Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

We propose an analytically tractable approach for studying the transient behavior of multi-server queueing systems and feed-forward networks. We model the queueing primitives via polyhedral uncertainty sets inspired by the limit laws of probability. These uncertainty sets are characterized by variability parameters that control the degree of conservatism of the model. Assuming the inter-arrival and service times belong to such uncertainty sets, we obtain closed-form expressions for the worst case transient system time in multi-server queues and feed-forward networks with deterministic routing. These analytic formulas offer rich qualitative insights on the dependence of the system times as a function of the variability parameters and the fundamental quantities in the queueing system. To approximate the average behavior, we treat the variability parameters as random variables and infer their density by using ideas from queues in heavy traffic under reflected Brownian motion. We then average the worst case values obtained with respect to the variability parameters. Our averaging approach yields approximations that match the diffusion approximations for a single queue with light-tailed primitives and allows us to extend the framework to heavy-tailed feed-forward networks. Our methodology achieves significant computational tractability and provides accurate approximations for the expected system time relative to simulated values.