Stopping problems for compound processes with applications to queues (Q1589681)

For a compound Poisson process (CPP) the author considers the first time at which the process crosses (i) an upper straight line or (ii) a lower straight line both of slope 1. The probability that (i) happens before (ii) is expressed in terms of the stationary distribution of the corresponding M/G/1 workload process. This yields elegant formulas for the maximum deviation of the CPP and the linear boundary before crossing in both cases (i) and (ii). The corresponding distributions for a compound renewal process with exponential upward jumps are derived by means of a duality argument. En passant it is also shown that the two formulas in the literature (due to Takacs and Cohen) for the distribution function of the maximum workload of an M/G/1 queue during a busy period both follow from the same level crossing argument.