Balking and Re-service in a Vacation Queue with Batch Arrival and Two Types of Heterogeneous Service

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1 Journal of Mathematics Research; Vol. 4, No. 4; 212 ISSN E-ISSN Publishe by Canaian Center of Science an Eucation Balking an Re-service in a Vacation Queue with Batch Arrival an Two Types of Heterogeneous Service 1 Brunel University, UK Monita Baruah 1, Kailash C. Maan 2 & Tillal Elabi 3 2 College of Information Technology, Ahlia University, Kingom of Bahrain 3 Brunel Business School, Brunel University, UK Corresponence: Monita Baruah, Department of Mathematics, University of Bahrain, Kingom of Bahrain. Monita.Baruah@brunel.ac.uk, monitabrh@gmail.com Receive: June 17, 212 Accepte: July 2, 212 Online Publishe: July 26, 212 oi:1.5539/jmr.v4n4p114 URL: Abstract In this paper we have consiere a single server queue with customers or units arriving in groups or batches in a system proviing two types of general heterogeneous service. The server provies type 1 or type 2 services on a first come first serve basis. At the beginning of a service, a customer has the option to choose either type 1 or type 2 service. Also we ae the concept of balking an re-service in this stuy. Balking is a kin of customer behavior. If a batch on arrival for service is reluctant or refuses to join the system for some reasons is sai to balk. Once a service is complete the customer may leave the system or he has the option to eman re-service for the same service taken. Further as soon as service of any type gets complete, the server may take vacation or continue staying at the system even if the system is empty. For this moel we have erive the steay state queue size istribution at ranom epoch an some particular cases have been evelope an compare with known results. Keywors: batch arrivals, general heterogeneous service, balking, vacations, queue size istribution, waiting time 1. Introuction Research stuies on queues with batch arrival an vacations have been increase tremenously an still many researchers have been eveloping on the theory of ifferent aspects of queuing. Queuing is an important phenomenon in the real life, be it human or virtual queues an eminent researchers have been eveloping theories on queuing moels since 195. Research authors like Baba (1986), Lee et al. (1994; 1995), Doshi (1986) an Takagi (199) were the pioneers in this fiel. We can fin vast literature on the extensive amount of stuies been one on ifferent queuing moels with server vacations, arrival pattern an customers behavior. Stuies on queues with vacations an breakown ha been one by authors like Borthakur an Chouhury (1997; 2; 24), Maan (1995; 2; 21), Maan an Dayyeh (22), Chouhury an Maan (27). Many of the authors here have stuie single server with ifferent vacation policies. Queues with vacations an restricte amissibility of customers have been stuie by Chouhury (27) an Maan (2). In our moel we assume customers arriving in batches an the system is proviing in parallel two types of general heterogeneous service, the customer can choose any one type of service. We can fin many real life applications of this moel. Such situations are mostly observe in fuel stations, banks, post offices etc. where the customer has option of choosing any one kin of service. The concept of balking was first stuie by Haight in the year 1957, since then consierable attention has been given on many queuing theories with customer impatience. In our moel we assume that arrival units may refuse