This Article is provided by the Brunel Open Access Publishing Fund - Copyright @ 2011 Hindawi PublishingWe study a queuing system which is equipped with a stand-by server in addition to the main server. The stand-by server provides service to customers only during the period of absence of the main server when either the main server is on a vacation or it is in the state of repairs due to a sudden failure from time to time. The service times, vacation times, and repair times are assumed to follow general arbitrary distributions while the stand-by service times follow exponential distribution. Supplementary variables technique has been used to obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers, and the average waiting time in the queue while the MathCad software has been used to illustrate the numerical results in this work

Khalaf, RF

Lucas, CA

Madan, KC

Journal of Probability and Statistics

English

We study a queuing system which is equipped with a stand-by server in addition to the main server. The stand-by server provides service to customers only during the period of absence of the main server when either the main server is on a vacation or it is in the state of repairs due to a sudden failure from time to time. The service times, vacation times, and repair times are assumed to follow general arbitrary distributions while the stand-by service times follow exponential distribution. Supplementary variables technique has been used to obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers, and the average waiting time in the queue while the MathCad software has been used to illustrate the numerical results in this work

Lukas, CA

Brunel University Research Archive

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2011, Article ID 812726, 11 pagesdoi:10.1155/2011/812726Research ArticleOn a Batch Arrival Queuing System Equippedwith a Stand-by Server during Vacation Periods orthe Repairs Times of the Main ServerRehab F. Khalaf,1 Kailash C. Madan,2 and Cormac A. Lukas11 School of Information Systems Computing and Mathematics, Brunel University,Middlesex UB83PH, UK2 College of Information Technology, Ahlia University, P.O. Box 10878, BahrainCorrespondence should be addressed to Rehab F. Khalaf, rihabfadil@gmail.comReceived 25 January 2011; Accepted 20 May 2011Academic Editor: Rongling WuCopyright q 2011 Rehab F. Khalaf et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.We study a queuing system which is equipped with a stand-by server in addition to the mainserver. The stand-by server provides service to customers only during the period of absenceof the main server when either the main server is on a vacation or it is in the state of repairsdue to a sudden failure from time to time. The service times, vacation times, and repair timesare assumed to follow general arbitrary distributions while the stand-by service times followexponential distribution. Supplementary variables technique has been used to obtain steady stateresults in explicit and closed form in terms of the probability generating functions for the numberof customers in the queue, the average number of customers, and the average waiting time in thequeue while the MathCad software has been used to illustrate the numerical results in this work.1. IntroductionDue to their wide applications in flexible manufacturing or computer communication sys-tems, MX/G/1 queueing system with vacations and MX/G/1 queueing system withbreakdowns have been studied by several authors including 1–8. Recently the authors of9 have studied some queueing systems with vacations and breakdowns.In this work, we study an MX/G/1 queueing system with Bernoulli schedulevacations and random breakdowns with an additional significant assumption that the systemdeploys a stand-by server during