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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to use without necessarily imposing stronger conditions on the controlled class. This discipline is a viable al- ternative for reducing waiting time expectations in such systems. For more on the beneﬁts of threshold control policies see Efrosinin and Rykov [6], [5] and Efrosinin and Breuer [4]. In a recent work 6 on the **M**/**G**/ 2 **queue** with a violation of the FCFS **queue** discipline and a ﬂex- ible customer dependent control policy, we have shown that one can obtain the equivalent of the FCFS waiting time expectations when the violation is minimal owing to some degree of control on the service and queuing disci- plines 7 . This work together with those of Krishnamoor- thy [2] and Efrosinin and Sztrik [7] motivate us to further still investigate the **M**/**G**/ 2 queuing system in the light of a complete control policy of the customer routines in the system to study the impact of the policy on the waiting time expectations for the beneﬁt of service systems. 8 As stated in section **1**, the **model** presented in this work is called the **M**/**M**, **G**/ 2 **queue** with one exponential and one general server working under a controlled service process. We initiate a control service policy under which the gen- eral server can be put to use to study the performance of the **M**/**M**, **G**/ 2 queuing **model** for use in owner-controlled service systems. Our contributions could be summarized as follows:

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This paper is organized as follows. The mathematical description of our **model** is given in section 2. Definitions and notations are given in section 3. Equations governing the **model** are given in section 4. In section 5 supplementary variables technique has been used to obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the **queue**. The mean **queue** size and mean waiting time are found in section 6. Some particular cases are discussed in section 7. Numerical results and conclusion are given in section 8 and 9 respectively.

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characterization and computation of optimal policies for operating an **M**/**G**/**1** queuing system with removable server was also given by Bell [4]. Courtois and Georges [5] discussed a single- server finite queuing **model** with state-dependent arrival and service processes. Herzog et al. [6] gave the solution of **M**/**G**/**1** queuing problem by a recursive technique. Gupta and Srinivasa Rao [7] discussed a recursive method for **M**/**G**/**1**/K **model** to compute the steady-state probabilities. Zhang and Love [8] suggested the threshold policy for the **M**/**G**/**1** **queue** with an exceptional first vacation. For optimal control of a removable and non- reliable server, Wang et al. [9] developed **M**/H 2 /**1**

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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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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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This paper is organized as follows. The mathematical descrip- tion of our **model** is given in section 2. Definitions and Equations governing the system are given in section 3. The time depen- dent solution have been obtained in section 4 and corresponding steady state results have been derived explicitly in section 5. Av- erage **queue** size and average waiting time are computed in sec- tion 6 and 7 respectively. Particular case is discussed in section 8. Conclusion are given in section 9.

This paper focuses on a facility location problem with stochastic customer demand and congestion where cus- tomers travel to facilities to receive services. We explain a **model** that located customers and facilities on network and provide services at each facility operating as an **M**/**G**/**1** queuing system. The objective of our **model** is to search an optimal facility location maximizing the pro- bability of waiting less than or equal to a certain time for a customer that is chosen randomly from all customers who arrive to the system. In order to search an optimal location, we propose three heuristic algorithms: GD, GD- T, and RAND-T. We conduct numerical experiments by using these algorithms and the COMB and discuss the properties of these algorithms.

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(2.4) • The system may breakdown at random and the breakdowns are assumed to occur according to a Poisson stream with mean breakdown rate α > 0. Further we assume that once the system breakdown, the customer whose service is interrupted comes back to the head of **queue**.

A **queue** is a line of people or things waiting in an order to get the service [**1**]. Queuing theory was introduced by A.K.Erlang in 1909.He published various articles in telephone traffic [2].Queueing Theory is mainly a branch of applied probability theory. It has many applications in different fields, namely, communication networks, computer systems, machine plants etc. Consider a service providing centre and a population of customers, which at some time intervals enter the service centre in order to get the service. Mostly, the service provider can only serve a limited number of customers. If a new customer arrives and the service is exhausted, he needs to stand in waiting line or **queue** until the service provider becomes free. So main three elements of a service providing centre are: a population of customers, the service facility and the waiting line. Also it can be considered that the several service providers are arranged in a network and a single customer can walk through this network at a specific path, visiting several service centres. With the help of Queueing Theory, many questions can be answered e.g. the mean waiting time in the **queue**, the mean system response time (waiting time in the **queue** plus service times), distribution of the number of customers in the **queue**, mean utilization of the service facility, distribution of the number of customers in the system etc.

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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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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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Kim and Lee (2014)[16] analysed an **M**/**G**/**1** queueing system with disasters and working breakdowns. In this **model** it is assumed that the breakdown server is sent to repair facility and is replaced by a slow server till the server is fixed. The author analysed a batch arrival queueing system MX/**G**/**1** with disasters and working breakdowns and derived the steady state system size distributions and some important performance measures for the **model**. It is verified that when the mean batch size E(X)=**1**, the results obtained is exactlycoincide with the results of Kim and Lee (2014).

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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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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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service when someone else arrives. Using the terminology of the previous section, he is the one, and only one, who has drawn a priority parameter **1**. Likewise, from the point of view of the customers who possesses a priority parameter less than or equal to p , the customer with parameter p is a stand-by customer. Note that a stand-by customer does not inﬂict any extra waiting on any of the other customers. An economist would say here that his decision whether or not to join the **queue** does not come with any externalities. Our interest in this section is in the LST of the time in the system under the PRP regime. Below we deal ﬁrst with the **M**/**M**/**1** case and then with the more general **M**/**G**/**1** cases. Admittedly, one can derive the former case from the latter but with a minimal cost in space we do the special case ﬁrst while utilizing its special features and then we switch to the more general case.

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In this paper we analyzed the vacation **model** of an **M** (X) / **G** / **1** **queue** with balking and variable vacations. This type of modeling and analysis can be applied in many situations. Queuing **model** can be used to the employment services system and its design and to optimize the actual system according to the specific requirements of the system. Queuing system is suitable for analyzing and studying random phenomenon such as the employment service system services.

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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system and some measures of effectiveness are obtained using iterative method, probability generating function. Some special cases are deduced. Finally, a simulation study has been considered to illustrate the numerical application for the **model**.

ABSTRACT: Batch arrival retrial **queue** with positive and negative customers is considered. If the server is idle upon the arrival of a batch, one of the customers in the batch receives service immediately and others join the orbit. If the server is busy, all the customers join the orbit. The arrival of negative customer brings the server down and removes the customer in service from the system. The server is subject to random breakdown while it is working. Using supplementary variable technique, expected number of customers in the orbit and expected number of customers in the system are derived. Stochastic decomposition property is established. Some special cases are discussed and numerical results are presented.

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