DETERMINING THE EXIT TIME DISTRIBUTION FOR A CLOSED CYCLIC NETWORK

Consider a closed, N-node, cyclic network, where each node has an inde pendent, exponential single server. Using lattice-Bessel functions, we can explicitly solve for the transition probabilities of events that occur prior to one of the nodes becoming empty. This calculation entai ls associating with this absorbing process a symmetry group that is th e semidirect product of simpler groups. As a byproduct, we are able to compute explicitly the entire spectrum for the finite-dimensional mat rix generator of this process. When the number of nodes exceeds 1, suc h a spectrum is no longer purely real. Moreover, we are also able to o btain the quasistationary distribution or the limiting behavior of the network conditioned on no node ever being idle.