to join the system (balks) by estimating the uration of waiting time for a service to get complete or by witnessing the long length of the queue. In queuing literature, queues with vacations an behavior of impatient customers have been stuie by many for mainly single server but balking in case of two parallel servers for a general istribution has still not been worke on. Re-service is also an important factor in the theory of queuing system an have many applications in the real worl. There may be situations where re-service is esire, for instance while visiting a octor, the patient may be 114

2 Journal of Mathematics Research Vol. 4, No. 4; 212 recommene for some investigations, after which he may nee to see the octor again while in some situations offering public service a customer may fin his service unsatisfactory an consequently eman re-service. We fin many real life applications of queues with balking an re-service in sectors as telecommunication, computer networking, call centers, inventory an prouction, maintenance an quality control in inustrial organisations etc. Though in the queuing literature we fin limite stuies on queuing system with re-service, we refer to some type of queuing networks incluing cyclic queuing systems with feeback stuie by authors like Glenbe an Pujolle (1987) an Maan (1988). In fact, authors like Maan et al. (24), Jeyakumar an Arumuganathan (211) stuie queues with re-service. Here we have iscusse the case of optional re-service. In this case a customer has the choice of selecting any one of the two types of heterogeneous service, subsequently has the option to repeat the service taken by him or epart from the system. The mathematical moel has been efine uner the following assumptions. 2. Mathematical Moel Here we assume that customers (units) arrive in batches of variable size accoring following a compoun Poisson Process. Let λc i t (i = 1, 2, 3,...) be the first orer probability of arrival of i customers in batches in the system at a short interval of time (t, t + t where c i 1, i=1 c i = 1, λ>is the mean arrival rate of batches. We consier the case when there is a single server proviing parallel service of two types on a first come first serve basis (FCFS). At the start of the service, each customer has the choice of choosing either first service with probability ξ 1 or can choose secon service with probability ξ 2 an ξ 1 + ξ 2 = 1. We assume that the ranom variable of service time s j ( j = 1, 2) of the jth kin of service follows a general probability law with istribution function H j (s j ), h j (s j ) is the probability ensity function an E(s K j )isthekth moment (k = 1, 2,...) of service time, j = 1, 2. Let μ j (x) be the conitional probability of type j service uring the perio (x, x + x given that elapse time is x, so that an μ j (x) = h j (x) = μ j (s j )exp h j (x), j = 1, 2 (1) 1 H j (x) s μ j (x)x, j = 1, 2. (2) Once the service of a customer is complete, the server may ecie to take vacation with probability q or may continue to serve the next customer with probability (1 q) or may remain ile in the system even if there is no customer requiring service. Further we assume that the vacation time ranom variable follow general probability law with istribution function V(x), v(x)the probability ensity an E(V K )asthekth moment, k = 1, 2,... Let φ(x) be the conitional probability of a vacation perio uring the interval (x, x + x given elapse time is x, so that an φ(x) = v(y) = φ(y)exp V(x) 1 V(x) v φ(x)x Also we assume that (1 a 1 )( a 1 1) is the probability that an arriving batch balks uring the perio when the server is busy (available on the system) an (1 a 2 )( a 2 1) is the probability that an arriving batch balks uring the perio when server is on vacation. As soon as service (of any one kin) is complete he has the option to leave the system or join the system for reservice, if necessary. We assume that probability of repeating type j service as r j an leaving the system without re-service as (1 r j ), j = 1, 2. We consier that either service maybe repeate only once. 3. Definitions an Notations Assuming that steay state exists, we efine (3) (4) 115

3 Journal of Mathematics Research Vol. 4, No. 4; 212 P n, j (x) = Probability that there are n( 1) customers in the system incluing one customer in type j service, j = 1, 2 an elapse service time is x. Thus P n, j = P n, j (x)x is the corresponing steay state probability irrespective of elapse time x. R n, j (x) = Probability that there are n( 1) customers in the system incluing one customer who is repeating type j service, j = 1, 2 an elapse service time is x. Thus R n, j = R n, j (x)x is the corresponing steay state probability irrespective of elapse service time x. W n (x) = probability that there are n( ) customers in the queue an server is on vacation an elapse vacation time is x. W n = W n (x)x is the corresponing steay state probability irrespective of elapse vacation time x. E = Steay state probability of the server is ile but available in the system an there is no customer requiring service. The Probability Generating Functions are efine as: P j (x, z) = z n P n, j (x); P j (z) = z n P n, j, z 1; j = 1, 2 (5) n=1 n=1 n=1 R j (x, z) = z n R n, j (x); R j (z) = z n R n, j ; z 1; j = 1, 2 (6) n= n=1 W(x, z) = z n W n (x); W(z) = z n W n ; z 1 (7) C(z) = 4. Equations Governing the System Let us efine the steay state equations for our moel as n= x P n,1(x) + (λ + μ 1 (x))p n,1 (x) = λ(1 a 1 )P n,1 (x) + a 1 λ x P n,2(x) + (λ + μ 2 (x)) P n,2 (x) = λ(1 a 1 )P n,2 (x) + a 1 λ x R n,1(x) + (λ + μ 1 (x))r n,1 (x) = λ(1 a 1 )R n,1 (x) + a 1 λ x R n,2(x) + (λ + μ 2 (x))r n,2 (x) = λ(1 a 1 )R n,2 (x) + a 1 λ x W n(x) + (λ + φ(x)) W n (x) = λ(1 a 2 )W n (x) + a 2 λ z i c i ; z 1 (8) i=1 n c i P n i,1 (x) (9) i=1 n c i P n i,2 (x) (1) i=1 c i R n i,1 (x) (11) i=1 c i R n i,2 (x) (12) i 1 n C i W n i (x) (13) x W (x) + (λ + φ(x))w (x) = λ(1 a 2 )W (x) (14) λe = λ(1 a 1 )E + (1 q) (1 r 1 ) P 1,1 (x)μ 1 (x)x + (1 r 2 ) P 1,2 (x)μ 2 (x)x +(1 q) R 1,1 (x)μ 1 (x)x + where P, j (x) =, R, j (x) =, j = 1, 2 for (9), (1), (11) an (12). R 1,2 (x)μ 2 (x)x + i=1 W (x)φ(x)x (15) The above equations is to be solve subject to the bounary conitions given below at x = : P n,1 () = (1 q)ξ 1 (1 r 1 ) P n+1,1 (x)μ 1 (x)x + (1 r 2 ) P n+1,2 (x)μ 2 (x)x 116

4 Journal of Mathematics Research Vol. 4, No. 4; 212 +(1 q)ξ 1 R n+1,1 (x)μ 1 (x)x + P n,2 () = (1 q)ξ 2 (1 r 1 ) +(1 q)ξ 2 R n+1,1 (x)μ 1 (x)x + W n () = q (1 r 1 ) an the normalizing conition R n+1,2 (x)μ 2 (x)x + ξ 1 W n (x)φ(x)x + λa 1 ξ 1 c n E, n 1 (16) P n+1,1 (x)μ 1 (x)x + (1 r 2 ) P n+1,2 (x)μ 2 (x)x R n+1,2 (x)μ 2 (x)x + ξ 2 W n (x)φ(x)x + λa 1 ξ 2 c n E, n 1 (17) R n,1 () = r 1 P n,1 (x)μ 1 (x)x (18) R n,2 () = r 2 P n,2 (x)μ 2 (x)x (19) P n+1,1 (x)μ 1 (x)x + (1 r 2 ) P n+1,2 (x)μ 2 (x)x (2) E + 2 j=1 n=1 P n, j (x)x + 2 j=1 n=1 R n, j (x)x + n= W n (x)x = 1 (21) 5. Queue Size Distribution at Ranom Epoch Now let us multiply Equations (9)-(13) by z n, an taking summations over all possible values