We have derived expressions for the mean waiting time of a **vacation** queueing system in which the server does not immediately take another **vacation** upon returning from a **vacation** and finding the system empty, as in the multiple **vacation** scheme, or wait indefinitely for a customer to arrive, as in the single **vacation** scheme. In the proposed **model**, if the server returns from a **vacation** and finds the system empty, it waits for a fixed time c. If at the expiration of this time, no customer arrives, the server will take a **vacation**; otherwise it servers arriving cus- tomers exhaustively before taking another **vacation**. The results of the analysis are consistent with those of the multiple vacations scheme (where c = 0 ), and the single **vacation** scheme (where c = ∞ ). Numerical results are also given that show the impact of the timeout period on the mean delay. A linear cost **model** was assumed to obtain the optimal value of for the assumed **model**. Numerical results are obtained under a set of parameter assumptions, and the results indicate that there is a very small range of values of the offered where meaning op- timization can be applied.

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Recently, Servi and Finn [4] introduced the working **vacation** policy, in which the server works at a different rate rather than completely stopping service during the **vacation**. They studied an **M**/**M**/**1** queue with working vacations, and obtained the transform formulae for the distribution of the number of customers in the system and sojourn time in steady state, and applied these results to performance analysis of gateway router in fiber commu- nication networks. During the working **vacation** models, the server can not come back to the regular busy period until the **vacation** period ends. Subsequently, Wu and Takagi [5] generalized the **model** in [4] to an **M**/**G**/**1** queue with general working vacations. Baba [6] studied a GI/**M**/**1** queue with working vacations by using the ma- trix analytic method. Banik et al. [7] analyzed the GI/ **M**/**1**/N queue with working vacations. Liu et al. [8] established a stochastic decomposition result in the **M**/**M**/**1** queue with working vacations. Li et al. [9] estab-

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The following assumptions briefly describe the mathematical **model** of our study. The arrival follows Poisson distribution with mean arrival rate λ (> 0). There is a single server who provides the first essential service (FES) to all arriving customers. The service time for FES follows a general distribution, with distribution function B **1** (x) and density function b **1** (x).

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Later Wang and Li (2008) analyzed a more general repairable **M**/**G**/**1** retrial Queueing **model** with general retrial times, Bernoulli schedule **vacation** and second multi-optional service in which the server is allowed to take **vacation** after finishing a service to a customer. It is also assumed that, the server soon after the **vacation** period, must spend some time for setup operation before starting a new service. The steady state distributions of the orbit and system size are discussed under a steady state condition and some reliability characteristics such as availability and failure frequency of the server are also obtained.

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Queuing systems with vacations have been studied by various researchers for their profound applications in many real life situations such as telecommunication, computer networks, production systems, and so on. The survey papers of Doshi[**1**] and Teghem[2] the monograph of Takagi [3] and Tian and Zhang [4] acts as a reference for the readers. In these studies, it is assumed that the server completely ceases service during a **vacation**. However, there are many situations where the server does not remain completely inactive during a **vacation**. But provides service to the queue at a lower rate. This idea was first utilized by Servi and Finn [5] and introduced a class of semi **vacation** polices. This type of **vacation** is called a working **vacation** (WV). Servi and Finn [5] analyzed **M**/**M**/**1** queue with multiple working vacations policy and derived the PGF for the number of customers in the system and LST for waiting time distributions and utilized results to analyze the system performance of gateway router in fiber communication networks. Later **M**/**M**/**1** with multiple working **vacation** **model** was also studied by Liu, Xu and Tian[6] to obtain explicit expressions of the performance measures and their stochastic decomposition by using the matrix-geometric method. Subsequently, by applying the same method, **M**/**M**/**1** queue with single working **vacation** was analyzed by Tian and Zhao [7] and obtained various steady state indicators. Moreover, Kim, Choi and Chae [8], Wu and Takagi [9] and Li et al. [10] extended the work of Finn [5] to an **M**/**G**/**1**/WV queue . Baba [11] first analyzed the GI/**M**/**1** queue with general arrival process and multiple working vacations by utilizing the matrix-geometric solution method. Later, Li and Tian [12] investigated the GI/**M**/**1** queue with single working **vacation**. Banik et al. [13] analyzed a GI/**M**/**1**/N queuing system with limited waiting space and working **vacation**.

