Metastability of Queuing Networks with Mobile Servers

We study symmetric queuing networks with moving servers and FIFO service discipline. The mean-field limit dynamics demonstrates unexpected behavior which we attribute to the meta-stability phenomenon. Large enough finite symmetric networks on regular graphs are proved to be transient for arbitrarily small inflow rates. However, the limiting non-linear Markov process possesses at least two stationary solutions. The proof of transience is based on martingale techniques

A Linear Programming Approach to Error Bounds for Random Walks in the Quarter-plane

We consider the approximation of the performance of random walks in the quarter-plane. The approximation is in terms of a random walk with a product-form stationary distribution, which is obtained by perturbing the transition probabilities along the boundaries of the state space. A Markov reward approach is used to bound the approximation error. The main contribution of the work is the formulation of a linear program that provides the approximation error

Power series approximations for two-class generalized processor sharing systems

We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability beta, and a customer of queue 2 is served with probability 1-beta. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z (1),z (2)) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z (1),z (2)) in beta. The first coefficient of this power series corresponds to the priority case beta=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Pad, approximation

On the speed of convergence to stationarity of the Erlang loss system

We consider the Erlang loss system, characterized by $N$ servers, Poisson arrivals and exponential service times, and allow the arrival rate to be a function of $N.$ We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and display some bounds for the total variation distance between the time-dependent and stationary distributions. We also pay attention to time-dependent rates