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

ISBN-10: 3540084088

ISBN-13: 9783540084082

ISBN-10: 3540371753

ISBN-13: 9783540371755

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Additional info for An Introduction to the Regenerative Method for Simulation Analysis

Sample text

R1 However 3 the portion of the process between epoch 0 and epoch ; while independent of the rest of the process~ is allowed to have a different distribution. Let 7j = K j + l " Rj ~ j ~ i . Then the sequence [7j~ J ~ I} represents the time spans between consecutive epochs of regeneration~ and is a sequence of independent and identically distributed variables. we make the mild assumption that El71} < ~ As before This assumption is certainly satisfied for the repairman model of Section 2~ and is also satisfied for any Markov chain in continuous time which is positive recurrent and irreducible.

Let [%s J ~ I} ~j = ~j + 1 - ~j for j ~ 1 . Note that the "sojourn times" between consecutive epochs of regeneration are independent and identically distributed. We will assume henceforth that E{~ I) < ~ not a restrictive assumption. This is For example; it holds in almost any queueing system of practical interest s and it certainly holds for the queueing and inventory examples of Section 2 s as well as for any positive recurrent irreducible Markov chain. The regenerative property is an extremely powerful tool for obtaining analytical results for the process [~n s n ~ I} Under very mild conditions the process has a limiting or steady-state distribution.

Random As applied to the failure times for the operating units~ this prop- erty tells us that the time to failure for a unit has the same (exponential) distribution regardless of how long the unit has been operating. words, an operating unit is always in a "like-new" In other condition until it fails; it is no more likely to fail soon if it has been operating for a long time. Now suppose we begin the simulation assuming we have a "like-new system," that is, I0 units are placed into operation at time 0 and there are 5 new 25 units in the spare pool.

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An Introduction to the Regenerative Method for Simulation Analysis by M. A. Crane, A. J. Lemoine (eds.)

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