Analysis of an Unreliable Retrial G-Queue with Working Vacations and Vacation Interruption Under Bernoulli Schedule

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ENGINEERING PHYSICS AND MATHEMATICS Analysis of an unreliable retrial G-queue with working vacations and vacation interruption under Bernoulli schedule

P. Rajadurai *, V.M. Chandrasekaran, M.C. Saravanarajan

Department of Mathematics, School of Advanced Sciences, VIT University, Vellore 632014, Tamilnadu, India

Received 10 August 2015; revised 15 February 2016; accepted 2 March 2016

KEYWORDS Abstract In this paper, we consider a single server retrial queueing system with working vacations. Retrial queue; Further vacation interruption is considering with the regular busy server is subjected to breakdown G-queue; due to the arrival of negative customers. When the orbit becomes empty at the time of service com- Working vacations; pletion for a positive customer, the server goes for a working vacation. The server works at a lower Vacation interruption; service rate during working vacation (WV) period. If there are customers in the system at the end of Supplementary variable each vacation, the server becomes idle and ready for serving new arrivals with probability p (single technique WV) or it remains on vacation with probability q (multiple WVs). By using the supplementary variable technique, we found out the steady state probability generating function for the system and its orbit. System performance measures, reliability measures and stochastic decomposition law are discussed. Finally, some numerical examples and cost optimization analysis are presented. Ó 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction queueing system, queues with repeated attempts are character- ized by an arriving customer who finds the server busy, leaves In queueing theory, vacation queues and retrial queues are the service area and repeats its demand after some time. active research topics for a long time. We can find general Between trials, the blocked customer joins a pool of unsatisfied models in vacation queues and retrial queues from Ke et al. customers called orbit. An arbitrary customer in the orbit who [1] and Artalejo and Gomez-Corral [2] respectively. In retrial repeats the request for service is independent of the rest of the customers in the orbit. Such a retrial queue plays a special role in computers, telecommunication systems, communication * Corresponding author. Tel.: +91 416 2202321; fax: +91 416 protocols and retail shopping queues, etc. 2243092. For the last two decades, many researchers studied queue- E-mail addresses: [email protected] (P. Rajadurai), [email protected] ing networks with the concept of positive and negative cus- (V.M. Chandrasekaran), [email protected] (M.C. Saravanarajan). tomers. Queues with negative customers (called G-queues) Peer review under responsibility of Ain Shams University. have concerned huge interests due to their extensive applications in computers, communication networks, neural networks and manufacturing systems [3,4]. The named G-queue has Production and hosting by Elsevier been adopted for the queue with negative customers in the

http://dx.doi.org/10.1016/j.asej.2016.03.008 2090-4479 Ó 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Rajadurai P et al., Analysis of an unreliable retrial G-queue with working vacations and vacation interruption under Bernoulli schedule, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.03.008 2 P. Rajadurai et al. acknowledgment of Gelenbe, who first introduced this type of model. Important special cases of this model are given in Sec- queue in [5,6]. Harrison [7] has studied the idea of composi- tion 6. Cost optimization analysis is discussed in Section 7.In tional reversed Markov processes with applications to Section 8, the effects of various parameters on the system per- G-networks. The positive customer arrives into the system formance are analyzed numerically. Conclusion and summary and gets service as ordinary queueing customers, but the of the work are presented in Section 9. negative customers enter into the system only at the service time of positive customers. This type of negative customers 2. Description of the model removes the positive customers being in service from the system and causes the server breakdown and the service channel will fail for a short interval of time. When the server fails, it In this section, we consider a single server retrial queueing sys- will be sent to repair immediately. After completion of repair, tem with both single and multiple working vacations and vaca- the server will treat as good as new. Do [8] has presented a tion interruption, where the regular busy server is subjected to survey on queueing systems with G-networks, negative cus- breakdown due to the arrival of negative customers. The tomers and applications. Choudhury and Ke [9], Rajadurai detailed description of our model is given as follows: et al. [10,11] and Singh et al. [12] have discussed the retrial queue with the concept of breakdown and repair. Recently, The arrival process: Customers arrive at the system accord- k Krishnakumar et al. [13], Gao and Wang [14], Peng et al. ing to a Poisson process with rate . [15] and Rajadurai et al. [16,17] have discussed different types The retrial process: If an arriving positive customer finds of queueing models operating with the simultaneous presence that the server is free, the customer begins his service imme- of negative arrivals. diately. Otherwise, the server is busy or working vacation or In working vacation period (WV), the server gives service to breakdown, the arrivals join pool of blocked customers customer at lower service rate, but the server stops the service called an orbit in accordance with FCFS discipline, which completely in the normal vacation period. This queueing sys- means that only one customer at the head of the orbit queue tem has major applications in providing network service, is allowed to access the server. We assume that inter-retrial web service, file transfer service and mail service, etc. In times follow a general random variable R with an arbitrary 2002, Servi and Finn [18] have introduced an M/M/1 queueing distribution RðtÞ having corresponding Laplace Stieltjes # system with working vacations. Wu and Takagi [19] have Transform (LST) R ð Þ. extended the M/M/1/WV queue to an M/G/1/WV queue. Very The Bernoulli working vacation process: The server begins a recently, Arivudainambi et al. [20] have introduced M/G/1 working vacation each time when the orbit becomes empty retrial queue with single working vacation. Liu and Song [21] and the vacation time follows an exponential distribution h have discussed a discrete time retrial queue with non- with parameter . During a vacation period if any customer persistent customers and working vacations. Furthermore if arrives, the server gives service at a lower speed service rate. there are customers in the system at the end of a lower speed If any customers in the orbit at a lower speed service com- service, the server can stop the vacation and come back to pletion instant in the vacation period then the server will the normal busy state. This policy is called vacation interrup- stop the vacation and come back to the normal busy period tion. Li and Tian [22] have presented an M/M/1 queueing which means that vacation interruption happens. If no cus- model with working vacations and vacation interruption. tomers are in the system at the end of the vacation, the ser- Some of the authors such as, Zhang and Hou [23], Gao and ver either remains idle to serve a new customer with Liu [24], Gao et al. [25], Zhang and Liu [26], Rajadurai et al. probability p (single working vacation) in regular mode or [27,28] have analyzed a single server retrial queue with working leaves for another working vacation with probability vacations and vacation interruptions. q =1 p (multiple working vacation). When a vacation In this paper, we have extended the work of Gao et al. [25] ends and if there are customers in the orbit then the server and Zhang and Liu [26] by incorporating the concept of nega- switches to the normal working level. During the tive customers (G-queues) in both single and multiple working working vacation period, the service time follows a vacations with breakdowns and repair. To the author’s best of general random variable Sv with distribution function # knowledge, there is no work published in the queueing litera- SvðtÞ havingR LST Sv ð Þ and the first moment is given by 0 h 1 hx ture with the combination of retrial queueing system with gen- Sv ð Þ¼ 0 xe dSvðxÞ. eral retrial times, negative customers, single and multiple The regular service process: Whenever a new positive cus- working vacations, vacation interruption and breakdowns by tomer or retry positive customer arrives at the server idle using the method of supplementary variable technique. The state then the server immediately starts normal service for mathematical results and theory of queues of this model pro- the arrivals. The service time follows a general distribution vide to serve a specific and convincing application in the com- and it is denoted by the random variable Sb with distribu- # puter processing system. Our model is helpful to managers tion function SbðtÞ havingR LST Sbð Þ and the first moment 0 d 1 dx who can design a system with economic management. is given by Sb ð Þ¼ 0 xe dSbðxÞ. The remainder of this work is given as follows. The detailed The removal rule and the repair process: The negative cus- mathematical description and practical applications of our tomers arrive from outside the system according to a Pois- model are given in Section 2. The steady state joint distribution son arrival rate d. These negative customers arrive only at of the server state and the number of customers in the system the regular service time of the positive customers. Negative and its orbit are obtained in Section 3. Some system perfor- customers cannot accumulate in a queue and do not receive mance measures, the mean busy period, the mean busy cycle service, will remove the positive customers being in service and reliability measures are obtained in Section 4. In Section 5, from the system. These types of negative customer’s cause conditional stochastic decomposition is shown good for our the server breakdown and the service channel will fail for

