State of a network when one node overloads

We delve into a couple of topics in the theory of Markov chains and stochastic networks. The properties of a stable Markov chain X = (Xi,X) will be investigated when X\ tends to infinity. We derive the distribution of X when X\ passes a threshold for the first time as the threshold tends to infinity. Moreover, the exact asymptotics of the mean time until X\ reaches the threshold is given. In addition, we present a new approach to determine the exact asymptotics of the X's steady state. The results are applied to an open modified Jackson network with two partially coupled processors. Finally, a ratio limit property is established for a Markovian kernel which has unbounded jumps.

[1]  Yiqiang Q. Zhao,et al.  The stationary tail asymptotics in the GI/G/1-type queue with countably many background states , 2004, Advances in Applied Probability.

[2]  Hui Li,et al.  Light-Tailed Behavior in QBD Processes with Countably Many Phases , 2009 .

[3]  Harry Kesten,et al.  Renewal Theory for Functionals of a Markov Chain with General State Space , 1974 .

[4]  G. Fayolle,et al.  Two coupled processors: The reduction to a Riemann-Hilbert problem , 1979 .

[5]  W. Veech.  The necessity of Harris’ condition for the existence of a stationary measure , 1963 .

[6]  Michel Mandjes,et al.  Large Deviations for Gaussian Queues: Modelling Communication Networks , 2007 .

[7]  Adam Shwartz,et al.  Large Deviations For Performance Analysis , 2019 .

[8]  Yutaka Sakuma,et al.  Decay rate for a PH/M/2 queue with shortest queue discipline , 2006, Queueing Syst. Theory Appl..

[9]  Rare events for stationary processes , 2000 .

[10]  A. Pakes,et al.  Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state , 1995, Advances in Applied Probability.

[11]  Servet Martínez,et al.  EXISTENCE OF QUASI-STATIONARY DISTRIBUTIONS. A RENEWAL DYNAMICAL APPROACH , 1995 .

[12]  D. P. Kroese,et al.  Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process , 2003 .

[13]  D. McDonald,et al.  Asymptotics of first passage times for random walk in an orthant , 1999 .

[14]  David R. McDonald,et al.  Bridges and networks: Exact asymptotics , 2005, math/0503488.

[15]  G. Högnäs.  On the quasi-stationary distribution of a stochastic Ricker model , 1997 .

[16]  D. McDonald,et al.  Large deviations of a modified Jackson network: Stability and rough asymptotics , 2005, math/0503487.

[17]  A. K. Erlang.  The theory of probabilities and telephone conversations , 1909 .

[18]  D. McDonald,et al.  Join the shortest queue: stability and exact asymptotics , 2001 .

[19]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[20]  Masaaki Kijima.  On the Existence of Quasi-Stationary Distributions in Denumerable R-Transient Markov Chains , 1992 .

[21]  Hui Li,et al.  Geometric Decay in a QBD Process with Countable Background States with Applications to a Join-the-Shortest-Queue Model , 2007 .

[22]  David D. Yao,et al.  Fundamentals of Queueing Networks , 2001 .

[23]  L. Flatto,et al.  Two parallel queues created by arrivals with two demands. II , 1984 .

[24]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .