By M. A. Crane, A. J. Lemoine (eds.)

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**Extra info for An Introduction to the Regenerative Method for Simulation Analysis**

**Example text**

1~) provides an approximate confidence interval for n. n Elf(X)) ~ again with the substitution In other words, the procedure for a fixed run length t is identical to the procedure for a fixed number of regeneration cycles, except that statistics are computed only for the cycles completed by time It is known that the ratio t -~ ~ , with probability one. E~(Xl}/t Thus, replacing t converges to the value N(t) by I as t/B(~ I) 3 we see that the confidence interval obtained in the manner above for a run of fixed duration t has a length which is approximately d ~-i(I - ~) tl/a' for large t,, where d = 2~/[E{~I)] I/2 is also unknown prior to the simulation.

We now go on from here to set down a simple framework in Section 3 for analyzing the output of any stochastic simulation having a "regenerative" property of the sort we observed in the examples of this section. of simulations is very broad indeed. We will see that this class Moreover~ we will give a simple procedure for constructing a confidence interval (from the simulation output) for a wide variety of steady-state parameters of interest in a "regenerative" simulation. 0 THE REGENERATIVE METHOD The examples of Section 2 suggest a unified approach toward analyzing the output of those simulations of stochastic systems which have the property of ,,regeneration" from time to time.

10) for the repairman model of Section 2. ~)- As before, the condition that E{If ( X )I} < ~ is not restrictive for applications. 10)~ the problem of estimating has been reduced to that of estimating E[ZI]/E{71} and identically distributed observations E{f(X)] based on the independent {(Zj~ 7j)~ J ~ 1} . 3) will apply in either case. As a final note~ the random variable integral of by f[ X (t)] tl~ t2~ . . tm~ Z. is defined above as the J over the interval from Rj to Rj + i If we denote say~ the time points in the interval from where the process has a "jump" and we let m g.