Applied Mathematical Sciences, Vol. 3, 2009, no. 24, 1203-1208 A Vacation Queue with Additional Optional Service in Batches S. Pazhani Bala Murugan Department of Mathematics, Annamalai University Annamalainagar-608 002, India spbmaths@yahoo.com R. Kalyanaraman Department of Mathematics, Annamalai University Annamalainagar-608 002, India r.kalyan24@rediff.com Abstract A single server infinite capacity queueing system with Poisson arrival and exponential service time distribution along with second optional service in batches is considered. The server takes single vacation each time the system becomes empty and the duration of the vacation follows an exponential distribution. In steady state, the probability generating function for queue length has been obtained. The average queue length have been found and numerical results are presented to test the feasibility of the queueing model. Mathematics Subject Classification: 60K25, 60K30 Keywords: Optional batch service, Steady state equations, Queue length, Waiting time, Single vacation 1 Introduction In this paper, an M/M/1 queueing model with server vacation is taken. The single server, apart from providing the usual service one by one, also provides an additional optional service to the customers in batches of fixed size b( 1). The customers are queued up for the first service, which is essential for all the customers. The second service is optional which is demanded by some of the units whereas the others leave the system after the first service is over.
1204 S. Pazhani Bala Murugan and R. Kalyanaraman The steady state and the transient solutions for the M/M/1 model in various forms have been dealt with by a wide range of authors. To mention a few references we would name Bailey[1], Chaudhry et al[2], Cohen[3], Conolley[4], Daley[5], Feller[7], Jaiswal[8], Medhi[13], Neuts[14], Saaty[15], Takacs[16]. The idea of optional service has been dealt with by several authors including Madan[10, 11, 12]. Queues with vacations have been studied extensively in the past: a comprehensive survey can be found in Teghem[17] and Doshi[6]. The organization of the paper is as follows: The model under consideration is described in section 2. In section 3, we analyze the model by deriving the system steady state equations. Using the equations the probability generating function of queue length are obtained in section 4. The operating characteristics are obtained in section 5 and a numerical study is carried out in section 6 to test the effect of the system performance. 2 Model Analysis Let us define the following probability for our subsequent analysis; P [0] 0,0 = P r {the server is idle} P m,n [1] = P r {there are m, n units waiting for the first and the second service respectively excluding one unit in the first service} m,0 = P r {there are m units waiting for the first service excluding a batch being provided the second service} Q m,0 = P r {there are m unit waiting for the first service, no unit waiting for second service and the server is on vacation} The system has the following set of steady state equations: (λ + μ 1 )P m,n [1] = λ m 1,n + μ 1 q m+1,n + μ 1 pp m+1,n 1; [1] m>0; 0 <n<b(1) (λ + μ 1 ) m,0 = λ m 1,0 + μ 1 q m+1,0 + μ 2 m+1,0 + θq m+1,0 ; m>0 (2) (λ + μ 1 ) 0,n = μ 1 q 1,n + μ 1 pp 1,n 1; [1] 0 <n<b(3) (λ + μ 1 ) 0,0 = μ 1 q 1,0 + μ 2 1,0 + θq 1,0 + λp [0] 0,0 (4) (λ + μ 2 ) m,0 = λ m 1,0 + μ 1 p m,b 1 ; m>0(5) (λ + μ 2 ) 0,0 = μ 1 p b 1 n=1 0,n (6) (λ + θ)q m,0 = λq m 1,0 ; m>0 (7) (λ + θ)q 0,0 = μ 1 q 0,0 + μ 2 0,0 (8) λp [0] 0,0 = θq 0,0 (9)
A vacation queue 1205 3 Probability Generating Functions of Queue Length We define the following probability generating functions: (α, β) = n (α) = 0 (α) = b 1 P m,nα [1] m ; P m [1] (β) = P m,nβ [1] n n=0 b 1 b 1 P m [1] (β)α m = n=0 n (α)β n = n=0 m,0α m and Q 0 (α) = m,nα m β n Q m,0 α m (10) Performing, α times equation (3)+ α m+1 times equation (1); α times equation (4) + α m+1 times equation (2); equation (6) + α m times equation (5) (for n = b) and rearranging the terms we have {[λ(1 α)+μ 1 ] α μ 1 q} n (α) = μ 1p n 1(α) μ 1 q 0,n μ 1 p 0,n 1 (0 <n b 1)) (11) {[λ(1 α)+μ 1 )] α μ 1 q} 0 (α) = μ 2 0 (α)+θq 0 (α) (λ + θ)q 0,0 λ(1 α)p [0] 0,0 (12) [λ(1 α)+μ 2 ] 0 (α) = μ 1 p Performing, equation (8) + Performing, equation (12) + b 1 n=1 terms and simplifying, we have (α, β) = b 1 (α)+μ 1p b 2 n=1 α m times equation (7), we have 0,n (13) [λ(1 α)+θ] Q 0 (α) = (λ + θ)q 0,0 (14) β n times equation (11), and rearranging the μ 2 0 (α)+θq 0 (α) (λ + θ)q 0,0 λ(1 α)p [0] 0,0 μ 1 (q + pβ) 0 (β)+μ 1 q 0,0 [λ(1 α)+μ 1 ] α μ 1 (q + pβ) (15)
