On Markov Jump Processes Imbedded at Jump Epochs and Their Queueing-Theoretic Applications

Let {X_t}_t>0 be a Markov jump process and {T_n}_n=0^∞ a subset of the jump epochs of the process {X_t}_t>0. The main results of this paper are computable formulae for the transient and invariant distributions of {X_Tn}_n=0^∞ and {X_Tn̄}_n=0^∞. We also characterize when the invariant distributions of {X_t}_t>0 and {X_Tn}_n=0^∞ are the same and show that in this case {T_n}_n=0^∞ is a Poisson process. The results are applied to a variety of discrete-slate queueing networks to obtain their state distribution as seen by customers in arrival streams, departure streams, and traffic streams on the arcs. The main conclusion drawn is that for many traffic streams, the invariant distribution seen by customers in that stream coincides with the invariant distribution of {X_t}_t>0 provided the customer in motion is excluded.