Variant impatient behavior of a Markovian queue with balking reserved idle time and working vacation

2020 ◽  
Vol 54 (3) ◽  
pp. 783-793
Author(s):  
Arumugam Azhagappan ◽  
Thirunavukkarasu Deepa
Keyword(s):  
Research Work ◽  
Fixed Time ◽  
System Size ◽  
Idle Time ◽  
Service Rate ◽  
Working Vacation ◽  
Trading Company ◽  
Transient System ◽  
The Impact ◽  
Vacation Period

The customers’ impatience and its effect plays a major role in the economy of a country. It directly affects the sales of products and profit of a trading company. So, it is very important to study various impatient behaviors of customers and to analyze different strategies to hold such impatient customers. This situation is modeled mathematically in this research work along with working vacation and reserved idle time of server, balking and re-service of customers. This paper studies the transient analysis of an M/M/1 queueing model with variant impatient behavior, balking, re-service, reserved idle time and working vacation. Whenever the system becomes empty, the server resumes working vacation. When he is coming back from the working vacation and finding the empty system, he stays idle for a fixed time period known as reserved idle time and waits for an arrival. If an arrival occurs before the completion of reserved idle time, the server starts a busy period. Otherwise, he resumes another working vacation after the completion of reserved idle time. During working vacation, the arriving customers may either join or balk the queue. The customers waiting in the queue for service, during working vacation period, become impatient. But, the customer who is receiving the service in the slow service rate, does not become impatient. After each service, the customer may demand for immediate re-service. The transient system size probabilities for the proposed model are derived using generating function and continued fraction. The time-dependent mean and variance of system size are also obtained. Finally, numerical illustrations are provided to visualize the impact of various system parameters.

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.

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 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 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.

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.

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 of Impatient Customers in Queues with Bernoulli Schedule Working Vacations and Vacation Interruption

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Veena Goswami
Keyword(s):  
Performance Measures ◽  
System Size ◽  
Stochastic Decomposition ◽  
Model Parameters ◽  
The Mean ◽  
Working Vacations ◽  
Vacation Interruption ◽  
The Impact ◽  
Vacation Period ◽  
Bernoulli Schedule

This paper analyzes customers’ impatience in Markovian queueing system with multiple working vacations and Bernoulli schedule vacation interruption, where customers’ impatience is due to the servers’ vacation. During the working vacation period, if there are customers in the queue, the vacation can be interrupted at a service completion instant and the server begins a regular busy period with probability 1-q or continues the vacation with probability q. We obtain the probability generating functions of the stationary state probabilities and deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures such as the mean system size, the proportion of customers served, the rate of abandonment due to impatience, and the mean sojourn time of a customer served are derived. We obtain the stochastic decomposition structures of the queue length and waiting time. Finally, some numerical results to show the impact of model parameters on performance measures of the system are presented.

scholarly journals The Evaluation of M/M/2 Queuing Framework with Two Heterogeneous Servers for Balking and Reneged Customers

2019 ◽  
Vol 9 (1) ◽  
pp. 5402-5408
Keyword(s):  
Quantitative Method ◽  
The Other ◽  
Scientific Models ◽  
Model Parameters ◽  
Heterogeneous Servers ◽  
Working Vacation ◽  
Clump Size ◽  
The Impact ◽  
Vacation Period ◽  
Low Rate

Queuing hypothesis is a quantitative method which comprises in building scientific models of different sorts of lining frameworks. Occupied time of the framework is broke down and mean holding up time in the stationary system processed. At long last, some numerical outcomes are introduced to demonstrate the impact of model parameters on the framework execution measures. The traveling server, nonetheless, comes back to landing which is used to offer at a low rate whereas the other server is occupied. At whatever point the framework ends up and the subsequent server leaves for a working excursion while the principal server stays inert in the framework. These models can be utilized for making expectations about how the framework can change with requests. The framework is examined in the enduring state utilizing lattice geometric strategy. The clients enter the line in the Poisson manner and the time of each bunch size is dared to be circulated exponentially as for mean ward clump size and clients may balk away or renege when the holding up the line of the clients, in general, be exceptionally enormous. This work exhibits the investigation of a recharging input different working excursions line with balking, reneging and heterogeneous servers. Queuing hypothesis manages the investigation of lines and lining conduct. Different execution proportions of the model, for example, anticipated framework length, anticipated balking rate and reneging rate have been talked about. The technique breaks down an M/M/2 lining framework with two heterogeneous servers, one of which is constantly accessible however the different travels without clients sitting tight for service. During a working vacation period, the subsequent server gives administration at a slower rate as opposed to totally ceasing service. The relentless state probabilities of the model are advantageous and recursive strategies.

Social optimization in M/M/1 queue with working vacation and N-policy

2018 ◽  
Vol 52 (2) ◽  
pp. 439-452 ◽  
Author(s):  
Qing-Qing Ma ◽  
Ji-Hong Li ◽  
Wei-Qi Liu
Keyword(s):  
Social Welfare ◽  
Queue Length ◽  
Service Rate ◽  
Working Vacation ◽  
Stationary Probability Distribution ◽  
The Social ◽  
The Mean ◽  
Working Vacations ◽  
Mean Queue Length ◽  
The Impact

This paper deals with the N-policy M/M/1 queueing system with working vacations. Once the system becomes empty, the server begins a working vacation and works at a lower service rate. The server resumes regular service when there are N or more customers in the system. By solving the balance equations, the stationary probability distribution and the mean queue length under observable and unobservable cases are obtained. Based on the reward-cost structure and the theory of Markov process, the social welfare function is constructed. Finally, the impact of several parameters and information levels on the mean queue length and social welfare is illustrated via numerical examples, comparison work shows that queues with working vacations(WV) and N-policy have advantage in controlling the queue length and improving the social welfare.

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