Analysis of repairable M [X] /(G 1,G 2 )/1 - feedback retrial G-queue with balking and starting failures under at most J vacations

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1 Available at Appl. Appl. Math. ISSN: Vol. 0, Issue 2 (December 205), pp Applicatios ad Applied Mathematics: A Iteratioal Joural (AAM) Aalysis of repairable M [X] /(G,G 2 )/ - feedback retrial G-queue with balkig ad startig failures uder at most J vacatios P Rajadurai, M C Saravaaraja ad V M Chadrasekara School of Advaced Scieces, VIT Uiversity, Vellore Tamiladu, Idia. psdurai7@gmail.com;mcsaravaaraja@vit.ac.i;vmchadrasekara@vit.ac.i Abstract Received: Jauary, 205; Accepted: March 6, 205 I this paper, we discuss the steady state aalysis of a batch arrival feedback retrial queue with two types of service ad egative customers. Ay arrivig batch of positive customers fids the server is free, oe of the customers from the batch eters ito the service area ad the rest of them joi ito the orbit. The egative customer, arrivig durig the service time of a positive customer, will remove the positive customer i-service ad the iterrupted positive customer either eters ito the orbit or leaves the system. If the orbit is empty at the service completio of each type of service, the server takes at most J vacatios util at least oe customer is received i the orbit whe the server returs from a vacatio. The busy server may breakdow at ay istat ad the service chael will fail for a short iterval of time. The steady state probability geeratig fuctio for the system size is obtaied by usig the supplemetary variable method. Numerical illustratios are discussed to see the effect of system parameters. Keywords: Bulkig; Feedback Balkig; G queue; Breakdow; Startig failures; Steady-state solutio MSC 200 No.: 60K25, 68M20, 90B22. Itroductio Queueig system is a powerful tool for modelig commuicatio etworks, trasportatio etworks, productio lies ad operatig systems. I recet years, computer etworks ad data commuicatio systems are the fastest growig techologies, which have led to sigificat developmet i applicatios such as swift advace i iteret, audio data traffic, video data traffic, etc. I the retrial literature, may of the researchers discussed a retrial queueig model i

2 may directios. Recetly, retrial queues have bee ivestigated extesively due to their applicatios i various fields, such as telephoe switchig systems, call ceters ad telecommuicatio etworks with retrasmissio ad computers. The characteristic of retrial queueig systems is that a customer who fids the server busy upo arrival is obliged to leave the service area ad repeat his demad after some time (retrial time). Betwee trials, a blocked customer who remais i a retrial group is said to be i orbit. For detailed overviews of the related literatures o retrial queues, readers are referred to the books of Fali ad Templeto (997), Artalejo ad Gomez-Corral (2008) ad the survey papers of Artalejo (200). May queueig situatios have the feature that the customers may be served repeatedly for a certai reaso. Whe the service of a customer is usatisfied, it may be retried agai ad agai util successful service completio. These queueig models arise i the stochastic modelig of may real-life situatios. For example, i data trasmissio, a packet trasmitted from the source to the destiatio may be retured ad it may go o like that util the packet is fially trasmitted. Krishakumar et al. (203) studied a model with the cocept of M/G/ feedback retrial queueig system with egative customers. Ke ad Chag (2009a) have discussed Modified vacatio policy for M/G/ retrial queue with balkig ad feedback. The cocept of balkig (customers decide ot to joi the lie at all if he fids the server is uavailable upo arrival) was first studied by Haight i 957. There are may situatios where the customers may be impatiet, such as impatiet telephoe switchboard customers ad the hospital emergecy rooms hadlig critical patiets, web access, icludig call ceters ad computer systems, etc. Ke (2007) studied the M [X] /G/ queue with variat vacatios ad balkig. Some of the authors like Wag ad Li (2009) ad Gao ad Wag (204) discussed the cocept balkig. Recetly, may researchers have studied queueig etworks with the cocept of positive ad egative customers. Queues with egative customers (also called G queues) have attracted cosiderable iterests due to their extesive applicatios, such as computer commuicatio etworks ad maufacturig system. The positive customers arrive to the system ad get their service i the ormal maer. The egative customers arrive ito the system oly at the service time of positive customer. These customers do ot joi i the queue ad do ot get ay service. The egative customers will vaish ad reduce oe positive customer i service. The the positive customer may joi the queue for aother regular service or may leave the system. Negative customers have bee regarded as virus, ihibitor sigals, operatio mistakes or system ad server disaster i eural ad computer commuicatio etworks. Some authors like, Wag ad Zhag (2009), Wag et al. (20), Yag et al. (203), Wu ad Lia (203), Krishakumar et al. (203), Gao ad Wag (204) discussed differet types of queueig models operatig with the simultaeous presece of G-queues. I a vacatio queueig system, the server may ot be available for a period of time due to may reasos like, beig checked for maiteace, workig at other queues, scaig for ew work (a typical aspect of may commuicatio systems) or simply takig a break. This period of time, whe the server is uavailable for primary customers is referred to as a vacatio. Krishakumar ad Arivudaiambi (2002) have ivestigated a sigle server retrial queue with Beroulli schedule, where the radom decisio whether to take a vacatio or ot are allowed oly at

3 istaces whe the system is ot empty (ad a service or vacatio has just bee completed). If the system is empty, the server must take a vacatio. That is the assumptio for their model. Chag ad Ke (2009) examied a batch retrial model with J vacatios i which if orbit becomes empty, the server takes at most J vacatios repeatedly util at least oe customer appears i the orbit upo returig from a vacatio. By applyig the supplemetary variable techique, system characteristics are derived. Later, Ke ad Chag (2009a) ad Che et al. (200), Rajadurai et al. (204) discussed the differet types of queueig model with J vacatio queueig models. The service iterruptios are uavoidable pheomea i may real life situatios. I most of the studies, it is assumed that the server is available i the service statio o a permaet basis ad service statio ever fails. However, these assumptios are practically urealistic. I practice we ofte fid the case where service statios may fail ad ca be repaired. Applicatios of these models are foud i the area of computer commuicatio etworks ad flexible maufacturig system etc. Ke ad Chag (2009b) have studied a batch arrival retrial queueig system with two phases of service uder the cocept of Beroulli vacatio schedules, where the server may meet a upredictable breakdow subject to startig failure whe a customer requires his service iitially. Ke ad Choudhury (202) discussed a batch arrival retrial queueig system with two phases of service uder the cocept of breakdow ad delayig repair. The busy server may breakdow at ay istat ad the service chael will fail for a short period of time. The repair process does ot start immediately after a breakdow ad there is a delay time for repair to start. Choudhury ad Deka (202) have discussed a sigle server queueig model with two phases of service, where the server is subject to breakdow. Some authors like Wag ad Li (2009), Che et al. (200), Choudhury et al. (200) ad Rajadurai et al. (205) discussed the retrial queueig systems with the cocept of breakdow ad repair. Recetly, Haghighi ad Mishev (203) discussed three possible stages for the hadlig of job applicatios i a hirig process as a etwork queuig model. However, there is o work that has bee published i the queueig literature with the combiatio batch arrival retrial queue, two types of service, G-queues, balkig, feedback, modified vacatio (at most J vacatios) ad breakdows. The mathematical results ad theory of queues of this model provide to serve a specific ad covicig applicatio i the trasfer model of a system. I Simple Mail Trasfer Protocol (SMTP) mail system, delivers the messages betwee mail servers. The rest of this work is orgaized as follows. I sectio 2, the detailed descriptio ad practical justificatio of this model are give. I sectio 3, we cosider the goverig equatios of the model ad also obtai the steady state solutios. Some performace measures are derived i sectio 4. I sectio 5, some special cases are discussed. I sectio 6 the effects of various parameters o the system performace are aalyzed umerically. Summary of the work is preseted i sectio Model Descriptio I this paper, we cosider a batch arrival feedback retrial queueig system with two types of service, egative customers uder modified vacatio policy where the server is subject to startig failure, breakdow ad repair. The detailed descriptio of the model is give as follows:

