On the Characterization of Product-Form Multiclass Queueing Models with Probabilistic Disciplines

A stochastic process, called reallocatable GSMP (RGSMP for short), is introduced in order to study insensitivity of its stationary distribution. RGSMP extends GSMP with interruptions, and is applicable to a wide range of queues, from the standard models such as BCMP and Kelly's network queues to new ones such as their modifications with interruptions and Serfozo's (1989) non-product form network queues, and can be used to study their insensitivity in a unified way. We prove that RGSMP supplemented by the remaining lifetimes is product-form decomposable, i.e. its stationary distribution splits into independent components if and only if a version of the local balance equations hold, which implies insensitivity of the RGSMP scheme in a certain extended sense. Various examples of insensitive queues are given, which include new results. Our proofs are based on the characterization of a stationary distribution for SCJP (self-clocking jump process) of Miyazawa (1991).

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On product-form stationary distributions for reflected diffusions with jumps in the positive orthant

Advances in Applied Probability ◽

10.1239/aap/1113402406 ◽

2005 ◽

Vol 37 (1) ◽

pp. 212-228 ◽

Cited By ~ 3

Author(s):

Francisco J. Piera ◽

Ravi R. Mazumdar ◽

Fabrice M. Guillemin

Keyword(s):

Sufficient Conditions ◽

Planck Equation ◽

Product Form ◽

Stationary Distributions ◽

Reflected Diffusions ◽

Positive Orthant ◽

Boundary Properties ◽

Necessary And Sufficient ◽

Oblique Boundary

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density inR+nthat satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

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On product-form stationary distributions for reflected diffusions with jumps in the positive orthant

Advances in Applied Probability ◽

10.1017/s0001867800000100 ◽

2005 ◽

Vol 37 (01) ◽

pp. 212-228 ◽

Cited By ~ 2

Author(s):

Francisco J. Piera ◽

Ravi R. Mazumdar ◽

Fabrice M. Guillemin

Keyword(s):

Sufficient Conditions ◽

Planck Equation ◽

Product Form ◽

Stationary Distributions ◽

Reflected Diffusions ◽

Positive Orthant ◽

Boundary Properties ◽

Necessary And Sufficient ◽

Oblique Boundary

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R + n that satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

Download Full-text