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A simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform. We extend this technique to 1) multiclass Σ\textrmGI/G/1 $ queues and 2) Markov Additive Processes (MAPs) whose background processes can be time-inhomogeneous or have an uncountable state-space. Both extensions are facilitated by a necessary and sufficient ordinary differential equation (ODE) condition for MAPs to admit continuous martingale transforms. Simulations show that the bounds on waiting time distributions are almost exact in heavy-traffic, including the cases of 1) heterogeneous input, e.g., mixing Weibull and Erlang-k classes and 2) Generalized Markovian Arrival Processes, a new class extending the Batch Markovian Arrival Processes to continuous batch sizes.

Two Extensions of Kingman's GI/G/1 Bound