Impatient customers in Markovian queue with Bernoulli feedback and waiting server under variant working vacation policy
01 Jan 2020-Operations Research and Decisions (Politechnika Wroclawska Oficyna Wydawnicza)-Vol. 30, Iss: 4
TL;DR: This paper deals with customers’ impatience behaviour for single server Markovian queueing system under K-variant working vacation policy, waiting server, Bernoulli feedback, balking, reneging, and retention of reneged customers and proves the stochastic decomposition properties.
Abstract: This paper deals with customers’ impatience behaviour for single server Markovian queueing system under K-variant working vacation policy, waiting server, Bernoulli feedback, balking, reneging, and retention of reneged customers. Using the probability generating function (PGF) technique, we obtain the steady-state solution of the system. Besides, we prove the stochastic decomposition properties. Useful performance measures of the considered queueing system are derived. A cost model is developed. Then, the parameter optimisation is carried out numerically, using a quadratic fit search method (QFSM). Finally, numerical examples are provided to visualise the analytical results.
Citations
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Journal Article•
TL;DR: A cost model is formulated to determine the optimal service rate during working vacation and regular busy period and some other performance measures are obtained and their monotonicity is discussed.
Abstract: T his paper analyzes a batch arrival infinite-buffer single server queueing system with variant working vacations wherein customers arrive according to a Poisson process As soon as the system becomes empty, the server takes working vacation The service times during regular busy period, working vacation period and vacation times are assumed to be exponentially distributed and are mutually independent During working vacations the customer may renege and the reneging time follows exponential distribution We derive the probability generating function of the steady-state probabilities and obtain the closed form expressions of the system size when the server is in different states In addition, we obtain some other performance measures and discuss their monotonicity A cost model is formulated to determine the optimal service rate during working vacation and regular busy period
2 citations
TL;DR: In this paper , the authors considered a preemptive priority queueing system with vacation, where the single server may break down with imperfect coverage and derived the stationary probability distribution by using the probability generating function approach.
Abstract: This work considers a preemptive priority queueing system with vacation, where the single server may break down with imperfect coverage. Various combinations of server vacation priority queueing models have been studied by many scholars. A common assumption in these models is that the server will only resume its normal service rate after the vacation is over. However, such speculation is more limited in real-world situations. Hence, in this study, the vacation will be interrupted if a customer waits for service in the system at the moment of completion of service during vacation. The stationary probability distribution is derived by using the probability generating function approach. We also develop varieties of performance measures and provide a simple numerical example to illustrate these measures. Optimization analysis is finally carried out, including cost optimization and tri-object optimization.
TL;DR: In this article , the authors model a single server finite buffer Markovian queuing model with discouraged arrival, balking, reneging and retention of reneged customers and obtain the steady state probabilities using Markov process method.
Abstract: Queuing models where arrival rates go down consequent to increase in the
number of customers are called systems with discouraged arrivals.
Discouraged arrivals are distinct from balking in the sense that balking
implies that arriving customers do not join. In this paper, we model a
single server finite buffer Markovian queuing model with discouraged
arrival, balking, reneging and retention of reneged customers. The steady
state probabilities are obtained using Markov process method. Closed form
expression of traditional as well as some freshly designed performance
measures are presented. We also perform sensitivity analysis to examine the
variations in performance measures with the variations in system parameters.
Our results are numerically illustrated through a field level problem with
design connotations.
References
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TL;DR: The classical single server vacation model is generalized to consider a server which works at a different rate rather than completely stops during the vacation period, which approximates a multi-queue system whose service rate is one of the two speeds for which the fast speed mode cyclically moves from queue to queue with an exhaustive schedule.
388 citations
TL;DR: In this paper, the distribution of the queue size as well as the Laplace-Stieltjes transform and the first two moments of the distribution function of the total time spent in the system by a customer are determined for a stationary process.
Abstract: Let us suppose that customers arrive at a counter in accordance with a Poisson process of density λ. The customers are served by a single server in order of arrival. The service times are identically distributed, mutually independent, positive random variables with distribution function H(x). Suppose that after being served each customer either immediately joins the queue again with probability p or departs permanently with probability q (p + q =1). In this paper we shell determine for a stationary process the distribution of the queue size as well as the Laplace-Stieltjes transform and the first two moments of the distribution function of the total time spent in the system by a customer.
247 citations
"Impatient customers in Markovian qu..." refers background in this paper
...A few interesting papers include Takacs [28], Krishnakumar et al....
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...A few interesting papers include Takacs [28], Krishnakumar et al. [19], Choudhury and Paul [9], Kalidass and Kasturi [16], Varalakshmi et al. [34], Varalakshmi et al. [33], Bouchentouf and Messabihi [7], Bouchentouf et al. [6] and the references therein....
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TL;DR: The distributions and the stochastic decomposition structures for the number of customers and the waiting time and some indices of systems are obtained and the quasi birth and death process and matrix-geometric solution method is obtained.
Abstract: In this paper, we study the M/M/1 queue with working vacations and vacation interruptions. The working vacation is introduced recently, during which the server can still provide service on the original ongoing work at a lower rate. Meanwhile, we introduce a new policy: the server can come back from the vacation to the normal working level once some indices of the system, such as the number of customers, achieve a certain value in the vacation period. The server may come back from the vacation without completing the vacation. Such policy is called vacation interruption. We connect the above mentioned two policies and assume that if there are customers in the system after a service completion during the vacation period, the server will come back to the normal working level. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions and the stochastic decomposition structures for the number of customers and the waiting time and provide some indices of systems.
96 citations
"Impatient customers in Markovian qu..." refers background in this paper
...In multiple working vacations, the server resumes several working vacation each time the system leads to an empty state [14, 20, 23]....
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TL;DR: It is shown that the general stochastic decomposition law for M/G/1 vacation models holds for the present system also and some special cases are also studied.
Abstract: This paper is concerned with the analysis of a single-server queue with Bernoulli vacation schedules and general retrial times. We assume that the customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed access to the server. We first present the necessary and sufficient condition for the system to be stable and derive analytical results for the queue length distribution, as well as some performance measures of the system under steady-state condition. We show that the general stochastic decomposition law for M/G/1 vacation models holds for the present system also. Some special cases are also studied.
91 citations
"Impatient customers in Markovian qu..." refers background in this paper
...[19], Choudhury and Paul [9], Kalidass and Kasturi [16], Varalakshmi et al....
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Journal Article•
TL;DR: In this article, an M/M/1 queue with single working vacation was studied and the distributions for the number of customers and the virtual time in the system in steady state were derived.
Abstract: In this paper, we study an M/M/1 queue with single working vacation. Using quasi birth and death process and matrix-geometric solution method, we give the distributions for the number of customers and the virtual time in the system in steady state. Furthermore, we obtain expected busy period and expected busy cycle. Finally, we get the stochastic decomposition structures of stationary indices.
64 citations
"Impatient customers in Markovian qu..." refers background in this paper
...are some customers in the system, the server immediately starts the service, otherwise, he remains idle in the system, waiting for a new arrival [27, 26, 22]....
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