scholarly journals M/M/1 Retrial Queue with Working Vacation Interruption and Feedback under N-Policy

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Li Tao ◽  
Liyuan Zhang ◽  
Shan Gao
Keyword(s):  
Retrial Queue ◽  
Stochastic Decomposition ◽  
Service Rate ◽  
Working Vacation ◽  
Stationary Probability Distribution ◽  
The Matrix ◽  
Vacation Interruption ◽  
Necessary And Sufficient ◽  
Vacation Period ◽  
Conditional Stochastic Decomposition

We consider an M/M/1 retrial queue with working vacations, vacation interruption, Bernoulli feedback, and N-policy simultaneously. During the working vacation period, customers can be served at a lower rate. Using the matrix-analytic method, we get the necessary and sufficient condition for the system to be stable. Furthermore, the stationary probability distribution and some performance measures are also derived. Moreover, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, we present some numerical examples and use the parabolic method to search the optimum value of service rate in working vacation period.

AN M/G/1 RETRIAL QUEUE WITH GENERAL RETRIAL TIMES, WORKING VACATIONS AND VACATION INTERRUPTION

2014 ◽  
Vol 31 (02) ◽  
pp. 1440006 ◽  
Author(s):  
SHAN GAO ◽  
JINTING WANG ◽  
WEI WAYNE LI
Keyword(s):  
Retrial Queue ◽  
Stochastic Decomposition ◽  
Supplementary Variable ◽  
Waiting Time Distribution ◽  
Working Vacation ◽  
Stationary Probability Distribution ◽  
Working Vacations ◽  
Vacation Interruption ◽  
Vacation Period ◽  
Conditional Stochastic Decomposition

We consider an M/G/1 retrial queue with general retrial times, and introduce working vacations and vacation interruption policy into the retrial queue. During the working vacation period, customers can be served at a lower rate. If there are customers in the system at a service completion instant, the vacation will be interrupted and the server will come back to the normal working level. Using supplementary variable method, we obtain the stationary probability distribution and some performance measures. Furthermore, we carry out the waiting time distribution and prove the conditional stochastic decomposition for the queue length in orbit. Finally, some numerical examples are presented.

An M/G/1 retrial queue with single working vacation under Bernoulli schedule

2020 ◽  
Vol 54 (2) ◽  
pp. 471-488
Author(s):  
Tao Li ◽  
Liyuan Zhang ◽  
Shan Gao
Keyword(s):  
Cost Optimization ◽  
Retrial Queue ◽  
Service Rate ◽  
Working Vacation ◽  
Optimization Analysis ◽  
Normal Period ◽  
Necessary And Sufficient ◽  
Number Of Customers ◽  
Vacation Period ◽  
Bernoulli Schedule

In this paper, an M/G/1 retrial queue with general retrial times and single working vacation is considered. We assume that the customers who find the server busy are queued in the orbit in accordance with a first-come-first-served (FCFS) discipline and only the customer at the head of the queue is allowed access to the server. During the normal period, if the orbit queue is not empty at a service completion instant, the server begins a working vacation with specified probability q (0 ≤ q ≤ 1), and with probability 1 − q, he waits for serving the next customer. During the working vacation period, customers can be served at a lower service rate. We first present the necessary and sufficient condition for the system to be stable. Using the supplementary variable method, we deal with the generating functions of the server state and the number of customers in the orbit. Various interesting performance measures are also derived. Finally, some numerical examples and cost optimization analysis are presented.

scholarly journals An M/M/1 Queue with Working Vacation and Vacation Interruption Under Bernoulli Schedule

2018 ◽  
Vol 7 (4.10) ◽  
pp. 448
Author(s):  
P. Manoharan ◽  
A. Ashok
Keyword(s):  
Geometric Approach ◽  
Stochastic Decomposition ◽  
Working Vacation ◽  
System Parameters ◽  
Mean Waiting Time ◽  
The Mean ◽  
Vacation Interruption ◽  
Mean Queue Length ◽  
Vacation Period ◽  
Bernoulli Schedule