4 Arrival Process: Positive customers arrive i batches accordig to a compoud Poisso process with rate λ. Let X k deote the umber of customers belogig to the k th arrival batch, where X k, k =, 2, 3 are with a commo distributio Pr[X k = ] = χ, =, 2, 3 ad X(z) deotes the probability geeratig fuctio of X. We deote X [k] as the k th factorial momet of X(z) for (k =,2). Retrial process: We assume that there is o waitig space ad therefore if a arrivig batch fids the server free, oe of the customers from the batch begis his service ad rest of them joi ito orbit. If a arrivig batch of customers fids the server beig busy, vacatio or breakdow, the arrivals either leave the service area with probability -b or joi the pool of blocked customers called a orbit with probability b. Iter-retrial times have a arbitrary distributio Rt () with correspodig Laplace-Stieltjes Trasform (LST) R ( ). Service process: A sigle server provides the two types of service. If ay batch of arrivig positive customers fids the server free, the oe of the customers from the batch is allowed to start the First Type Service (FTS) with probability p or Secod Type Service (STS) with probability p 2 (p +p 2 =) ad others joi the orbit. It is assumed that the i th (i =, 2) type service times follows a geeral radom variable S i with distributio fuctio Si() t ad LST Si ( ). Startig failure repair process: Whe ay batch of arrivig positive customers fids the server free, oly the customer at the head of the batch arrivig is allowed to start (tur o) the server ad the others leave the service area ad joi the orbit. O the other had, the service disciplie for the customers i the orbit is to first retry success first service (FRSFS). A returig customer that fids the server free (retry successfully) must start (tur o) the server. The startup time of server could be egligible. Moreover, the server may be started from failure with a probability α = α. If the server is started successfully, the customer gets service immediately. Otherwise, the server is repaired immediately ad the customer must leave the service area ad make a retrial at a later time. That is, the probability of successful commecemet of service is for a ew ad returig customer. Note that the repair time of the failure server is of radom legth H with distributio fuctio Ht, () LST H * ( ) ad fiite k th ( k) momet h (k =,2). Feedback rule: After completio of type service (or type2 service) for each positive customer, the usatisfied positive customers may rejoi the orbit as a feedback customer for receivig aother regular service with probability p (0 p ) or may leave the system with probability q where p + q =. Negative arrival process: The egative customers arrive from outside the system accordig to a Poisso arrival rate δ. These egative customers arrive oly at the service time of the positive customers. Oce the egative customer arrived ito the system it will remove the positive customer i service ad the iterrupted positive customer either eters ito the orbit with probability θ (0 θ ) or leaves the system forever with probability. Vacatio process: Wheever the orbit is empty, the server leaves for a vacatio of radom legth V. If o customer appears i the orbit whe the server returs from a vacatio, it leaves

5 agai for aother vacatio with the same legth. Such patter cotiues util it returs from a vacatio to fid at least oe customer i the orbit or it has already take J vacatios. If the orbit is empty at the ed of the J th vacatio, the server remais idle for ew arrivals i the system. If at a vacatio completio epoch the orbit is oempty, the server waits for the customers i the orbit or for a ew arrival. The vacatio time V has distributio fuctio Vt, () LST V ( ) ad fiite k th ( k) momet (k =,2). Breakdow process: While the server is workig with ay types of service, it may breakdow at ay time ad the service chael will fail for a short iterval of time, i.e., server is dow for a short iterval of time. The breakdows, i.e., server s life times are geerated by exogeous Poisso processes with rates a for FTS ad a 2 for STS, which we may call some sort of disaster durig FTS ad STS periods respectively. Repair process: As soo as a breakdow occurs the server is set for repair, durig which time it stops providig service to the primary customers util the service chael is repaired. The customer who was just beig served before server breakdow waits for the remaiig service to be completed. The repair time (deoted by G for FTS ad G 2 for STS) distributios of the server for both types of service are assumed to be arbitrarily distributed with distributio fuctio Gi ( t ), Laplace-Stieltjes Trasform Gi ( ) ad fiite k th ( k) momet g i (for i =,2 ad k =,2). Various stochastic processes ivolved i the system are assumed to be idepedet of each other. 2.. Practical justificatio of the suggested model The suggested model has potetial applicatio i the trasfer model of a system. I Simple Mail Trasfer Protocol (SMTP) mail system, delivers the messages betwee mail servers. Whe a mail trasfer program cotacts a server o a remote machie, it forms a Trasmissio Cotrol Protocol (TCP) coectio over which it commuicates. Oce the coectio is i place, the two programs follow SMTP that allows the seder to idetify it, specify a recipiet ad trasfer a message. For receivig a group of messages, cliet applicatios usually use either the Post Office Protocol (POP) or the Iteret Message Access Protocol (IMAP) to access their mail box accouts o a mail server. Typically, cotactig a group of messages arrive at the mail server followig the Poisso stream. Whe messages arrive at the mail server, it will be free. The oe of the messages from the group is selected to access successfully (i POP or IMAP) ad the rests will go to the buffer. I the buffer, each message waits ad requires its service agai after some time. If the server is iitially failed, all the arrivig group of messages will joi the buffer ad try this service after some time. The target server is the same as the seder s mail server ad the sedig message will be possibly retrasmitted to the server to request the receivig service oce agai from the buffer. The mail server may be subjected to electroic failure durig the service period ad receive repair immediately. Meawhile, the workig server may receive additioal task as a result of triggers/viruses. Upo arrival at the servers system, the trigger istataeously displaces the message beig served at the receiver mail server from the receiver mail to the buffer or is forced to leave the system.