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Abstract. A batch arrival retrial queue with general retrial times, state dependent admission, modified **vacation** policy and feed back is analysed in this paper. Arrivals are controlled according to the state of the server. If the orbit is empty, the server takes at most J vacations until at least one customer is received in the orbit when the server returns from the **vacation**. At the service completion epoch, the test customer may either enter the orbit for another service with probability p or leave the system with probability **1**-p. By supplementary variable technique some analytical results for the system size distribution as well as some other performance measures of the system are derived. Keywords: Retrial queue, state dependent admission control, modified **vacation**, feedback.

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Abstract— This paper characterizes a non markovian queuing **model** in which arrival takes after a poisson procedure. All the arriving clients are rendered the administration following a general distribution. What's more, system may interrupt aimlessly because of different reasons. Immediately, the server gets into a repair procedure. Amid the season of breakdown, not all the arriving clients are permitted to join the framework. An idea of limited tolerability is connected over amid the service intrusion. After the consummation of the service, if there are no clients in the framework, the server experiences a mandatory **vacation**. At the season of get-away, support work are completed for the server which causes the system to keep running with superior along with insignificant intrusion. By the use of supplementary variable strategy, probability creating function of the queue size and the various execution measures are resolved. For the legitimization of the **model**, numerical delineation is completed .

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We consider an **M**/**G**/**1** queue with K-phase of **vacation** and with second optional service. The service policy is after completion of essential service, the customer chooses an optional service with probability p or leaves the system with probability (**1**-p). Both the essential service and optional service follows general distributions. In addition, after completion of essential service or second optional service, if there are no customers in the system, the server takes **vacation** consisting of K-phases. After completing the K th phase of **vacation**, the server enters into the service station independent of the number of customers in the system. The **vacation** periods follows general distribution. For this **model** the supplementary variable technique has been applied to obtain the probability generating functions of number of customers in the queue at different server states. Some particular models are obtained, and a numerical study is also carried out.

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• After completion of service, if the customer is not satisfied with the service for certain reason or if customer received unsuccessful service, the customer may immediately join the tail of the original queue as a feedback customer for receiving another regular service with probability p(0 < p < **1**). Otherwise the customer may depart forever from the system with probability q(= **1** − p). The service discipline for feedback and newly customers are first come first served. Also service time for a feedback customer is independent of its previous service times.

Queueing models with working vacations have several applications in many fields such as in Educational Institutions, Industrial sectors, agricaltural fields and Transportations etc. Queueing systems with server vacations by using pgf technique was first introduced by Levy and Yechiali. Servi and Finn introduced a class of semi-**vacation** policies, one which is working **vacation** (WV), they studied an **M**/**M**/**1** queue with working vacations. Working **vacation** queues typically refer to queues where the server utilizes its time on **vacation** for other purposes. Xu, X and Tian, N (2009) work on Performance analysis of an **M**/**M**/**1** working **vacation** queue with setup times. G.Kannadasan and N. Sathiyamoorth anlyze the **M**/**M**/**1** queue with working **vacation** in fuzzy environment. Joshuo Patterson and Andrzej korzeniowski works on **M**/**M**/**1** **model** with unreliable service and a working **vacation**. Yue et al (2011) discussed a two-phase **M**/**M**/**1** queueing system with impatient customers, multiple vacations, customer balking and reneging. Avi-Itzhak, B. and Naor, p. (1963) discussed some queueing problems with the service station subject to breakdowns in 1963. V.N.Rama Devi and Dr.K.Chandan studied the optimal of a two-phase **M**/**M**/**1** queueing system with server startup, N-Policy, unreliable server and Balking. Queueing systems with impatient customers have been studied by Ancker C.J., Gafarian A., ―Some queueing problems with balking and reneging: I‖ and Rakesh Kumar, Sumeet Kumar Sharma (2012), ―An **M**/**M**/**1**/N Queueing **Model** with Retention of Reneged Customers and Balking‖. The main objective of this paper is to analyze an **M**/**M**/**1** queueing **model** with working **vacation**, server failure and customer’s impatience. Also carried out the numerical results for various parameter effects on the system.

