Abstract
The variance of a time average is important for planning, running and interpreting experiments. This paper derives a simple method to find this variance for the case of a Markov process. This method is then applied to obtain the variance of a time average for the case of a birth-death process.
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References
N. Blomqvist, The covariance function of the M/G/1 queueing system, Skandinavisk Aktuarietidskrift 50(1967)157.
D. Burman, A functional central limit theorem for birth and death processes, Talk given at the ORSA/TIMS Meeting in Wahington D.C. (1980).
W. Feller,An Introduction to Probability Theory and Its Applications, 2nd ed., 7th printing (Wiley, New York, 1962).
A.V. Gafarian and C.J. Ancker, Mean value estimation from digital computer simulation, Oper. Res. 14(1966)25.
R.F. Gebhard, A limiting distribution of an estimate of mean queue length, Oper. Res. 11(1963)1000.
P.W. Glynn, Some asymptotic formulas for Markov chains with applications to simulation, J. Statist. Comput. Simul. 19(1984)97.
W.K. Grassmann, The optimal estimation of the expected number in an M/D/∞ queueing system, Oper. Res. 29(1981)1208.
W.K. Grassmann, Initial bias and estimation error in discrete event simulation, Proc. 1982 Winter Simul. Conf., p. 377.
S. Halfin, Linear estimators for a class of stationary queueing processes, Oper. Res. 30(1982)515.
J.H. Jenkins, The relative efficiency of direct and maximum likelihood estimates of mean waiting time in the simple queue M/M/1, J. Appl. Prob. 9(1972)396.
J. Keilson and S.S. Rao, A process with chain dependent growth rate, J. Appl. Prob. 7(1970)699.
J.G. Kemeny and J.L. Snell,Finite Markov Chains (reprint of the 1960 edition published by Van Nostrand, Princeton) (Springer, New York, 1976).
P.M. Morse, Stochastic properties of waiting lines, Oper. Res. 3(1955)255.
B. Pagurek and C.M. Woodside, An expression for the sum of serial correlations of times in GI/G/1 queues with rational arrival processes, Oper. Res. 27(1979)755.
J.F. Reynolds, The covariance structure of queues and related processes — a survey of recent work, Adv. Appl. Prob. 7(1975)383.
J.F. Reynolds, Asymptotic properties of mean length estimators for finite Markov queues, Oper. Res. 20(1972)52.
J.F. Reynolds, Some theorems on the covariance structure of Markov chains, J. Appl. Prob. 9(1972)214.
J.R. Wilson, The need for improved efficiency in discrete-event simulations, Proc. 1982 Winter Simul. Conf., San Diego, CA, p. 604.
C.M. Woodside, P. Pagurek and G.F. Newell, A diffusion approximation for correlation in queues, J. Appl. Prob. 17(1980)1033.
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Grassmann, W.K. The asymptotic variance of a time average in a birth-death process. Ann Oper Res 8, 165–174 (1987). https://doi.org/10.1007/BF02187089
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DOI: https://doi.org/10.1007/BF02187089