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 .