# Using Multiple Runs in the Simulation of Stochastic Systems for Estimating Equilibrium Expectations

### Winfried Grassmann

#### Abstract

In complex stochastic systems, Monte-Carlo simulation is often the only way to estimate equilibrium expectations. The question then arises what is better: a single run of length T, or n runs, each of length T/n. In this paper, it is argued that if there is a good state to start the simulation in, multiple runs may be advantageous. To illustrate this, we use numerical examples. These examples are obtained by using deterministic methods, that is, methods based on probability theory not using Monte-Carlo methods. The results of our numerical calculations forced us to make a sharp distinction between the time to reach equilibrium and the appropriate length of the warm-up period, and this distinguishes our study from earlier investigations.

#### References

- Alexopoulos, C. and Goldman, D. (2004). To batch or not to batch? ACM Transaction on Modeling and Computer Simulation, 14(4):76-214.
- Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis, volume 57 of Stochastic Modelling and Applied Probability. Springer Verlag, New York.
- Banks, J., Carson, J. S., Nelson, B. L., and Nicol, D. (2005). Discrete Event Simulation. Prentice Hall, Englewood Cliffs, NJ.
- Grassmann, W. K. (1982). Initial bias and estimation error in discrete event simulation. In Highland, H. G., Chao, Y. W., and Madrigal, O., editors, Proceedings of the 1982 Winter Simulation Conference, pages 377-384, Piscataway, NY. The Institute of Electrical and Electronics Engineers, Inc.
- Grassmann, W. K. (1987). Means and variances of time averages in Markovian environments. European Journal of Operationonal Research, 31:132-139.
- Grassmann, W. K. (2008). Warm-up periods in simulation can be detrimental. Probabilitiy in Engineering and Informational Sciences, 22:415-429.
- Grassmann, W. K. (2011). Rethinking the initialization bias problem in steady-state discrete event simulation. In Jain, S., Creasey, R. R., Himmelspach, J., White, K. P., and Fu, M., editors, Proceedings of the 2011 Winter Simulation Conference, pages 593-599, Piscataway, NY. The Institute of Electrical and Electronics Engineers, Inc.
- Grassmann, W. K. (2014). Factors affecting warm-up periods in discrete event simulation. Simulation, 90(1):11-23.
- Kelton, W. D. (1989). Random initialization methods in simulation. IIE Transactions, 21(4):355-367.
- Kelton, W. D. and Law, A. M. (1985). The transient behavior of the M/M/c queue, with implication for steadystate simulation. Operations Research, 33:378-396.
- Law, A. M. and Kelton, W. D. (2000). Simulation Modelling and Analysis. McGraw Hill, New York, third edition.
- Madansky, A. (1976). Optimal conditions for a simulation problem. Operations Research, 24:572-577.
- Newell, G. F. (1982). Applications of Queueing Theory. Chapman and Hall, London, second edition.
- Pasupathy, R. and Schmeiser, B. (2010). The initial transient in steady state point estimation: Context, a biography, the MSE criterium, and the MSER statistic. In Johansson, B., Jain, S., Montaya-Torres, J., Hugan, J., and YĆ¼cesan, E., editors, Proceedings of the 2010 Winter Simulation Conference, pages 184-197, Piscataway, NY. The Institute of Electrical and Electronics Engineers, Inc.
- Pawlikowski, K. (1990). Steady-state simulation of queueing processes: a survey of problems and solutions. ACM Computing Surveys, 22(2):123-170.
- Tocher, K. D. (1963). The Art of Simulation. English University Press, London.
- Whitt, W. (1991). The efficiency of one long run versus independent replications in steady-state simulation. Management Science, 37(6):645-666.
- Wilson, J. R. and Prisker, A. A. B. (1978). Evaluation of startup policies in simulation experiments. Simulation, 31:79-88.

#### Paper Citation

#### in Harvard Style

Grassmann W. (2015). **Using Multiple Runs in the Simulation of Stochastic Systems for Estimating Equilibrium Expectations** . In *Proceedings of the 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,* ISBN 978-989-758-120-5, pages 138-144. DOI: 10.5220/0005535501380144

#### in Bibtex Style

@conference{simultech15,

author={Winfried Grassmann},

title={Using Multiple Runs in the Simulation of Stochastic Systems for Estimating Equilibrium Expectations},

booktitle={Proceedings of the 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},

year={2015},

pages={138-144},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005535501380144},

isbn={978-989-758-120-5},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,

TI - Using Multiple Runs in the Simulation of Stochastic Systems for Estimating Equilibrium Expectations

SN - 978-989-758-120-5

AU - Grassmann W.

PY - 2015

SP - 138

EP - 144

DO - 10.5220/0005535501380144