Tamkag Joural of Sciece ad Egieerig, Vol., No. 3, pp. 8392 (27) 83 A Feedback Retrial Queuig System with Startig Failures ad Sigle Vacatio G. S. Mokaddis, S. A. Metwally ad B. M. Zaki Ai Shams Uiversity, Faculty of Sciece, Departmet of Pure Mathematics, Cairo, Egypt Abstract The M/G/ retrial queue with Beroulli feedback ad sigle vacatio is studied i this paper, where the server is subjected to startig failure. The retrial time is assumed to follow a arbitrary distributio ad the customers i the orbit access the server uder FCFS disciplie. The server leaves for a vacatio as soo as the system becomes empty. Whe the server returs from the vacatio ad fids o customers, he waits free for the first customer to arrive from outside the system. The system size distributio at radom poits ad various performace measures are derived. The geeral decompositio law is show to hold good for this model also. Some of the eistig results i [7] are deced as special cases from our results. Key Words: Feedback, Vacatio, Startig Failures, Retrial Queues, Steady State. Itroctio Retrial queues have bee widely used to model may problems i telephoe switchig systems, telecommuicatio etworks ad computers competig to gai service from a cetral processor uit. Most of the papers o retrial queues have cosidered the system without feedback. A more practical retrial queue with feedback occurs i may practical situatios: for eample, multiple access telecommuicatio systems, where messages tured out as errors are set agai, ca be modeled as retrial queue with feedback. Choi ad Kulkari [] have studied M/G/ retrial queue with feedback. A remarkable ad uavoidable pheomeo i the service facility of a queuig system is its breakdow. Kulkari ad Choi [2] have aalysed the M/G/ retrial queue with server subjected to repairs ad breakdows. Further, Choi et al. [3] have ivestigated the M/G/ retrial queue with customer collisios. Recetly, Yag ad Li [4] have studied the M/G/ retrial queue with the server subject to startig Correspodig author. E-mail: gmokaddis@yahoo.com failures. They obtaied aalytical results for the queue legth distributio ad the stochastic decompositio law, where the retrial time is assumed to be epoetially distributed. Fayolle [5] has ivestigated a M/M/ retrial queue where the customers i the retrial group form a queue ad oly the customer at the head of the queue ca request a service to the server after epoetially distributed retrial time with costat rate. A retrial queuig system with FCFS disciplie ad geeral retrial times has bee etesively discussed by Gomez-Cortal [6]. Krisha, Pavai ad Vijayakumar [7] discussed a retrial queue with feedback ad startig failures. I this paper, we cosider a M/G/ retrial queue with Beroulli feedback ad sigle vacatio where the server is subjected to startig failures. We also assume that the retrial time is govered by a arbitrary distributio ad that the customer at the head of the orbit queue is allowed for access to the server. 2. Model Descriptio We cosider a sigle-server retrial queue with the
