More Storage Models

Abstract

In the preceding chapter we investigated storage models in which the net input process Y was of the form Y(t) = X(t) − t, X being a non-negative Lévy process. Although such an input would seem to be rather special, we found it to be appropriate for a wide variety of situations, in particular, for queues with first come, first served discipline as well as queues with priorities. However, there still remains the general case of the insurance risk problem and the simple queue in which the net input is of the form Y(t) = A(t) − D(t) with A, D two independent simple Poisson processes. In order to investigate these and further storage models we need to develop some new concepts concerning Lévy processes. Accordingly, let Y be a Levy process with the c.f.

$$ E{e^{i\omega Y\left( t \right)}} = {e^{ - t\phi \left( \omega \right)}}, $$

(1)

where φ(ω) is as given in Chapter 3, Section 2. Let us define the random variables

$$ T = \inf \left\{ {t:Y\left( t \right) > 0} \right\},\;\;\;\bar T = inf\left\{ {t:Y\left( t \right) < 0} \right\}. $$

(2)