Closed queueing networks under congestion: non-bottleneck independence and bottleneck convergence

We analyze the behavior of closed product-form queueing networks when the number of customers grows to infinity and remains proportionate on each route (or class). First, we focus on the stationary behavior and prove the conjecture that the stationary distribution at non-bottleneck queues converges weakly to the stationary distribution of an ergodic, open product-form queueing network. This open network is obtained by replacing bottleneck queues with per-route Poissonian sources whose rates are determined by the solution of a strictly concave optimization problem. Then, we focus on the transient behavior of the network and use fluid limits to prove that the amount of fluid, or customers, on each route eventually concentrates on the bottleneck queues only, and that the long-term proportions of fluid in each route and in each queue solve the dual of the concave optimization problem that determines the throughputs of the previous open network.Comment: 22 page

The Mx/G/1 queue with queue length dependent service times

We deal with the MX/G/1 queue where service times depend on the queue length at the service initiation. By using Markov renewal theory, we derive the queue length distribution at departure epochs. We also obtain the transient queue length distribution at time t and its limiting distribution and the virtual waiting time distribution. The numerical results for transient mean queue length and queue length distributions are given.Bong Dae Choi, Yeong Cheol Kim, Yang Woo Shin, and Charles E. M. Pearc

Transient bayesian inference for short and long-tailed GI/G/1 queueing systems

In this paper, we describe how to make Bayesian inference for the transient behaviour and busy period in a single server system with general and unknown distribution for the service and interarrival time. The dense family of Coxian distributions is used for the service and arrival process to the system. This distribution model is reparametrized such that it is possible to define a non-informative prior which allows for the approximation of heavytailed distributions. Reversible jump Markov chain Monte Carlo methods are used to estimate the predictive distribution of the interarrival and service time. Our procedure for estimating the system measures is based in recent results for known parameters which are frequently implemented by using symbolical packages. Alternatively, we propose a simple numerical technique that can be performed for every MCMC iteration so that we can estimate interesting measures, such as the transient queue length distribution. We illustrate our approach with simulated and real queues