Tail asymptotics of the stationary distribution for M/M-JSQ with k parallel queues (abstract only)

2012 ◽  
Vol 39 (4) ◽  
pp. 28-28
Author(s):  
Masahiro Kobayashi ◽  
Yutaka Sakuma ◽  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Tail Asymptotics ◽  
Parallel Queues

scholarly journals Two-node fluid network with a heavy-tailed random input: the strong stability case

2014 ◽  
Vol 51 (A) ◽  
pp. 249-265
Author(s):  
Sergey Foss ◽  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Heavy Tails ◽  
Sample Path ◽  
Strong Stability ◽  
Tail Asymptotics ◽  
Two Dimensional ◽  
Batch Arrivals ◽  
One Dimensional ◽  
Fluid Network ◽  
Heavy Tailed

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

scholarly journals Two-node fluid network with a heavy-tailed random input: the strong stability case

2014 ◽  
Vol 51 (A) ◽  
pp. 249-265 ◽  
Author(s):  
Sergey Foss ◽  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Heavy Tails ◽  
Sample Path ◽  
Strong Stability ◽  
Tail Asymptotics ◽  
Two Dimensional ◽  
Batch Arrivals ◽  
One Dimensional ◽  
Fluid Network ◽  
Heavy Tailed

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

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.

scholarly journals Exact tail asymptotics for a three-dimensional Brownian-driven tandem queue with intermediate inputs

2021 ◽  
Author(s):  
Hongshuai Dai ◽  
Donald A. Dawson ◽  
Yiqiang Q. Zhao
Keyword(s):  
Stationary Distribution ◽  
Kernel Method ◽  
Three Dimensional ◽  
Value Theory ◽  
Tail Asymptotics ◽  
Tandem Queue ◽  
Intermediate Inputs ◽  
Exact Tail Asymptotics ◽  
Reflection Matrix ◽  
Joint Stationary Distribution

In this paper, we consider a three-dimensional Brownian-driven tandem queue with intermediate inputs, which corresponds to a three-dimensional semimartingale reflecting Brownian motion whose reflection matrix is triangular. For this three-node tandem queue, no closed form formula is known, not only for its stationary distribution but also for the corresponding transform. We are interested in exact tail asymptotics for stationary distributions. By generalizing the kernel method, and using extreme value theory and copula, we obtain exact tail asymptotics for the marginal stationary distribution of the buffer content in the third buffer and for the joint stationary distribution.

Sign in / Sign up

Export Citation Format

Share Document