An M/G/1 Retrial Queue with a Single Vacation Scheme and General Retrial Times

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1 America Joural of Operatioal Research 23, 3(2A): 7-6 DOI:.5923/s.ajor A M/G/ Retrial Queue with a Sigle Vacatio Scheme ad Geeral Retrial Times K. Lakshmi,, Kasturi Ramaath 2 Departmet of Mathematics, A.M. Jai College, Meeambakkam, Cheai-4, Tamil Nadu, Idia 2 School of Mathematics, Madurai Kamaraj Uiversity, Madurai-2, Tamil Nadu, Idia Abstract Let us cosider a service facility where a sigle server provides some service. It could be a plumber lookig after repair ad maiteace of the plumbig work i the apartmet complexes situated ear the shop or could be a electricia or a paiter. Requests for service arrive i accordace with a oisso process. Whe the server is away with the service of a customer, ay other requests for service ca be recorded ad the customer caot wait i a queue but has to leave ad try for service after some time. The server after completio of the work o had decides to take a break before attedig to the ext chore. This is a example of a retrial queue i which the server takes a vacatio after the completio of each service. Motivated by this example, we have studied a M/G/ retrial queue with server vacatios. I this paper, we assume that the retrial times are geerally distributed ad that the retrial policy is costat. We have derived probability geeratig fuctios of the system size ad the orbit size. We have ivestigated the coditios uder which the steady state exists. Some useful performace measures are also obtaied. Numerical examples are provided to illustrate the sesitivity of the performace measures to chages i the parameters of the system. Keywords Retrial Queues, Costat Retrial olicy, Sigle Vacatios, Steady State Behaviour. Itroductio I this paper, we cosider a sigle-server retrial queue with server vacatios. We have assumed that the iter-retrial times are geerally distributed with a costat retrial policy. After the completio of each service the server goes o a vacatio of radom legth. After the completio of each vacatio, the server waits for the ext customer. This could be either a primary arrival or the server could try to cotact the first customer i the orbit. This is i cotrast to the multiple vacatio schemes, where the server could take a vacatio every time he fids o customers i the system. A example of such a retrial queuig system is as follows; a service provider (plumber, paiter, electricia etc.,) is provided with some mechaism to record the details of the customers requestig service, whe he is away attedig to some job. After the completio of each service, the provider may take a break before goig back to atted to the ext customer. Customer requests which arrive durig his absece do ot wait i a queue but after recordig their requiremets go away to come back after some time to repeat the request for service. Motivated by this example, we have therefore cosidered Correspodig author: lakshmikrisha8@yahoo.com (K. Lakshmi) ublished olie at Copyright 23 Scietific & Academic ublishig. All Rights Reserved a retrial queue with a limited sigle vacatio scheme. This scheme is differet from the oe based o the Beroulli schedule vacatios ad applied i the literature by authors like Wag ad Li[5], Ebeesar Aa Bagyam ad Udaya Chadrika[]. The rest of the paper is orgaized as follows: I sectio-2, we preset a brief review of related works i the queueig theory literature. I Sec-3; we preset the mathematical model of the system. I Sec-4, we preset a steady state aalysis of the system ad we derive the joit probability geeratig fuctios (.G.F s) of the server state ad orbit size ad server state ad system size. I Sec-5, we derive the ecessary ad sufficiet coditio for the existece of the steady state. I sec-6, we cosider some useful performace measures of the system. I Sec-7, umerical results are used to compare umerically the system performace measures with respect to chages of parameters. 2. Literature Review Retrial queueig systems form a active research area due to their wide applicability i telephoe switchig system, telecommuicatio etworks ad computer etworks. Retrial queueig systems are those systems i which arrivig customers, who fid the server busy, joi the retrial queue (orbit) to try agai for their requests after sometime. Retrial queues are useful i modelig may problems i telephoe switchig systems, computer ad commuicatio