6 To keep the mail server fuctioig well, virus scaig is a importat maiteace activity for the mail server. It ca be performed whe the mail server is idle. This type of maiteace ca be programmed to perform o a regular basis. However, these maiteace activities do ot repeat cotiuously. Whe these activities are fiished, the mail server will eter the idle state agai ad wait for the cotact messages to arrive. Because there is o mechaism to record how may cotactig messages are from various seders curretly, it is appropriate for desigig a program to collect iformatio of cotactig messages for the reaso of efficiecy. I this queueig sceario, the buffer i the seder mail server, the receiver mail server, the POP ad IMAP, iitial failure, the retrasmissio policy, flow of triggers/viruses ad the maiteace activities correspod to the orbit, the server, the Type ad Type2 service, the startig failure repair process, the feedback policy, the arrival of egative customers ad the vacatio policy, respectively. This model fids aother practical applicatio i Verteiler Espritz Pumps maufacturig, i computer etworkig systems, maufacturig systems ad commuicatio systems etc., For example, i the process of cell trasfer if the iterferece of a virus causes iformatio elemet trasmissio failure, the some kids of virus ca be see as egative customers. I computer etworkig systems, if virus eters a ode, oe or more files may be ifected. A virus may origiate from outside the etwork, e.g., through a floppy disk, or by a electroic mail, productio lies, i the operatioal model of WWW server for HTTP requests, call ceters, ivetory ad productio, maiteace ad quality cotrol i idustrial orgaizatios etc. 3. System aalysis I this sectio, we first develop the steady state differece-differetial equatios for the retrial system by treatig the elapsed retrial times, the elapsed service times, the elapsed vacatio times ad the elapsed repair times as supplemetary variables. The we derive the probability geeratig fuctio (PGF) for the server states, the PGF for umber of customers i the system ad orbit by usig the supplemetary variable method. I steady state, we assume that R(0) = 0, R() =, S i (0) = 0, S i () =, V j (0) = 0, V j () =, H(0) = 0, H() = (for i =,2 ad j =,2,J ) are cotiuous at x = 0 ad G i (0) = 0, G i () = (for i =,2) are cotiuous at x = 0 ad y = 0. So that the fuctios ax ( ), i( x), ( x), ( x) ad i( y) are the coditioal completio rates for retrial, service o both types, o vacatio ad repair o both types respectively (for i =, 2). dr( x) dsi( x) dv ( x) dh ( x) dgi( y) a( x) dx, i( x) dx, ( x) dx, ( x) dx ad i( y) dy. R( x) S ( x) V ( x) H( x) G ( y) i I additio, let R ( t), Si ( t), Vj ( t), H ( t) ad Gi ( t ) be the elapsed retrial times, service times o both types, vacatio times, repair times o startig failure server or repair times o both types (for i=,2 ad j =,2,3,4, J + 3) respectively at time t. We also ote that the states of the system at time t ca be described by the bivariate Markov process C( t), N( t); t 0 where C(t) deotes the server state (0,,2,3,4,, J + 3) depedig o the server is idle, busy o FTS or STS, repair o startig i

7 failure server, repair o FTS or STS ad st vacatio, 2 d vacatio,,j th vacatio. N(t) deotes the umber of customers i the orbit. If C(t) = 0 ad N(t) > 0, the R 0 () t represet the elapsed retrial time, if C(t) = ad Nt ( ) 0 the S 0 i () t correspods to the elapsed time of the customer beig served o FTS (STS) (for i =,2). If C(t) = 2 ad Nt ( ) 0, the H 0 () t correspods to the elapsed time of the failure server beig repaired. If C(t)=3 ad Nt ( ) 0, the G 0 i () t correspods to the elapsed time of the server beig repaired o FTS (STS) (for i =,2). If C(t) = 4 ad Nt ( ) 0, 0 the V () t correspods to the elapsed st vacatio time. If C(t) = j+4 ad Nt ( ) 0, the V 0 j () t correspods to the elapsed j st vacatio time. Let { t ; =,2,...} be the sequece of epochs at which either a type or type 2 service completio occurs, a vacatio period eds or a repair period eds. The sequece of radom vectors Z Ct, N t forms a Markov chai which is embedded i the retrial queueig system. It follows from the Appedix that system will be stable, where X [] ( R ( )) Z N ; ad is ergodic if ad oly if, the the [] [] () bx () () ( p) ps ( ) p2s2 ( ) bx h + p g S ( ) p2 2g2 S2 ( ) For the process N( t), t 0, we defie the probabilities P ( t) 0 P C( t) 0, N( t) 0 ad the probability desities for ( i (,2), t 0, ( x, y) 0 ad 0) P x t dx P C t N t x R t x dx 0 (, ) ( ) 0, ( ), ( ) ; 0 i, ( x, t) dx P C( t), N( t), x Si ( t) x dx ; Q x t dx P C t N t x H t x dx 0 (, ) ( ) 2, ( ), ( ) ; R x y t dy P C t N t G t y dy S t x 0 0 i, (,, ) ( ) 3, ( ), y i ( ) i ( ) ; 0 ( x, t) dx P C( t) j 3, N( t), x V ( t) x dx,for ( j J ) j, The followig probabilities are used i subsequet sectios: P0 () t is the probability that the system is empty at time t. P ( x, t ) is the probability that at time t there are exactly customers i the orbit ad the elapsed retrial time of the test customer udergoig retrial lyig i betwee x ad x + dx. i, ( xt, ), (i =,2) is the probability that at time t there are exactly customers i the orbit ad the elapsed service time of the test customer udergoig service lyig i betwee x ad x+dx i their respective types. Q ( x, t ) is the probability that at time t there are exactly customers i the orbit ad the elapsed repair time of server lyig i betwee x ad x + dx o the failure server. Ri, ( x, y, t ),(i =,2) is the probability that at time t there are exactly customers i the orbit, the elapsed service time of the test customer udergoig service is x ad the elapsed repair time of server lyig i betwee y ad y + dy i their respective types. j, ( xt, ), (j =,2,, J) is the probability that at time t there are exactly customers i the orbit ad the elapsed vacatio time of the vacatio lyig i betwee x ad x + dx. j We assume that the stability coditio is fulfilled ad so that we ca set for t 0, x 0, ad (i =, 2 ad j=,2,, J)

