Product form stationary distributions for the M|G|k group-arrival group-departure loss system under a general acceptance policy

1999 ◽  
Vol 112 (1) ◽  
pp. 196-206 ◽  
Author(s):  
A. Economou ◽  
D. Fakinos
Keyword(s):  
Product Form ◽  
Loss System ◽  
Stationary Distributions ◽  
General Acceptance ◽  
Group Arrival

scholarly journals Product-form distributions for an M/G/k group-arrival group-departure loss system

1989 ◽  
Vol 21 (3) ◽  
pp. 721-724 ◽  
Author(s):  
D. Fakinos ◽  
K. Sirakoulis
Keyword(s):  
Equilibrium Distribution ◽  
Product Form ◽  
Loss System ◽  
Group Arrival

In this work we investigate under what circumstances the equilibrium distribution of the numbers of groups of various sizes in a certain M/G/k group-arrival group-departure loss system can be obtained in a closed product form.

scholarly journals Product-form distributions for an M/G/k group-arrival group-departure loss system

1989 ◽  
Vol 21 (03) ◽  
pp. 721-724
Author(s):  
D. Fakinos ◽  
K. Sirakoulis
Keyword(s):  
Equilibrium Distribution ◽  
Product Form ◽  
Loss System ◽  
Group Arrival

In this work we investigate under what circumstances the equilibrium distribution of the numbers of groups of various sizes in a certain M/G/k group-arrival group-departure loss system can be obtained in a closed product form.

scholarly journals On Truncation of the Matrix-Geometric Stationary Distributions

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 798 ◽  
Author(s):  
Naumov ◽  
Gaidamaka ◽  
Samouylov
Keyword(s):  
Finite Number ◽  
Stationary Distribution ◽  
Queueing Systems ◽  
Death Process ◽  
Birth And Death Process ◽  
Loss System ◽  
Stationary Distributions ◽  
Open Network ◽  
The Matrix

In this paper, we study queueing systems with an infinite and finite number of waiting places that can be modeled by a Quasi-Birth-and-Death process. We derive the conditions under which the stationary distribution for a loss system is a truncation of the stationary distribution of the Quasi-Birth-and-Death process and obtain the stationary distributions of both processes. We apply the obtained results to the analysis of a semi-open network in which a customer from an external queue replaces a customer leaving the system at the node from which the latter departed.

Queueing networks with instantaneous movements: a unified approach by quasi-reversibility

2000 ◽  
Vol 32 (01) ◽  
pp. 284-313 ◽  
Author(s):  
Xiuli Chao ◽  
Masakiyo Miyazawa
Keyword(s):  
Queueing Networks ◽  
Product Form ◽  
Unified Approach ◽  
Stationary Distributions ◽  
Batch Arrivals ◽  
Unified View ◽  
Batch Services ◽  
Product Form Queueing Networks

In this paper we extend the notion of quasi-reversibility and apply it to the study of queueing networks with instantaneous movements and signals. The signals treated here are considerably more general than those in the existing literature. The approach not only provides a unified view for queueing networks with tractable stationary distributions, it also enables us to find several new classes of product form queueing networks, including networks with positive and negative signals that instantly add or remove customers from a sequence of nodes, networks with batch arrivals, batch services and assembly-transfer features, and models with concurrent batch additions and batch deletions along a fixed or a random route of the network.

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

2005 ◽  
Vol 37 (1) ◽  
pp. 212-228 ◽  
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.

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

2005 ◽  
Vol 37 (01) ◽  
pp. 212-228 ◽  
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.

scholarly journals Jackson networks with unlimited supply of work

2005 ◽  
Vol 42 (03) ◽  
pp. 879-882 ◽  
Author(s):  
Gideon Weiss
Keyword(s):  
Joint Distribution ◽  
Product Form ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Marginal Distributions ◽  
Flow Rates ◽  
Jackson Network ◽  
Jackson Networks ◽  
Unlimited Supply ◽  
Infinite Supply

We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.

The M/G/k group-arrival group-departure loss system

1982 ◽  
Vol 19 (4) ◽  
pp. 826-834 ◽  
Author(s):  
D. Fakinos
Keyword(s):  
Poisson Process ◽  
Group Size ◽  
Loss System ◽  
Group Arrival ◽  
Service Times ◽  
Departure Processes

This paper considers the equilibrium behaviour of the M/G/k group-arrival group-departure loss system. Such a system has k servers whose customers arrive in groups, the arrival epochs of groups being points of a Poisson process. The duration of a service can be characteristic of the group size; however, customers who belong to the same group have equal service times. The customers of a group start being served immediately upon their arrival, unless their number is greater than the number of idle servers. In this case the whole group leaves and does not return later (i.e. is lost). Among other things, a generalization of the Erlang B-formula is given and it is shown that the arrival and departure processes are statistically indistinguishable.

Equilibrium results for the M/G/k group-arrival loss system

Top ◽  
2010 ◽  
Vol 21 (1) ◽  
pp. 163-181
Author(s):  
Spiros Dimou ◽  
Demetrios Fakinos
Keyword(s):  
Loss System ◽  
Group Arrival
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