of n an simplifying we get x P 1(x, z) + {a 1 (λ λc(z)) + μ 1 (x)}p 1 (x, z) = (22) x P 2(x, z) + {a 1 (λ λc(z)) + μ 2 (x)}p 2 (x, z) = (23) x R 1(x, z) + {a 1 (λ λc(z)) + μ 1 (x)}r 1 (x, z) = (24) x R 2(x, z) + {a 1 (λ λc(z)) + μ 2 (x)}r 2 (x, z) = (25) x W(x, z) + {a 2(λ λc(z)) + φ(x)}w(x, z) = (26) We now integrate Equations (22)-(26) between limits an x an obtain P 1 (x, z) = P 1 (, z)e a 1λ(1 C(z)) x μ 1(t)t (27) P 2 (x, z) = P 2 (, z)e a 1λ(1 C(z)) x μ 2(t)t R 1 (x, z) = R 1 (, z)e a 1λ(1 C(z)) x μ 1(t)t R 2 (x, z) = R 2 (, z)e a 1λ(1 C(z)) x μ 2(t)t (28) (29) (3) W(x, z) = W(, z)e a 2λ(1 C(z)) x φ(t)t,λ> (31) Next we multiply Equation (16) with appropriate powers of z, take summation over all possible values of n an using relations (15) an (27), after simplifying we obtain zp 1 (, z) = (1 q) ξ 1 (1 r1 ) +(1 q) ξ 1 P 1 (x, z) μ 1 (x) x + (1 r 2 ) P 2 (x, z) μ 2 (x, z) x R 1 (x, z) μ 1 (x) x + R 2 (x, z) μ 2 (x) x + zξ 1 W(x, z) φ(x) x + za 1 ξ 1 λ ( C(z) 1)E 117 (32)

5 Journal of Mathematics Research Vol. 4, No. 4; 212 We perform the same operations on Equations (17)-(2) an thus obtain zp 2 (, z) = (1 q)ξ 2 (1 r1 ) +(1 q) ξ 2 P 1 (x, z)μ 1 (x)x + (1 r 2 ) P 2 (x, z)μ 2 (x, z)x R 1 (x, z)μ 1 (x)x + R 2 (x, z)μ 2 (x)x (33) zw(, z) = q + z ξ 2 W(x, z) φ (x) x + za 1 ξ 2 λ ( C(z) 1)E (1 r 1 ) R n,1 () = r 1 P 1 (x, z)μ 1 (x)x (34) R n,2 () = r 2 P 2 (x, z)μ 2 (x)x (35) P 1 (x, z)μ 1 (x)x + (1 r 2 ) P 2 (x, z)μ 2 (x)x We now multiply Equations (27) an (29) by μ 1 (x) an Equations (28) an (3) by μ 2 (x), integrate by parts w.r.t. x, use Equation (2) to obtain (36) P 1 (x, z)μ 1 (x)x = P 1 (, z)h 1 (a 1(λ λc(z))) (37) P 2 (x, z)μ 2 (x)x = P 2 (, z) H 2 (a 1(λ λc(z))) (38) R 1 (x, z)μ 1 (x)x = R 1 (, z)h 1 (a 1(λ λc(z))) (39) R 2 (x, z)μ 2 (x)x = R 2 (, z)h 2 (a 1(λ λc(z))) (4) Similarly multiplying Equation (31) by φ(x)an applying the same operations as above, we get W(x, z)φ(x)x = W(, z)v (a 2 (λ λc(z))) (41) where H j (a 1(λ λc(z))) = e a1(λ λc(z) H j (x) is the Laplace-Transform of jth type of service, j = 1, 2 an V (a 2 (λ λc(z))) = e a2(λ λc(z)) V(x) is the Laplace-Transform of vacation time. Let us substitute relations (37)-(41) in Equations (32) an (33), thus we get or zp 1 (, z) = (1 q) ξ 1 (1 r1 )P 1 (, z)h 1 (a 1(λ λc(z)) ) + (1 r 2 )P 2 (, z)h 2 (a 1(λ λc(z)) ) + +(1 q) ξ 1 R1 (, z)h 1 (a 1(λ λc(z)) ) + R 2 (, z)h 2 (a 1(λ λc(z))) + zξ 1 W(, z) V (a 2 (λ λc(z)) ) +z λ a 1 ξ 1 ( C(z) 1) ) E zp 1 (, z) = (1 q) ξ 1 (1 r1 )P 1 (, z)h 1 (a 1(λ λc(z)) ) + (1 r 2 ) P 2 (, z)h 2 (a 1(λ λ C(z))) + (1 q) ξ 1 P1 (, z)(h 1 (a 1(λ λc(z)) ) 2 + P 2 (, z)(h 2 (a 1 (λ λ C(z)) ) 2 + z ξ 1 q ( 1 r 1 ) P 1 (, z) H 1 (a 1(λ λc(z)) ) + (1 r 2 ) P 2 (, z) H 2 (a 1 (λ λc(z))) ) V (a 2 (λ λc(z))) +z λ a 1 ξ 1 (C(z) 1 ) E (42) 118

6 Journal of Mathematics Research Vol. 4, No. 4; 212 Similarly zp 2 (, z) = (1 q)ξ 2 (1 r1 )P 1 (, z)h 1 (a 1(λ λc(z)) ) + (1 r 2 )P 2 (, z)h 2 (a 1(λ λc(z)) ) +(1 q)ξ 2 P1 (, z)(h 1 (a 1(λ λc(z))) 2 + P 2 (, z)(h 2 (a 1(λ λc(z))) 2 + z ξ 2 (1 r1 ) P 1 (, z) H 1 (a 1(λ λc(z))) + (1 r 2 )P 2 (, z)h 2 (a 1(λ λc(z))) V (a 2 (λ λc(z))) (43) +z λa 1 ξ 2 (C(z) 1)E R 1 (, z) = r 1 P 1 (, z) H1 (a 1(λ λc(z))) (44) R 2 (, z) = r 2 P 1 (, z)h2 (a 1(λ λc(z))) (45) zw(, z) = qp 1 (, z)h1 (a 1(λ λc(z))) + P 2 (, z)h2 (a 1(λ λc(z))) (46) Now solving (42) an (43) we get the following P 1 (, z) = zλa 1ξ 1 (1 C(z))E P 2 (, z) = zλa 1ξ 2 (1 C(z))E R 1 (, z) = zλa 1r 1 ξ 1 (1 C(z))H 1 (a 1(λ λc(z)))e R 2 (, z) = zλa 1r 2 ξ 2 (1 C(z))H2 (a 1(λ λc(z)e W(, z) = qλa 1 (1 