The caudal characteristic curve of queues

Many queues and related stochastic models, and in particular those that have a matrix-geometric stationary probability vector, have steady-state queue-length densities that are asymptotically geometric. The graph of the asymptotic rate η of these densities as a function of the traffic intensity ρ is the caudal characteristic curve. This is an informative graph from which a number of qualitative inferences about the behavior of the queue may be drawn. The caudal characteristic curve may be computed (by elementary algorithms) for several useful models for which a complete exact numerical solution is not practically feasible. These include queues with certain types of superimposed arrival processes and/or multiple non-exponential servers. The necessary theorems which lead to the algorithmic procedures as well as the interpretation of several numerical examples are discussed.