A batch arrival unreliable Bernoulli vacation model with two phases of service and general retrial times
13 May 2015-International Journal of Mathematics in Operational Research (Inderscience Publishers (IEL))-Vol. 7, Iss: 3, pp 318-347
TL;DR: The system size distribution at a departure epoch and the probability generating function of the joint distributions of the server state and orbit size are derived and proved, and the decomposition property is proved.
Abstract: This paper deals with the steady state behaviour of an Mx/G/1 retrial queue with two successive phases of service and general retrial times under Bernoulli vacation schedule for an unreliable server. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. The primary customers finding the server busy, down, or on vacation are queued in the orbit in accordance with first come, first served (FCFS) retrial policy. After the completion of the second phase of service, the server either goes for a vacation of random length with probability p or may serve the next unit, if any, with probability (1 - p). For this model, we first obtain the condition under which the system is stable. Then, we derive the system size distribution at a departure epoch and the probability generating function of the joint distributions of the server state and orbit size, and prove the decomposition property.
Citations
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TL;DR: In this investigation, a single server M/M/1/N feedback queueing system with vacation, balking, reneging and retention of reneged customers is analyzed and the steady state probabilities of the number of customers in the system are derived.
Abstract: In this investigation, a single server M/M/1/N feedback queueing system with vacation, balking, reneging and retention of reneged customers is analyzed. By considering the mathematical modeling, we derive the steady state probabilities of the number of customers in the system. We obtain important measures of effectiveness of the model by using the stationary distribution, and develop a cost model of the queueing system. Further, a numerical study and a cost profit analysis are carried out.
23 citations
01 Jan 2017
TL;DR: Batch arrival and bulk service queueing systems are quite common in many real life situations such as the arrival of aircraft passengers, elevators, Manufacturing systems, communication network, giant wheel, tourism, etc.
Abstract: Batch arrival and bulk service queueing systems are quite common in many real life situations such as the arrival of aircraft passengers, elevators, Manufacturing systems, communication network, giant wheel, tourism, etc. Bulk service queueing model was initially originated with Bailey [1]. Neuts [2] proposed the “General Bulk Service Rule” in which service initiates only when a certain number of customers in the queue is available. Some general bulk service results have discussed by Holman, Chaudhry, and Ghosal [3]. Chaudhry and Templeton [4] have worked on bulk service rule by using the supplementary variable technique. Banerjee et.al [5] has considered a queueing model with variable service capacity and batch-size-dependent service. Sikdar and Samanta [6] have analyzed a finite buffer queueing model with bulk service variable server capacity. Recently, bulk service queueing model with multiple working vacations have studied by Jeyakumar and Senthilnathan [7].
2 citations
References
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TL;DR: This survey gives an overview of some general decomposition results and the methodology used to obtain these results for two vacation models and attempts to provide a methodological overview to illustrate how the seemingly diverse mix of problems is closely related in structure and can be understood in a common framework.
Abstract: Queueing systems in which the server works on primary and secondary (vacation) customers arise in many computer, communication, production and other stochastic systems. These systems can frequently be modeled as queueing systems with vacations. In this survey, we give an overview of some general decomposition results and the methodology used to obtain these results for two vacation models. We also show how other related models can be solved in terms of the results for these basic models. We attempt to provide a methodological overview with the objective of illustrating how the seemingly diverse mix of problems is closely related in structure and can be understood in a common framework.
1,136 citations
"A batch arrival unreliable Bernoull..." refers background in this paper
...The literature on vacation models recognises this property as one of the most interesting features in this mater, e.g., see Doshi (1986), and Fuhrmann and Cooper (1985)....
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18 Aug 1985847 citations
"A batch arrival unreliable Bernoull..." refers background in this paper
...…( ( )) ( ( )),B z q pγ b z λ z λ z= + β β i.e., the PGF of a batch of customers who arrived during our generalised service time, in the well-known result of Gross and Harris (1985) which is of the form (see Section 5.1.9, p.237): [ ] [ ] 0 0 ( ) ( ) Ψ ( ) ; ( ) C B z za z z B z z − = − (38) where…...
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..., the PGF of a batch of customers who arrived during our generalised service time, in the well-known result of Gross and Harris (1985) which is of the form (see Section 5....
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TL;DR: In this article, the authors considered a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time and showed that the stationary number of customers present in the system at a random point in time is distributed as the sum of two or more independent random variables.
Abstract: This paper considers a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time. As has been noted by other researchers, for several specific models of this type, the stationary number of customers present in the system at a random point in time is distributed as the sum of two or more independent random variables, one of which is the stationary number of customers present in the standard M/G/1 queue i.e., the server is always available at a random point in time. In this paper we demonstrate that this type of decomposition holds, in fact, for a very general class of M/G/1 queueing models. The arguments employed are both direct and intuitive. In the course of this work, moreover, we obtain two new results that can lead to remarkable simplifications when solving complex M/G/1 queueing models.
664 citations
01 Jul 1955
TL;DR: In this paper, it is shown that certain stochastic processes with discrete states in continuous time can be converted into Markov processes by the well-known method of including supplementary variables, and conditions under which it is possible to obtain a solution for arbitrary distributions are examined.
Abstract: Certain stochastic processes with discrete states in continuous time can be converted into Markov processes by the well-known method of including supplementary variables. It is shown that the resulting integro-differential equations simplify considerably when some distributions associated with the process have rational Laplace transforms. The results justify the formal use of complex transition probabilities. Conditions under which it is likely to be possible to obtain a solution for arbitrary distributions are examined, and the results are related briefly to other methods of investigating these processes.
569 citations
"A batch arrival unreliable Bernoull..." refers background in this paper
...3.1 The steady state equations The Kolmogorov forward equations to govern the system under steady state conditions (e.g., see Cox, 1955) can be written as follows: [ ]( ) ( ) ( ) 0; 1n n d ψ x λ η x ψ x n dx + + = ≥ (1) [ ] ( ),0 1 ( ) ( ) ( ) 1 ( ); 0 n n n n k n k k d Q y λ γ y Q y λ δ a Q y n dy…...
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..., see Doshi (1986), and Fuhrmann and Cooper (1985)....
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TL;DR: A survey of the main results and methods of the theory of retrial queues, concentrating on Markovian single and multi-channel systems, as well as models with batch arrivals, multiclasses, customer impatience, double connection, control devices, two-way communication and buffer.
Abstract: We present a survey of the main results and methods of the theory of retrial queues, concentrating on Markovian single and multi-channel systems. For the single channel case we consider the main model as well as models with batch arrivals, multiclasses, customer impatience, double connection, control devices, two-way communication and buffer. The stochastic processes arising from these models are considered in the stationary as well as the nonstationary regime. For multi-channel queues we survey numerical investigations of stationary distributions, limit theorems for high and low retrial intensities and heavy and light traffic behaviour.
485 citations
"A batch arrival unreliable Bernoull..." refers methods in this paper
...For a review of main results and methods, the reader is referred to the survey papers by Yang and Templeton (1987), Falin (1990) and the book by Falin and Templeton (1997)....
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