the vacation periods and the repair periods of the mainserver.Madan 10 studied the steady state behavior of a queuing system with a stand-byserver which provides service to costumers only during the repair period. In that work, repair2 Journal of Probability and Statisticstimes were assumed to follow an exponential distribution. The present paper considers bothvacations and breakdowns with additional assumptions of deployment of a standby duringthe vacation periods and repair periods. We generalize results obtained not only by Madan10 but also the results obtained by Maraghi et al. 9. Most importantly, we assume that theservice times, vacation times, and repair times have diﬀerent general arbitrary distributionswhile the stand-by service times follow exponential distribution. Out of five distributions inthis model we assume that four are a generally distributed, this is very important because allthe other distributions such as exponential, deterministic, and Erlang-k distributions will beincluded.The rest of this paper is arranged as follow. Section 2 gives the assumptions underlyingthe considered queueing system. Related definitions and used notations are given inSection 3. Equations governing the system are formulated in Section 4. In Section 5, we givethe solution of the equations formulated in the previous section to find the queue sizedistribution at a random epoch. The average queue size and the average waiting time aregiven in Section 6. In Section 7, we consider a numerical example to illustrate application ofour results.2. AssumptionsCustomers arrive at the system in batches of variable size in a compound Poisson process.Let λciΔt i  1, 2, 3, . . . be the first order probability that a batch of i customers arrives at thesystem during a short interval of time t, t 	 Δt, where 0 ≤ ci ≤ 1 and∑∞i1 ci  1 and λ > 0 isthe mean arrival rate of batches. The customers provided service one by one on a “first come-first served basis.”The service times of the main server follow a general arbitrary distribution with dis-tribution function Gs and density function gs. Let μxΔx be the conditional probabilitydensity of service completion during the interval x, x 	 Δx, given that the elapsed servicetime is x, so thatμx gx1 −Gx , 2.1and, thereforegs  μse−∫s0 μxdx. 2.2On completion of a service, the server may take a vacation of random length with pro-bability P , or may stay in the system providing service with probability 1−P , where 0 ≤ P ≤ 1.The server’s vacation times follow a general arbitrary distribution with distributionfunction Bv and density function bv. Let βxΔx be the conditional probability of acompletion of a vacation during the interval x, x 	 dx given that the elapsed vacation timeis x, so thatβx bx1 − Bx , 2.3Journal of Probability and Statistics 3and, thereforebv  βve−∫v0 βxdx. 2.4The system may break down at random, and breakdowns are assumed to occuraccording to a Poisson stream with mean breakdown rate α > 0. Further we assume thatonce the system breaks down, the customer whose service is interrupted comes back to thehead of the queue but it is instantly taken up for service by the stand-by server.Once the system breaks down, its repairs start immediately and the duration of repairsfollows a general arbitrary distributionwith distribution function Fr and density functionfr. Let γxΔx be the conditional probability density of repair completion during theinterval x, x 	 Δx, given that the elapsed repair time is x, so thatγx fx1 − Fx , 2.5and, thereforefr  γre−∫r0 γxdx. 2.6The stand-by server starts serving the customers as soon as the main server breaksdown or as soon as the main server leaves for a vacation after completing a service. Thestand-by service times follow an exponential distribution with stand-by service rate δ > 