On a Continuous-Time Martingale and Two Applications

We construct a continuous-time martingale to analyze two queueing systems: One is the GI(X)/M/1 queue with light-tailed batch arrivals for which we obtain the first exact result in closed-form when.. has a Geometric distribution, and stochastic bounds otherwise. The second application concerns queues with Semi-Markovian (SM) inter-arrival times; of particular interest is the scenario with multiplexed SM sources whose aggregate loses the SM property. Unlike existing exact but implicit solutions which rely on numerical transform methods, even in the M-r/M/1 case, our stochastic bounds are in closed-form and shown to be (mostly) numerically accurate.