r1 )ξ 1 H1 (a 1(λ λc(z))) + (1 r 2 )ξ 2 H2 (a 1(λ λc(z))) E where is given by the following relation = {(1 q) + qv (a 2 (λ λc(z)))}{(1 r 1 )ξ 1 H1 (a 1(λ λc(z))) + (1 r 2 )ξ 2 H2 (a 1(λ λc(z)))} (47) (48) (49) (5) (51) +(1 q) { r 1 ξ 1 (H 1 (a 1(λ λc(z))) 2 + r 2 ξ 2 (H 2 (a 2(λ λc(z))) 2} z Further we integrate Equations (27)-(31) w.r.t x an use Equation (2) to obtain P 1 (z) = zξ 11 H 1 (a 1(λ λc(z)))e (52) P 2 (z) = zξ 21 H 2 (a 1(λ λc(z)))e R 1 (z) = zr 1ξ 1 H 1 (a 1(λ λc(z))1 H 1 (a 1(λ λc(z))e R 2 (z) = zr 2ξ 2 H 2 (a 1(λ λc(z))1 H 2 (a 1(λ λc(z))e W(z) = q a 1 a 2 1 V (a 2 (λ λc(z))(1 r 1 )ξ 1 H1 (a 1(λ λc(z)) + (1 r 2 )ξ 2 H2 (a 1(λ λc(z))e (56) Let us now efine P Q (z) as the probability generating function of the queue size irrespective of the type of service the server is proviing, such that aing Equations (52)-(56) we get (53) (54) (55) P Q (z) = P 1 (z) + P 2 (z) + R 1 (z) + R 2 (z) + W(z) (57) 119

7 Journal of Mathematics Research Vol. 4, No. 4; 212 We etermine the unknown probability E by using the relation of the normalizing conition E + P 1 (1) + P 2 (1) + R 1 (1) + R 2 (1) + W(1) = 1 We now use L Hopital s Rule as the Equations (52)-(56) is ineterminate of the zero/zero form at z = 1 an simplifying we obtain P 1 (1) = lim P 1 ( z) z 1 ξ = 1 a 1 λ E(I)E(S 1 )E 1 a 1 λe(i){(1 r 1 )ξ 1 E(S 1 ) + (1 r 2 )ξ 2 E(S 2 )} + qa 2 λ E(I)E(V) {(1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } (58) 2(1 q) a 1 λ E(I){ r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E (S 2 )} P 2 (1) = lim P 2 (z) z 1 = = = = ξ 2 a 1 λ E(I) E(S 2 ) E 1 a 1 λe(i){(1 r 1 )ξ 1 E(S 1 ) + (1 r 2 )ξ 2 E(S 2 )} + qa 2 λ E(I) E(V) {(1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } 2(1 q) a 1 λ E(I) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 ) } R 1 (1) = lim z 1 R 1 (z) r 1 ξ 1 a 1 λ E(I) E(S 1 ) E 1 a 1 λe(i){(1 r 1 )ξ 1 E(S 1 ) + (1 r 2 )ξ 2 E(S 2 )} + qa 2 λ E(I) E(V) {(1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } 2(1 q) a 1 λ E(I) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 ) } R 2 (1) = lim z 1 R 2 (z) r 2 ξ 2 a 1 λ E(I) E(S 2 ) E 1 a 1 λe(i){(1 r 1 )ξ 1 E(S 1 ) + (1 r 2 )ξ 2 E(S 2 )} + qa 2 λ E(I)E(V) {(1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } 2(1 q) a 1 λ E(I) {r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 ) } W(1) = lim z 1 W(z) qa 1 λ E(I) E(V) { (1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 }E 1 a 1 λ E(I)(1 r 1 ) ξ 1 E(S 1 ) + (1 r 2 ) ξ 2 E(S 2 ) } + qa 2 λ E(I) E(V) { } (1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 2(1 q) a 1 λ E(I){ r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} where E(I) is the mean size of batch of arriving customers, E(S 1 ), E(S 2 ) are the mean service times of type 1 an type 2 server respectively, E(V) is the mean of vacation time an H () = 1, V () = 1. The RHS of the results (58)-(62) respectively give the steay state probability that the server is busy proviing type 1 service, type 2 services, repeating service of type 1, repeating type 2 service an server being on vacation. Now aing Equations (58)-(62) we get a1 λ E(I) { (1 + r 1 ) ξ 1 E(S 1 ) + (1 + r 2 ) ξ 2 E(S 2 )} + qa 1 λ E(I) E(V) { (1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } E P Q (1) = 1 a 1 λ E(I) {(1 r 1 ) ξ 1 E(S 1 ) + (1 r 2 ) ξ 2 E(S 2 ) } + qa 2 λ E(I) E(V) { } (1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 2(1 q) a 1 λ E(I){ r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} (63) which gives the steay state probability that the server is busy, irrespective