Extended generator and associated martingales for M/G/1 retrial queue with classical retrial policy and general retrial times

2022 ◽  
pp. 1-8
Author(s):  
S. Meziani ◽  
T. Kernane
Keyword(s):  
Markov Process ◽  
Retrial Queue ◽  
Transient Regime ◽  
Expected Number ◽  
Piecewise Deterministic Markov Process ◽  
Number Of Customers

Abstract A retrial queue with classical retrial policy, where each blocked customer in the orbit retries for service, and general retrial times is modeled by a piecewise deterministic Markov process (PDMP). From the extended generator of the PDMP of the retrial queue, we derive the associated martingales. These results are used to derive the conditional expected number of customers in the orbit in the transient regime.

scholarly journals Sojourn Times in A Queueing System with Breakdowns and General Retrial Times

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2882
Author(s):  
Ivan Atencia ◽  
José Luis Galán-García
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Queueing System ◽  
Retrial Queue ◽  
Stochastic Decomposition ◽  
Discrete Time System ◽  
Main Research ◽  
Extensive Analysis ◽  
Complete Study ◽  
Number Of Customers

This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M/G/1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper.

scholarly journals The properties of the queuing model with the parallel structure

2020 ◽  
pp. 79-82
Author(s):  
O. A. Chechelnitsky
Keyword(s):  
Stationary Regime ◽  
Parallel Structure ◽  
Stochastic Dynamic ◽  
General Function ◽  
Expected Number ◽  
Queuing Model ◽  
Poisson Input ◽  
Infinite Server Queues ◽  
Result Analysis ◽  
Number Of Customers

The present article is devoted to research the multi-channelk model with the parallel structure. It means that we consider the model which consists of two infinite-server queues. The service time in the each system has general function of distribution. In this case the stochastic dynamic of our model cannot be defined by Markov chain. As a result, analysis of such models is much more difficult than that of the corresponding Markovian queueing models. Besides we assume that customers arrive to our model according a bivariate Poisson input flow. This input process is characterized by the fact that customers arrive according to a bivariate Poisson flow simultaneously. We consider the number of customers in the systems at time t. This stochastic process describes the state of our model. In present paper we find the limit joint distribution of the number of customers in the systems. In a general way (by differentiating the corresponding generating function.) we obtain the main characteristics of this distribution, such as the expected number of customers in the nodes, its variance and correlation. In the case when parameters of our model dependent on the parameter n (number of series) the limit normal distribution was obtained for the service process in the stationary regime.

scholarly journals Stochastic Flips on Dimer Tilings

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Thomas Fernique ◽  
Damien Regnault
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Upper Bound ◽  
Numerical Experiments ◽  
Triangular Grid ◽  
Expected Number ◽  
Worst Case ◽  
Average Case ◽  
International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

Performance Analysis of an M/G/1 Feedback Retrial Queue with Two Types of Service and Bernoulli Vacation

2016 ◽  
pp. 38-54
Author(s):  
Arivudainambi D ◽  
Gowsalya Mahalingam
Keyword(s):  
Retrial Queue ◽  
Single Server ◽  
Supplementary Variable ◽  
Number Of Customers ◽  
Bernoulli Schedule ◽  
Schedule Mechanism ◽  
Fcfs Discipline ◽  
Bernoulli Vacation

This chapter is concerned with the analysis of a single server retrial queue with two types of service, Bernoulli vacation and feedback. The server provides two types of service i.e., type 1 service with probability??1 and type 2 service with probability ??2. We assume that the arriving customer who finds the server busy upon arrival leaves the service area and are queued in the orbit in accordance with an FCFS discipline and repeats its request for service after some random time. After completion of type 1 or type 2 service the unsatisfied customer can feedback and joins the tail of the retrial queue with probability f or else may depart from the system with probability 1–f. Further the server takes vacation under Bernoulli schedule mechanism, i.e., after each service completion the server takes a vacation with probability q or with probability p waits to serve the next customer. For this queueing model, the steady state distributions of the server state and the number of customers in the orbit are obtained using supplementary variable technique. Finally the average number of customers in the system and average number of customers in the orbit are also obtained.