8 3.. The steady state equatios P lim P ( t), P ( x) lim P ( x, t), ( x) lim ( x, t), Q ( x) lim Q ( x, t), 0 0 i, i, t t t t ( x) lim ( x, t) ad R ( x, y) lim R ( x, y, t). j, j, i, i, t t By the method of supplemetary variable techique, we obtai the followig system of equatios that gover the dyamics of the system behavior. (3.) bp ( x) ( x) dx 0 J,0 0 dp ( x) [ a( x)] P ( x) 0, (3.2) dx di,0( x) [ i i ( x)] i,0 ( x) ( b) i,0 ( x) i ( y) Ri,0( x, y) dy, 0, for ( i,2) dx (3.3) di, ( x) [ ( x)] ( x) ( b) ( x) b ( x) dx i i i, i, k i, k k 0 0 ( y) R ( x, y) dy,, for ( i,2) i i, (3.4) dq0 ( x) [ ( x)] Q0 ( x) ( b) Qi,0( x), 0, (3.5) dx dq ( x) [ ( x)] Q ( x) ( b) Q ( x) b Q ( x), dx (3.6) i, k i, k k dri,0( x, y) [ i ( y)] Ri,0( x, y) ( b) Ri,0( x, y), 0, for( i,2) (3.7) dy dri, ( x, y) [ ( y)] R ( x, y) ( b) R ( x, y) R ( x, y),, for( i,2) dy (3.8) i i, i, k i, k k d j,0( x) [ ( x)] j,0 ( x) ( b) j,0( x), 0, for( j,2,..., J) (3.9) dx d j, ( x) [ ( x)] ( x) ( b) ( x) b ( x),, for( j,2,..., J) dx (3.0) j, j, k j, k k The steady state boudary coditios at x = 0 ad y = 0 are J P (0) j, ( x) ( x) dx Q ( x) ( x) dx, ( x) dx 2, ( x) dx j ( ) ( x) dx ( x) dx q ( x) ( x) dx ( x) ( x) dx, 2,, 2, p x x dx x x dx, ( ) ( ) 2, ( ) 2( ), (3.) 0 0 i,0(0) pi P ( x) a( x) dx bp0, 0, for( i=,2) (3.2) 0

9 i, (0) pi P ( x) a( x) dx k P k( x) dx bp0,, for( i=,2) 0 k 0 (3.3) Q P x a x dx P x dx b P (0) ( ) ( ) k k ( ) 0, 2, (3.4) 0 k 0 (0) q ( x) ( x) dx ( x) ( x) dx ( ) ( x) dx ( x) dx, 0 (3.5),,0 2,0 2,0 2, j, ( x) ( x) dx, 0, j 2,3..., J j, (0) 0 (3.6) 0, R ( x,0) ( x),, for( i,2) (3.7) i, i i, The ormalizig coditio is J 2 P0 P ( x) dx Q ( x) dx j, ( x) dx i, ( x) dx Ri, ( x, y) dxdy j 0 0 i The steady state solutio (3.8) The probability geeratig fuctio techique is used here to obtai the steady state solutio of the retrial queueig model. To solve the above equatios, we defie the geeratig fuctios for z, for (i=,2) as follows: i i, i i, 0 0 P( x, z) P ( x) z ; P(0, z) P (0) z ; ( x, z) ( x) z ; (0, z) (0) z ; Q( x, z) Q ( x) z ; j j, j j, i i, Q(0, z) Q (0) z ; ( x, z) = ( x) z ; (0, z) (0) z ; R ( x, y, z) R ( x, y) z ; i(,0, ) i, (,0) ad ( ) 0 R x z R x z X z z Multiplyig the steady state equatio ad steady state boudary coditio (3.) - (3.7) by z ad summig over, ( = 0,,2...) for (i =,2 ad j =,2, J) Pxz (, ) [ a( x)] P( x, z) 0 x (3.9) i ( xz, ) [ b( X ) i i ( x)] i ( x, z) i ( y) Ri ( x, y, z) dy, x (3.20) Q( x, z) [ b( X ) ( x)] Q( x, z) 0 x Ri ( x, y, z) [ b( X ) i( y)] Ri( x, y, z) 0 y j ( xz, ) [ b( X ) ( x)] j ( x, z) 0 x 0 (3.2) (3.22) (3.23)

10 J P(0, z) j ( x, z) ( x) dx z ( x, z) dx 2( x, z) dx Q( x, z) ( x) dx j (3.24) J pz q ( x, z) ( x) dx 2( x, z) 2 ( x) dx j,0(0) bp0, 0 0 j ( ) i(0, z) pi P( x, z) a( x) dx P( x, z) dx, for( i=,2) z z z X z bx () z P0 (3.25) 0 0 Q(0, z) P( x, z) a( x) dx X P( x, z) dx bx P0, 0 0 (3.26) R ( x,0, z) ( x, z) (3.27) i i i Solvig the partial differetial equatios (3.9)-(3.23), it follows that P( x, z) P(0, z)[ R( x)]exp x (3.28) i ( x, z) i (0, z)[ Si ( x)]exp Ai x, for( i=,2) (3.29) Q( x, z) Q(0, z)[ H( x)]exp b x, (3.30) j( x, z) j(0, z)[ V( x)]exp b x,for( j,2,... J) (3.3) Ri ( x, y, z) Ri ( x,0, z)[ Gi ( y)]exp b y, for( i=,2) (3.32) where A b( X ) G b ad b b X. i i i From (3.9) we obtai, bx ( x) (0)[ V( x)] e, j,2,..., J. (3.33) j,0 j,0 Multiplyig with equatio (3.33) by (x) o both sides for j = J ad itegratig with respect to x from 0 to, the from (3.) we have, 0,0 (0) bp J V ( b). (3.34) From the equatios (3.34) ad solvig (3.6), (3.33) over the rage j = J-, J-2,,, we get o simplificatio j 0,0(0) bp, j,2,..., J [ V ( b)] J j (3.35) From (3.6), (3.34) ad (3.35), we get bp0 j (0, z), j,2,..., J J j [ V ( b)] (3.36) Itegratig the equatio (3.33) from 0 to ad usig (3.34) ad (3.35) agai, we fially obtai j,0 J j P0 V ( b) (0, z), j,2... J [ V ( b)] (3.37) Note that, j,0 represets the steady-state probability that o customers appear while the server is o the j th vacatio. Let us defie 0 as the probability that o customers appear i the system while the server is o vacatio. The,

