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

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))_t≥0 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 G_u and G_d (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of τ_y = inf{t ≥ 0|X(t) ∉ (0,y)},y > 0, and X(τ _y) as well as that of T = inf{t ≥ 0|X(t) ≤ 0} and X(T). We also determine the distribution of sup{X(t)[0 ≤ t ≤ T}.