1206 S. Pazhani Bala Murugan and R. Kalyanaraman We note that for β =0,P (1) (α, β) =P (1) 0 (α) and P (1) 0 (0) = P (1) 0,0. Thus for β =0, equation (15) gives 0 (α) = μ 2 0 (α)+θq 0 (α) (λ + θ)q 0,0 λ(1 α)p [0] 0,0 [λ(1 α)+μ 1 ]α μ 1 q (16) Equation (13) for b = 1 gives 0 (α) = μ 1p 0 (α) (17) λ(1 α)+μ 2 Substituting equations (14) and (17) in (16) and simplifying we have 0 (α) = λ [λ(1 α)+μ 2] {(λ + θ)q 0,0 +[λ(1 α)+θ]p 0,0} [0] [λ(1 α)+θ][λ 2 α 2 λ(λ + μ 1 + μ 2 )α + μ 1 (λq + μ 2 )] (18) Using equation (18) in (17), we have 0 (α) = λμ 1 p{(λ + θ)q 0,0 +[λ(1 α)+θ]p [0] 0,0} [λ(1 α)+θ][λ 2 α 2 λ(λ + μ 1 + μ 2 )α + μ 1 (λq + μ 2 )] (19) We define P q (α) = P [0] 0,0 + Q 0 (α)+ 0 (α)+ 0 (α) (20) as the probability generating function of the number of customers in the queue.substituting equations (18),(19)and (14) into (20) and use equation (9),we have where P q (α) = N(α) [0] [P D(α) 0,0] (21) N(α) = { λ θ (λ + θ)+[λ(1 α)+θ]}{λ2 (α 2 2α +1)+λ(μ 1 + μ 2 )(1 α)+μ 1 μ 2 } D(α) = [λ(1 α)+θ][λ 2 α 2 λ(λ + μ 1 + μ 2 )α + μ 1 (λq + μ 2 )] Now, we determine the only unknown P 0,0, [0] by substituting α = 1 in (21), we have [ P [0] 0,0 = 1 λ ( pμ ][ 1 θ 2 ] +1) (22) μ 1 μ 2 λ(λ + θ)+θ 2
A vacation queue 1207 ( which implies that the utilisation factor is ρ = λ μ1 p μ 1 μ 2 +1 ) and the steady state condition is therefore ( ) λ μ1 p +1 < 1 (23) μ 2 μ 1 We observe that on setting p = 0 in equation (23) reduces to ρ = λ μ 1 < 1 which is the usual well known steady state condition for the M/M/1 queue with single vacation. Particular case If there is no vacation,then the above system coincides with the system that was considered by Madan(1992). 4 Operating Characteristics In this section, we derive the average queue length and the average waiting time of a customer in the queue. Let E(L q ) denote the mean number of customers in the queue and E(W q ) denote the average waiting time of a customer in the queue. The average queue length is E(L q )= d dα [P q(α)] α=1 = D(1)N (1) (1) N(1)D (1) (1) [D(1)] 2 [P [0] 0,0] (24) using Little s formula E(L q )=λe(w q ), we can find that E(W q ), the average waiting time of a customer in the queue, is given by where E[W q ] = D(1)N (1) (1) N(1)D (1) (1) λ[d(1)] 2 [P [0] 0,0] (25) [ ] λ N(1) = (λ + θ)+θ μ 1 μ 2 θ [ D(1) = θμ 1 μ 2 1 λ ( )] μ1 p +1 μ 1 μ 2 [ ] λ N (1) (1) = ( λ)(μ 1 μ 2 )+ (λ + θ)+θ [ λ(μ 1 + μ 2 )] θ [ D (1) (1) = λμ 1 μ 2 1 λ ( )] μ1 p +1 + θλ(λ μ 1 μ 2 ) μ 1 μ 2 5 Numerical Study In this section we numerically analyse the model defined in this paper for the arbitrary values of λ, μ 1,μ 2, p and various values of θ. The dependence is
1208 S. Pazhani Bala Murugan and R. Kalyanaraman then shown graphically in the following figures. These figures shows respectively the dependence of E(L q ) and E(W q ) versus θ by decreasing curves. E(Lq) Vacation rate verses E(Lq) 6 5 4 3 2 1 0 0 10 20 30 Vacation rate E(Wq) Vacation rate verses E(Wq) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Vacation rate References [1] J.W. Cohen, The single server queue, North Holland Publications, Amsterdam, 1969. [2] B.T. Doshi, Single server queues with vacations, stochastic analysis of computer and communications systems, ed, H. Takagi, Elsevier, Amsterdam, 1990. [3] W. Feller, An introduction to probability theory and its applications, Vol. 1, 3rd ed., Wiley, New York, 1968. [4] N.K. Jaiswal, Time-dependent solution of the bulk service queueing problem.oper. Res., 8 (1960), 773-781. [5] J.D.C. Little, A proof of the queueing formula: L = λw, Operations Res., 9(1961) No. 3, 383-387. [6] K.C. Madan, An M/M/1 queueing system with additional optional service in batches. IAPQR Transactions. 17(1992) No. 1. 23-33. [7] J. Medhi, Waiting time distribution in a Poisson queue with a general bulk service rule, Management Sci., 21(1975), 777-782. [8] M.F. Neuts, A general class of bulk queues with Poisson input, Ann. Math. Statist., 38(1967), 759-770. Received: October, 2008