On the maximal content of a dam and logarithmic concave renewal functions

J. W. Cohen

Stochastic Processes and their Applications, 1978, vol. 6, issue 3, 291-304

Abstract: For the classical dam model the distribution of the supermum of the dam content between two successive downcrossings of level x>0 by the content process is studied. The result generalizes previous results for the M/G/1 queueing system. The derivation which is rather simple is based on some recent results concerning up- and downcrossings. The resulting distribution is a functional of the solution of a renewal type integral equation occurring frequently in applied probability models. It is shown that this solution is logarithmic concave and that its reciprocal is an infinitely divisible p-function, thus leading to a number of hitherto unknown properties of a certain class of renewal functions.

Date: 1978
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