This work deals with M/M/1 queue with Vacation and Vacation Interruption Under Bernoulli schedule. When there are no customers in the system, the server takes a classical vacation with probability p or a working vacation with probability 1-p, where . At the instants of service completion during the working vacation, either the server is supposed to interrupt the vacation and returns back to the non-vacation period with probability 1-q or the sever will carry on with the vacation with probability q. When the system is non empty after the end of vacation period, a new non vacation period begins. A matrix geometric approach is employed to obtain the stationary distribution for the mean queue length and the mean waiting time and their stochastic decomposition structures. Numerous graphical demonstrations are presented to show the effects of the system parameters on the performance measures.  

scholarly journals Performance of an M/M/1 Retrial Queue with Working Vacation Interruption and Classical Retrial Policy

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Tao Li ◽  
Liyuan Zhang ◽  
Shan Gao
Keyword(s):  
Generating Functions ◽  
System Performance ◽  
Cost Optimization ◽  
Retrial Queue ◽  
Stable Condition ◽  
Working Vacation ◽  
Optimization Analysis ◽  
Vacation Interruption ◽  
Number Of Customers ◽  
Vacation Period

An M/M/1 retrial queue with working vacation interruption is considered. Upon the arrival of a customer, if the server is busy, it would join the orbit of infinite size. The customers in the orbit will try for service one by one when the server is idle under the classical retrial policy with retrial ratenα, wherenis the size of the orbit. During a working vacation period, if there are customers in the system at a service completion instant, the vacation will be interrupted. Under the stable condition, the probability generating functions of the number of customers in the orbit are obtained. Various system performance measures are also developed. Finally, some numerical examples and cost optimization analysis are presented.

scholarly journals Performance Analysis of Retrial Queueing Model with Working Vacation, Interruption, Waiting Server, Breakdown and Repair

2021 ◽  
Vol 13 (3) ◽  
pp. 833-844
Author(s):  
P. Gupta ◽  
N. Kumar
Keyword(s):  
Cost Optimization ◽  
Probability Generating Function ◽  
Function Technique ◽  
Service Rate ◽  
Queueing Model ◽  
Working Vacation ◽  
Retrial Queueing ◽  
Server Breakdown ◽  
Vacation Interruption ◽  
Vacation Period

In this present paper, an M/M/1 retrial queueing model with a waiting server subject to breakdown and repair under working vacation, vacation interruption is considered. Customers are served at a slow rate during the working vacation period, and the server may undergo breakdowns from a normal busy state. The customer has to wait in orbit for the service until the server gets repaired. Steady-state solutions are obtained using the probability generating function technique. Probabilities of different server states and some other performance measures of the system are developed.  The variation in mean orbit size, availability of the server, and server state probabilities are plotted for different values of breakdown parameter and repair rate with the help of MATLAB software. Finally, cost optimization of the system is also discussed, and the optimal value of the slow service rate for the model is obtained.

scholarly journals Analysis and optimisation of a M/M/1/WV queue with Bernoulli schedule vacation interruption and customer’s impatience

2021 ◽  
Vol 13 (2) ◽  
pp. 367-395
Author(s):  
Shakir Majid ◽  
Amina Angelika Bouchentouf ◽  
Abdelhak Guendouzi
Keyword(s):  
Function Technique ◽  
Service Rate ◽  
Single Server ◽  
Working Vacation ◽  
Vacation Interruption ◽  
The Impact ◽  
Number Of Customers ◽  
Vacation Period ◽  
System Characteristics ◽  
Bernoulli Schedule

Abstract In this investigation, we establish a steady-state solution of an infinite-space single-server Markovian queueing system with working vacation (WV), Bernoulli schedule vacation interruption, and impatient customers. Once the system becomes empty, the server leaves the system and takes a vacation with probability p or a working vacation with probability 1 − p, where 0 ≤ p ≤ 1. The working vacation period is interrupted if the system is non empty at a service completion epoch and the server resumes its regular service period with probability 1 − q or carries on with the working vacation with probability q. During vacation and working vacation periods, the customers may be impatient and leave the system. We use a probability generating function technique to obtain the expected number of customers and other system characteristics. Stochastic decomposition of the queueing model is given. Then, a cost function is constructed by considering different cost elements of the system states, in order to determine the optimal values of the service rate during regular busy period, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Further, by taking illustration, numerical experiment is performed to validate the analytical results and to examine the impact of different parameters on the system characteristics.

scholarly journals Semigroup Methods for the M/G/1 Queueing Model with Working Vacation and Vacation Interruption