84 G. S. Mokaddis et al. server beig subjected to startig failures ad sigle vacatio. New customers arrive from outside the system accordig to a Poisso process with rate. We assume that there is o waitig space ad therefore if a arrivig customer fids the server busy or dow, the customer is obliged to leave the service area ad repeat his request for service after some radom time. Betwee trials, the blocked customer jois a pool of usatisfied customers called orbit i accordace with a FCFS disciplie. That is, oly the customer at the head of the orbit queue is allowed for access to the server. Successive iter-retrial times of ay customer are govered by a arbitrary probability distributio fuctio A(), with correspodig desity fuctio a() ad Laplace-Stieltjes trasform (). If the server is free, a arrivig (primary or retrial) customer must start (tur o) the server, which takes egligible time. If the server is started successfully (with a certai probability), the customer gets service immediately. Otherwise, the repair for the server commeces immediately ad the customer must leave for the orbit ad make a retrial at a later time. Successive service times are idepedet with commo probability distributio fuctio B(), desity fuctio b(), Laplace- Stieltjes trasform () ad first three momets, 2 ad 3. Similarly, the successive repair times are idepedet ad idetically distributed with probability distributio fuctio D(), desity fuctio d(), Laplace- Stieltjes trasform () ad first three momets, 2 ad 3. As soo as the system becomes empty, the server leaves the system for a vacatio period of radom legth. O returig from the vacatio, the server either fids o customers ad waits free util the arrival of a primary customer, or fids at least oe customer ad begis service. Vacatio times are idepedet with commo probability distributio fuctio G(), desity fuctio g(), Laplace-Stieltjes trasform () ad first three momets, 2 ad 3. After the customer is served completely, he will decide either to joi the retrial group agai for aother service with probability p or to leave the system forever with probability q (= p). It is assumed that the probability of successful commecemet of service is for a ew customer who fids the server free ad sees o other customer i the orbit (the couterpart of a customer who starts a busy period i the stadard M/G/ system) ad is for all other ew ad returig customers. Iterarrival times, retrial times, service times, vacatio times ad breakdow times are assumed to be mutually idepedet. From this descriptio, it is clear that at ay service completio, the server becomes free ad i such a case, a possible ew (primary) arrival ad the oe (if ay) at the head of the orbit queue, compete for service. The state of the system at time t ca be described by the Markov process {N(t); t } = {(C(t), X(t), (t), (t), 2 (t), 3 (t)), t }, where C(t) deotes the server state (,, 2 or 3 accordig as the server beig free, busy, repair or o vacatio out of the service facility, respectively) ad X(t) correspods to the umber of customers i orbit at time t. If C(t) = ad X(t) >, the (t) respresets the elapsed retrial time, if C(t) =, the (t) correspods to the elapsed time of the customer beig served, if C(t) = 2 ad X(t) >, the 2 (t) represets the elapsed repair time at time t, if C(t) = 3 ad X(t), the 3 (t) represets the elapsed vacatio time at time t. The fuctios r(), (), () ad () are the coditioal completio rates (at time ) for repeated attempts, for service, for repair ad for vacatio, respectively; i.e., r() = a() ( A()), () = b() ( B()), () = d() ( D()) ad () = g() ( G()). 3. Steady State Distributio For the process {X(t); t } we defie the probability P (t) = p {C(t) =, X(t) = }, P = Lim P (t); t ad the probability desities P (, t) = p {C(t) =, X(t) =, (t)<+}, for t, ad, P () = Lim P (, t), for ad ; t Q (, t) = p {C(t) =, X(t) =, (t)<+}, for t, ad, Q () = Lim Q (, t), for ad ; t R (, t) = p {C(t) = 2, X(t) =, 2 (t)<+}, for t, ad, R () = Lim R (, t), for ad ; t