0and mean stand-by service time 1/δ.We further assume that the main server joins the system immediately after the com-pletion of its vacation or completion of its repairs, and the customer being served by thestand-by server is immediately transferred to the main server to start a service afresh.All stochastic processes involved in the system are independent of each other.3. NotationsWe leti Pnt, x: probability that at time t, there are n ≥ 0 customers in the queue excludingone customer in the service served by the main server, and the elapsed service timeof this customer is x. Accordingly, Pnt ∫∞0 Pnt, xdx denotes the probabilitythat there are n ≥ 1 customers in the queue excluding one customer in serviceirrespective of the value of x;ii Vnt, x: probability that at time t, there are n ≥ 0 customers in the queue andone customer is being served by the stand-by server, and the main server is onvacation with elapsed vacation time x. Accordingly, Vnt ∫∞0 Vnt, xdx denotesthe probability that at time t, there are n ≥ 0 customers in the queue and the server ison vacation irrespective of the value of x. As soon as the vacation starts the stand-byserver starts serving the customers in the system;4 Journal of Probability and Statisticsiii Rnt, x: Probability that at time t, there are n n ≥ 0 customers in the queue andone customer is being served by the stand-by server while the system is underrepair with elapsed repair time x. Accordingly, Rnt ∫∞0 Rnt, xdx denotes theprobability that at time t, there are n ≥ 0 customers in the queue and the server isunder repair irrespective of the value of x;iv Qt: probability that at time t, there are no customers in the system and the serveris idle but available in the system.Assuming that the steady state exists, we letlimt→∞Ant, x  Anx, limt→∞Ant  limt→∞∫∞0Ant, xdx  An,limt→∞dAntdt 0, where A  P, V, R,limt→∞Qt  Q.3.14. Equations Governing the SystemAccording to the assumptions mentioned above, we have four possible states of our systemduring a short time interval t, t	Δt: the first state is that themain server is providing service,the second is that the main server is on vacation and the stand-by server is providing service,the third state is that the main server is inactive due to a system breakdown and is underrepair and the stand-by server is providing service, the last possible state is that the serveris idle there are no customers in the system but available in the system. By discussing theprobabilities of every situation and finding the limit as Δt → 0, we obtain the following setof diﬀerential-diﬀerence equations∂∂xPnx  −(λ 	 μx 	 α)Pnx 	 λn−1∑i1ciPn−ix, n ≥ 1, 4.1∂∂xP0x  −(λ 	 μx 	 α)P0x, 4.2∂∂xVnx  −(λ 	 βx 	 δ)Vnx 	 λn∑i1ciVn−ix 	 δVn	1x, n ≥ 1, 4.3∂∂xV0x  −(λ 	 βx 	 δ)V0x 	 δV1x, 4.4∂Rnx∂x −(λ 	 γx 	 δ)Rnx 	 λn∑i1ciRn−ix 	 δRn	1x, n ≥ 1, 4.5∂R0x∂x −(λ 	 γx 	 δ)R0x 	 δR1x, 4.6λQ ∫∞0R0xγxdx 	(1 − p)∫∞0P0xμxdx 	 1 − r∫∞0V0xβxdx. 4.7Journal of Probability and Statistics 5The following boundary conditions will be used to solve the above equations:Pn0 (1 − p)∫∞0Pn	1xμxdx 	∫∞0Vn	1xβxdx	∫∞0Rn	1xγxdx 	 λcn	1Q, n ≥ 0,4.8Vn0  p∫∞0Pnxμxdx, n ≥ 0, 4.9Rn0  α∫∞0Pn−1xdx, n ≥ 1, 4.10R00  0. 4.115. Queue Size Distribution at a Random EpochDefining the following probability generating functionsAqx, z ∞∑n0znAnx, Aqz ∞∑n0znAn,A  P, V, R,Cz ∞∑i1zici,5.1we multiply 4.1 by zn, take summation over n from 1 to ∞, adding to 4.2 then by sim-plifying and using 5.1we get∂∂xPqx, z 	(λ − λCz 	 μx 	 α)Pqx, z  0. 5.2Using the same process, from 4.3, 4.4 and 4.5, 4.6 we get, respectively,∂∂xVqx, z 	(λ − λCz 	 βx 	 δ − δz)Vqx, z  0, 5.3∂∂xRqx, z 	(λ − λCz 	 γx 	 δ − δz)Rqx, z  0. 