Laplace transform and moments of waiting times in Poisson driven (max,+) linear systems

(Max,+) linear systems can be used to represent stochastic Petri nets belon ging to the class of event graphs. This class contains various instances of queueing networks like 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. It also contains some b asic manufacturing models such as kanban networks, assembly systems and so forth. In their 1997 paper, Baccelli, Hasenfuss and Schmidt provide explicit expre ssions for the expected value of the waiting time of the nth customer in a given subarea of a (max,+) linear system. Using similar analysis, we presen t explicit expressions for the moments and the Laplace transform of transie nt waiting times in Poisson driven (max,+) linear systems. Furthermore, sta rting with these closed form expressions, we also derive explicit expressio ns for the moments and the Laplace transform of stationary waiting times in a class of (max,+) linear systems with deterministic service times. Exampl es pertaining to queueing theory are given to illustrate the results.