2 8 K. Lakshmi et al.: A M/G/ Retrial Queue with a Sigle Vacatio Scheme ad Geeral Retrial Times systems. A review of the literature o retrial queues ad their applicatios ca be foud i Fali[] ad Yag ad Temp leto[23], ad Artalejo[6]. Two moographs etirely derived to the topic exist ad are writte by Fali ad Templeto[2] ad Artalejo ad Gomez-Co rrel[5]. I retrial queueig literature, retrial customers are usually assumed to behave idepedetly ad retur without regard to other retrial customers. However, Farahmad[3] proposed a special type of retrial system with FCFS retrial queue, that is, if a customer fids the server busy, he may joi the tail of a retrial queue i accordace with a first-come-first-served (FCFS) disciplie ad the customer at the head of the retrial queue would the try agai to eter the system, competig with ew arrivals. Go me z-corral[4] cosidered a M/G/ retrial queue with a FCFS retrial queue ad geeral retrial times. Sigle-server queues with vacatios have bee studied extesively i the past. Comprehesive surveys ca be foud i Doshi[9] ad Takagi[22] ad recet developmets i vacatio queueig Models was itroduced by Ke et.al[6], Krishakumar ad Arivudaiambi[7]. I our model the server takes a break every time he completes the service. Most of the aalyses for retrial queues with vacatios cocer the exhaustive service schedule (Artalejo[] ad Artalejo ad Rodrigo[4]) ad the gated service policy (Lagaris[2]). Recetly, several authors cosidered retrial queues with Beroulli schedule, as ca be see i Atecia ad Moreo[7], Kumar et al.[8] ad Zhou[24]. I this paper we have cosidered a limited vacatio policy. Kasturi Ramaath ad K.Kalidass[9] have cosidered a two phase service M/G/ vacatio queue with geeral retrial times ad o-persistet customers. I the cotext of the service provider the limited vacatio scheme wherei, the server takes the vacatio after each service completio is appropriate. Also, the server caot take a vacatio each time he fids the system empty. The FCFS orbit also makes sese sice the server has a record of all the requests that have arrived durig his absece. With these ideas we have icluded a FCFS orbit, a limited sigle vacatio scheme ito our model. 3. The Mathematical Model I this sectio, we cosider a Sigle-server retrial queuig system with a o-expoetial retrial time distributio ad with server vacatios. Arrivig customers to the system are i accordace with a oisso process with a rate λ.the service time is geerally distributed with a distributio fuctio B(x), Laplace Stieljes trasform (LST) sx B ( s) e µ ( x) = ad the hazard rate fuctio ( x) B µ ( x) =. B x A customer upo arrival, who fids the server free, immed iately proceeds for service. Otherwise, the customer leaves the system ad jois a orbit from where he/she makes repeated attempts to gai service. The time itervals betwee two successive retrials are assumed to be geerally distributed with a distributio fuctio R(x), hazard rate fuctio η ( x) ad LST R(s). We assume that oly the customer at the head of the orbit is allowed to make repeated attempts. We also assume that the elapsed retrial time is measured from the momet the server becomes available for service. After completio of a service, the server is allowed to take a vacatio. The duratio of the server vacatio is assumed to be geerally distributed with a distributio fuctio V(x), a LST V(s) ad a hazard rate fuctio ξ ( x). We assume that the iter-arrival times, service times, iter-retrial times ad the duratio of the server vacatio are all idepedet of each other. Let N (t) deote the umber of customers i the orbit at ay istat of time t. Let C (t) deote the state of the server at time t: C t, if the server is idle =, if the server is busy 2, if the server is o vacatio I order to make the stochastic process ivolve ito a cotiuous time Markov process, we employ the supplemetary variable techique. Th is was first itroduced by Cox[7]. See Medhi[2] for a detailed explaatio of the techique. We defie the supplemetary variable X (t) as follows: X ( t) =, if C(t) =,N(t) =, elapsed retrial time if C(t) =,N(t) =, elapsed service time if C(t) =,N(t) =, elapsed vacatio time if C(t) = 2,N(t) =. We defie the followig probability fuctios: { = = }, ( t) C( t) N( t), ( x, t) = C( t) = ; N( t) = ; x X ( t) x + i, ( x, t) = Ct = in ; ( t) = ; x X ( t) x + The,, = rob ; rob, rob,, i =, 2. Markov process. { } C t N t X t is a cotiuous time