A Feedback Retrial Queuig System with Startig Failures ad Sigle Vacatio 85 ad W (, t) = p {C(t) = 3, X(t) =, 3 (t)<+}, for t, ad, W () = Lim W (, t), for ad. t By the method of supplemetary variable techique, we obtai the followig system of equatios that gover the dyamics of the system behavior: Q (,) t P (,) t P (,)() t r, R (, t) P ( t) P(, t) r (3.) (3.2) dp () t P() t W(,) t dt P(, t) P(, t) [ r ( )] P ( t, ), t Q(, t) Q(, t) [ ] Q (, t) t (3.) (3.2) (3.3) R (,) t P (,) t P (,)() t r, 2 W (, t) q Q (, t) (3.3) (3.4) W (,t)=, (3.5) Q(, t) Q(, t) [ ] Q (, t) t Q (, t), R(, t) R(, t) [ ] R (, t) t R(, t) R(, t) [ ] R (, t) t R (, t), 2 W(, t) W(, t) [ ] W (, t) t (3.4) (3.5) (3.6) (3.7) ad the ormalizig coditio P () t P (,) t Q (,) t Lettig t i equatios (3.)(3.6), oe has R (, t) W (, t) P W (3.6) (3.7) W(, t) W(, t) [ ] W (, t) t W (, t), with the boudary coditios (3.8) dp [ r ( )] P, dq [ ] Q (3.8) (3.9) P (,) t q Q (,) t () p Q (,) t () R (, t) W (, t), Q (, t) P ( t) P(, t) r (3.9) (3.) dq [ ] Q Q, dr [ ] R dr [ ] R R, 2 (3.2) (3.2) (3.22)
86 G. S. Mokaddis et al. dw [ ] W dw [ ] W W, The steady-state boudary coditios are P() q Q ( ) p Q ( ) R W, Q () P P r Q() P( ) P ( r ) ( ), (3.23) (3.24) (3.25) (3.26) (3.27) Pz (, ) zp, Qz (, ) zq, R(, z) z R ad W(, z) z W The followig theorem discusses the steady state distributio of the system. Theorem 3.: The joit steady state distributio of {X(t); t } is obtaied as P z{( pzq) ( ( z)) z ( ( z)) } [ ( ( z))] zw(, z) P(,z) = z{( pzq) ( ( z)) z ( ( z))} { z( z) ( )} e r( u) (3.33) R () P P r (3.28) R() P ( ) P( r ) ( ), 2 (3.29) W () q Q (3.3) W ()=, (3.3) ad the ormalizig coditio is P P( ) Q( ) R W (3.32) To solve equatios (3.7)(3.32), we defie the geeratig fuctios Qz (, ) P P[ z( z) ( )] [( pz q) ( ( z)) z ( ( z)) ] [ ( ( z))][ z( z) ( )] W(, z) z[( pzq) ( ( z)) z ( ( z))] [ z( z) ( )] e ( z) ( u) Rz (, ) Pz Pzz [ ( z) ( )] [( pz q) ( ( z)) z ( ( z)) ] z[ ( ( z))][ z( z) ( )] W(, z) z[( pzq) ( ( z)) z ( ( z))] [ z( z) ( ) e W(, z) W(, z) e where ( z) ( u) ( z) ( u) (3.34) (3.35) (3.36)
A Feedback Retrial Queuig System with Startig Failures ad Sigle Vacatio 87 W(, z) q Q P / ( ) (see[8]) ( u ) ( z) ( ( z)) e e ( u ) ( z) ( ( z)) e e ( u ) ( z) ( ( z)) e e ru ( ) ( ) re ( ) e ad the probability P ca be determied from the ormalizatio coditio. Proof: Multiplyig equatios (3.7)(3.3) by z ad summig over, =,,2,...oecaobtai the followig equatios: Pz (, ) [ r ( )] Pz (, ) Qz (, ) [ ( z) ] Q(, z) Rz (, ) [ ( z) ] R(, z) W(, z) [ ( z) ] W(, z) (3.37) (3.38) (3.39) (3.4) P(, z) ( pzq) Qz (, ) ( ) Rz (, ) ( ) W (, z) q Q P (3.4) Q(, z) P P(, z) P(, z) r z (3.42) R(, z) z P z P(, z) P(, z) r W(, z) q Q (3.43) (3.44) Solvig the partial differetial equatios (3.37)(3.4), to obtai Pz (, ) P(, ze ) Qz (, ) Q(, ze ) Rz (, ) R(, ze ) W(, z) W(, z) e r( u) ( z) ( u) (3.45) (3.46) (3.47) (3.48) Usig (3.46)(3.48) i (3.4) ad solvig for P(,z), after some maipulatios, oe ca get [ z( z) ( )] (3.49) Combiig (3.45), (3.49) ad (3.42) ad o simplificatio we have ( z) ( u) ( z) ( u) P z{( pzq) ( ( z)) z ( ( z)) } [ ( ( z))] zw(, z) P(, z) z[( pzq) ( ( z)) z ( ( z))] Q(, z) P P[ z( z) ( )] [( pz q) ( ( z)) z ( ( z)) ] [ ( ( z))][ z( z) ( )] W(, z) z[( pzq) ( ( z)) z ( ( z))] [ z( z) ( )] (3.5)
