The M/G/1 Retrial Queue With Retrial Rate Control Policy

1993 ◽  
Vol 7 (1) ◽  
pp. 29-46 ◽  
Author(s):  
Bong Dae Choi ◽  
Kyung Hyune Rhee ◽  
Kwang Kyu Park
Keyword(s):  
Retrial Queue ◽  
Control Policy ◽  
Single Server ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Virtual Waiting Time ◽  
The Laplace Transform ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Number Of Customers

We consider a single-server retrial queueing system where retrial time is inversely proportional to the number of customers in the system. A necessary and sufficient condition for the stability of the system is found. We obtain the Laplace transform of virtual waiting time and busy period. The transient distribution of the number of customers in the system is also obtained.

scholarly journals STABILITY ANALYSIS AND SIMULATION OF N-CLASS RETRIAL SYSTEM WITH CONSTANT RETRIAL RATES AND POISSON INPUTS

2014 ◽  
Vol 31 (02) ◽  
pp. 1440002 ◽  
Author(s):  
K. AVRACHENKOV ◽  
E. MOROZOV ◽  
R. NEKRASOVA ◽  
B. STEYAERT
Keyword(s):  
Queueing System ◽  
Stability Domain ◽  
Single Server ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Single Orbit ◽  
Transmission Control Protocols ◽  
Regenerative Approach ◽  
The Stability ◽  
Multi Access

In this paper, we study a new retrial queueing system with N classes of customers, where a class-i blocked customer joins orbit i. Orbit i works like a single-server queueing system with (exponential) constant retrial time (with rate [Formula: see text]) regardless of the orbit size. Such a system is motivated by multiple telecommunication applications, for instance wireless multi-access systems, and transmission control protocols. First, we present a review of some corresponding recent results related to a single-orbit retrial system. Then, using a regenerative approach, we deduce a set of necessary stability conditions for such a system. We will show that these conditions have a very clear probabilistic interpretation. We also performed a number of simulations to show that the obtained conditions delimit the stability domain with a remarkable accuracy, being in fact the (necessary and sufficient) stability criteria, at the very least for the 2-orbit M/M/1/1-type and M/Pareto/1/1-type retrial systems that we focus on.

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. 

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 On the virtual waiting time for an M/G/1 retrial queue with two types of calls

1993 ◽  
Vol 6 (1) ◽  
pp. 11-23 ◽  
Author(s):  
Bong Dae Choi ◽  
Dong Hwan Han ◽  
Guennadi Falin
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Retrial Queue ◽  
Arrival Rate ◽  
Switching System ◽  
Stieltjes Transform ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Virtual Waiting Time ◽  
Incoming Call

We consider an M/G/1 retrial queueing system with two types of calls which models a telephone switching system. In the case that arriving calls are blocked due to the channel being busy, the outgoing calls are queued in priority group whereas the incoming calls enter the retrial group in order to try service again after a random amount of time. In this paper we find the Laplace-Stieltjes transform of the distribution of the virtual waiting time for an incoming call. When the arrival rate of outgoing calls is zero, it is shown that our result is consistent with the known result for a retrial queueing system with one type of call.

scholarly journals An M/G/1 Retrial Queue with Fluctuating Modes of Services

2018 ◽  
Vol 7 (4.10) ◽  
pp. 762
Author(s):  
P. Rajadurai ◽  
S. Venkatesh ◽  
K. Parameswari
Keyword(s):  
Steady State ◽  
Queueing System ◽  
Retrial Queue ◽  
Probability Generating Function ◽  
Single Server ◽  
Supplementary Variable ◽  
Working Vacation ◽  
Variable Technique ◽  
Retrial Queueing System ◽  
Retrial Queueing

In this paper, we consider a single server retrial queueing system with working vacation and two classes of customers, which are priority customers and ordinary customers. The single server provides fluctuating modes (optional phases) of services. Using the method of Probability Generating Function (PGF) approach and supplementary variable technique, the steady state results are obtained. 

Stochastic analysis of a preemptive retrial queue with orbital search and multiple vacations

2020 ◽  
Vol 54 (1) ◽  
pp. 231-249
Author(s):  
Shan Gao ◽  
Jinting Wang
Keyword(s):  
Cost Optimization ◽  
Retrial Queue ◽  
Length Distribution ◽  
Stochastic Decomposition ◽  
Embedded Markov Chain ◽  
Multiple Vacations ◽  
Retrial Queueing System ◽  
Queue Length Distribution ◽  
Retrial Queueing ◽  
Necessary And Sufficient

This paper deals with a preemptive priority M/G/1 retrial queue with orbital search and exhaustive multiple vacations. By using embedded Markov chain technique and the supplementary variable method, we discuss the necessary and sufficient condition for the system to be stable and the joint queue length distribution in steady state as well as some important performance measures and the Laplace–Stieltjes transform of the busy period. Also, we establish a special case and the stochastic decomposition laws for this preemptive retrial queueing system. Finally, some numerical examples and cost optimization analysis are presented.

A Single Server Retrial Queueing System with Two Types of Batch Arrivals

2016 ◽  
pp. 55-70
Author(s):  
Kalyanaraman Rathinasabapathy
Keyword(s):  
Queueing System ◽  
Numerical Study ◽  
Priority Queue ◽  
Single Server ◽  
Type I ◽  
Type Ii ◽  
Operating Characteristics ◽  
Batch Arrivals ◽  
Retrial Queueing System ◽  
Retrial Queueing

A retrial queueing system with two types of batch arrivals is considered. The arrivals are called type I and type II customers. The type I customers arrive in batches of size k with probability c_k and type II customers arrive in batches of size k with probability d_k. Service time distributions are identical independent distributions and are different for both type of customers. If the arriving customers are blocked due to server being busy, type I customers are queued in a priority queue of infinity capacity whereas type II customers entered into retrial group in order to seek service again after a random amount of time. For this model the joint distribution of the number of customers in the priority queue and in the retrial group in closed form is obtained. Some particular models and operating characteristics are obtained. A numerical study is also carried out.

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.

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