Consider a closed, N-node, cyclic network, where each node has an independent, 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 entails associating with this absorbing process a symmetry group that is the semidirect product of simpler groups. As a byproduct, we are able to compute explicitly the entire spectrum for the finite-dimensional matrix generator of this process. When the number of nodes exceeds 1, such a spectrum is no longer purely real. Moreover, we are also able to obtain the quasistationary distribution or the limiting behavior of the network conditioned on no node ever being idle.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Science(all)