First-exit times for compound Poisson processes for some types of positive and negative jumps

Abstract

We consider the one-sided and the two-sided first-exit problem for a compound Poisson process with linear deterministic decrease between positive and negative jumps. This process (X(t))(tgreater than or equal to0) occurs as the workload process of a single-server queueing system with random workload removal, which we denote by M/G(u)/G(d)/1, where G(u)(G(d)) stands for the distribution of the upward (downward) jumps; other applications are to cash management, dams, and several related fields. Under various conditions on Gu and Gd (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of tau(y) = inf{t greater than or equal to 0X(t) is not an element of (0,y)}, y > 0, and X(tau(y)) as well as that of T = inf{t greater than or equal to 0X(t) less than or equal to 0} and X(T). We also determine the distribution of sup {X(t)0 less than or equal to t less than or equal to T}.