a short interval of time. When the server fails, it will be sent messages (customers) arrive into the system one by one, and to repair immediately. After completion of repair, the server the processor (server) is in charge of processing messages. will treat as good as new. The repair time (denoted by G)of The working mail server may be affected by virus (negative the server is assumed to be arbitrarily distributed with dis- customers), and the system may be subjected to electronic fails tribution function GðtÞ having LST Gð#Þ and the first and (breakdowns) during service period and receive repair immedi- second moments are denoted by gð1Þ and gð2Þ. ately. If the processor is available indicating that it is not cur- We assume that all the random variables (inter-arrival rently working on a task and then a message is processed. The times, retrial times, service times, working vacation times messages are temporarily stored in a buffer to be served some and repair times) defined above are independent of each time later (retrial time) according to FCFS if the processor is other. unavailable. To enhance the computer performance, whenever all messages are processed and no new messages arrive, the processor will perform a sequence of maintenance jobs, such 2.1. Practical application of the proposed model as virus scan (working vacations). During the maintenance period, the processor can deal with the messages at the slower The potential practical application of this model is in the oper- rate to economize the cost (working vacation period). Upon ational model of stochastic production and inventory systems completion of the each maintenance, the processor checks with a multipurpose production facility. The production facil- the messages and decides whether or not to resume the normal ity performs other additional tasks using the time between suc- service rate (single working vacation). At this moment, if no ceeding productions. We consider a practical problem related message is in the system then the processor may decide to go to a production to order system. In production order system, for another maintenance activity (multiple working vacations). the customer orders for the product (positive customers) and This type of working vacation discipline is a good approxima- some customers cancel orders (negative customers) due to tion of such computer processing system. financial crisis or disaster, etc. Let us assume that order system follows the Poisson processes. At the time of arriving customer 3. Steady state analysis orders the production facility is busy with other order, and a new arriving order will form a waiting line which corresponds In this section, we develop the steady state difference–differen- to the retrial queue. Otherwise, the order is processed immedi- tial equations for the retrial queueing system by treating the ately. After the completion of order, the management policy elapsed retrial times, the elapsed service times, the elapsed sets up the facility and begins product to the next order on working vacation times and the elapsed repair times as supple- the list; otherwise, another external order arrives before the mentary variables. Then we derive the probability generating order is made. The order time is assumed to be generally dis- function (PGF) for the server states and for the number of cus- tributed (general retrial time). During the service time of order- tomers in the system and orbit. ing, if any canceling order exists then the major production will be stopped (i.e. server breakdown). To enhance the production 3.1. The steady state equations facility performance, the management policy is to set up the optional job facility (working vacation). After completion of service for last order and there is no order in system (system In steady state, we assume that R(0) = 0, Rð1Þ =1, empty), the facility stops the major production and is available Sbð0Þ =0, Sbð1Þ =1, Svð0Þ =0, Svð1Þ =1, G(0) = 0, to perform optional jobs (single working vacation) with lower Gð1Þ = 1 are continuous at x = 0. So that the function l l n production rate (lower speed service rate). After the result of aðxÞ, bðxÞ, vðxÞ and ðxÞ are the conditional completion disaster or an optional job completion, if there are no orders, rates (hazard rate) for retrial, normal service, lower rate service the production facility will continue to perform the optional and repair respectively. jobs at lower production rate (multiple working vacations). Otherwise, it performs the major production. If there are : :; dRðxÞ ; l dSbðxÞ ; i e aðxÞdx ¼ bðxÞdx ¼ orders at the instant of the optional jobs completion, the pro- 1 RðxÞ 1 SbðxÞ duction facility will perform the major production (vacation l dSvðxÞ ; n dGðxÞ : vðxÞdx ¼ ðxÞdx ¼ interruption). This type of system is very useful to increase 1 SvðxÞ 1 GðxÞ the performance of the production facility and to stop the pro- 0 0 0 0 duction facility from becoming overloaded. In addition, let R ðtÞ, SbðtÞ, Sv ðtÞ and G ðtÞ be the elapsed Another practical application of this model is in the area of retrial time, elapsed normal service time, elapsed working computer processing system. In a computer processing system, vacation time and elapsed repair time respectively at time t. the buffer size (orbit) used to store messages is finite and the Further, we introduce the random variable,

8 > ; ; > 0 if the server is free and in working vacation period > <> 1; if the server is free and in regular service period; ; ; XðtÞ¼> 2 if the server is busy and in regular service period at time t > > 3; if the server is busy and in working vacation period at time t; :> 4; if the server is under repair period at time t:

0 0 0 Thus the supplementary variables R ðtÞ, SbðtÞ, Sv ðtÞ and We assume that the stability condition is fulﬁlled in the 0 G ðtÞ are introduced in order to obtain a bivariate Markov sequel and so that we can set Q0 ¼ limt!1Q0ðtÞ; P P process fgXðtÞ; NðtÞ; t P 0 , where XðtÞ denotes the server state P0 ¼ limt!1P0ðtÞ and limiting densities for t 0, x 0 and (0,1,2,3,4) depending on the server is free on both regular n P 1. busy period and working vacation period, regular busy, on PnðxÞ¼limPnðx; tÞ; Pb;nðxÞ¼limPb;nðx; tÞ; working vacation and under repair. If XðtÞ = 1 and t!1 t!1 > 0 ; ; : NðtÞ 0, then R ðtÞ represent the elapsed retrial time, if Qv;nðxÞ¼limQv;nðx tÞ and RnðxÞ¼limRnðx tÞ P 0 t!1 t!1 XðtÞ = 2 and NðtÞ 0 then SbðtÞ corresponding to the elapsed time of the customer being served in regular busy period. If By using the method of supplementary variable technique, P 0 XðtÞ = 3 and NðtÞ 0, then Sv ðtÞ corresponding to the we formulate the system of governing equations of this model elapsed time of the customer being served in lower rate service as follows: period. If XðtÞ = 4 and NðtÞ P 0, then G0ðtÞ corresponding to k h : the elapsed time of the server being repaired. P0 ¼ pQ0 ð3 1Þ We analyze the ergodicity of the embedded Markov chain Z 1 at departure, vacation or repair epochs. Let {t ; n ðÞk þ h Q ¼ hqQ þ P ; ðxÞl ðxÞdx ... 0 0 b 0 b n =1,2, } be the sequence of epochs of either normal ser- Z 0 Z vice completion times or working vacation completion times 1 1 l n : or repair period ends. The sequence of random vectors þ Qv;0ðxÞ vðxÞdx þ R0ðxÞ ðxÞdx ð3 2Þ 0 0 Zn ¼ fgXtðÞnþ ; NtðÞnþ forms a Markov chain which is embedded in the retrial queueing system. dP ðxÞ n þ ðÞk þ aðxÞ P ðxÞ¼0; n P 1 ð3:3Þ dx n Theorem 3.1. The embedded Markov chainfg Zn; n 2 N is P ergodic if and only if q < R ðkÞ for our system to be stable, d b;0ðxÞ k d l P ; ; : þ ðÞþ þ bðxÞ b;0ðxÞ¼0 n ¼ 0 ð3 4Þ q k=d d d ð1Þ dx where ¼ ðÞ1 Sbð Þ 1 þ g