88 G. S. Mokaddis et al. Similarly, substitutig (3.45) ad (3.49) ito (3.43) we get (3.5) Combiig (3.45)(3.5), we obtai the required results (3.33)(3.35). For the limitig probability geeratig fuctios P(,z), Q(,z), R(,z) ad W(,z) defie P( z) P(, z), Q( z) Q(, z), R( z) R(, z), ad W(z) = Wz (, ) ad the probability geeratig fuctio of the umber of customers i the system uder steady state is K(z) = P + P(z) + zq(z) + R(z) + W(z). The the mai result is give by Theorem 3.2: I steady state, R(, z) Pz Pz[ z( z) ( )] [( pz q) ( ( z)) z ( ( z)) ] z[ ( ( z))][ z( z) ( )] W(, z) z[( pzq) ( ( z)) z ( ( z))] [ z( z) ( )] Pz ( ) z[ ( )] P [( pzq) ( ( z)) ( ( )) ] [ ( ( ))] (, ) z z z W z z[( pzq) ( ( z)) ( ( ))][ ( ) ( )] z z z z (3.52) Qz ( ) [ ( ( z))] P [ z{ z( z) ( )} ( ) z ( ( z)) }] [ ( ( z))] [ z( z) ( )] W(, z) ad where Rz ( ) z[ ( ( z))] P z{ z( z) ( )} pz q z {( )( ) ( ( )) } z[ ( ( z))][ z ( z) ( )] W(, z) z [( pz q) ( ( z)) (3.54) (3.55) (3.56) (3.57) Proof: Itegratig the equatios (3.33)(3.36) from to with respect to, we obtai respectively (3.52)(3.55). At this poit, the oly ukow is P which ca be determied usig the ormalizig coditio, P + P() + Q() + R() + W() =. Thus, settigz=i(3.52)(3.55) ad applyig L Hopital s rule wheever ecessary, we get after usig the ormalizig coditio ad rearragemet. ( ( ))][ ( ) ( )] z z z z [ ( ( z))] W( z) W(, z) ( z) K( z) P [ z( z) ( )][ q ( ( z)) { z ( ( z))( ) }] zq ( ( z)) q ( ( z))[ ( ( z))] [ z( z) ( )] W(, z) z[( pzq) ( ( z)) ( ( ))][ ( ) ( )] z z z z q ( ) q( )( ) P ( ) ( p) ( ) qw(, z) ( ) ( p) ( ) P[ q ( ) q( )( )] q W(, z) ( z) z[( pzq) ( ( z)) ( ( ))][ ( ) ( ) z z z z (3.53) which yields (3.57). Fially, the probability geeratig fuctio K(z) = P +
A Feedback Retrial Queuig System with Startig Failures ad Sigle Vacatio 89 P(z) + zq(z) + R(z) + W(z) of the umber of customers i the system is obtaied by usig (3.52)(3.55) ad some mathematical maipulatios yield (3.56). Remark 3.. If = =q=,the (3.56) ad (3.57) become ad K( z) ( ( z)) ( z)[ ( ) W(, z)] (3.58) (3.59) If the vacatio do ot eist, the results (3.58), (3.59) will rece to the same results i [7] which agree with Gomez-Corral [6] without feedback ad startig failures. z z z W z [ ( ( ))][ ( ) ( )] (, ) ( )( z) ( ( z)) z[ ( ( z))] P W(, z) ( ) ad D, the steady probability that the server is dow or free or o vacatio. From Theorem 3.2, we obtai: ad U = Q() = /q, I P() q ( ) q( )( ) [ ( )][ q ] q [ ( )] W(, z) R R() q q { ( ) ( )( p )} q ( ) W(, z) ( ) ( ) ( ) W = W() = W(, z) D=P + P() + R() + W()= q The mea umber of customers i the system L S uder steady state coditios is obtaied by differetiatig (3.56) with respect to z ad evaluatio atz=. Remark 3.2. Further, if the retrial time distributio is epoetial with parameter, the () =/( + ). I this case (3.58) yields K( z) ( ( z)) ( z)[ ( ) W z z z W z ( ) (, )] [ ( ( ))]( ) (, ) [ ( ( z)) z] z[ ( ( z)) ] (3.6) If the vacatio do ot eist equatio (3.6) coicides with the equatio (2.3) i [9]. We ow obtai some performace measures for the system i steady state. Let U be the server utilizatio, (or the steady state probability that the server is attedig a customer) that is the server is busy, I, the steady state probability that the server is idle rig the retrial time, R, the steady state probability that the server is uder repair, W, the steady state probability that the server is o vacatio L S The orbit characteristics are of cosiderable iterest i retrial queues. For the model uder cosideratio, we obtai the probability of orbit beig empty as V=P +Q +W 2 2 ( )[ 2 2( ){ ( )}] 2 ( ) 2[ ( ) ( )( )] 2( ( ))[ ( p) ( )] 2 2 ( p) ( 2 2) 2[ ( ) ( p) ( )] 2 2 2[ ( ) ( p) ( )] [ ( ) ( )( )] q W(, z){( )[2 2( ) [ ( )]] 2 ( )} 2 p q W(, z) 2 q [ ( )] W(, z) 2{ ( ) ( ) ( )}
9 G. S. Mokaddis et al. where Q is the probability that the orbit is empty while the server is busy, P is the probability of a empty system ad W is the probability that the orbit is empty while the server is o vacatio out of the system, we observe that ad Q W P [ ( ( z))] q ( ) [ ( ( z))][ ( ( z))] W(, z) q ( ) where P is give i (3.57) ad [ ( )] W(, z) [ p ( )][ ( ) ( p) ( )] V q ( )[ q ( ) ( ) q( )] [ p ( )] W(, z) ( )[ q ( ) q( )( )] [ p ( )][ ( )] W(, z) q ( ) Defie H(z) = P + P(z) + Q(z) + R(z) + W(z). The H(z) represets the probability geeratig fuctio for the umber of customers i the orbit. Usig (3.52) (3.55) ad simplifyig, we get H( z) P [ z( z) ( )] z ( ( z)) [ p ( ( z))] pz ( ( z)) z ( ) P ( z) ( )[ p ( ( z)) ] [ ( ( z))][ p ( ( z))] [ z( z) ( )] W(, z) z[( pzq) ( ( z)) Hece, the mea umber of customers i the orbit is give by ( ( ))][ ( ) ( ) z z z z q p 2 p q q ( )[2( ){ ( ( )) } q(2 )] 2 ( ) Lq H`() 2[ ( ) ( ) ( )] 2{ ( )}[ ( p) ( )] 2 ( 2 2) 2 ( p) 2[ ( ) ( p) ( )] (2 2)} 2 p ( )] 2{ ( ) ( p) ( )} { q ( ) ( ) q( )} q W(, z)[( ){2( )[ q( ( )) p] q 2q W z W o z q p 2{ ( ) ( p) ( )} (, ) 2 (, )[ ( ( )) ] Let W S be the average time a customer speds i the system uder steady state. Due to little s formula, we have W S = L S 4. Stochastic Decompositio Stochastic decompositio has bee widely observed amog M/G/ type queues with server vacatios (see, for eample, [3]). A key result, i these aalyses is that the umber of customers i the system i steady state at a radom poit i time is distributed as the sum of two idepedet radom variables, oe of which is the umber of customers i the correspodig stadard M/G/ queuig system (i steady state) at a radom poit i time. The other radom variable may have differet probabilistic iterpretatios i specific cases depedig o how the vacatios are scheled. Stochastic decompositio has also bee observed to hold for some M/G/ retrial queues [4,5]. Let (z) be the probability geeratig fuctio of the umber of customers i the M/G/ queuig system with Beroulli feedback (see [6]) i steady state at a radom poit i time, (z) be the probability geeratig fuctio of the umber of customers i the vacatio system at a radom poit i time give that the server is idle ad K(z) be the probability geeratig fuctio of the ra-
