Simulation Conference. Piscataway, NJ: The Institute of Electrical and Electronics Engineers, Inc., 1982, pp. 377–384.
33.
RobinsonS. A statistical process control approach to selecting a warm-up period for a discrete-event simulation. Eur J Operat Res2007; 176: 332–346.
34.
GrassmannW. Rethinking the initialization bias problem in steady-state discrete event simulation. In: JainSCreaseyRRHimmelspachJWhiteKPFuM (eds), Proceedings of the 2011 Winter Simulation Conference. Piscataway, NJ: The Institute of Electrical and Electronics Engineers, Inc., 2011, pp. 593–599.
35.
KeltonWDLawAM. The transient behavior of the M/M/c queue, with implication for steady-state simulation. Operat Res1985; 33: 378–396.
36.
GrassmannWK. Optimal estimation of the expected number in an M/D/∞ system. Operat Res1981; 29(6): 1208–1211.
