Taylor series expansions for Poisson-driven $(\max,+$)-linear systems

Abstract

We give a Taylor series expansion for the mean value of the canonical stationary state variables of open (max +)-linear stochastic systems with Poisson input process. Probabilistic expressions are derived for coefficients of all orders, under certain integrability conditions. The coefficients in the series expansion are the expectations of polynomials, known in explicit form, of certain random variables defined from the data of the (max +)-linear system. These polynomials are of independent combinatorial interest: their monomials belong to a subset of those obtained in the multinomial expansion; they are also invariant under cyclic permutation and under translations along the main diagonal. The method for proving these results is based on two ingredients: (1) the (max +)-linear representation of the stationary state variables as functionals of the input point process; (2) the series expansion representation of functionals of marked point processes and, in particular, of Poisson point processes. Several applications of these results are proposed in queueing theory and within the framework of stochastic Petri nets. It is well known that (max +)-linear systems allow one to represent stochastic Petri nets belonging to the class of event graphs. This class contains various instances of queueing networks like acyclic or cyclic fork-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking (manufacturing and communication), synchronized queueing networks and so on. It also contains some basic manufacturing models such as Kanban networks, Job-Shop systems and so forth. The applicability of this expansion method is discussed for several systems of this type. In the M/D case (i.e., all service times are deterministic), the approach is quite practical, as all coefficients of the expansion are obtained in closed form. In the M/GI case, the computation of the coefficient of order k can be seen as that of joint distributions in a stochastic PERT graph of an order which is linear in k .