11 Isertig equatio (3.29)-(3.3) ad (3.36) i (3.37) J P0 [ V ( )] 0, (3.38) J bv [ ( )] * 0 2 z A (0, z) S A A (0, z) S A P(0, z) bp N Q(0, z) H b ( pz q) (0, z) S A (0, z) S A 2 A A where [ V ( b)] N V b( X ) J [ V ( b)] V ( b) Isertig equatio (3.28), (3.29) i (3.25) ad make some maipulatio, fially we get, J P(0, z) bx i(0, z) pi R ( ) X ( R ( )) P0, for ( i=,2) z z Isertig equatio (3.28), (3.29) i (3.26) ad make some maipulatio, fially we get, (3.39) (3.40) 0 (3.4) Q(0, z) P(0, z) R ( ) X ( R ( )) bx P, Usig the equatio (3.29) i (3.27), we get, Ri ( x,0, z) i i(0, z) Si( x) exp Ai x, for ( i=,2) (3.42) Usig (3.40)-(3.4) i (3.39), we get Nr() z P(0, z) (3.43) Dr() z * zn X ( pz q) ps A p2s2 A2 zh b A A2 Nr() z bp0 X ( z) p A2 S A p2a S2 A2 * z ( pz q) ps A p2s2 A2 zh b R ( ) X ( R ( )) A A2 Dr() z R ( ) X ( R ( )) ( z) p A2 S A p2a S2 A2 Usig (3.43) i (3.40), we get Usig (3.43) i (3.4), we get Usig (3.44) i (3.42), we get i(0, z) bp0 p i N R ( ) X ( R ( )) X A A2 Dr,for ( i=,2) Q(0, z) zbp 0 N R ( ) X ( R ( )) X A A2 Dr, N R ( ) X ( R ( )) X A A2 Ri ( x,0, z) ibp0 pi Dr,for ( i=,2) Ai x Si( x) e Usig (3.43)-(3.46) i (3.28)-(3.32), the we get the probability geeratig fuctio P x z ( x, z), H( x, z), R ( x, y, z) ad R ( x, y, z). 2 2 (3.44) (3.45) (3.46) (, ), ( x, z), Next we are iterested i ivestigatig the margial orbit size distributios due to system state of the server.

12 Theorem 3.. Uder the stability coditio ρ <, the joit distributios of the umber of customers i the system whe server beig idle, busy o both types, o vacatio, uder repair o startig failure server ad uder repair o both types (for i=, 2) are give by Pz () Nr() z Dr() z (3.47) * z N X ( pz q) ps A p2s2 A2 zh b A A2 Nr b R ( ) P0 X ( z) p A2 S A p2 A S2 A2 * z ( pz q) ps A p2s2 A2 zh b R ( ) X ( R ( )) A A2 Dr() z R ( ) X ( R ( )) ( z) p A2 S A p2a S2 A2 * bp0 p S A N R ( ) X ( R ( )) X A 2 Dr, * 2 bp0 p2 S2 A2 N R ( ) X ( R ( )) X A Dr, Q zbp0 H b N R ( ) X ( R ( )) X A A2 b Dr, R bp0 p b Dr, N R ( ) X ( R ( )) X A 2 * * S A G b N R ( ) X ( R ( )) X A * * S2 A2 G2 b R2 2bP0 p2 b Dr, P 0 V b( X ) j, j,2,..., J, J j X [ V ( b)] (3.48) (3.49) (3.50) (3.5) (3.52) (3.53) where Proof: [] ( ( )). P 0 X R (3.54) [] N() () b () () X ( R ( ))( b ) bh p [] g S ( ) p2 2g2 S2( ) X N() ( p) ps ( ) p2s2 ( ) b( R ( )) [] X J [] () V ( b) bx N(), Ai b i Gi b ad b b X. J V ( b) V ( b) Itegratig the above equatios (3.28) - (3.3) with respect to x ad defie the partial probability geeratig fuctios as,

13 P P( x, z) dx, ( x, z) dx, Q Q( x, z) dx, ( x, z) dx for ( i,2) i i j j Itegratig the above equatios (3.32) with respect to x ad y defie the partial probability geeratig fuctios as, R ( x, z) R ( x, y, z) dy, R R ( x, z) dx for ( i,2). i i i i 0 0 Sice, the oly ukow is P 0 the probability that the server is idle whe o customer i the orbit ca be determied usig the ormalizig coditio (j =,2,, J). Thus, by settig z = i (3.47) (3.53) ad applyig L Hôpital s rule wheever ecessary ad we get Theorem 3.2. J P P() () () Q() R () R () () Uder the stability coditio ρ <, probability geeratig fuctio of umber of customers i the system ad orbit size distributio at statioary poit of time is j j Nr Nr2 Nr3 ( ) ( ) ( ) Ks() z P0 Dr z Dr z Dr z * ps A p2s2 A2 z H b A A2 Nr z p A2 S A p2a S2 A2 N R ( ) X ( R ( )) X * za A2 ( pz q) ps A p2s2 A2 zh b R ( ) X ( R ( )) Nr2 N p A2 S A R ( ) X ( R ( )) ( z) p2a S2 A 2 za A2 b( R ( ))( N ) R ( ) X ( b)( R ( )) * Nr3 X ( pz q) ps A p2s2 A2 zh b A A2 + ( z) p A2 S A p2a S2 A2 * za A2 ( pz q) ps A p2s2 A2 zh b R ( ) X ( R ( )) A A2 Dr X p A2 S A R ( ) X ( R ( )) ( z) p2a S2 A 2 Nr4 Nr2 Nr3 ( ) ( ) ( ) Ko() z P0 Dr z Dr z Dr z (3.55) (3.56)

14 where P 0 is give i equatio (3.54). Proof: * ps A p2s2 A2 z H b A A2 Nr4 p A2 S A p2a S2 A2 N R ( ) X ( R ( )) X The probability geeratig fuctio of the umber of customers i the system K s (z) ad the probability geeratig fuctio of the umber of customer i the orbit K o (z) are obtaied by usig 2 J 2 J Ks P0 P z Q i Ri j ad Ko P0 P Q i Ri j i j i j Substitutig (3.47) (3.54) i the above results, the the equatios (3.55) ad (3.56) ca be obtaied by direct calculatio. 4. Performace Measures I this sectio, we obtai some iterestig probabilities, whe the system is i differet states. We also derive system performace measures. Our results are umerically validated. Note that (3.54) gives the steady state probability that the server is idle but available i the system. It follows from (3.47)-(3.53) that the probabilities of the server state are as follows i Theorem 4.. Theorem 4.. If the system satisfies the stability coditio ρ <, the we get the followig probabilities, (i) Let P be the steady state probability that the server is idle durig the retrial time, [] P P() b R ( ) N () X (ii) Let Π be the steady-state probability that the server is busy o first type service with positive customer, S ( ) [] () bp N() X R ( )) (iii) Let Π 2 be the steady-state probability that the server is busy o secod type service with positive customer, S2 ( ) [] 2 2() bp 2 N() X R ( ))