5.46 Journal of Probability and StatisticsMultiply 4.8 by zn	1, sum over n from 0 to ∞, and use the generating functions defined in5.1, we getzPq0, z (1 − p)∫∞0Pqx, zμxdx 	∫∞0Vqx, zβxdx 	∫∞0Rqx, zγxdx 	 λCzQ−[(1 − p)∫∞0P0xμxdx 	 1 − r∫∞0V0x, zβxdx 	∫∞0R0x, zγxdx].5.5From 4.7, we havezPq0, z (1 − p)∫∞0Pqx, zμxdx 	∫∞0Vqx, zβxdx 	∫∞0Rqx, zγxdx	 λQCz − 1.5.6Multiply 4.9 by zn and sum over n from 0 to∞, we getVq0, z  p∫∞0Pqx, zμxdx. 5.7Similarly, from 4.10 we getRq0, z  αz∫∞0Pqx, zdx  αzPqz, n ≥ 0. 5.8Integrating 5.2 from 0 to x yieldsPqx, z  Pq0, ze−λ−λCz	αx−∫x0 μtdt, 5.9where Pq0, z is given by 5.6. Let us consider a  λ − λCz 	 α.Integrating equation 5.9 by parts with respect to x yieldsPqz  Pq0, z(1 −G∗aa), 5.10where G∗a ∫∞0 e−axdGx is the Laplace-Stieltjes transform of the service time Gx.Now multiplying both sides of 5.9 by μx and integrating over x we get∫∞0Pqx, zμxdx  Pq0, zG∗a. 5.11Journal of Probability and Statistics 7Using 5.11, from 5.7 we getVq0, z  pPq0, zG∗a. 5.12Similarly, we integrate 5.3 from 0 to x, we getVqx, z  Vq0, ze−λ−λCz	δ−δ/xx−∫x0 βtdt. 5.13Substituting by the value of Vq0, z from 5.12 in 5.13we getVqx, z  pPq0, zG∗ae−λ−λCz	δ−δ/zx−∫x0 βtdt. 5.14Let us consider b  λ − λCz 	 δ − δ/z now integrating 5.14 by parts with respect tox we getVqz pPq0, zG∗a1 − B∗bb, 5.15where B∗b ∫∞0 e−bxdBx is the Laplace-Stieltjes transform of the vacation time Bx.Now multiplying both sides of 5.14 by βx and integrating over x we get∫∞0Vqx, zβxdx  pPq0, zG∗aB∗b. 5.16Now integrating 5.4 from 0 to x, yieldsRqx, z  Rq0, ze−λ−λCz	δ−δ/zx−∫x0 γtdt. 5.17Substituting by the value of Rq0, z from 5.8 in 5.17 we getRqx, z αzPq0, z1 −G∗aae−λ−λCz	δ−δ/zx−∫x0 γtdt, 5.18integrating 5.18 by parts with respect to x we getRqz  αzPq0, z(1 −G∗a1 − F∗bab), 5.19where F∗b ∫∞0 e−λ−λCz	δ−δ/zxdFx is the Laplace-Stieltjes transform of the repair timeFx.8 Journal of Probability and StatisticsNow multiplying both sides of 5.18 by γx and integrating over x we get∫∞0Rqx, zγxdx  αzPq0, z(1 −G∗aa)F∗b. 5.20Now using 5.11, 5.16 and 5.20, 5.6 becomesPq0, z −acQa(z −G∗a(1 − p 	 pB∗b)) − αz1 −G∗aF∗b , 5.21where c  λ − λCz, from 5.21 equations 5.10, 5.15 and 5.19 become, respectively,Pqz −cQ1 −G∗aa(z −G∗a(1 − p 	 pB∗b)) − αz1 −G∗aF∗b ,Vqz −acQpG∗a1 − B∗bab(z −G∗a(1 − p 	 pB∗b)) − αzb1 −G∗aF∗b ,Rqz −αzcQ1 −G∗a1 − F∗bab(z −G∗a(1 − p 	 pB∗b)) − αzb1 −G∗aF∗b .5.22Let Sqz denote the probability generating function of the queue size irrespective of the stateof the system. That is, Sqz  Pqz 	 Vqz 	 Rqz.Then adding 5.22 we obtainSqz −cQ1 −G∗a{b 	 αz1 − F∗b} − acQpG∗a1 − B∗bab(z −G∗a(1 − p 	 pB∗b)) − αzb1 −G∗aF∗b . 5.23In order to find Q, we use the normalization conditionSq1 	Q  1. 5.24Note that if z  1 then b  0 and c  0, so Sq1 is indeterminate of 0/0 form. Therefore, weapply L’Hopitals Rule twice on 5.23, we getSq1  limz→ 1N ′′zD′′z, 5.25Journal of Probability and Statistics 9where Nz and Dz are the numerator and denominator of the right hand side of 5.23respectively. Double primes in 5.25 denote the second derivative at z  1. Carrying out thederivatives at z  1 we haveN ′′1  −2QλEIλEI − δ{1 −G∗α{1 	 αER} 	 αpG∗aEV }, 5.26D′′1  2λEI − δ{1 −G∗aλEI 	 α{1 	 λEI − δER}−α(1 − pλEI − δG∗aEV )},5.27where C1  1, C′1  EI is the mean batch size of the arriving customers, B∗0  1, andB∗′0  −EV  the mean vacation time, and F∗0  1, and F∗′0  −ER is the mean repairtime.Therefore, adding Q to 5.25 and equaling to 1 and simplifying we getQ α(1 − pλEI − δG∗aEV ) − 1 −G∗aλEI 	 α{1 	 λEI − δER}α{δER1 −G∗α 	G∗α(1 	 pδEV )} .5.28From 5.28 we can find the utilization factor, ρ, where ρ  1 −Q.As a particular case if we assume there is no stand by server this means that δ  0,b  c  λ − λEI using this in the main results of this paper, we get,Sqz −Q1 −G∗a{b 	 αz1 − F∗b} − aQpG∗a1 − B∗ba(z −G∗a(1 − p 	 pB∗b)) − αz1 −G∗aF∗b , 5.29Q  1 − λEI{1αG∗a− 1α	ERG∗a− ER 	 pEV }. 