FLUCTUATIONS OF RENEWAL-REWARD PROCESS
CALIFORNIA UNIV BERKELEY OPERATIONS RESEARCH CENTER
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Fluctuation theory is concerned with the study of extreme values of sums of independent, arbitrary-valued random variables. Simple but powerful combinatorial methods due chiefly to E. S. Andersen, F. Spitzer, and W. Feller have recently provided an easy method of attack on these problems. However, operations research models are concerned with fluctuations of various economic returns which are earned at random points in time, and whose increments are correlated with the interval since the last payoff. Our generalization considers the fluctuations of a cumulative reward process, defined on an underlying renewal process. Most of the classical results carry through, including Weiner-Hopf type factorization, an Andersen-Pollaczek-Spitzer type identity, and certain Waldian-Pollaczek results. As applications, we find the distribution of the maximum return over a mixed index-epoch horizon, and show how certain general results for the GIG1 queue follow directly from the various three-dimensional ladder distributions.
- Operations Research