15 (iv) Let F Loss be the frequecy of the customer loss due to arrival of egative customers is give, [] FLoss ( ) b( ) N () X R ( ) p S ( ) p S ( ) (v) Let Q be the steady state probability that the server is o startig failure, () [] Q Q() bp h N() X R ( )) (vi) Let Ω be the steady state probability that the server is o vacatio, J j N() [] j () X R ( ) [] X (vii) Let R be the steady state probability that the server is uder repair time o first type service, S ( ) () [] R R () bp g N() X R ( )) (viii) Let R 2 be the steady state probability that the server is uder repair time o secod type service, S2 ( ) () [] R2 R2 () 2bp2g 2 N() X R ( )). The stated formula follows by direct calculatio. Theorem 4.2. Let L s, L q, W s ad W q be the mea umber of customers i the system, the mea umber of customers i the orbit, average time a customer speds i the system ad average time a customer speds i the orbit, respectively. The uder the stability coditio, we have L q Nrq () Drq () Drq () Nrq () P Drq() 2 [] [] () R ( ) N () b X X ( b ) bh [] + N() X b R ( ) ( p) ps ( ) p2s2 ( ) p A S p2 A2 S2 [] [] 3 N() X ( b)( R ( )) ( p) ps ( ) p2s2 ( ) [] [2] [] ( R ( )) ( b) 2X X 2 p X ( R ( )) [] [2] 2 px X 2 Nr() 2 2 () ( ) () ( ) Nr () 6 X ( b) R ( ) A() A () p S ( ) p S ( ) 3 N () p S ( ) A () p S ( ) A () R ( ) p S ( ) p S ( )

16 [] b X ( R ( )) 2 A() A2() [] N p X ( R ( )) ps ( ) p2s2 ( ) [] 2 X R ( ) p A () S ( ) p2 A2() S2 ( ) [] 3 X b R ( ) p A2 () S ( ) p2 A () S2 ( ) [] 6 X b R ( ) p A2() S ( ) p2 A () S2 ( ) [] X b R A A2 A A2 ps p2s2 [] A A2 X b R X ps ( ) A () p2s2 ( ) A2() bh X [] A () A2 () ps ( ) p2s2 ( ) [2] [] R X X b p A S p2 A2 S 2 [] X b R p ps p2s2 bh X [2] X b R 2 [2] [] b R X X p ps A p2s2 A2 3 () 6 ( ) () () () () ( ) ( ) 2 () () 3 ( ) ( ) [] () [] 3 X ( b) R ( ) 3 ( ) 2 ( ) () ( ) () ( ) 2 ( ) ( ) 2 2 () [] 3 ( ) ( ) ( ) ( ) 2 [] [2] [] [] () [] (2) [2] () 3 X ( b)( R ( )) 2X 2X bx 2h bx h bx h 2 p S ( ) A () S ( ) A () 2 p2 S2 ( ) A2 () S2 ( ) A2 () 3 ( ) 2 ( ) ( ) () ( ) () 2 [] 2 [2] () 3 R ( ) X bx h [] [] Dr() 2X X ( R ( )) [] 2 2 () 3 ( ) () ( ) () Dr X p S A S A [] X p2 S2 ( ) A2 () S2 ( ) A2 () [] () [] (2) [2] () bx 2 h bx h X bh ( R ( )) [] [] () 2 X ( R ( )) bx h [] 2 [] 6 X p X ( R ( )) p S ( ) A () p S ( ) A () [] 2 [] [2] 3X R ( ) p S ( ) p S ( ) 2pX X [] 2 3X ps ( ) A () p2s2 ( ) A2() [] 2 p X ( R ( )) ps ( ) p2s2 ( ) [] () [] bx h X ( R ( )) [] 6 X A () A ()

17 Proof: L s [] 3X p S ( ) p S ( ) A () A () 2 A () A () [2] [] 3X X ( R ( )) Nrs () Drq () Drq () Nrq () P Drq() 2 [] Nrs () Nrq () 6 N() X R ( ) p A () S ( ) p2 A () S2 ( ) bx h L L s q Ws ad Wq E( X ) E( X ) 2 2 ( b) E( V ) [ V ( b)] where N() ; J V ( b) [ V ( b)] x 2 x i i i i S ( ) xe ds ( x), S ( ) x e ds ( x), for( i=,2), 0 0 () () 2 2 [2] () (2) [] [2] () (2) [] 2 2 [] [] (), 2() 2 A bx g A bx g A() bx g g bx ad A() bx g g bx J 2 [] () The mea umber of customers i the orbit (L q ) uder steady state coditio is obtaied by differetiatig (3.56) with respect to z ad evaluatig at z =, ad we get d Lq Ko () lim Ko. z dz The mea umber of customers i the system (L s ) uder steady state coditio is obtaied by differetiatig (3.55) with respect to z ad evaluatig at z = ad we get d Ls Ks () lim Ks. z dz The average time a customer speds i the system (W s ) ad the average time a customer speds i the queue (W q ) are foud by usig the Little s formula. 5. Special cases Ls Ws ad Lq Wq. I this sectio, we aalyze briefly some special cases of our model, which are cosistet with the existig literature. Case : Sigle type, No batch arrival, No Vacatio, No balkig, No startig failure, No breakdow ad Expoetial retrial.

18 Let Pr[X k = ] = ; Pr [S 2 = 0] = ; b = = ; Pr [V = 0] = ; p = 0 ad 2 0. Our model ca be reduced to a sigle server feedback retrial queueig system with egative customers. The followig expressio is coicided with the result of Krishakumar et al. (203). Case 2: Au ( ) S ( ( z) ) Dr Ko exp du B( u) Dr 0 Dr z ( r rz) S ( ( z) ) ( ) ( r) S ( ( z) ), S ( ) where ( S ( )) ( ) rs Sigle type, No batch arrival, No retrial, No feedback, No egative customer, No balkig ad No breakdow Let p 2 = 0, Pr[S 2 = 0] =, p = δ = 0; b =, R * () ad 2 0. The we get a batch arrival queueig system with balkig ad modified vacatios. K z S b( X z S b( X J J [ V ( b)] V b( X ( X ) [ V ( b)] V ( b) J () J b [ V ( b)] V ( b) V ( b) X The above result coicides with the result of Ke (2007). Case 3: Sigle type, No batch arrival, No egative arrival ad No breakdow Let Pr[X k = ] = ; p =, Pr [S 2 = 0] =, δ = 0 ad 2 0. Our model ca be reduced to a modified vacatio for a M/G/ retrial queueig system with balkig ad feedback. I this case, the followig expressio is coicided with the result i Ke ad Chag (2009a). Case 4: z S A N R ( ) z R ( ) N( R ( )) Ks P0 N z ( pz q) R ( ) z( R ( )) S A b( z) z R ( ) N( R ( )) ( pz q) R ( ) S A b( z) z ( pz q) R ( ) ( ( )) z R SA ( z ) Sigle type, No batch arrival, No Vacatio, No feedback, No balkig, No egative customers ad No breakdow Let Pr[X k = ] = ; Pr [S 2 = 0] = ; p = 0; r = 0; δ = 0; b = ; Pr[V = 0] = ad 2 0. The we get a sigle server retrial queueig system with geeral retrial times. The followig result coicides with the result of Gomez-Corral (999). J ;