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Maragathasundari[27] inspected a mass queuing models with opened up experience and stages in repair. Kendall. D.G [14] concentrated the stochastic Processes occurring in the speculation of lines and their examination by the system for introduced Markov chains. Suthersan and Maragathasundari[28]investigated a queueing **model** of restricted admissibility. Murugeswari. N and Maragathasundari.S[29] studied the queuing **model** of compulsory **vacation** with stand by server service. In perspective of all the above research work, another queuing **model** has been enveloped for the above adaptable correspondence framework. In this paper we consider a Priority queuing framework with a solitary server serving two lines **M**/G1/**1** and **M**/G2/**1** with confined suitability and mandatory server get-away in view of depleted administration of the need units. The administration time takes after a general appropriation. Server takes an obligatory excursion of irregular length when the administration towards the last client in the priority unit gets over. In the event that the server occupied or in the midst of some recreation an arriving non priority client join the line with the probability υ .In addition to that, not all the arriving clients are permitted to join the line. An imperative idea of confined admissibility is considered over the landing of clients. For the above arranged **model**, we decide the time dependent generating function for both priority unit and non require units to the extent Laplace changes.

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Relating with the server of a queuing system, the server may be assumed as a reliable one, but this is not the case in most of real scenarios that the server will not be last forever in order to provide service. So in this context, numerous papers of the server may be assumed unreliable, which can encounter breakdown. Thus queuing **model** with server break down is a remarkable and unavoidable phenomenon and the study of queues with server breakdowns and repairs has importance not only in the point of theoretic view but also in the engineering applications. Avi-Itzhak(1963) considered some queuing problems with the servers subject to breakdown. Kulkarni et al.(1990) studied retrial queues with server subject to breakdowns and repairs. Tang(1997) studied **M**/**G**/**1** **model** with server break down and discussed reliability of the system. Madan et al.(2003) obtained the steady state results of single server Markovian **model** with batch service subject to queue models with random breakdowns. Queuing systems with random break downs and **vacation** have also been keenly analyzed by many authors including Grey (2000) studied **vacation** queuing **model** with service breakdowns. Madan and Maraghi (2009) have obtained steady state solution of batch arrival queuing system with random breakdowns and Bernoulli schedule server vacations having general **vacation** time. Thangaraj(2010) studied the transient behaviour of single server with compulsory **vacation** and random break downs.

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This paper pays attention on a single server non-Monrovian retrial queuing **model** with non persistent customers, two stage heterogeneous service, Bernoulli **vacation**. Customers arrive according to Poisson stream with arrival rate λ and are served one by one with first come first served basis. In this **model** the server provides two phases of service in which after the first phase service is complete, second phase service starts and both the service time follow general (arbitrary) distribution. The customer, who finds the server busy upon arrival, can either join the orbit with probability p or he/she can leave the system with probability **1**-p. On completion of a service the server may go for a **vacation** with probability θ or stay back in the system to serve a next customer with probability **1**-θ, if any. Also when the **vacation** period is over the server is assumed as it must spend some time to get ready for giving proper service, called set up time and which is arbitrarily distributed. We obtain the steady solutions of the **model** by using supplementary variable technique. Also we derive the system performance measures and reliability indices are obtained.

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In this investigation, we have derived the approximate formulae for the expected system size and expected waiting time for an **M** x /**G**/**1** queueing **model** with Bernoulli schedule vacations and repairable server. The organization of the paper is as follows. The **model** under consideration is described along with assumptions and limiting probabilities and governing equations in section 2. Various queueing characteristics along with queue length have been derived in section 3. Section 4 is devoted to Maximum Entropy Method (MEM). The approximate results for the expected queue size and expected waiting time are established in sections 5 and 6, respectively. A comparative analysis between the approximate and exact results is also performed. Numerical illustrations have been provided in next section 7. In the last section 8, the conclusions are made.

