Two extensions of Kingman's GI/G/1 bound

A simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform. We extend this technique to 1) multiclass $\Sigma\textrm{GI/G/1}$ 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