dPb;nðxÞ Proof. To prove the sufficient condition of ergodicity, it is very þ ðÞk þ d þ l ðxÞ Pb;nðxÞ¼kPb;n1ðxÞ; n P 1; dx b convenient to use Foster’s criterion (see Pakes [29]), which ð3:5Þ states that the chain fgZn; n 2 N is an irreducible and aperi- odic Markov chain is ergodic if there exists a non-negative dQ ; ðxÞ ; e > v 0 k h l ; : function fðjÞ j 2 N and 0, such that mean drift þ ðÞþ þ vðxÞ Qv;0ðxÞ¼0 n ¼ 0 ð3 6Þ w = dx j ¼ Ef½ðznþ1ÞfðznÞ zn ¼ j is finite for all j 2 N and w 6 e for all j 2 N, except perhaps for a finite number j’s. j dQv;nðxÞ þ ðÞk þ h þ l ðxÞ Q ; ðxÞ¼kQ ; ðxÞ; n P 1 ð3:7Þ In our case, we consider the function fðjÞ¼j. then we have dx v v n v n1 q 1; if j ¼ 0; w dR0ðxÞ j ¼ þ ðÞk þ nðxÞ R ðxÞ¼0; n ¼ 0: ð3:8Þ q R ðkÞ; if j ¼ 1; 2; ... dx 0 Clearly the inequality q < R k is sufficient condition for ð Þ dR ðxÞ Ergodicity. n þ ðÞk þ nðxÞ R ðxÞ¼kR ðxÞ; n P 1: ð3:9Þ dx n n1 To prove the necessary condition, as noted in Sennott et al. ; P [30], if the Markov chain fgZn n 1 satisfies Kaplan’s To solve the Eqs. (3.2)–(3.9), the steady state boundary w < P condition, namely, j 1 for all j 0 and there exits conditions at x = 0 are followed, w P P j0 2 N such that j 0 for j j0. Notice that, in our case, Z Z Kaplan’s condition is satisfied because there is a k such that 1 1 P ð0Þ¼ P ; ðxÞl ðxÞdx þ R ðxÞnðxÞdx m ¼ 0 for j < i k and i > 0, where M ¼ðm Þ is the one step n b n b n ij ij 0 Z 0 transition matrix of fgZ ; n 2 N . Then q P RðkÞ implies the 1 n l ; P : non-Ergodicity of the Markov chain. þ Qv;nðxÞ vðxÞdx n 1 ð3 10Þ 0 ; Z Z Let us define the limiting probabilities Q0ðtÞ¼PXf ðtÞ¼0 1 1 N t 0 ; P t PXt 1; N t 0 and the probability P h k ; ; ð Þ¼ g 0ð Þ¼ fgð Þ¼ ð Þ¼ b;0ð0Þ¼ P1ðxÞaðxÞdx þ Qv;0ðxÞdx þ P0 n ¼ 0 densities are 0 0 : 0 ð3 11Þ Pnðx; tÞdx ¼ PXðtÞ¼1; NðtÞ¼n; x 6 R ðtÞ < x þ dx ; Z Z for t P 0; x P 0 and n P 1: 1 1 P k 0 b;nð0Þ¼ Pnþ1ðxÞaðxÞdx þ PnðxÞdx P ; ðx; tÞdx ¼ PXðtÞ¼2; NðtÞ¼n; x 6 S ðtÞ < x þ dx ; b n b 0Z 0 for t P 0; x P 0; n P 0: 1 þ h Q ; ðxÞdx ; n P 1; ð3:12Þ ; ; ; 6 0 < ; v n Qv;nðx tÞdx ¼ PXðtÞ¼3 NðtÞ¼n x Sv ðtÞ x þ dx 0 for t P 0; x P 0 and n P 0: k ; ; ; ; 6 0 < ; Q0 n ¼ 0 Rnðx tÞdx ¼ PXðtÞ¼4 NðtÞ¼n x G ðtÞ x þ dx Q ; ð0Þ¼ ð3:13Þ v n 0; n P 1 for t P 0; x P 0 and n P 0:

Please cite this article in press as: Rajadurai P et al., Analysis of an unreliable retrial G-queue with working vacations and vacation interruption under Bernoulli schedule, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.03.008 Analysis of an unreliable retrial G-queue 5 Z 1 Rðx; zÞ¼Rð0; zÞ½1 GðxÞebðzÞx; ð3:27Þ Rnð0Þ¼d Pb;nðxÞdx; n P 0 ð3:14Þ 0 where AbðzÞ¼ðÞd þ kð1 zÞ ; AvðzÞ¼ðÞh þ kð1 zÞ and The normalizing condition is Z Z bðzÞ¼kbð1 zÞ. X1 X1 1 1 Inserting Eqs. (3.24)–(3.27) in (3.21) and make some manip- P þ Q þ P ðxÞdx þ P ; ðxÞdx 0 0 n b n ulation, ﬁnally we get, Z n¼1 0 Z n¼0 0 1 1 ; Pð0 zÞ : Pbð0; zÞ¼ ½þR ðkÞþzðÞ1 R ðkÞ kP0 þ kQ VðzÞ þ Qv;nðxÞdx þ RnðxÞdx ¼ 1 ð3 15Þ z 0 0 0 ð3:28Þ where VðzÞ¼ h 1 SðÞA ðzÞ . 3.2. The steady state solution hþkð1zÞ v v Using Eqs. (3.25)–(3.27) in (3.20), we get ; P ; ; The steady state solution of the retrial queueing model is Pð0 zÞ¼ bð0 zÞSbðÞþAbðzÞ Qvð0 zÞSv ðÞAvðzÞ obtained by using the probability generating function tech- ; k h : þ Rð0 zÞG ðÞbðzÞ ðÞþ p Q0 ð3 29Þ nique. To solve the above equations, the PGFs are deﬁned Using Eqs. (3.25)–(3.27) in (3.23), we get for jzj 6 1 as follows: X1 X1 1 S A z ; dP ; bðÞbð Þ ; : P x; z P x zn; P 0; z P 0 zn; Rð0 zÞ¼ bð0 zÞ ð3 30Þ ð Þ¼ nð Þ ð Þ¼ nð Þ AbðzÞ n¼1 n¼1 X1 X1 Using Eqs. (3.22), (3.28), and (3.30) in (3.29), we get n n P ðx; zÞ¼ P ; ðxÞz ; P ð0; zÞ¼ P ; ð0Þz ; b b n b b n NrðzÞ n¼0 n¼0 Pð0; zÞ¼ ð3:31Þ X1 X1 DrðzÞ ; n; ; n; Qvðx zÞ¼ Qv;nðxÞz Qvð0 zÞ¼ Qv;nð0Þz n¼0 n¼0 k h k h NrðzÞ¼zQ0 Sv ðÞAvðzÞ 1 p þ ðÞVðzÞþ p X1 X1 ; n ; n Rðx zÞ¼ RnðxÞz and Rð0 zÞ¼ Rnð0Þz d G ðÞbðzÞ 1 SbðÞAbðzÞ n 0 n 0 ¼ ¼ SbðÞþAbðzÞ n AbðzÞ On multiplying the Eqs. (3.2)–(3.14) by z and summing over n, (n =0,1,2,...), we get DrðzÞ¼ z ðÞRðkÞþzð1 RðkÞÞ @Pðx; zÞ k a x P x; z 0 3:16 @ þ ðÞþ ð Þ ð Þ¼ ð Þ d x G ðÞbðzÞ 1 SbðÞAbðzÞ SbðÞþAbðzÞ AbðzÞ @P ðx; zÞ b þ ðÞkð1 zÞþd þ l ðxÞ P ðx; zÞ¼0; ð3:17Þ @x b b Using Eq. (3.21) in (3.28), we get P ð0; zÞ¼Q k SðÞA ðzÞ 1 hp ðÞRðkÞþzð1 RðkÞÞ @Q ðx; zÞ b 0 v v v þ ðÞkð1 zÞþh þ l ðxÞ Q ðx; zÞ¼0 ð3:18Þ k h = : @x v v þ zðÞgVðzÞþ p DrðzÞð3 32Þ @ ; Using Eq. (3.32) in (3.30), we get Rðx zÞ k n ; : þ ðÞð1 zÞþ ðxÞ Rðx zÞ¼0 ð3 19Þ ; d k h k @x Rð0 zÞ¼ Q0 1 SbðÞAbðzÞ Sv ðÞAvðzÞ 1 p ðR ð Þ Z Z k k h = ; : 1 1 þ zð1 R ð ÞÞÞ þ zðÞgVðzÞþ p AbðzÞDrðzÞ ð3 33Þ ; P ; l ; l Pð0 zÞ¼ bðx zÞ bðxÞdx þ Qvðx zÞ vðxÞdx Using the Eqs. (3.22) and (3.31)–(3.33) in (3.24)–(3.27),thenwe 0 Z 0 ; P ; ; 1 get the results for the following PGFs Pðx zÞ, bðx zÞ, Qvðx zÞ ; n k h h : þ Rðx zÞ ðxÞdx ðÞððÞþ Q0 qP0 3 20Þ and Rðx; zÞ. Next we are interested in investigating the marginal 0 orbit size distributions due to system state of the server. Z Z 1 1 1 P ð0; zÞ¼ Pðx; zÞaðxÞdx þ k Pðx; zÞdx b z Theorem 3.2. The marginal probability distributions of the 0Z 0 1 number of customers in the orbit when server being idle, busy, on h ; ; : þ Qvðx zÞdx ð3 21Þ working vacation and under repair are given by 0 NrðzÞ P z Q ð0; zÞ¼kQ ð3:22Þ ð Þ¼ v 0 DrðzÞ Z 1 1 R k ð Þ k h NrðzÞ¼zQ0 S ðÞAvðzÞ 1 p Rð0; zÞ¼d Pbðx; zÞdx ð3:23Þ k v 0 dG ðÞbðzÞ 1S ðÞAbðzÞ Solving the partial differential Eqs. (3.20)–(3.23), it follows that þðÞkVðzÞþhp SðÞþA ðzÞ b b b A z k bð Þ Pðx; zÞ¼Pð0; zÞ½1 RðxÞe x ð3:24Þ k k DrðzÞ¼ zðÞR ð Þþzð1R ð ÞÞ SbðÞAbðzÞ P ; P ; AbðzÞx; : bðx zÞ¼ bð0 zÞ½1 SbðxÞe ð3 25Þ dG ðÞbðzÞ 1S ðÞAbðzÞ þ b ; ; AvðzÞx; : AbðzÞ Qvðx zÞ¼Qvð0 zÞ½1 SvðxÞe ð3 26Þ ð3:34Þ

(), k SðÞA ðzÞ 1 hp ðÞþRðkÞþzð1 RðkÞÞ zðÞkVðzÞþhp d v v ; : RðzÞ¼ Q0 bðzÞAbðzÞDrðzÞ ð3 37Þ 1 SbðÞAbðzÞ ðÞ1 G ðÞbðzÞ where the Eqs. (3.34)–(3.39) in the above results, then the Eqs. (3.40) and (3.41) can be obtained by direct calculation. h RðkÞq Q ¼ ð3:38Þ 0 k=h h k h =k q h ðÞ1 Sv ð Þ þ R ð ÞðÞ1 þ ðÞp Sv ð Þ 4. System performance measures hpRðÞðkÞq P ¼ 0 kk=h h k h =k q h ðÞ1 Sv ð Þ þ R ð ÞðÞ1 þ ðÞp Sv ð Þ In this section, we derive some system probabilities, mean ð3:39Þ number of customers in the system and its orbit, reliability analysis, mean busy period and mean busy cycle of this model. q k=d d d ð1Þ ; d k ; ¼ ðÞ1 Sbð Þ 1 þ g AbðzÞ¼ðÞþ ð1 zÞ 4.1. System state probabilities AvðzÞ¼ðÞh þ kð1 zÞ and bðzÞ¼kbð1 zÞ