5.30These results agree with results given by 9.6. The Average Queue Size and the Average Waiting TimeLet Lq denote the mean number of customers in the queue under the steady state. ThenLq ddzSqz∣∣∣∣z1. 6.110 Journal of Probability and StatisticsSince this formula gives 0/0 form, then using the L’Hopital’s rule four times we obtainLq  limz→ 1D′′zN ′′′z −N ′′zD′′′z3D′′z2, 6.2whereN ′′1 and D′′1 are given in 5.26 and 5.27, respectively, andN ′′′1  −3Qny1 −G∗α1 	 αER− 6Qym2G∗′α1 	 αER− 3Qm1 −G∗a[x1 	 αER 	 2αyER 	 αy2E(R2)]	 3pQEV yG∗a{2m2 − αn}	 6αpQm2yEV G∗′a− 3αpQmG∗α(y2E(V 2)	 xEV ),6.3D′′′1  3(1 −G∗apEV y){2my − αx}	 6m2yG∗′α 	 31 −G∗a{ny 	mx}− 3αyp(2G∗′amyEV  −G∗a(E(V 2)y2 	 xEV ))	 6αy21 −G∗aER 	 3α{1 	 yER}{x1 −G∗a 	 2myG∗′α}	 3αy1 −G∗a(y2E(R2)	 xER),6.4where λEII − 1 	 2δ  x, λEI − δ  y, λEII − 1  n, and λEI  m.7. Numerical ExampleTo illustrate the results of this chapter numerically we consider that the service times, vacationtimes, stand-by service times, and repair times are exponentially distributed. All the valueswere chosen so that the steady state condition is satisfied. In Table 1 we will show the eﬀectof the new contribution of this paper, where we will show the influence of the new parameterδ stand-by service rate on the trends. We choose the following values μ  7, γ  3, λ  2,β  7, α  2, P  0.5, EI  1, and EII − 1  0, we consider that δ takes the values 0, 1, 4,6, 8, and 10.Journal of Probability and Statistics 11Table 1: Computed values of various queue performance measures μ  7, λ  2, α  2, P  0.5, γ 3, β  7.δ ρ Q Lq Wq L W0 0.619 0.381 1.4048 0.7024 2.0238 1.01191 0.5306 0.4694 0.8064 0.4032 1.3371 0.66854 0.3714 0.6286 0.3432 0.1716 0.7147 0.35736 0.3095 0.6905 0.2807 0.1403 0.5902 0.29518 0.2653 0.7347 0.2617 0.1309 0.527 0.263510 0.2321 0.7679 0.2591 0.1295 0.4912 0.2456Table 1 shows that increasing the value of δ decreases the value of utilization factor,the mean queue size, and the mean waiting time of the customers while the server idle timeincreases. All the trends shown by the table are as expected.AcknowledgmentsThe authors would like to express their deep appreciation and thanks to the referees fortheir comments and suggestions, which led to improvement of the paper in its present form.References1 Y. Levy and U. Yechiali, “AnM/M/s queue with servers vacations,” INFOR Journal, vol. 14, no. 2, pp.153–163, 1976.2 S. W. Fuhrmann and R. B. Cooper, “Stochastic decomposition in M/G/1 queue with generalizedvacations,” Operations Research, vol. 33, no. 5, pp. 1117–1129, 1985.3 B. T. Doshi, “Queueing systems with vacations—a survey,” Queueing Systems, vol. 1, no. 1, pp. 29–66,1986.4 A. Aissani and J. R. Artalejo, “On the single server retrial queue subject to breakdowns,” QueueingSystems, vol. 30, no. 3-4, pp. 309–321, 1998.5 J. Wang, J. Cao, and Q. Li, “Reliability analysis of the retrial queue with server breakdowns andrepairs,” Queueing Systems, vol. 38, no. 4, pp. 363–380, 2001.6 W. J. Gray, P. P. Wang, and M. Scott, “A queueing model with multiple types of server breakdowns,”Quality Technology and Quantitative Management, vol. 1, no. 2, pp. 245–255, 2004.7 Q. Xu, S. Bao, Z. Ma, and N. Tian, “MX/G/1 queue with multiple vacations,” Stochastic Analysis andApplications, vol. 25, no. 1, pp. 127–140, 2007.8 K. C. Madan, W. Abu-Dayyeh, and M. Gharaibeh, “Steady state analysis of two Mx/Ma,b/1 queuemodels with random breakdowns,” International Journal of Information and Management Sciences, vol.14, no. 3, pp. 37–51, 2003.9 F. A. Maraghi, K. C. Madan, and K. Darby-Dowman, “’Batch arrival queueing system with randombreakdowns and Bernoulli schedule server vacations having general vacation time distribution,”International Journal of Information and Management Sciences, vol. 20, no. 1, pp. 55–70, 2009.10 C. K. Madan, “A bulk input queue with a stand-by,” The South African Statistical Journal, vol. 29, pp.1–7, 1995.