3 America Joural of Operatioal Research 23, 3(2A): Steady State Aalysis Now, aalysis of our queueig model ca be performed with the help of the followig Kolmogorov forward equatios:, ( t) = -λ, ( t) 2, ( x, t) ξ( x) For,, ( xt, ) +, ( xt, ) = x t λ+ η Fo r,, ( xt, ) +, ( xt, ) = x t λ + µ x xt, + () ( x) ( xt, ), { }, + ( ) ( xt, ) δ λ, 2, ( xt, ) + 2, ( xt, ) = x t λ+ ξ( x) 2, ( xt, ) +,, ( δ, ) 2, ( xt) The boudary coditios are as follows, for, For, (, ) (, ) ξ, 2, (2) (3) (4) t = x t x (5) (, ) = λ η t t + x x (6),,, (, ) λ (, ) + (, ) η t x t x t x,,, + = (7) µ, t = x, t x,for 2,, (8) Assumig that the system reaches the steady state, the equatios () to (8) become, 2, ξ λ = x x (9) d ( x, ) = λ+ η( x ), ( x ),for. () d, ( x, t ) = { λ + µ ( x )}, ( x,) () + δ λ xt,, (, ), d 2, ( x ) = - ( x ) 2, ( x ) λ+ ξ +, ( δ, ) 2, - ( x ), The boudary coditios are as follows for For, ξ, 2, (2) = x (3) ( ) = λ η + x x (4),,, ( ) λ + η x x x,,, + = (5) µ = x x for 2,, (6) The solutio of equatio () is give by λx =, x e R x, We defie the followig partial probability geeratig fuctios, for z, Let i (, ) =, (7) = xz xz xz x (, ) = z, = xz x (8) (, ) = z 2 2, = z (9) = x, z,where i=,,2 (2) i Theorem: 4. I the steady state, the probability geeratig fuctio of the orbit size is give by z = ( z) R ( λ) λ( β + µ ) ( ) ( ) zv λ z R λ B λ z + V z R B z z B = µ = β ad the λ( ) ( λ) λ( ) where ( ) ad V (2) probability geeratig fuctio of the system size is give by = ( z) B λ( z) R ( λ) λ( β + µ ) λ( ) ( λ) λ( ) K z { } zv z R B z + V λ( z) R ( λ) B λ( z) z (22)

4 K. Lakshmi et al.: A M/G/ Retrial Queue with a Sigle Vacatio Scheme ad Geeral Retrial Times roof: Multiplyig equatio () by z ad summig over from to, the solutio of the resultig equatio is λ( z) x xz, =, z e B x (23) Similarly equatio (2) gives, λ( ) z x 2 xz, = 2, z e V x Similarly equatios (3), (4), (5) ad (6), we get, = λ( ) ( z) λ + ( z) R ( λ), z, z V z λ 2,, =,, +, z R z 2(, z) = (, z) B λ ( z) ( λ ) (24) (25) (26) (27) After some algebraic calculatios, we get the followig values, ( z), = λ, z V ( λ( z) ) B ( λ( z) ) zv ( λ( z) ) B ( λ( z) ) R ( λ) + V ( λ( z) ) B ( λ( z) ) R ( λ) z, z = λ, zv λ z B ( λ z ) R ( λ) + V ( λ z ) B ( λ( z) ) R ( λ) z R ( λ ) ( z) ( ) ( ) (, z) = λ 2, (28) (29) R ( λ) ( z) B ( λ( z) ) (3) zv ( λ( z) ) B ( λ( z) ) R ( λ) + V ( λ( z) ) B ( λ( z) ) R ( λ) z Now, fro m (2),we get, R ( λ ) ( z) = (, z) (3) λ B λ ( z) ( z) = (, z ) (32) λ ( z) Hece, z = λ ( z) ( z) V 2( z) = 2(, z ) λ V ( λ( z) ) B ( λ( z) ) + ( λ) ( λ) ( λ ) ( λ( )) ( λ( )) ( λ) ( λ( )) ( λ( )) ( λ), (33) z R V z B z (34) z V z B z R + V z B z R z z =, R ( λ) B ( λ( z) ) (35) z V ( λ( z) ) B ( λ( z) ) R ( λ) + V ( λ( z) ) B ( λ( z) ) R ( λ) z ( z) = 2, R B z R B z V z The GF (z) of the orbit size is give by ( λ) ( λ( )) + ( λ) ( λ( )) ( λ( )) V ( λ( z) ) B ( λ( z) ) R ( λ) + V ( λ( z) ) B ( λ( z) ) R ( λ) z = z z z z, 2 ( z) R ( λ ) = z V ( λ( z) ) B ( λ( z) ) R ( λ) + V ( λ( z) ) B ( λ( z) ) R ( λ) z (36), (37) To obtai the value of,, we use the ormalizig coditio () =. Applyig L Hospital s rule i a appropriate place we get R ( λ) + λ V B, = + R ( λ ) R ( λ) λ[ β + µ ], = R ( λ ) Hece ( z) = + ( z) + ( z) + ( z), 2 (38)