19 ( ) ( ) ( ) 2 ( ) ( ) K z L K z 2 2 R E S z S z E S E S R s ( ) ; q lim o ( ) z z R ( ) z( R ( )) S z 2 R ( ) E( S) 6. Numerical illustratio I this sectio, we preset some umerical examples usig MATLAB i order to illustrate the effect of various parameters i the system performace measures. We cosider retrial times, service times, vacatio times ad repair times are expoetially, Erlagialy ad hyperexpoetially distributed. Further, we assume that customers are arrivig oe by oe, so X [] =, X [2] = 0. The arbitrary values to the parameters are chose like µ = 8; µ 2 = 0; ξ = 6; ξ 2 = 8; = 2; 2 = ; = 5; c = 0.8 such that they satisfy the stability coditio. The followig tables give the computed values of various characteristics of our model such as probability that the server is idle P 0, the mea orbit size L q, probability that server is idle durig retrial rime (P), busy o both types phases (Π, Π 2 ), o vacatio (Ω), repair o failure server (Q), F Loss probability ad uder repair o both types (R, R 2 ), respectively. The expoetial distributio is x f ( x) e 2 x, x 0, Erlag-2stage distributio is f ( x) xe, x 0 ad hyper-expoetial distributio is 2 x 2 x. f ( x) c e ( c) e, x 0 Table shows that whe egative arrival rate (δ) icreases, the the probability that server is idle P 0 icreases, the mea orbit size L q icreases ad the probability that the frequecy of the customer loss due to arrival of egative customer server (F Loss ) also icreases. Table 2 shows that whe service loss probability (θ) icreases, the the probability that server is idle P 0 decreases, the mea orbit size L q icreases ad the probability that server is idle durig retrial time P also icreases. Table 3 shows that whe the successful arrival probability (α) icreases, the probability that server is idle P 0 icreases, the mea orbit size L q decreases ad probability that server is idle durig retrial time P also decreases. Table 4 shows that whe the umber of vacatios (J) icreases, the probability that server is idle P 0 decreases, the probability that server is idle durig retrial time P icreases ad the probability that server is o vacatio Ω also icreases. Table 5 shows that whe repair rate o FTS (ξ ) icreases, the probability that server is idle P 0 icreases, the the mea orbit size L q decreases ad probability that server is uder repair o FTS R also decreases. For the effect of the parameters, δ, p, α, a, θ, p, ad o the system performace measures i two dimesioal graphs are draw i Figures ad 2. Figure shows that the mea orbit size L q icreases for icreasig value of the egative arrival rate (δ). Figure 2 shows that the idle probability P 0 decreases for icreasig value of the service loss probability (θ). Three dimesioal graphs are illustrated i Figures 3 Figure 6. I Figure 3, the surface displays a upward tred as expected for icreasig value of the arrival rate () ad egative arrival rate (δ) agaist the mea orbit size L q. The mea orbit size L q decreases for icreasig value of the feedback probability (p) ad balkig probability (b) i Figure 4. The surface displays a upward tred as expected for icreasig value of the successful service probability (α) ad vacatio rate (γ) agaist the idle probability P 0 i Figure 5. I Figure 6, the mea orbit size L q decreases for icreasig value of the first type probability (p ) ad repair rate o FTS (ξ ). Table. The effect of egative arrival probability (δ) o P 0, Lq ad F Loss.

20 Retrial distributio Expoetial Erlag-2 stage Hyper-Expoetial δ P 0 L q F Loss P 0 L q F Loss P 0 L q F Loss Negative arrival rate Table 2. The effect of service loss probability (θ) o P 0, Lq ad P Retrial distributio Expoetial Erlag-2 stage Hyper-Expoetial θ P 0 L q P P 0 L q P P 0 L q P Service loss probability Table 3. The effect of failure probability ( ) o P 0, Lq ad P. Repair distributio Expoetial Erlag-2 stage Hyper-Expoetial α P 0 L q P P 0 L q P P 0 L q P Successful probability Table 4. The effect of umber of vacatios (J) o P 0, P ad Ω. Vacatio distributio Expoetial Erlag-2 stage Hyper-Expoetial J P 0 P Ω P 0 P Ω P 0 P Ω Number of vacatios

21 Table 5. The effect of repair rate o FTS (ξ ) o P 0, Lq ad R. Repair distributio Expoetial Erlag-2 stage Hyper-Expoetial ξ P 0 L q R P 0 L q R P 0 L q R Repair rate o FTS Figure. L q versus δ Figure 2. P 0 versus θ Figure 3. L q versus λ ad δ Figure 4. L q versus p ad b

