Tail probability of transient and stationary waiting times in (max,+)-linear systems

(Max,+) linear systems can be used to represent a class of queueing networks, such as acyclic or cyclic fork-and-join queueing networks, finite- or infinite-capacity tandem queueing networks with various types of blocking, synchronized queueing networks, and so on. In this paper, we derive explicit expressions for the tail probability of transient and stationary waiting times in Poisson-driven (max,+) linear systems. As an application of our results, we consider the problem of maximizing the long-run average throughput subject to a probabilistic constraint on stationary waiting times in (max,+) linear systems with deterministic service times