Inequalities for the M/G/∞ queue and related shot noise processes

Suppose that pulses arrive according to a Poisson process of rate λ with the duration of each pulse independently chosen from a distribution F having finite mean. Let X ( t ) be the shot noise process formed by the superposition of these pulses. We consider functionals H ( X ) of the sample path of X ( t ). H is said to be L- superadditive if for all functions f and g. For any distribution F for the pulse durations, we define H ( F ) = EH ( X ). We prove that if H is L- superadditive and for all convex functions ϕ, then . Various consequences of this result are explored.