From Eqs. (3.34)–(3.37), by setting z 1 and applying l’Hoˆpi- Proof. Integrating the Eqs. (3.24)–(3.27)R with respect to x,we ! P z 1 P x; z dx P z tal’s rule whenever necessary, then we get the following results, deﬁneR the PGFs as,R ð Þ¼ 0 ð Þ R , bð Þ¼ 1 P ðx; zÞdx, Q ðzÞ¼ 1 Q ðx; zÞdx, RðzÞ¼ 1 Rðx; zÞdx. 0 b v 0 v 0 (i) The probability that the server is idle during the retrial By using the normalized condition, we can determine the probability that the server is idle (P ). Thus, by setting z =1in P ¼ Q ðÞ1 RðkÞ 0 ()0 (3.34)–(3.37) and applying de l’Hoˆpital’s rule whenever necessary ðÞk=h 1 SðhÞ þ k 1 SðhÞ þ hp 1 SðdÞ 1 þ dgð1Þ =d v v b and we get P þ Q þ Pð1ÞþP ð1ÞþQ ð1ÞþRð1Þ¼1. h k k d d ð1Þ =d 0 0 b v R ð Þ 1 Sbð Þ ðÞ1 þ g (ii) The probability that the server is regular busy, Theorem 3.3. The probability generating function of number of () k2=h 1 SðhÞ þ RðkÞ k 1 SðhÞ þ hp customers in the system and orbit size distribution at stationary P ¼ Q 1 SðdÞ =d vv b 0 b R k k 1 S d 1 dgð1Þ =d point of time is ð Þ bð Þ ðÞþ

NrsðzÞ P : KsðzÞ¼ ¼ P0 þ Q0 þ PðzÞþzðÞþbðzÞþQvðzÞ RðzÞð3 40Þ DrsðzÞ 8 2 39 k=h h k h k k d > ðÞðÞðÞþþ zVðzÞ p zAbðzÞðÞR ð Þþzð1 R ð ÞÞ AbðzÞSbðÞþAbðzÞ G ðÞbðzÞ 1 SbðÞAbðzÞ > > 6 () 7> < 6 7= ðÞ1 z 4 ðÞkVðzÞþhp AbðzÞS ðÞþAbðzÞ dG ðÞbðzÞ 1 S ðÞAbðzÞ 5 Nr ðzÞ¼Q þzðÞ1 RðkÞ b b s 0> k h > > þAbðzÞ Sv ðÞAvðzÞ 1 p > :> ;> d k h k k k h þ 1 SbðÞAbðzÞ ðÞzbðzÞþ ðÞ1 G ðÞbðzÞ Sv ðÞAvðzÞ 1 p ðÞþR ð Þþzð1 R ð ÞÞ zðÞVðzÞþ p k k d DrsðzÞ¼bðzÞ zAbðzÞðÞR ð Þþzð1 R ð ÞÞ AbðzÞSbðÞþAbðzÞ G ðÞbðzÞ 1 SbðÞAbðzÞ

NroðzÞ P : : KoðzÞ¼ ¼ P0 þ Q0 þ PðzÞþ bðzÞþQvðzÞþRðzÞ ð3 41Þ DrsðzÞ 8 2 39 k=h h k h k k d > ðÞðÞðÞþþ VðzÞ p zAbðzÞðÞR ð Þþzð1 R ð ÞÞ AbðzÞSbðÞþAbðzÞ G ðÞbðzÞ 1 SbðÞAbðzÞ > > 6 () 7> < 6 7= ðÞ1 z 4 ðÞkVðzÞþhp AbðzÞS ðÞþAbðzÞ dG ðÞbðzÞ 1 S ðÞAbðzÞ 5 Nr ðzÞ¼Q þzðÞ1 RðkÞ b b 0 0> k h > > þAbðzÞ Sv ðÞAvðzÞ 1 p > :> ;> d k h k k k h þ 1 SbðÞAbðzÞ ðÞbðzÞþ ðÞ1 G ðÞbðzÞ Sv ðÞAvðzÞ 1 p ðÞþR ð Þþzð1 R ð ÞÞ zðÞVðzÞþ p

where Q0 is given in Eq. (3.38).

(iii) The probability that the server is on working vacation 4.3. Reliability measures Q ¼ kQ 1 SðhÞ =h v 0 v In the queueing system with unreliable server, the reliability measures will provide the information which is required for (iv) The probability that the server is under repair, the improvement of the system. To justify and validate the () 2 A k =h 1 S ðhÞ þ RðkÞ k 1 S ðhÞ þ hp analytical results of this model, the availability measure ð vÞ R ¼ Q gð1Þ 1 SðdÞ vv 0 b k k d d ð1Þ =d and failure frequency (Ff) are obtained as follows: R ð Þ 1 Sbð Þ ðÞ1 þ g

(i) The steady state availability Av, which is the probability that the server is either working for a positive customer 4.2. Mean system size and orbit size or in an idle period such that the steady state availability of the server is given by If the system is in steady state condition, Av ¼ 1 lim ðÞ¼RðzÞ 1 Rð1Þ 8z!1 9 > k2 > (i) The expected number of customers in the orbit (L )is <>gð1Þ 1 S d 1 S h R k k 1 S h hp => q bð Þ h v ð Þ þ ð Þ v ð Þ þ obtained by differentiating (3.41) with respect to z and ¼ 1 > ðÞk=h 1 SðhÞ þ RðkÞðÞ1 þ ðÞhp=k qSðhÞ > evaluating at z =1 :> v v ;> 2 3 (ii) The steady state failure frequency is obtained as d 6Nr000ð1ÞDr00ð1ÞDr000ð1ÞNr00ð1Þ7 0 4 q qq q 5 d P Lq ¼ Koð1Þ¼lim KoðzÞ¼Q0 2 Ff ¼ bð1Þ z!1 dz 00 () 3 Dr ð1Þ 2 q Q 1 SðdÞ k =h 1 SðhÞ þ RðkÞ k 1 SðhÞ þ hp ¼ 0 b v v k k d d ð1Þ =d R ð Þ 1 Sbð Þ ðÞ1 þ g 00 k d d ð1Þ h k k h Nrqð1Þ¼2 1 Sbð Þ 1 þ g ðÞ p Sv ð Þ 4.4. Mean busy period and busy cycle dk2=h k h 2 ðÞ1 R ð Þ 1 Sv ð Þ Let EðT Þ and EðT Þ be the expected length of busy period and Nr000ð1Þ¼3ksSðhÞ2 k3=h 1 þ dgð1Þ 1 SðhÞ b c q v v busy cycle under the steady state conditions. The results follow h k 0 h k=h h þ R ð ÞSv ð Þ ðÞ1 Sv ð Þ þ 1 directly by applying the argument of an alternating renewal process [9] which leads to 6 dkgð1Þ 1 SðdÞ kSðdÞ fðÞkV00ð1Þ b b k k=h h k h h EðT0Þ 1 1 ðÞ1 R ð Þ ðÞþ 1 Sv ð Þ þ p P ¼ ; EðT Þ¼ 1 and 0 EðT ÞþEðT Þ b k P k d k d ð1Þ h b 0 0 6 1 Sbð Þ ðÞ1 R ð Þ 1 þ g ð p 1 : þk 1 SðhÞ þ 6kkðÞ=h h þ k 1 SðhÞ þ hp EðTcÞ¼k ¼ EðT0ÞþEðTbÞð4 1Þ v v P0 6k2 d=h V0 1 1 R k 1 S h kS0 h ðÞð ÞþðÞ ð Þ v ð Þ þ v ð Þ where T0 is length of the system in empty state and EðT0Þ¼ðÞ1=k . Substituting the Eq. (3.39) into (4.1) and use 00 k k q Drqð1Þ¼2 ðÞR ð Þ the above results, then we can get ðÞk=h 1 SðhÞ þ RðkÞ 1 SðdÞ =d 1 þ dgð1Þ kSðhÞþhp 000 ks kdq k k dk ð1Þ d k d EðT Þ¼ v b v Drq ð1Þ¼3 ðÞþ 2 6 R ð Þ g 1 Sbð Þ Sbð Þ b hpRðÞðkÞq ð4:2Þ where V0ð1Þ¼k 1 SðhÞþhS0ðhÞ ; q ¼ ðÞk=d h v R v d d ð1Þ 0 h 1 hx 0 d ðÞk=h 1 SðhÞ þ RðkÞðÞ1 þ ðÞhp=k SðhÞ k 1 SðdÞ =d 1þ dgð1Þ 1 Sbð Þ 1 þ g ; Sv ð Þ¼ xe dSvðxÞ; Sb ð Þ¼ E T v v b R 0 ð cÞ¼ h k q 1 dx pRðÞð Þ 0 xe dSbðxÞ. ð4:3Þ s k2 0 d d ð1Þ d ð2Þ d ¼ 2Sb ð Þ 1 þ g þ g 1 Sbð Þ (ii) The expected number of customers in the system (Ls)is 5. Conditional stochastic decomposition property obtained by differentiating (3.40) with respect to z and evaluating at z =1 2 3 In this section, we study the stochastic decomposition property of the system size distribution. The number of customers in the d 6Nr000ð1ÞDr00ð1ÞDr000ð1ÞNr00ð1Þ7 0 4 s qq q 5 Ls ¼ Ksð1Þ¼lim KsðzÞ¼Q0 2 system is distributed as the sum of two independent random z!1 dz 00 3 Drqð1Þ variables. In particular, in the context our system, we will dis- cuss the conditional stochastic decomposition of the number of where customers in the orbit given that the server is busy. Let Nb is the conditional orbit size of our retrial queuing system given 000 000 k k d h k h Nrs ð1Þ¼Nrq ð1Þ6 R ð Þ 1 Sbð Þ p þ 1 Sv ð Þ that server is busy and N is the conditional orbit size of the 0 d ð1Þ k3=h d h M/G/1 retrial queueing system with negative customers is þ 6 g 1 Sbð Þ 1 Sv ð Þ given that the server is busy which is discussed in Theorem 5.1,