5 America Joural of Operatioal Research 23, 3(2A): 7-6 z = ( z) R ( λ) λ( β + µ ) ( ) ( ) zv λ z R λ B λ z + ( ) ( ) V λ z R λ B λ z z (39) Let K (z) be the GF of the system size =, K( z) = ( z) B λ( z) R ( λ) λ( β + µ ) zv λ( z) R ( λ) B λ( z) K z z z z z { } + V λ( z) R ( λ) B λ( z) z 5. The Embedded Markov Chai (4) I this sectio, we derive the ecessary ad sufficiet coditio for stability of our system. We cosider the embedded Markov chai of the process. Let X N( η ) = + be the umber of customers i the orbit immediately after the η.the X X + = if the th service completio epoch + customer is a X = X + A + + B, where th retrial customer, otherwise A is the umber of arrivals ito the orbit durig the vacatio time of the server ad B is the umber of customers arrivig ito the orbit durig the service time of + customer. th the By our assumptio, the arrival process is idepedet of the service mechaism ad of the vacatios of the server ad of the retrial processes iitiated by the customers i the orbit. Therefore { : } X is a Markov chai. The system is ergodic if ad oly if the embedded Markov chai { : } X is ergodic. To prove that the embedded Markov chai is ergodic,we employ Foster s criterio whose statemet is give below: Foster s criterio: For a irreducible ad aperiodic Markov chai ξ with state space S, a sufficiet coditio j for ergodocity is the existece of a o-egative fuctio f(s), s S ad ε > such that the mea drift xs = E( f ( ξi+ ) f ( ξi) / ξi = s) is fi ite for all s S ad xs ε for all s S except perhaps a fiite umber. Theorem 5.: The ecessary ad sufficiet coditio for the system cosidered i the previous sectio to be ergodic is give by λ( β µ ) R ( λ) + <. 6. erformace Measures I this sectio, we examie some useful performace measures of the system. (a) The expected umber of customers i the system is give by d E( L) = lim K( z) = z dz 2λ( β + µ ) R ( λ) λ β + µ + 2βµ + λµ 2 R ( λ) λ( β + µ ) (b) The expected umber of customers i the orbit E L = = ( q ) ( + ) R 2λ β µ λ ( 2) ( 2) ( 2 ) 2 λ β µ βµ 2 R ( λ) λ( β + µ ) The steady state distributio of the server state is give by q =rob {server is id le} = + = ( + ). q =rob {server is busy} = λ β µ, = λµ. 2 q =rob {server is o vacatio} = 7. Numerical Illustratios 2 = λβ. I order to verify the efficiecy of our aalytical results, we perform umerical experimets by usig MALE. I this sectio, we preset some umerical examples to illustrate the effect of varyig the parameters o the followig performace characteristics of our system. We cosider the performace measures: the probabilities q, q, q 2 are the probabilities of the server beig idle, the busy ad o vacatio respectively. We also cosider the expected umber of customers i the orbit, expected umber of customers i the system ad,. Moreover, for the purpose of umerical illustratios, we assume that the arrival process is oisso with parameter λ varyig from. to.7, the service time distributio fuctio is expoetial with mea µ =.3, the retrial times follow a expoetial distributio with LST R ( λ ) = φ λ + φ with parameter φ =.8.The vacatio time is also expoetially distributed with mea β =.25. I all the cases, the parametric values are chose to satisfy the stability coditio. Example :

6 2 K. Lakshmi et al.: A M/G/ Retrial Queue with a Sigle Vacatio Scheme ad Geeral Retrial Times I this example, we study the effect of varyig the arrival rate λ. Fro m table, we observe that if the value of λ icreases, the probabilities of idle time- q ad, decrease. The probabilities of busy time, expected o. of customers i the system ad the expected o. of customers i the orbit also icrease. The graph for the data give i table is give below; Ta ble. Effect of varyig λ o various performace measures λ, q q q ( ) K ( ) (.) (.2) (.3) (.4)

