Two finite-capacity loss queues in series, with Poisson arrival at the first queue and with Erlang service time distribution at both queues are considered. It is assumed that the service rate at the second queue is held fixed and that a maximum buffer space is globally available. For this model, the problem of optimally choosing the service rate at the first queue and the buffer space distribution over the two queues is considered in order to minimize the steady-state total network loss rate. It is shown that a decrease of the service rate at the first queue can reduce the total network loss rate only if this lower service rate decreases the variability of the process feeding the second queue. Numerical results show that, when optimal values are assigned to the parameters of the model, a significant reduction of the network loss rate can be achieved, especially when the network load is moderate and when the service time distributions have a small variance.>

A method is presented to compute, for the steady-state conditional probabilities of a given Markov subchain, the best lower and upper bounds derivable from the submatrix of transition probabilities between the states of that subchain only. The bounds can be improved when additional information, even fragmentary, on the entire chain is available. When the submatrix has a special structure, analytical expressions of the bounds can be obtained. The method is shown to be useful and economical to bound performance measures in large nonproduct-form queueing network models of computer communication systems.

This paper deals with the positive eigenvectors of nonnegative irreducible matrices that are merely characterized by a given upper bound ), on their spectral radius and by a given matrix L of lower bounds for their elements.For any such matrix, the normalized positive left (right) eigenvector is shown to belong to the polyhedron the vertices of which are given by the normalized rows (columns) of the matrix (~,1 -L) -~.This polyhedron is proved to be also the smallest dosed convex set that is guaranteed to contain the positive left (right) normalized eigenvector; its vertices are therefore the best bounds one can obtain.These results are then used to obtain componentwise upper and lower bounds on the error that is made when the positive e~genvectors of a large nonnegative irreducible matrix have to be approximated by a block decomposition and aggregation technique.The computation of these bounds can itself be regarded as a new approximation technique, called here bounded aggregation.Finally, the particular case of stochastic matrices is analyzed and a numerical example is gwen.

A steady-state analysis of theM/G/1finite capacity queue with delays is being made. In this model every busy period is followed by the execution of a noninterruptable task other than the servicing of ordinary customers; the duration of this task, called a delay, is a random variable with general distribution. Closed form expressions, easy to evaluate, are given for the distribution of the queue length and the first two moments of the queueing time distribution. A variant is also studied in which a busy period is followed by as many delays as possible, new delays being reinitiated as long as no customer has arrived.