of whether it is proviing type 1 or type 2 service an server being on vacation. Let us simplify the normalizing conition E + P Q (1) = 1 to get E an thus we have (59) (6) (61) (62) E = 1 a 1 λ E(I){ (1 r 1 ) ξ 1 E(S 1 ) + (1 r 2 ) ξ 2 E(S 2 ) } qa 2 λe(i) E(V) {(1 r 1 )ξ 1 + (1 r 2 ) ξ 2 } 2(1 q) a 1 λ E(I){r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} 1 + qa 1 λ E(I)E(V) {(1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } qa 2 λe(i)e(v) {(1 r 1 )ξ 1 + (1 r 2 ) ξ 2 } 2(1 q)a 1 λe(i){ r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 ) } (64) Also from (64) we obtain ρ as the utilization factor of the system as ρ = 1 E. 12

8 Journal of Mathematics Research Vol. 4, No. 4; 212 Now let P S (z) enote the probability generating function of the system size irrespective of the state of the system Thus P S (z) = P S (z) = P 1 (z) + P 2 (z) + R 1 (z) + R 2 (z) + W(z) + E {(1 q) + qv (a 2 (λ λc(z))+ q a 1 E a 2 1 V (a 2 (λ λc(z)) z}{(1 r 1 ) ξ 1 H1 (a 1(λ λc(z)) )+ (1 r 2 ) ξ 2 H2 (a 1(λ λc(z)) )}+{(1 q) z}{ r 1 ξ 1 (H1 (a 1(λ λc(z)) ) 2 + r 2 ξ 2 (H2 (a 1 (λ λc(z)) )2 } {(1 q) + qv (a 2 (λ λc(z))}{(1 r 1 ) ξ 1 H 1 (a 1(λ λc(z)) + (1 r 2 ) ξ 2 H 2 (a 1(λ λc(z))} + (1 q) { r 1 ξ 1 (H 1 (a 1(λ λc(z))) 2 + r 2 ξ 2 (H 2 (a 1(λ λc(z)) ) 2 } z 6. The Average Queue Size Let Lq enote the mean queue size at ranom epoch. Then L q = lim z 1 z P Q(z) Since this formula is also of zero/zero form, we use L Hopital s Rule twice an obtain D (z)n (z) N (z)d (z) L S = lim (66) z 1 2(D (z)) 2 where primes an ouble primes in (66) enote first an secon erivatives at z =1 respectively. N (1) = E a 1 λ E(I) { (1 + r 1 ) ξ 1 E(S 1 ) + (1 + r 2 ) ξ 2 E(S 2 )} qa 1 λe(i) E(V) {(1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } (65) 2 a 1 λ E(I){ ( 1 r 1 )ξ 1 E(S 1 ) + (1 r 2 ) ξ 2 E(S 2 ) } + a 1 λ E(I/I 1){ ξ 1 E(S 1 ) + ξ 2 E(S 2 )} ) + a 1 λ E(I) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 ) } + 2(a 1 λ E(I)) 2 { r 1 ξ 1 E(S 2 1 ) + r 2 ξ 2 E(S 2 2 ) } N (1) = E +(a 1 λ E(I/I 1) ) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 ) } (a 1 λ E(I)) 2 { r 1 ξ 1 ( E(S 1 )) 2 + r 2 ξ 2 ( E(S 2 )) 2 } +2q (a λ E(I)) 2 E(V) { (1 r 1 )ξ 1 E(S 1 ) + (1 r 2 ) ξ 2 E(S 2 )} +{ q ( a 1 λ E(I) ) 2 E(V 2 ) + a 1 λ E(I/I 1)E(V) } { (1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } D (1) = a 1 λ E(I) (1 r 1 ) ξ 1 E(S 1 ) + (1 r 2 ) ξ 2 E(S 2 ) + qa 2 λ E(I) E(V) { (1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } +2(1 q) a 1 λ E(I) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} 1 { (a 1 λ E(I)) 2 { (1 r 1 ) ξ 1 E(S 2 1 ) + (1 r 2) ξ 2 E(S 2 2 ) D (1) = a 1 λ E(I/I 1) } (1 r 1 ) ξ 1 E(S 1 ) + (1 r 2 ) ξ 2 E(S 2 ) } + a 1 a 2 ( λ E(I)) 2 { (1 r 1 ) ξ 1 E(S 1 ) + (1 r 2 ) ξ 2 E(S 2 )} + q ( a 2 λ E(I)) 2 E(V 2 ) { (1 r 1 ) ξ 1 + (1 r 2 ) ξ 2 } + 2(1 q) (a 1 λ E(I)) 2 { r 1 ξ 1 E(S 2 1 ) + r 2 ξ 2 E(S 2 2 ) +a 1 λ E(I/I 1){ r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} + (a 1 λ E(I)) 2 { r 1 ξ 1 (E(S 1 )) 2 + r 2 ξ 2 (E(S 2 )) 2 } where E(S 1 2), E(S 2 2), E(V2 ) are the secon moments of service times of type 1, type 2 an vacation time respectively. E(I/I 1) is the secon factorial moment of the batch of arriving customers, H1 () = 1, H 2 () = 1, V () = 1 an E has been obtaine in (64). We can also obtain L = L q + ρ by using the above relations, which gives the steay-state average size of the queue system. 121