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Many papers have been published about queueing systems with server vacations. But most works on **vacation** models have been limited to the analysis of steady-states. There are few treatments of transient behavior, see Welch [], Minh [], Takagi [], Gupur [, ] for instance. In , Takagi [] ﬁrst established the mathematical **model** of the **M**/**G**/ queueing system with exceptional service time for the ﬁrst customer in each busy period by using the supplementary variable technique, then studied the time-dependent solution of the **model** by using probability generating functions and got the Laplace transform of the probability generating function. Roughly speaking, he obtained the existence of a time- dependent solution of the **model**. In , by using C -semigroup theory in functional

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J.C and Pearn, W. L [8] discussed the optimal management policy of heterogenous arrival **M**/**M**/**1** queue with server’s break down and vacations. Later Ke, J.C[6] studied the operating system characteristics of **M** X /**G**/**1** queueing system under **vacation** policies with start up and break down times in which the server may breakdown according to a Poisson process while working and his repair time has general distribution. With the help of available literature in this paper we are analysing **M**/**M**(a,b)/**1**/MWV queueing **model**, for an unreliable server.

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This paper clearly analyses a two phase queueing system with optional service and optional **vacation**. An application of this **model** can be found in mobile network where the messages are in batch form, the service may have two phases such that first phase is essential but the second phase may be chosen randomly as optional. The probability generating function of the number of customers in the queue is found using the supplementary variable technique. Also by some numerical approaches the validity of the results are examined.

Queuing systems with vacations have been studied extensively by various researchers for their profound appli- cations in many real life situations such as telecommunication, computer networks, production systems, and so on. The details have been reported in the survey of Doshi (1986), the monographs of Takagi (1991) and Tian and Zhang (2006). In these studies, it is assumed that the customers service is completely stopped during a **vacation**. However, there are numerous examples, where the server serves the customers at a lower rate during a **vacation**, rather than being completely inactive. This type of semi **vacation** policy was first introduced by Servi and Finn (2002). This type of **vacation** policy is called a working **vacation** (WV). Servi and Finn (2002) analyzed **M**/**M**/**1** queue with multiple working **vacation** policy and derived the PGF of the queue length and LST of the waiting time, and utilized their results to analyze the system performance of gateway router in fiber communication networks. Later **M**/**M**/**1** with multiple working **vacation** **model** was also studied by Liu, Xu and Tian (2007) to obtain explicit expressions of the performance measures and their stochastic decomposition by using the matrix-geometric method. Subsequently, by applying the same method, **M**/**M**/**1** queue with single working **vacation** was analyzed by Tian and Zhao (2008) and obtained various steady state indicators. Kim, Choi and Chae (2003), Wu and Takagi (2006) and Li et al. (2011) ex- tended the study in Servi and Finn (2002) to an **M**/**G**/**1**/WV queue. Baba (2005) applied the matrix-geometric solution method to generalize the work of Servi and Finn (2002) to a GI/**M**/**1** queue with general arrival process and multiple working vacations. Later, Li and Tian (2011) investigated the GI/**M**/**1** queue with single working **vacation**.

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By using generating function method, the queue size distribution has been obtained in section IV. Some queueing indices to predict the behaviour of the system are derived in section V. In section VI, the results for some well- established models as special cases of our **model**, are deduced. To validate the analytical results and to facilitate the sensitivity analysis, we present some numerical results for system performance indices in section VII. Finally, we have concluded our work in section VIII.

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Many authors have taken considerable efforts to study about Bernoulli feedback queueing systems Takacs was the first to study such a **model**, where the customers who completed their services feedback instantaneously to the tail of the queue with probability p or leaves the system forever with probability q = **1** – p. Many results with Bernoulli feedback are found [**1** – 6].Several contributions have been made by dealing queueing systems of **M**/**G**/**1** type.[7,9-15].Much research [8, 16 – 20]has studied queueing models under vacations. For complete reference on **vacation** models, one may refer to Doshi [17]and Tagagi [3].**M**/**G**/**1** **vacation** models under the various service disciplines have been investigated[8,18,21-24].

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