Two networks of queues models, presented initially by Jackson, in the open case, and
Gordon and Newell, in the closed case, stochastic processes are presented and studied in some of their details and problems. The service times are exponentially distributed and there is only one class of customers.
The main target of this paper is to present the Markov chain C that, not giving explicitly the queue lengths stationary probabilities, has the necessary information to its determination for open networks of queues with several classes of customers and exponential service times, allowing to overcome ingeniously this problem. The situation for closed networks, in the same conditions, much easier is also presented.
A network belongs to the monotone separable class if its state variables are
homogeneous and monotone functions of the epochs of the arrival process. This
framework contains several classical queueing network models, including
generalized Jackson networks, max-plus networks, polling systems, multiserver
queues, and various classes of stochastic Petri nets. We use comparison
relationships between networks of this class with i.i.d. driving sequences and
the $GI /GI /1/1$ queue to obtain the tail asymptotics of the stationary
maximal dater under light-tailed assumptions for service times. The exponential
rate of decay is given as a function of a logarithmic moment generating
function. We exemplify an explicit computation of this rate for the case of
queues in tandem under various stochastic assumptions.; Comment: 15 pages, shortened version, case of (max,plus)-networks handled in a