MARTINGALE APPROACH FOR TAIL ASYMPTOTIC PROBLEMS IN THE
GENERALIZED JACKSON NETWORK
Abstract: We study the tail asymptotic of the stationary joint queue length distribution
for a generalized Jackson network (GJN for short), assuming its stability. For the
two-station case, this problem has recently been solved in the logarithmic sense for the
marginal stationary distributions under the setting that arrival processes and service
times are of phase-type. In this paper, we study similar tail asymptotic problems on
the stationary distribution, but problems and assumptions are different. First, the
asymptotics are studied not only for the marginal distribution but also the stationary
probabilities of state sets of small volumes. Second, the interarrival and service
times are generally distributed and light tailed, but of phase-type in some cases.
Third, we also study the case that there are more than two stations, although the
asymptotic results are less complete. For them, we develop a martingale method,
which has been recently applied to a single queue with many servers by the author.
2010 AMS Mathematics Subject Classification: Primary: 60K25, 90B15; Secondary:
60G44, 60F10.
Keywords and phrases: Generalized Jackson network, stationary distribution, tail
asymptotic, piecewise deterministic Markov process, martingale, change of measure,
instability.