2016 ◽  
Vol 8 (5) ◽  
pp. 56 ◽  
Author(s):  
Ehmet Kasim
Keyword(s):  
Semigroup Theory ◽  
Steady State Solution ◽  
Linear Operators ◽  
Time Dependent ◽  
Queueing Model ◽  
Working Vacation ◽  
Positive Time ◽  
Vacation Interruption ◽  
Vacation Period ◽  
Time Dependent Solution

By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with working vacation and vacation interruption has a unique positive time dependent solution which satisfies probability conditions. When the both service completion rate in a working vacation period and in a regular busy period are constant, by investigating the spectral properties of an operator corresponding to the model we obtain that the time-dependent solution of the model strongly converges to its steady-state solution.

scholarly journals Finite Buffer GI/M(n)/1 Queue with Bernoulli-Schedule Vacation Interruption under N-Policy

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
P. Vijaya Laxmi ◽  
V. Suchitra
Keyword(s):  
Performance Measures ◽  
Finite Buffer ◽  
Supplementary Variable ◽  
Working Vacation ◽  
Variable Technique ◽  
Special Cases ◽  
Vacation Interruption ◽  
System Length ◽  
Vacation Period ◽  
Bernoulli Schedule

We study a finite buffer N-policy GI/M(n)/1 queue with Bernoulli-schedule vacation interruption. The server works with a slower rate during vacation period. At a service completion epoch during working vacation, if there are at least N customers present in the queue, the server interrupts vacation and otherwise continues the vacation. Using the supplementary variable technique and recursive method, we obtain the steady state system length distributions at prearrival and arbitrary epochs. Some special cases of the model, various performance measures, and cost analysis are discussed. Finally, parameter effect on the performance measures of the model is presented through numerical computations.

scholarly journals Performance Characteristics of Discrete-Time Queue With Variant Working Vacations

2020 ◽  
Vol 11 (2) ◽  
pp. 1-21
Author(s):  
P. Vijaya Laxmi ◽  
Rajesh P.
Keyword(s):  
Discrete Time ◽  
Cost Model ◽  
Probability Generating Function ◽  
System Size ◽  
Service Rate ◽  
Single Server ◽  
Working Vacation ◽  
Optimal Service ◽  
Working Vacations ◽  
Vacation Period

This article analyzes an infinite buffer discrete-time single server queueing system with variant working vacations in which customers arrive according to a geometric process. As soon as the system becomes empty, the server takes working vacations. The server will take a maximum number K of working vacations until either he finds at least on customer in the queue or the server has exhaustively taken all the vacations. The service times during regular busy period, working vacation period and vacation times are assumed to be geometrically distributed. The probability generating function of the steady-state probabilities and the closed form expressions of the system size when the server is in different states have been derived. In addition, some other performance measures, their monotonicity with respect to K and a cost model are presented to determine the optimal service rate during working vacation.

ON MARKOVIAN QUEUES WITH SINGLE WORKING VACATION AND BERNOULLI INTERRUPTIONS

2021 ◽  
pp. 1-28
Author(s):  
Ruiling Tian ◽  
Zhe George Zhang ◽  
Siping Su
Keyword(s):  
Optimal Strategies ◽  
Service Rate ◽  
Single Server ◽  
Working Vacation ◽  
Markovian Queue ◽  
Rate Period ◽  
System Information ◽  
Vacation Period ◽  
Maintenance Process ◽  
Different Levels

This paper considers the customers’ equilibrium and socially optimal joining–balking behavior in a single-server Markovian queue with a single working vacation and Bernoulli interruptions. The model is motivated by practical service systems where the service rate can be adjusted according to whether or not the system is empty. Specifically, we focus on a single-server queue in which the server's service rate is reduced from a regular to a lower one when the system becomes empty. This lower rate period is called a working vacation for the server which may represent that part of the service facility is under a maintenance process or works on other non-queueing job, or simply for saving the energy (for a machine server case). In this paper, we assume that the working vacation period is terminated after a random period or with probability p after serving a customer in a non-empty system. Such a system is called a queue with single working vacation and Bernoulli interruptions. Customers are strategic and can make choice of joining or balking based on different levels of system information. We consider four scenarios: fully observable, almost observable, almost unobservable, and fully unobservable queue cases. Under a reward-cost structure, we analyze the customer's equilibrium and social-optimal strategies. In addition, the effects of system parameters on optimal strategies are illustrated by numerical examples.

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