Theorem 5.1. The conditional orbit size Nb is given that the From above stochastic decomposition law, we observe that server is busy can be decomposed into the sum of two NbðzÞ¼N0ðzÞNcðzÞ which conform that the decomposition independent random variables results of Gao et al. [25], also valid for this special vacation system.

Nb ¼ N0 þ Nc where N0 has the generating function N0ðzÞ as follows,

ð1Þ d 1 S ðÞAbðzÞ ðÞz 1 R ðkÞ k 1 S ðdÞ 1 þ dg d N ðzÞ¼ b b 0 k k d d zAbðzÞðÞR ð Þþzð1 R ð ÞÞ AbðzÞSbðÞþAbðzÞ G ðÞbðzÞ 1 SbðÞAbðzÞ 1 Sbð Þ

and Nc is the additional queue length due to vacations with the 6. Special cases probability generating function NcðzÞ as follows,

( ) ðÞk=h VðzÞ zAbðzÞðÞR ðkÞþzð1 R ðkÞÞ AbðzÞS ðÞþAbðzÞ dG ðÞbðzÞ 1 S ðÞAbðzÞ b b 1 SðdÞ k h k k k h b þ 1 SbðÞAbðzÞ Sv ðÞAvðzÞ 1 p ðÞþR ð Þþzð1 R ð ÞÞ zðÞVðzÞþ p NcðzÞ¼ d k k=h h d k h h k2=h ð1Þ d h 1 SbðÞAbðzÞ ðÞz 1 R ð Þ ðÞ1 Sv ð Þ þ 1 Sbð Þ 1 Sv ð Þ þ p g 1 Sbð Þ 1 Sv ð Þ

Proof. The mathematical version of the stochastic decomposi- In this section, we analyze brieﬂy some special cases of our tion law is N ðzÞ¼N ðzÞN ðzÞ. b 0 c model, which are consistent with the existing literature. In an M/G/1 retrial queueing system with negative customers, the marginal function of the number of customers Case (i): No retrial and Multiple working vacations in the orbit when the server is busy is given by Let RðkÞ!1, our model can be reduced to an

kP0R ðkÞ 1 S ðÞAbðzÞ ðÞz 1 UðzÞ¼ b k k d zAbðzÞðÞR ð Þþzð1 R ð ÞÞ AbðzÞSbðÞþAbðzÞ G ðÞbðzÞ 1 SbðÞAbðzÞ and the probability that the server is busy is given by M/G/1 G-queue with working vacations and () vacation interruption. In this case, KsðzÞ can be found as follows, kP0R ðkÞ 1 S ðdÞ Uð1Þ¼ b ; d k k d d ð1Þ =d R ð Þ 1 Sbð Þ ðÞ1 þ g then for the probability generating function N0ðzÞ, we have

ð1Þ UðzÞ d 1 S ðÞAbðzÞ ðÞz 1 R ðkÞ k 1 S ðdÞ 1 þ dg =d N ðzÞ¼ ¼ b b 0 U k k d d ð1Þ zAbðzÞðÞR ð Þþzð1 R ð ÞÞ AbðzÞSbðÞþAbðzÞ G ðÞbðzÞ 1 SbðÞAbðzÞ 1 Sbð Þ

From the Eqs. (3.35) and (3.36), we know that for our retrial system the probability generating function of Nb is given by

PbðzÞþQðzÞ NbðzÞ¼P (bð1ÞþQð1Þ ) k=h V z zA z R k z 1 R k A z S A z dG b z 1 S A z 1 S A z ðÞð Þ bð ÞðÞð Þþ ð ð ÞÞ bð Þ bðÞbð Þ þ ðÞð Þ bðÞbð Þ þ bðÞbð Þ ð1Þ k S ðÞAvðzÞ 1 hp ðÞþR ðkÞþzð1 R ðkÞÞ zðÞkVðzÞþhp R ðkÞ k 1 S ðdÞ 1 þ dg =d ¼ ( v b ) zA ðzÞðÞRðkÞþzð1 RðkÞÞ A ðzÞSðÞþA ðzÞ dGðÞbðzÞ 1 SðÞA ðzÞ b b b b b b k k=h h d k h h k2=h ð1Þ d h R ð Þ ðÞ1 Sv ð Þ þ 1 Sbð Þ 1 Sv ð Þ þ p g 1 Sbð Þ 1 Sv ð Þ

NbðzÞ¼N0ðzÞNcðzÞ

( ) 1 z zA z A z S A z dG b z 1 S A z kzV z =h 1 ðÞ bð Þbð Þ bðÞþbð Þ ðÞð Þ bðÞbð Þ ðÞðÞþð Þ d þ 1 SbðÞAbðzÞ Sv ðÞAvðzÞ 1 þ zVðzÞ ðÞzbðzÞþ ðÞ1 G ðÞbðzÞ K ðzÞ¼P s 0 d ðÞ1 z zAbðzÞ AbðzÞSbðÞþAbðzÞ G ðÞbðzÞ 1 SbðÞAbðzÞ

Case (v): Single working vacation This coincides with the result of Zhang and Let p = 1, our model can be reduced to a sin- Liu [26]. gle server retrial queueing system with nega- Case (ii): No negative arrival, No vacation interruption tive customers, single working vacation and and Single working vacation vacation interruption. Let (d,h) ? (0,0), our model can be reduced to Case (vi): Multiple working vacations M/G/1 retrial queue with single working vaca- Let p = 0, our model can be reduced to a sin- tion. In this case, KsðzÞ coincides with the gle server retrial queueing system with nega- result of Arivudainambi et al. [20] as follows, tive customers, multiple working vacation and vacation interruption.

