| (Max,+)-linear systems can be used to represent stochastic Petri nets belonging to the class of event graphs which admits a representation in terms of an evolution equation involving only maximization and addition operations. This class contains 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 basic manufacturing models such as Kanban networks, assembly systems and so forth.; Baccelli and Schmidt (1996) provided Taylor series expansions with respect to arrival rate for the expected stationary waiting times in a given subarea of Poisson driven (max,+)-linear system. Baccelli, Hasenfuss and Schmidt (1997) showed explicit expressions for the expected value of waiting time of n-th customer in such a system. Using similar analysis, in this thesis we develop Taylor series expansions for certain characteristics such as moments, Laplace transform and tail probability of transient and stationary waiting times in Poisson driven (max,+)-linear systems. As an application of our results, we develop a release strategy that maximizes the long run average throughput of a system subject to a probabilistic constraint on stationary waiting times. Since it is usually difficult to compute coefficients beyond a certain order due to the computational complexity in evaluating the expectations of multi-dimensional integrals, we consider approximating these coefficients using Monte Carlo simulation. Finally, we also develop bounds on moments of transient and stationary waiting times in (max,+)-linear systems. |
