scholarly journals On the single-server retrial queue

2006 ◽  
Vol 16 (1) ◽  
pp. 45-53 ◽  
Author(s):  
Natalia Djellab
Keyword(s):  
Approximate Solution ◽  
Retrial Queue ◽  
Retrial Queues ◽  
Stochastic Decomposition ◽  
Single Server ◽  
Queue Size ◽  
Decomposition Property ◽  
The Subject ◽  
Exponential Case ◽  
Number Of Customers

In this work, we review the stochastic decomposition for the number of customers in M/G/1 retrial queues with reliable server and server subjected to breakdowns which has been the subject of investigation in the literature. Using the decomposition property of M/G/1 retrial queues with breakdowns that holds under exponential assumption for retrial times as an approximation in the non-exponential case, we consider an approximate solution for the steady-state queue size distribution.

A NOTE ON INFINITE-SERVER MARKOV MODULATED AND SINGLE-SERVER RETRIAL QUEUES

2014 ◽  
Vol 31 (02) ◽  
pp. 1440003
Author(s):  
ZHE DUAN ◽  
MELIKE BAYKAL-GÜRSOY
Keyword(s):  
Queueing Systems ◽  
Retrial Queue ◽  
System Size ◽  
Retrial Queues ◽  
Stochastic Decomposition ◽  
Applied Probability ◽  
Single Server ◽  
The Matrix ◽  
Service Rates ◽  
Markov Modulated

We reconsider the M/M/∞ queue with two-state Markov modulated arrival and service processes and the single-server retrial queue analyzed in Keilson and Servi [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471]. Fuhrmann and Cooper type stochastic decomposition holds for the stationary occupancy distributions in both queues [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471; Baykal-Gürsoy, M and W Xiao (2004). Stochastic decomposition in M/M/∞ queues with Markov-modulated service rates. Queueing Systems, 48, 75–88]. The main contribution of the present paper is the derivation of the explicit form of the stationary system size distributions. Numerical examples are presented visually exhibiting the effect of various parameters on the stationary distributions.

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.

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 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.

Stability condition for a single-server retrial queue

1993 ◽  
Vol 25 (03) ◽  
pp. 690-701 ◽  
Author(s):  
Huei-Mei Liang ◽  
V. G. Kulkarni
Keyword(s):  
Stability Condition ◽  
Service Time ◽  
Retrial Queue ◽  
Arrival Rate ◽  
Retrial Queues ◽  
Single Server ◽  
Sufficient Stability Condition ◽  
Sufficient Stability ◽  
The Mean ◽  
The Stability

A single-server retrial queue consists of a primary queue, an orbit and a single server. Assume the primary queue capacity is 1 and the orbit capacity is infinite. Customers can arrive at the primary queue either from outside the system or from the orbit. If the server is busy, the arriving customer joins the orbit and conducts a retrial later. Otherwise, he receives service and leaves the system. We investigate the stability condition for a single-server retrial queue. Let λ be the arrival rate and 1/μ be the mean service time. It has been proved that λ / μ < 1 is a sufficient stability condition for the M/G /1/1 retrial queue with exponential retrial times. We give a counterexample to show that this stability condition is not valid for general single-server retrial queues. Next we show that λ /μ < 1 is a sufficient stability condition for the stability of a single-server retrial queue when the interarrival times and retrial times are finite mixtures of Erlangs.

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.

scholarly journals On the stochastic decomposition property of single server retrialqueuing systems

2017 ◽  
Vol 41 ◽  
pp. 918-932
Author(s):  
Nawel ARRAR ◽  
Natalia DJELLAB ◽  
Jean-Bernard BAILLON
Keyword(s):  
Stochastic Decomposition ◽  
Single Server ◽  
Decomposition Property

Stability condition for a single-server retrial queue

1993 ◽  
Vol 25 (3) ◽  
pp. 690-701 ◽  
Author(s):  
Huei-Mei Liang ◽  
V. G. Kulkarni
Keyword(s):  
Stability Condition ◽  
Service Time ◽  
Retrial Queue ◽  
Arrival Rate ◽  
Retrial Queues ◽  
Single Server ◽  
Sufficient Stability Condition ◽  
Sufficient Stability ◽  
The Mean ◽  
The Stability

A single-server retrial queue consists of a primary queue, an orbit and a single server. Assume the primary queue capacity is 1 and the orbit capacity is infinite. Customers can arrive at the primary queue either from outside the system or from the orbit. If the server is busy, the arriving customer joins the orbit and conducts a retrial later. Otherwise, he receives service and leaves the system.We investigate the stability condition for a single-server retrial queue. Let λ be the arrival rate and 1/μ be the mean service time. It has been proved that λ/μ < 1 is a sufficient stability condition for the M/G/1/1 retrial queue with exponential retrial times. We give a counterexample to show that this stability condition is not valid for general single-server retrial queues. Next we show that λ /μ < 1 is a sufficient stability condition for the stability of a single-server retrial queue when the interarrival times and retrial times are finite mixtures of Erlangs.

scholarly journals Equilibrium Customer Strategies in the Single-Server Constant Retrial Queue with Breakdowns and Repairs

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Zhengwu Zhang ◽  
Jinting Wang ◽  
Feng Zhang
Keyword(s):  
Retrial Queue ◽  
Profit Maximization ◽  
Optimal Strategies ◽  
The State ◽  
Equilibrium Analysis ◽  
Arrival Process ◽  
Single Server ◽  
Retrial Queueing System ◽  
Exact Number ◽  
Number Of Customers

We consider a single-server constant retrial queueing system with a Poisson arrival process and exponential service and retrial times, in which the server may break down when it is working. The lifetime of the server is assumed to be exponentially distributed and once the server breaks down, it will be sent for repair immediately and the repair time is also exponentially distributed. There is no waiting space in front of the server and arriving customers decide whether to enter the retrial orbit or to balk depending on the available information they get upon arrival. In the paper, Nash equilibrium analysis for customers’ joining strategies as well as the related social and profit maximization problems is investigated. We consider separately the partially observable case where an arriving customer knows the state of the server but does not observe the exact number of customers waiting for service and the fully observable case where customer gets informed not only about the state of the server but also about the exact number of customers in the orbit. Some numerical examples are presented to illustrate the effect of the information levels and several parameters on the customers’ equilibrium and optimal strategies.

scholarly journals Equilibrium joining strategies in the single-server constant retrial queues with Bernoulli vacations

2019 ◽  
Author(s):  
Ke Sun ◽  
Jinting Wang
Keyword(s):  
Queue Length ◽  
Numerical Experiments ◽  
Retrial Queue ◽  
The State ◽  
Cost Structure ◽  
Retrial Queues ◽  
Single Server ◽  
Equilibrium Behavior ◽  
Information Level

We consider the equilibrium joining strategies in an M/M/1 constant retrial queue with Bernoulli vacations. There is no buffer in front of the server, thus an arriving customer will be served immediately if the server is available, and blocked ones wait in a queue if the server is busy or under vacation. The queue length information of orbit is observable to customers upon their arrivals. Then, blocked customers decide whether to join the orbit or not based on a reward-cost structure and their information level. After completing service, the server begins a vacation or remains available and it becomes available again when a vacation ends. The available server seeks to serve the customer in the head of the orbit queue. During the seeking process, an external arrival can interrupt it and obtain service. Our goal is to explore equilibrium behavior of customers in two information cases, fully observable case and almost observable case, which corresponding to whether blocked arrivals can differentiate the state of unavailable server. We obtain the threshold strategies of blocked customers in two information cases and provide numerical experiments to characterize the influence of different parameters on the equilibrium joining strategies.

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