8 9() ð1Þ < k k kb = Sv ð Þ R ð Þ S ðÞk kz 1 ðÞþR ðkÞþzð1 R ðkÞÞ ðÞ1 z R ðkÞS ðkÞ S ðÞk kz K ðzÞ¼ v v b s :k k k ; k k k k k EðSvÞR ð ÞSv ð Þ Sv ð Þ z ðÞR ð Þþzð1 R ð ÞÞ SbðÞ z

Case (iii): No retrial, No negative arrival and Multiple working vacation 7. Cost optimization analysis Let d ? 0 and RðkÞ!1, suppose that there is no retrial time in the system then we get an M/ In order to carry out cost analysis, the optimum design of a G/1 queue with working vacations and vaca- retrial queueing system is to determine the optimal system tion interruption. In this case, K ðzÞ can be s parameters, such as optimal mean service rate or optimal obtained as follows,

1 kbð1Þ ðÞ1 z z SðÞA ðzÞ ðÞðÞþkzVðzÞ=h 1 þ z 1 SðÞA ðzÞ SðÞþA ðzÞ zVðzÞ1 b b b b v v KsðzÞ¼ k=h h kbð1Þ h 1 þ ðÞ1 Sv ð Þ Sv ð Þ ðÞ1 z z SbðÞAbðzÞ

This coincides with the result of Zhang and number of servers (see in [31,32]). In this section, the optimal Hou [23]. design of the single server retrial G-queue with working Case (iv): No negative arrival and Multiple working vaca- vacations and vacation interruption under Bernoulli schedule tions is addressed. Based on the deﬁnitions of cost elements d Let = p = 0, our model can be reduced to a (Ch; Co; Cs and Ca) and cost structure listed below, the total single server retrial queueing system with expected cost function per unit time is given by working vacations. In this case, KsðzÞ can be yielded as follows,

( "# ) z ðÞR ðkÞþzð1 R ðkÞÞ S ðÞAbðzÞ ðÞðÞþkVðzÞ=h 1 ðÞ1 z b þ 1 SðÞA ðzÞ SðÞA ðzÞ 1 ðÞþRðkÞþzð1 RðkÞÞ zVðzÞ k b b v v þzðÞ1 R ð Þ Sv ðÞþAvðzÞ VðzÞSbðÞAbðzÞ 1 K ðzÞ¼P s 0 k k ðÞ1 z z ðÞR ð Þþzð1 R ð ÞÞ SbðÞAbðzÞ

This coincides with the result of Gao et al. [25].

Table 1 Effects of (Ch; Co) on the expected cost function TC with Cs = $1000 and Ca = $100.

(Ch; Co) (5,100) (5,110) (5,120) (10,100) (15,100) TC 301.7446 303.3765 305.0084 303.4891 305.2337

EðTbÞ 1 EðT0Þ TC ¼ ChLs þ Co þ Cs þ Ca EðTcÞ EðTcÞ EðTcÞ

¼ ChLs þ CoðÞþ1 P0 Csk þ CaP0 where Ch is the holding costs per unit time for each customer present in the system, Co is the cost per unit time for keeping the server on and in operations, Cs is setup cost per busy cycle and Ca is the startup cost per unit time for the preparation work of the server before starting the service. If we assume exponential retrial times, service times, working vacation times and repair times then for the following values of the cost elements and other parameters like: k =1; Figure 1 TC versus k. l l n h d b =5; v =2; a =2; =3; =3; = 0.1; p = 0.5; Ch =5,Co = 100, Cs = 1000 and Ca = 100, we ﬁnd the total expected cost per unit of time TC = 301.7446; Also, in this case, the steady-state availability of the server Av = 98.94% while the steady-state failure frequency of the server is Ff = 3.18%. Moreover, we can examine the behavior of the expected cost function under different values of the cost parameters. Let us ﬁx the system parameters values as follows: k =1; l l n h d b =5; v =2;a =2; =3; =3; = 0.1; p = 0.5. Tables 1–3 illustrate the effects of (Ch; Co), (Co; Ca) and (Cs; Ca) on the expected cost function, respectively. It can be see that the expected cost function shows a linearly increasing trend with increasing cost parameters. Similarly, a sensitivity analysis of some of the parameters on the system can be conducted. Fixing the base values given above, one parameter can be varied at a time and the corresponding objective function value can be computed. The graphs (from Figs. 1–4) show the effect of some of the system parameters (k; d, a, p) on the total expected cost per unit of time. Figure 2 TC versus d.

8. Numerical examples arbitrary values to the parameters such that the steady state In this section, we present some numerical examples to study condition is satisﬁed. MATLAB software has been used to the effect of various parameters in the system performance illustrate the results numerically. Note that the exponential dis- t tx; > measures of our system where all retrial times, service times, tribution is fðxÞ¼ e x 0, Erlang-2 stage distribution is 2 tx working vacation times and repair times are exponentially, fðxÞ¼t xe ; x > 0 and hyper-exponential distribution is 2 Erlangianly and hyper-exponentially distributed. We assume fðxÞ¼ctetx þð1 cÞt2et x; x > 0.

Table 2 Effects of (Co; Ca) on the expected cost function TC with Ch = $5 and Cs = $1000.

(Co; Ca) (100,100) (125,100) (150,100) (100,110) (100,120) TC 301.7446 305.8243 309.9041 310.1127 318.4808

Table 3 Effects of (Ca; Cs) on the expected cost function TC with Ch = $5 and Co = $100.

