The tail asymptotic behavior of the stationary distribution of a double M/G/1 process and their applications to a batch arrival Jackson network

2009 ◽  
Author(s):  
Masahiro Kobayashi ◽  
Masakiyo Miyazawa
Keyword(s):  
Asymptotic Behavior ◽  
Stationary Distribution ◽  
Batch Arrival ◽  
Jackson Network ◽  
Tail Asymptotic

The asymptotic behavior of a stochastic multigroup SIS model

2018 ◽  
Vol 11 (03) ◽  
pp. 1850037 ◽  
Author(s):  
Chunyan Ji ◽  
Daqing Jiang
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Simulations ◽  
Stationary Distribution ◽  
Endemic Equilibrium ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Sis Model ◽  
Stochastic Perturbations ◽  
Long Time ◽  
Theoretical Results

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

scholarly journals Tail asymptotic behavior of the supremum of a class of chi-square processes

2019 ◽  
Vol 154 ◽  
pp. 108551 ◽  
Author(s):  
Lanpeng Ji ◽  
Peng Liu ◽  
Stephan Robert
Keyword(s):  
Asymptotic Behavior ◽  
Chi Square ◽  
Tail Asymptotic

Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

2006 ◽  
Vol 38 (02) ◽  
pp. 522-544 ◽  
Author(s):  
Allan J. Motyer ◽  
Peter G. Taylor
Keyword(s):  
Stationary Distribution ◽  
Decay Rates ◽  
Transition Structure ◽  
Birth And Death Processes ◽  
Jackson Network ◽  
Qbd Processes ◽  
Generator Matrices ◽  
Level Process

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

scholarly journals Martingale approach for tail asymptotic problems in the gener­alized Jackson network

2018 ◽  
Vol 37 (2) ◽  
pp. 395-430 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Length Distribution ◽  
Martingale Method ◽  
Asymptotic Results ◽  
Martingale Approach ◽  
Jackson Network ◽  
Queue Length Distribution ◽  
Phase Type ◽  
Service Times ◽  
Asymptotic Problems ◽  
Tail Asymptotic

MARTINGALE APPROACH FOR TAIL ASYMPTOTIC PROBLEMS IN THE GENERALIZED JACKSON NETWORKWe study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network GJN for short, assumingits 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.

scholarly journals Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

2014 ◽  
Vol 46 (02) ◽  
pp. 365-399 ◽  
Author(s):  
Masahiro Kobayashi ◽  
Masakiyo Miyazawa
Keyword(s):  
Random Walk ◽  
Stationary Distribution ◽  
Queueing Network ◽  
Tail Asymptotics ◽  
Two Dimensional ◽  
Batch Arrivals ◽  
Jump Size ◽  
Light Tails ◽  
Upward Jump ◽  
Tail Asymptotic

We consider a two-dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming that the upward jump size distributions have light tails, we find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the skip-free reflecting random walk in Miyazawa (2009). We exemplify these results for a two-node queueing network with exogenous batch arrivals.

scholarly journals Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

2014 ◽  
Vol 46 (2) ◽  
pp. 365-399 ◽  
Author(s):  
Masahiro Kobayashi ◽  
Masakiyo Miyazawa
Keyword(s):  
Random Walk ◽  
Stationary Distribution ◽  
Queueing Network ◽  
Tail Asymptotics ◽  
Two Dimensional ◽  
Batch Arrivals ◽  
Jump Size ◽  
Light Tails ◽  
Upward Jump ◽  
Tail Asymptotic

We consider a two-dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming that the upward jump size distributions have light tails, we find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the skip-free reflecting random walk in Miyazawa (2009). We exemplify these results for a two-node queueing network with exogenous batch arrivals.

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