22 Figure 5. P 0 versus α ad γ Figure 6. L q versus p ad From the above umerical examples, we ca fid the ifluece of parameters o the performace measures i the system ad kow that the results are coicidet with the practical situatios. 7. Coclusio I this paper, we have studied a batch arrival feedback retrial G-queueig system with balkig uder modified vacatio policy ad startig failures, where the server provides two types of service. The probability geeratig fuctios of the umber of customers i the system ad orbit are foud by usig the supplemetary variable techique. The explicit expressios for the average queue legth of orbit/system ad the average waitig time of customer i the system/orbit have bee obtaied, which provide a isight to the system desigers ad maagemet for reducig the waitig time ad the queue size of the cocered orgaizatio uder uavoidable techo-ecoomic costraits. The aalytical results are validated with the help of umerical illustratios. This model fids potetial applicatio i packet switched etwork to forward the packets withi a etwork for trasmissio ad Simple Mail Trasfer Protocol (SMTP) to deliver the messages betwee mail servers. The ovelty of this ivestigatio is the itroductio of feedback retrial queueig system with egative customers, balkig ad modified vacatio where the server is subject to breakdow. Moreover, our model ca be cosidered as a geeralized versio of may existig queueig models equipped with may features. Hopefully, this ivestigatio will be of great help for the system maagers to make decisios regardig the size of the system ad other factors i a well-to-do maer. Ackowledgemet The authors are thakful to the aoymous referees for their valuable commets ad suggestios for the improvemet of the paper. REFERENCES Artalejo, J.R. ad Gomez-Corral, A. (2008). Retrial queueig systems: a computatioal approach. Spriger, Berli. Artalejo, J.R. (200). Accessible bibliography o retrial queues: Progress i , Mathematical ad Computer Modellig, 5(9-0): Chag, F. M. ad Ke, J.C. (2009). O a batch retrial model with J vacatios, Joural of Computatioal ad Applied Mathematics, 232(2): Che, P., Zhu, Y. ad Zhag, Y. (200). A Retrial Queue with Modified Vacatios ad Server Breakdows, IEEE : Choudhury, G. ad Deka, K. (202). A sigle server queueig system with two phases of service subject to server breakdow ad Beroulli vacatio, Applied Mathematical Modellig, 36(2):

23 Choudhury, G., Tadj, L. ad Deka, K. (200). A batch arrival retrial queueig system with two phases of service ad service iterruptio, Computers ad Mathematics with Applicatios, 59(): Fali, G.I. ad Templeto, J.G.C. (997). Retrial Queues, Chapma & Hall, Lodo. Gao, S. ad Wag, J. (204). Performace ad reliability aalysis of a M/G/-G retrial queue with orbital search ad o-persistet customers, Europea Joural of Operatioal Research, 236(2): Gomez-Corral, A. (999). Stochastic aalysis of a sigle server retrial queue with geeral retrial times, Naval Res. Logist., 46(5): Haghighi, A.M. ad Mishev, D.P. (203). Stochastic three-stage hirig model as a tadem queueig process with bulk arrivals ad Erlag phase-type selectio, M X /M (k,k) /-M Y /E r /-. It. J. Mathematics i Operatioal Research, 5(5): Ke, J.C. (2007). Operatig characteristic aalysis o the M [X] /G/ system with a variat vacatio policy ad balkig, Applied Mathematical Modellig, 3(7): Ke, J.C. ad Chag, F.M. (2009a). Modified vacatio policy for M/G/ retrial queue with balkig ad feedback. Computers ad Idustrial Egieerig, 57(): Ke, J.C. ad Chag, F.M. (2009b). M [X] /(G, G 2 )/ retrial queue uder Beroulli vacatio schedule with geeral repeated attempts ad startig failures, Applied Mathematical Modellig, 33(7): Ke, J.C. ad Choudhury, G. (202). A batch arrival retrial queue with geeral retrial times uder Beroulli vacatio schedule for ureliable server ad delayig repair, Applied Mathematical Modellig, 36(): Krishakumar, B. ad Arivudaiambi, D. (2002). The M/G/ retrial queue with Beroulli schedules ad geeral retrial times, Comp. ad Math. with Applicatios, 43(-2), Krishakumar, B., Pavai Madheswari, S. ad Aatha Lakshmi, S.R. (203). A M/G/ Beroulli feedback retrial queueig system with egative customers, Oper. Res. It. J., 3(2): Rajadurai, P., Saravaaraja, M.C. ad Chadrasekara, V.M. (204). Aalysis of a M [X] /(G,G 2 )/ retrial queueig system with balkig, optioal re-service uder modified vacatio policy ad service iterruptio, Ai Shams Egieerig Joural, 5(3): Rajadurai, P., Varalakshmi, M., Saravaaraja, M.C. ad Chadrasekara, V.M. (205) Aalysis of M [X] /G/ retrial queue with two phase service uder Beroulli vacatio schedule ad radom breakdow, It. J. Mathematics i Operatioal Research, 7():9 4. Seott, L.I., Humblet, P.A. ad Tweedi, R.L. (983). Mea drifts ad the o ergodicity of Markov chais, Operatio Research, 3(4): Wag, J. Huag, Y. ad Dai, Z. (20). A discrete-time o-off source queueig system with egative customers, Computers ad Idustrial Egieerig, 6(4): Wag, J. ad Li, J. (2009). A sigle server retrial queue with geeral retrial times ad two phase of service, Jrl. Syst. Sci. & Complexity, 22(2): Wag, J. ad Zhag, P. (2009a). A discrete-time retrial queues with egative customers ad ureliable server, Computers ad Idustrial Egieerig, 56(): Wu, J. ad Lia, Z. (203). A sigle-server retrial G-queue with priority ad ureliable server uder Beroulli vacatio schedule, Computers & Idustrial Egieerig, 64(): Yag, S., Wu, J. ad Liu, Z. (203). A M [X] /G/ retrial G-queue with sigle vacatio subject to the server breakdow ad repair, Acta Mathematicae Applicatae Siica, Eglish Series. 29(3):

24 The embedded Markov chai stable, where X [] ( R ( )) Proof: Z N ; ad APPENDIX is ergodic if ad oly if for our system to be () bx[] () () ( p) ps ( ) p2s2 ( ) bx[] h + p g S ( ) p2 2g2 S2( ) From Gomez-Corral (999), it is ot difficult to see that Z ; N is a irreducible ad a aperiodic Markov chai. To prove Ergodicity, we shall use the followig Foster s criterio: a irreducible ad a aperiodic Markov chai is ergodic if there exists a oegative fuctio f(j), j N ad ε > 0, such that mea drift j Ef ( z) f ( z) / z j is fiite for all j N ad j for all j N, except perhaps for a fiite umber j s. I our case, we cosider the fuctio f(j) = j. The we have, j 0, j X[] ( R ( )), j,2... Clearly the iequality X [] () ( p) ps ( ) p2s2 ( ) bx[] h ( R ( )) bx [] () () + p g S ( ) p2 2g2 S2 ( ) is a sufficiet coditio for ergodicity. The same iequality is also ecessary for ergodicity. As oted i Seot et al. (983), we ca guaratee o-ergodicity, if the Markov chai Z ; satisfies Kapla s coditio, amely, j < for all j 0 ad there exits j 0 N such that j 0 for j j 0. Notice that, i our case, Kapla s coditio is satisfied because there is a k such that m ij = 0 for j < i - k ad i > 0, where = (m ij ) is the oe step trasitio matrix of Z ; N. The X [] () ( p) ps ( ) p2s2 ( ) bx[] h ( R ( )) bx [] () () + p g S ( ) p2 2g2 S2( ) implies the o-ergodicity of the Markov chai.