7 America Joural of Operatioal Research 23, 3(2A): (.5) (.6) Fi gure. Effect of varyig λ o various performace measures Fig (.) shows how, (i.e.) the probability of o customers i the system ad the server is idle decreases with icreasig values of λ.similarly, Fig(.2) shows how the proportio of idle time of the server decreases with icreasig values of λ. Fig (.3), Fig (.4), Fig (.5) ad Fig (.6) show how the server s busy period ad vacatio period probabilities, the expected umber of customers i the system ad the expected umber of customers i the orbit icrease with icreasig values of λ. Next, we assume that the arrival process is expoetially distributed with parameter λ =., the service time distributio fuctio is expoetial with mea µ varyig from.3 to.9, the retrial times follow a expoetial distributio with LST R ( λ ) = φ λ + φ with parameter φ =.8.The vacatio time is also expoetially distributed with a mea β =.25. Example 2: I this example, we study the effect of varyig the service rate µ. From the Table-2, we observe that if the value of µ icreases, the probabilities of idle time- q ad, are decrease ad the probabilities of busy time, expected o. of customers i the system ad the,expected o. of customers i the orbit icrease. Fig(2.) shows how, (i.e.) the probability of o customers i the system ad the server is idle decreases with icreasig values of µ.similarly, Fig(2.2) shows how the idle time probability of the server decreases with icreasig values of µ. Fig (2.3), Fig (2.4), Fig (2.5) shows how the server s busy period probability, vacatio time probability ad the expected umber of customers i the orbit icreases with icreasig values of µ. Fig (2.6) shows how the expected umbers of customers i the system icreases. Ta ble 2. Effect of varyig µ o various performace measures µ, q q q ( ) K ( )

8 4 K. Lakshmi et al.: A M/G/ Retrial Queue with a Sigle Vacatio Scheme ad Geeral Retrial Times (2.) (2.2) (2.3) (2.4) (2.5) (2.6) Fi gure 2. Effect of varyig µ o various performace measures

9 America Joural of Operatioal Research 23, 3(2A): Coclusios I this paper, we have studied a M/G/ retrial queue with geeral retrial times ad a costat retrial policy with server vacatios uder a limited vacatio disciplie ad a sigle vacatio scheme. Existig results i the literature o retrial queues with vacatios are cocered either with the exhaustive disciplie or the gated disciplie. Our aim is to icrease the scope for the by cosiderig a limited vacatio disciplie, where the server takes a vacatio after completio of k ( ) services. ACKNOWLEDGEMENTS The authors are very grateful to the aoymous reviewers for their valuable suggestios to improve the paper i the preset form. Appedix We first preset the derivatio of the equatios () to (8). Equatio (): t+ t = rob C t+ t =, N t+ t = { } = ( t)( λ t) C( t+ t) = 2, rob N( t t), x < X ( t) x = rob vacatio eds i ( tt, + t) ( λ ) (, ) ξ = t t + x t x t 2 Dividig through out by Δt ad takig limits as Δt, we get equatio (). Equatio (2): ( x+ tt, + t). t C( t+ t) =, N( t+ t) =, = rob x+ t < X ( t) x+ 2 t = xt, λ t t., { o ( t t + t) } (, ) (, )( λ )( η ) rob retrials i, x+ tt+ t t = xt t x t Dividig through out by (Δt) 2 ad takig limits as Δt, we obtai equatio (2).Equatios (3) to (8) are obtaied by usig similar argumets to those give above. roof of theorem 4.: From the above equatios () & (2) d, ( x, t ) = { λ + µ ( x )}, ( x,) +,, ( δ, ) λ, ( xt) roof of Theorem 5.: I order to prove the sufficiecy of the coditio, we use the test fuctio f(s) =s. The mea drift is the defied as ( ( ξ + ) ( ξ ) / ξ ) xs = E f i f i i = s For m k, xk= fg r m rr( λ )( k+ m k ) r= m + ( m ) R( λ) + m( R( λ) ) fg r m= r= m ( ( λ )) + R m+ k k fg Where time. gm r m r r m r r= f r = robability of r arrivals durig the vacatio ( λx) r λx fr e dv x r! = = probability of m-r arrivals durig the service of a customer. m r λx ( λx) m r = ( m r)! xk R ( λ) λ ( µ β) + ( R ( λ) ) λ ( µ + β) λ ( µ β) R ( λ) If λ ( µ + β) < R ( λ). g e db x = + = + <. Foster s criterio, the embedded Markov chai ad hece our process is stable if λ ( µ β) R ( λ) λ ( µ β) R ( λ) + <.The coditio + < is also a ecessary coditio. This ca be observed from equatio (38), sice, otherwise,. REFERENCES [] Artalejo, J.R., Aalysis of a M /G / queue with costat repeated attempts ad server vacatios. Computers ad Operatios Research, 24(6), , (997). [2] Artalejo, J.R. Accessible bibliography o retrial queues. Mathematical ad Computer Modellig, vol-3, -6, (999). [3] Artalejo, J.R. A classified bibliography of research o retrial queues: rogress i Top, 7,87-2, (999).

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