A Feedback Retrial Queuig System with Startig Failures ad Sigle Vacatio 9 dom variable beig decomposed. The, the mathematical versio of the stochastic decompositio law is K(z) = (z) (z). (4.) We ow verify that the decompositio law applied to the retrial model aalyzed i this paper. For the M/G/ queue with Beroulli feedback queuig system [6], we have ( q)( z) ( ( z)) ( z) ( pz q) ( ( z)) z (4.2) To obtai a epressio for (z) we first defie idle i our cotet. We say that the server is idle if the server is either uder repair or free or o vacatio. (Note that i retrial queues, there may be customers i the system eve whe the server is idle). Uder this defiitio, we have P P( z) R( z) W( z) ( z) P P() R() W() Usig the results (3.52), (3.54), (3.55) ad (3.57) of theorem 3.2, we obtai (4.3) From (2.56) we observe that K(z) = (z) (z) which cofirms that the decompositio law of [6] is also valid for this special vacatio system. Refereces [] Choi, B. D. ad Kulkari, V. G., Feedback Retrial P {[ z( z) ( )][ z ( ( z))( ) ] z}{( pzq) ( ( z)) z} ( z) ( )( z){ z[( pzq) ( ( z)) q z ( ( z))][ z( z) ( )]} [ ( ( z))][ z( z) ( )] [( pz q) ( ( z)) z] W (, z) ( )( z){ z[( pzq) ( ( z)) q z ( ( z))][ z( z) ( )]} Queuig Systems, Stochastic Model Relat. Field, pp. 935 (992). [2] Kulkari, V. G. ad Choi, B. D., Retrial Queue with Server Subject to Breakdow ad Repairs, Queuig Syst., Vol. 7, pp. 928 (99). [3] Choi, B. D., Shi, Y. W. ad Ah, W. C., Retrial Queues with Collisio Arisig from Uslotted CSMA/ CD Protocol, Queuig syst., Vol., pp. 335356 (992). [4] Yag, T. ad Li, H., The M/G/ Retrial Queue with the Server Subject to Startig Failures, Queuig Syst., Vol. 6, pp. 8396 (994). [5] Fayolle, G., A Simple Telephoe Echage with Delayed Feedbacks, i: O. J. Boma, J. W. Cohe, H. C. Tijms (Eds.), Traffic Aalysis ad Computer Performace Evaluatio, Elsevier, Amsterdam, pp. 245253 (986). [6] Gomez-Corral, A., Stochastic Aalysis of a Sigle Server Retrial Queue with Geeral Retrial Times, Nav. Res. Logis., Vol. 46, pp. 5658 (999). [7] Krisha Kumar, B., Pavai Madheswari, S. ad Vijaya- Kumar, A., The M/G/ Retrial Queue with Feedback ad Startig Failures, Applied Mathematical Modellig, Vol. 26, pp. 5775 (22). [8] Lee, H. W., Lee, S. S., Chae, K. C. ad Nadaraja, R., O a Batch Service Queue with Sigle Vacatio, Appl. Math Modelig, Vol. 6, pp. 3642 (992). [9] Choi, B. D., Rhee, K. H. ad Park, K. K., The M/G/ Retrial Queue with Retrial Rate Cotrol Policy, Prob. Eg. Ifo. Sci., Vol. 7, pp. 2646 (993). [] Cooper, R. B., Queues Served i Cyclic Order: Waitig Times, Bell Syst. Tech., Vol. 49, pp. 33943 (97). [] Doshi, B. T., A Note o Stochastic Decompositio i a GI/G/ Queue with Vacatio or Setup Times, J. Appl. Prob., Vol. 22, pp. 49428 (985). [2] Fuhrma, S. W. ad Cooper, R. B., Stochastic Decompositio i the M/G/ Queue with Geeralized Vacatios, Ops. Res., Vol. 33, pp. 729 (985). [3] Levy, Y. ad Yechiali, U., Utilizatio of Idle Time i a M/G/ Queuig System, Maage. Sci., Vol. 22, pp. 222 (975). [4] Artalejo, J. R. ad Gomez-Corral, A., Steady State
92 G. S. Mokaddis et al. Solutio of a Sigle Server Queue with Liear Request Repeated, J. Appl. Prob., Vol. 34, pp. 223233 (997). [5] Yag, T. ad Templeto, J. G. C., A Survey o Retrial Queue, Queuig Syst., Vol. 2, pp. 2233 (987). [6] Takacs, L., A Sigle Server Queue with Feedback, Bell Syst. Tech. J., Vol. 42, pp. 5559 (963). Mauscript Received: Nov. 2, 24 Accepted: Apr. 7, 25