9 Journal of Mathematics Research Vol. 4, No. 4; Particular Cases Case 1. Re-service in a Vacation queue with Batch arrival an Two types of General Heterogeneous Service. Here we consier the situation when there is no balking i.e. all customers join the system, an in that case, we take a 1 = a 2 = 1; thus Equation (65) becomes P S (z) = E (1 z ) { (1 r 1 ) ξ 1 H1 (λ λ C(z))+ (1 r 2 ) ξ 2 H2 (λ λ C(z))} +{ (1 q) z} {r 1 ξ 1 (H1 (λ λ C(z)))2 + r 2 ξ 2 ( H2 (λ λ C(z)))2 } { (1 q) + qv (λ λ C(z))}{(1 r 1 ) ξ 1 H1 (λ λ C(z)) + (1 r 2) ξ 2 H2 (λ λ C(z)) } + (1 q) { r 1 ξ 1 ( H1 (λ λc(z)))2 + r 2 ξ 2 ( H2 (λ λ C(z)) )2 } is the queue size istribution of a batch arrival queuing system with two types of general heterogeneous service with vacation an optional re-service where ile time E E = 1 λ E(I) { ξ 1 E(S 1 ) + ξ 2 E(S 2 )} λ E(I) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} qλ E(I) E(V) {(1 r 1 )ξ 1 + (1 r 2 ) ξ 2 } + 2q λ E(I) {r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} 1 2(1 q) λ E(I) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} Case 2. Balking an Re-service with Batch arrival with an two types of general Heterogeneous Service. Here we assume that the server oes not take vacation, i.e. q = ; an obtain { (1 r1 ) ξ 1 H1 (a 1(λ λc(z)) ) + (1 r 2 ) ξ 2 H2 (a 1(λ λc(z)) ) } E (1 z) + { r 1 ξ 1 ( H1 P S (z) = (a 1(λ λc(z)) ) 2 + r 2 ξ 2 ( H2 (a 1(λ λc(z)) ) 2} { (1 r1 ) ξ 1 H1 (a 1(λ λc(z)) ) + (1 r 2 ) ξ 2 H2 (a 1(λ λc(z)) ) } + { r 1 ξ 1 ( H 1 (a 1(λ λc(z)) ) 2 + r 2 ξ 2 (H 2 (a 1(λ λc(z)) ) 2} z is the queue size istribution of Batch arrival queuing system with two types of general heterogeneous service with balking an optional re-service an E = 1 a 1 λ E(I) { ξ 1 E(S 1 ) + ξ 2 E(S 2 )} a 1 λ E(I) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 ) } 1 2a 1 λ E(I) { r 1 ξ 1 E(S 1 ) + r 2 ξ 2 E(S 2 )} Case 3. Balking in a Vacation queue with batch arrival an two types of Heterogeneous service. Here we assume that there is no re-service, then r 1 = r 2 =, our queue size istribution reuces to P S (z) = E { (1 q) + qv (a 2 (λ λc(z)) ) + q a 1 a 2 1 V (a 2 (λ λc(z))) z }{ ξ 1 H 1 (a 1(λ λc(z)) ) + ξ 2 H 2 (a 1(λ λc(z)) ) } {(1 q) + qv (a 2 (λ λc(z)))}{ξ 1 H 1 (a 1(λ λc(z)) ) + ξ 2 H 2 (a 1(λ λc(z)) ) } z (69) E = 1 a 1 λ E(I) {ξ 1 E(S 1 ) + ξ 2 E(S 2 ) } qa 2 λ E(I) E(V). 1 + qa 1 λ E(I) E(V) qa 2 λ E(I) E(V) Case 4. If we assume that there is two types of heterogeneous service, optional re-service, no vacation, no balking, then q = ; a 1 = a 2 = 1, the PGF in (65) reuces to {1 z}{(1 r1 )ξ E 1 H1 (λ λc(z)) + (1 r 2)ξ 2 H2 (λ λc(z))} +(1 z){r 1 ξ 1 (H1 P S (z) = (λ λc(z)))2 + r 2 ξ 2 (H2 (λ λc(z)) )2 } {(1 r1 )ξ 1 H1 (λ λc(z))+ (1 r (7) 2)ξ 2 H2 (λ λc(z))} {r 1 ξ 1 (H1 (λ λc(z)) )2 + r 2 ξ 2 (H2 (λ λc(z)) z )2 E = 1 λ E(I) {(1 + r 1 )ξ 1 E(S 1 ) + (1 + r 2 )ξ 2 E(S 2 ))} ( ) Which tallies with the result obtaine by Maan et al. (24) for a M X G1 / 1 queue with optional re-service. G 2 Case 5. Again if we consier the situation when there is no case of optional re-service an no balking (i.e. all arriving batches join the system) with two types of heterogeneous service an vacation then, r 1 = r 2 = ; a 1 = a 2 = 1, then (65) becomes (67) (68) P S (z) = E (1 z){ξ 1 H1 (λ λc(z)) + ξ 2H2 (λ λc(z))} {(1 q) + qv (λ λc(z)}{ξ 1 H1 (λ λc(z) + ξ 2H2 (λ λc(z)} z (71) 122

