scholarly journals Laplace symbols and invariant distributions

2018 ◽  
Vol 137 ◽  
pp. 217-223
Author(s):  
Anita Behme ◽  
Alexander Schnurr
Keyword(s):  
Invariant Distributions

Limiting conditional and conditional invariant distributions for the Poisson process with negative drift

1999 ◽  
Vol 36 (04) ◽  
pp. 1194-1209 ◽  
Author(s):  
Raúl Fierro ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Initial Point ◽  
Time Process ◽  
Conditional Distributions ◽  
Exponential Law ◽  
Invariant Distributions ◽  
Virtual Waiting Time ◽  
The Family ◽  
Waiting Time Process ◽  
Negative Drift ◽  
Limiting Behaviour

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queue M/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues λ ∊ (0, λ c ], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.

scholarly journals Diffusion-Scale Tightness of Invariant Distributions of a Large-Scale Flexible Service System

2015 ◽  
Vol 47 (01) ◽  
pp. 251-269 ◽  
Author(s):  
A. L. Stolyar
Keyword(s):  
Diffusion Process ◽  
Service Time ◽  
Large Scale ◽  
Service System ◽  
Arrival Rate ◽  
Asymptotic Regime ◽  
Invariant Distributions ◽  
Multiple Customer Classes ◽  
The Mean ◽  
Diffusion Scale

A large-scale service system with multiple customer classes and multiple server pools is considered, with the mean service time depending both on the customer class and server pool. The allowed activities (routeing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a leaf activity priority (LAP) policy, introduced by Stolyar and Yudovina (2012). An asymptotic regime is considered, where the arrival rate of customers and number of servers in each pool tend to ∞ in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. We prove tightness of diffusion-scaled (centered at the equilibrium point and scaled down by r −1/2) invariant distributions. As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.

scholarly journals Diffusion-Scale Tightness of Invariant Distributions of a Large-Scale Flexible Service System

2015 ◽  
Vol 47 (1) ◽  
pp. 251-269 ◽  
Author(s):  
A. L. Stolyar
Keyword(s):  
Diffusion Process ◽  
Service Time ◽  
Large Scale ◽  
Service System ◽  
Arrival Rate ◽  
Asymptotic Regime ◽  
Invariant Distributions ◽  
Multiple Customer Classes ◽  
The Mean ◽  
Diffusion Scale

A large-scale service system with multiple customer classes and multiple server pools is considered, with the mean service time depending both on the customer class and server pool. The allowed activities (routeing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a leaf activity priority (LAP) policy, introduced by Stolyar and Yudovina (2012). An asymptotic regime is considered, where the arrival rate of customers and number of servers in each pool tend to ∞ in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. We prove tightness of diffusion-scaled (centered at the equilibrium point and scaled down by r−1/2) invariant distributions. As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.

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