Stopping problems for compound processes with applications to queues

We consider three problems for dual versions of statistical control processes. In the first problem each of the dual processes fluctuates between two linear boundaries; in the second, above the lower boundary; in the third, below the upper boundary. In the primal version the process is a compound Poisson process with positive general jumps and in the dual version it is a compound renewal process with exponential jumps. The duality is based on time reversal theory. For each problem we define a relevant functional. The relevant functional in the first problem is the probability of hitting the lower boundary before up-crossing the upper boundary. The relevant functionals in both the second and the third problems are the maximal distance from the boundary during the lifetime of the process. By exploiting queueing theory we introduce close-form expressions for the three functionals.