10 Journal of Mathematics Research Vol. 4, No. 4; 212 E = 1 λe(i) { ξ 1 E(S 1 ) + ξ 2 E(S 2 )} q λ E(I) E(V) Thus our result (71) is equivalent to( the result ) obtaine by Maan et al. (25) for the PGF of a queue size istribution at ranom epoch of M X G1 / /1/G(BS )/V G S with batch arrivals, two kins of general heterogeneous 2 service an general vacation time with Bernoulli Scheules. Case 6. Now further we assume that in aition to the particular cases in (5), if we take ξ 2 = ; i.e. ξ 1 = 1, which means the server is proviing only one type of service an without re-service an balking, we obtain P S (z) = E(1 z)h1 (λ λc(z)) { (1 q) + qv (λ λ C(z))} H1 (λ λ C(z)) z (72) E = 1 λe(i) ξ 1 E(S 1 ) q λ E(I) E(V) Thus the result (72) reuces to the probability generating function (PGF) of a queue size istribution at a ranom epoch for M X /G/1/G(BS )/V S queue with batch arrivals, general service an general vacation first stuie by Maan (24) who consiere first service general an secon service exponential. Case 7. Now if we consier a system with two types of general heterogeneous service but no optional re-service, no vacation an no balking, i.e. r 1 = r 2 =, q =, a 1 = a 2 = 1, then we obtain the queue size P S (z) = E(1 z){ξ 1H1 (λ λc(z)) + ξ 2H2 (λ λc(z))} {ξ 1 H1 (λ λc(z) + ξ 2H2 (λ λc(z))} z (73) E = 1 λe(i)ξ 1 E(S 1 ) λe(i)ξ 2 E(S 2 ) Case 8. Again in aition to particular cases in (73), if we assume that there is only one type of service then P S (z) = E(1 z)h 1 (λ λc(z)) H1 (λ λc(z)) z (74) E = 1 λe(i) ξ 1 E(S 1 ) is the queue size istribution of a M X /G/1 batch arrival proviing single service follows the result obtaine by Gaver (1959). References Baba, Y. (1986). On the M X /G/1 queue with vacation time. Operation Research Letters, 5, Borthakur, A., & Chouhury, G. (1997). On a batch arrival queue with generalize Vacation. Sankhya-Series B, 59, Chouhury, G. (22). Analysis of the M X /G/1 queuing system with vacation times. Sankhya-Series B, 64(1), Chouhury, G., & Maan, K. C. (27). A batch arrival Bernoulli vacation queue with ranom set up time uner restricte amissibility policy. International Journal of Operation Research (USA), 2(1), Doshi, B. T. (1986). Queuing Systems with Vacation - a survey. Queuing Systems 1, Gaver, D. P. (1959). Imbee markov chain analysis of a waiting line process in continuous time. Ann. Mathematical Statistics, 3, Glenbe, G., & Pujolle, G. (1987). Introuction to Queueing Networks. New York: John Wiley an Sons. Lee, H. W., & Lee, S. S. (1993). Correcte istribution of ile perio in M/G/1 queue with multiple vacations. Research Report, Department of Inustrial Engineering, Sung Kyon University, Suwon, Korea. Jeyakumar, S., & Arumuganathan. (211). A Non-Markovian Queue with Multiple Vacations an Control Policy on Request for Re-service. Quality Technology of Quantitative Management (QTQM), 8(3),

11 Journal of Mathematics Research Vol. 4, No. 4; 212 Maan, K. C. (1988). A cyclic queuing system with three servers an optional two-way feeback. Microelectron Rel., 28(6), Maan, K. C. (2). An M/G/1 queue with secon optional service. Queuing Systems Theory Appl., 34, 1-4, Maan, K. C., & Abu-Dayyeh, W. (22). Restricte amissibility of batches into an M/G/1 type bulk queue with moifie Bernoulli scheule server vacations. ESAIM Probability Statistics, 6, Maan, K. C., & Chouhury, G. (24). Steay state analysis of an M X /(G 1 G 2 )/1 queue with restricte amissibility an moifie Bernoulli scheule server vacation. Journal of Probability an Statistical Science, 2(2), Maan, K. C., Al-Rawi, Z. R., & Al-Nasser, A. D. (25). On a M X /G 1 G 2 )/1/G(BS )/V s vacation queue with two types of general heterogeneous service. Journal of Applie Mathematics an Decision Sciences, 3, Takagi, H. (199). Time-epenent analysis of an M/G/1 vacation moels with exhaustive service. Queueuing Systems, 6(1),