A bulk input queue with a stand-by,”

A queueing model with multiple types of server breakdowns,”

An M/M/s queue with servers vacations,”

Batch arrival queueing system with random breakdowns and Bernoulli schedule server vacations having general vacation time distribution,”

On the single server retrial queue subject to breakdowns,”

queue with multiple vacations,”

Queueing systems with vacations—a survey,”

Reliability analysis of the retrial queue with server breakdowns and repairs,”

Steady state analysis of two Mx/Ma,b/1 queue models with random breakdowns,”

Stochastic decomposition in M/G/1 queue with generalized vacations,”

On a batch arrival queuing system equipped with a stand-by server during vacation periods or the repairs times of the main server

We study a queuing system which is equipped with a stand-by server in addition to the main server. The stand-by server provides service to customers only during the period of absence of the main server when either the main server is on a vacation or it is in the state of repairs due to a sudden failure from time to time. The service times, vacation times, and repair times are assumed to follow general arbitrary distributions while the stand-by service times follow exponential distribution. Supplementary variables technique has been used to obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers, and the average waiting time in the queue while the MathCad software has been used to illustrate the numerical results in this work.</jats:p

Rehab F. Khalaf

Kailash C. Madan

Cormac A. Lukas

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On a Batch Arrival Queuing System Equipped with a Stand-by Server during Vacation Periods or the Repairs Times of the Main Server

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On a batch arrival queuing system equipped with a stand-by server during vacation periods or the repairs times of the main server

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