(Ca; Cs) (100,1000) (110,1000) (120,1000) (100,1050) (100,1100) TC 301.7446 310.1127 318.4808 311.7446 321.7446

decreases and probability that server is idle during retrial time (P) also decreases for the values of k =1; h =2; l =9; n d l p = 0.7; =5; = 0.2; v =4;c = 0.7. Table 5 shows that when negative arrival rate (d) increases, the mean orbit size (Lq) increases, probability that server is idle during retrial time (P) increases and the servers failure frequency (Ff) also increases for the values of k =1; h =2; l =9; p = 0.7; n l =5;a =2; v =4;c = 0.7. Table 6 shows that when vacation rate (h) increases, the idle probability (P0) increases, then the mean orbit size (Lq) decreases and probability that server is

under working vacation (Qv) also decrease for the values of k d l n l =1; = 0.2; =9; p = 0.7; =5; a =2; v =4; c = 0.7. Table 7 shows that when lower speed service rate l ( v) increases, the idle probability (P0) increases, then the mean orbit size (Lq) decreases and probability that server is on work- k ing vacation (Qv) also decrease for the values of =1; d = 0.2; l =9;p = 0.7; n =5;a =2;h =2;c = 0.7. For the effect of the parameters k, a, p, d; h; l, and l on the Figure 3 TC versus a. v system performance measures, three dimensional graphs are illustrated in Figs. 5–8.InFig. 5, the surface displays an Table 4 shows that when retrial rate (a) increases, then the upward trend as expected for increasing the value of arrival k d idle probability (P0) increases, the mean orbit size (Lq) rate ( ) and negative arrival rate ( ) against the mean orbit size

Table 4 The effect of retrial rate (a)onP0, Lq and P. Retrial distribution Exponential Erlang-2 stage Hyper-exponential

aP0 Lq PP0 Lq PP0 Lq P Retrial rate 2.00 0.4583 0.3400 0.0715 0.2394 0.7152 0.3276 0.4103 0.4323 0.1187 3.00 0.4715 0.2269 0.0479 0.3033 0.4580 0.2100 0.4322 0.2913 0.0794 4.00 0.4782 0.1699 0.0361 0.3337 0.3438 0.1540 0.4432 0.2207 0.0596 5.00 0.4822 0.1356 0.0289 0.3514 0.2791 0.1215 0.4498 0.1783 0.0477

Table 5 The effect of negative arrival rate (d)onLq, P and Ff. Retrial distribution Exponential Erlang-2 stage Hyper-exponential

d Lq PFf Lq PFf Lq PFf Negative arrival rate 0.30 0.3575 0.0718 0.0076 0.8118 0.3323 0.0176 0.4570 0.1188 0.0103 0.40 0.3749 0.0721 0.0133 0.9138 0.3369 0.0309 0.4816 0.1189 0.0181 0.50 0.3922 0.0725 0.0206 1.0218 0.3413 0.0476 0.5063 0.1190 0.0279 0.60 0.4094 0.0728 0.0294 1.1363 0.3456 0.0676 0.5308 0.1191 0.0396

h Table 6 The effect of vacation rate ( )onP0, Lq and Qv. Vacation distribution Exponential Erlang-2 stage Hyper-exponential h P0 Lq Qv P0 Lq Qv P0 Lq Qv Vacation rate 1.00 0.5421 0.0561 0.0282 0.4355 0.1140 0.0416 0.5163 0.0777 0.0336 2.00 0.6708 0.0558 0.0150 0.5381 0.0992 0.0206 0.6390 0.0720 0.0173 3.00 0.7285 0.0524 0.0095 0.5845 0.0845 0.0124 0.6940 0.0650 0.0107 4.00 0.7612 0.0487 0.0066 0.6111 0.0730 0.0083 0.7253 0.0589 0.0073

l Table 7 The effect of lower speed service rate ( v)onP0, Lq and Qv. Vacation distribution Exponential Erlang-2 stage Hyper-exponential l v P0 Lq Qv P0 Lq Qv P0 Lq Qv Lower speed service rate 2.00 0.1111 0.3721 0.1389 0.0489 0.8189 0.0916 0.0962 0.4802 0.1383 3.00 0.1184 0.3572 0.1184 0.0540 0.8174 0.0864 0.1035 0.4658 0.1221 4.00 0.1239 0.3473 0.1032 0.0588 0.8053 0.0817 0.1092 0.4592 0.1092 5.00 0.1281 0.3410 0.0915 0.0631 0.7910 0.0773 0.1139 0.4581 0.0988

l l Figure 6 P0 versus v and b. Figure 4 TC versus p.

Figure 5 Lq versus k and d. Figure 7 P0 versus p and h.

(Lq). In Fig. 6, we examine the behavior of the idle probability (Lq) decreases for increasing the value of lower speed service l (P0) increases for increasing the value of the lower service rate rate ( v) and retrial rate (a). l l ( v) and regular service rate ( b). Fig. 7 shows that the idle From the above numerical examples, we observed that the probability (P0) increases for increasing the value of single inﬂuence of parameters on the performance measures in the working vacation idle probability (p) and vacation rate (h). system and know that the results are coincident with the prac- In Fig. 8, we examine the behavior of the mean orbit size tical situations.

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[25] Gao S, Wang J, Li W. An M/G/1 retrial queue with general retrial V.M. Chandrasekaran is working as Senior times, working vacations and vacation interruption. Asia–Paciﬁc J Professor in Department of Mathematics, Oper Res 2014;31:6–31. School of Advanced Sciences, VIT University. [26] Zhang M, Liu Q. An M/G/1 G-queue with server breakdown, He was awarded Ph.D. degree by Madras working vacations and vacation interruption. OPSEARCH University in the year 1997. He has 20 years of 2015;52:256–70. teaching and research experience. His area of [27] Rajadurai P, Yuvarani S, Saravanarajan MC. Performance research is algebra and queueing theory. He analysis of preemptive priority retrial queue with immediate has published more than 60 research articles Bernoulli feedback under working vacations and vacation inter- in reputed national and international journals. ruption. Songkl J Sci Tech [in press]. Available from: http://rdo. He has guided four Ph.D. and ﬁve M.Phil. psu.ac.th/sjstweb/ArticleInPress.php. scholars. At present, He is guiding six Ph.D [28] Rajadurai P, Saravanarajan MC, Chandrasekaran VM. Analysis scholars. of an M/G/1 retrial queue with balking, negative customers, working vacations and server breakdown. Int J Appl Eng Res 2015;10(55):4130–5. M.C. Saravanarajan is working as an Assis- [29] Pakes AG. Some conditions for Ergodicity and recurrence of tant Professor (Senior) in Department of Markov chains. Operat Res 1969;17:1058–61. Mathematics, School of Advanced Sciences, [30] Sennott LI, Humblet PA, Tweedi RL. Mean drifts and the non- VIT University. He has 14 years of teaching Ergodicity of Markov chains. Operat Res 1983;31:783–9. experience. He has completed his Ph.D. degree [31] Tadj L, Chodhury G. Optimal design and control of queues. Top in VIT University under the guidance of Dr. 2005;13:359–414. V.M. Chandrasekaran in queueing theory. He [32] Yang DY, Wu CH. Cost-minimization analysis of a working has published more than 30 research papers in vacation queue with N-policy and server breakdowns. Comp international journals and international con- Indust Eng 2015;82:151–8. ferences. He is currently guiding four Ph.D scholars. His area of specialization is Stochastic processes and its applications, in P. Rajadurai holds his Master of Science degree particular Queueing theory. in Mathematics from Anna University, Chen- nai. He has completed his Master of Philosophy in VIT University. At present, he is doing his Ph.D. research program under the guidance of Dr. V.M. Chandrasekaran and Dr. M.C. Saravanarajan in School of Advanced Sciences at VIT University. He has published more than 25 research papers in international journals and international conferences. His area of specialization is Stochastic processes and its applications, in particular Queueing theory.