A stochastic integral in storage theory

On time reversal of piecewise deterministic Markov processes

“…Piecewise deterministic paths can be observed for Markov processes in a variety of applications. We mention risk process ( [1,48,19,20,25]), growth collapse and stress release models ( [9,53,58,12,41]), queueing models ( [14,11]), earthquake models ( [46]), repairable systems ( [38]), storage models ( [15,27,28]) and TCP data transmission ( [39,40,24]). The mathematical framework that is used in this paper was introduced by Davis [22,21].…”

Viscosity solutions associated with switching game for piecewise-deterministic processes

“…See also Yamada [32] and Ishii and Yamada [I81 for a treatment of systems of second order partial differential equations with bilateral obstacles from the viewpoint of viscosity solutions. P D processes have been studied from several viewpoints such as queueing processes in Rosenkrantz [28], storage processes in Cinlar and Pinsky [3], Gnedenko and Kovalenko [13], Harrison and Resnick [15], and renewal processes in Cox [6]. For other control problems involving P D processes see Davis [22,23] and Gatarek [12].…”

Continuous feedback fluid queues

“…which can be obtained by similar methods as those used in the Appendix to derive (5). Next, we take the limit…”

“…the rate at which the buffer is depleted) is constant, see [14]. Early extensions of this storage process are considered in [5,9], and references therein, in which the release rate is state-dependent; in fact it is a strictly positive piecewise continuous function of the current buffer content. Another paper worth mentioning in this context is [3], where not only the release rate, but also the arrival rate of customers is state-dependent.…”