scholarly journals Diffusion Limit of Multi-Server Retrial Queue with Setup Time

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2232
Author(s):  
Anatoly Nazarov ◽  
Alexander Moiseev ◽  
Tuan Phung-Duc ◽  
Svetlana Paul
Keyword(s):  
Asymptotic Methods ◽  
Retrial Queue ◽  
Power Saving ◽  
Setup Time ◽  
Diffusion Limit ◽  
Steady State Probability ◽  
Retrial Queueing System ◽  
Limiting Condition ◽  
Number Of Customers ◽  
Multi Server

In the paper, we consider a multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival of a customer. Customers that find all the servers busy join the orbit and retry for service after an exponentially distributed time. For this model, we derive the stability condition which depends on the setup time and turns out to be more strict than that of the corresponding model with an infinite buffer which is independent of the setup time. We propose asymptotic methods to analyze the system under the condition that the delay in the orbit is extremely long. We show that the scaled-number of customers in the orbit converges to a diffusion process. Using this diffusion limit, we obtain approximations for the steady-state probability distribution of the number of busy servers and that of the number of customers in the orbit. We verify the accuracy of the approximations by simulations and numerical analysis. Numerical results show that the retrial system under the limiting condition consumes more energy than that with an infinite buffer in front of the servers.

scholarly journals A Discrete-TimeGeo/G/1 Retrial Queue with Two Different Types of Vacations

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Feng Zhang ◽  
Zhifeng Zhu
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Retrial Queue ◽  
System Size ◽  
Stochastic Decomposition ◽  
Continuous Time Model ◽  
Time Model ◽  
Different Types ◽  
Number Of Customers ◽  
The Relationship

We analyze a discrete-timeGeo/G/1 retrial queue with two different types of vacations and general retrial times. Two different types of vacation policies are investigated in this model, one of which is nonexhaustive urgent vacation during serving and the other is normal exhaustive vacation. For this model, we give the steady-state analysis for the considered queueing system. Firstly, we obtain the generating functions of the number of customers in our model. Then, we obtain the closed-form expressions of some performance measures and also give a stochastic decomposition result for the system size. Moreover, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided to illustrate the effect of nonexhaustive urgent vacation on some performance characteristics of the system.

scholarly journals An M/G/1 Retrial Queueing System with Phase Type Services and Working Vacations

2018 ◽  
Vol 7 (4.10) ◽  
pp. 758
Author(s):  
P. Rajadurai ◽  
R. Santhoshi ◽  
G. Pavithra ◽  
S. Usharani ◽  
S. B. Shylaja
Keyword(s):  
Queueing System ◽  
Retrial Queue ◽  
Probability Generating Function ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Multiple Working Vacations ◽  
Phase Type ◽  
Working Vacations ◽  
Multi Phase ◽  
Number Of Customers

A multi phase retrial queue with optional re-service and multiple working vacations is considered. The Probability Generating Function (PGF) of number of customers in the system is obtained by supplementary variable technique. Various system performance measures are discussed. 

APPROXIMATION OF PH/PH/c RETRIAL QUEUE WITH PH-RETRIAL TIME

2014 ◽  
Vol 31 (02) ◽  
pp. 1440010 ◽  
Author(s):  
YANG WOO SHIN ◽  
DUG HEE MOON
Keyword(s):  
Time Distribution ◽  
Retrial Queue ◽  
Retrial Queues ◽  
General Distribution ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Phase Type ◽  
Sojourn Time Distribution ◽  
The Mean ◽  
Number Of Customers

We consider the PH/PH/c retrial queues with PH-retrial time. Approximation formulae for the distribution of the number of customers in service facility, sojourn time distribution and the mean number of customers in orbit are presented. We provide an approximation for GI/G/c retrial queue with general retrial time by approximating the general distribution with phase type distribution. Some numerical results are presented.

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