Priority Queueing System with a Single Server Serving Two Queues M [X 1], M [X 2] /G 1, G 2 /1 with Balking and Optional Server Vacation

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1 Available at Appl. Appl. Math. ISSN: Vol. 11, Issue 1 (June 216), pp Applications and Applied Mathematics: An International Journal (AAM) Priority Queueing System with a Single Server Serving Two Queues M [X 1], M [X 2] /G 1, G 2 /1 with Balking and Optional Server Vacation G. Ayyappan and P. Thamizhselvi Department of Mathematics Pondicherry Engineering College Puducherry, India ayyappanpec@hotmail.com; tmlselvi972@pec.edu Received: October 12, 215; Accepted: February 9, 216 Abstract In this paper we study a vacation queueing system with a single server simultaneously dealing with an M [X1] /G 1 /1 and an M [X2] /G 2 /1 queues. Two classes of units, priority and non-priority, arrive at the system in two independent compound Poisson streams. Under a non-preemptive priority rule, the server provides a general service to the priority and non-priority units. We further assume that the server may take a vacation of random length just after serving the last customer in the priority unit present in the system. If the server is busy or on vacation, an arriving non-priority customer either join the queue with probability b or balks(does not join the queue) with probability (1 b). The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also the average number of customer in the priority and the non-priority queue and the average waiting time are derived. Numerical results are computed. Keywords: Non-Preemptive Priority Queueing systems; Batch Arrival; Modified Server Vacations Transient Solution; Average Queue Size; Average Waiting Time MSC 21 No.: 6K25, 68M3, 9B22 61

2 62 G. Ayyappan and P. Thamizhselvi 1. Introduction The study on queuing models has become an indispensable area due to its wide applicability in real-life situations; all the models considered have had the property that units proceed to service on a first-come, first-served basis. This is obviously not only the manner of service, and there are many alternatives, such as last-come, first-served, selection in random order, and selection by priority. In order to offer different qualities of service for different kinds of customers, we often control a queueing system by priority mechanism. This phenomenon is common in practice. For example, in telecommunication transfer protocol, for guaranteeing different layers of service for different customers, priority classes control may appear in the header of an IP package or in an ATM cell. Priority control is also widely used in production practice, transportation management, etc. A few papers appear on bulk arrival priority queueing system. Hawkes (1956) considered the time dependent solution of a priority queue with bulk arrivals. Haghighi and Mishev (26) have studied a parallel priority queueing system with finite buffers. Vacation queues have been studied by several authors including Doshi (1986), Takagi (199), and Chae et al. (21). Ayyappan and Muthu Ganapathi Subramanian (29) have studied single server retrial queueing system with non-pre-emptive priority service and single vacation exhaustive service type. Madan (211) studied a Non-preemptive priority queueing system with a single server serving two queues M/G/1 and M/D/1 with optional server vacations based on the exhaustive service of the priority units. Haghighi and Mishev (213) have studied a Stochastic Three-stage Hiring Model as a Tandem Queueing Process with Bulk Arrivals and Erlang Phase-Type Selection. Jain and Charu Bhargava (28) have studied bulk arrival retrial queue with unreliable server and priority subscribers. Jinbio and Lian (213) have studied a single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule. Thangaraj and Vanitha (21) have studied an M/G/1 queue with two-stage heterogeneous service compulsory server vacation and random breakdowns, and Jau-Chauan and Fu-Min (29) have studied modified vacation policy for M/G/1 retrial queue with balking and feedback. In this paper we consider a priority queueing system with a single server serving two queues M [X1] /G 1 /1 and M [X2] /G 2 /1 with balking and optional server vacation based on exhaustive service of the priority units. The service time of the priority and non-priority customers follows general (arbitrary) distribution. We assume that the server may take a vacation of random length but no vacation is allowed if there is even a single priority unit present in the system. Thus the server may take an optional vacation of random length just after completing the service of the last customer in the priority unit present in the system with probability θ or else may just continue serving the non-priority units if present in the system with probability (1 θ). If the server is busy or on vacation, an arriving non-priority customer either joins the queue with probability b or balks (does not join the queue) with probability (1 b) and the priority units are not allowed to balk the queue. Here we derive time dependent probability generating functions for both priority and non-priority units in terms of Laplace transforms. We also derive the average queue size and average waiting

3 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 63 time in the queue for both priority and non-priority units. Some particular cases and numerical results are also discussed. The rest of the paper is organized as follows: Mathematical description of our model in Section (2). Definitions, equations governing of our model and the time dependent solution have been obtained in Sections (3) and (4). The corresponding steady state results have been derived explicitly in Section (5). Average queue size and the average waiting time are computed in Sections (6) and (7). Some particular cases are discussed in Section (8). In Section (9), we consider a numerical example to illustrate application of our results. 2. Mathematical description of our model (1) Priority and non-priority units arrive at the system in batches of variable size in a compound Poisson process. Let λ 1 c i dt(i = 1, 2, 3,...) and λ 2 c j dt(j = 1, 2, 3,...) be the first order probability that a batch of i and j customers arrives at the system during a short interval of time (t, tdt), where c i 1, c i = 1, c j 1, c j = 1 and λ 1 >, λ 2 > are the average arrival rates for priority and non-priority customers entering into the system and forming two queues. The server must serve all the priority units present in the system before taking up non-priority unit for service. In other words, there is no priority unit present in the system at the time of starting service of a non-priority unit. Further, we assume that the server follows a non-preemptive priority rule, which means that if one or more priority units arrive during the service time of a non-priority unit, the current service of a non-priority units is not stopped and a priority unit will be taken up for service only after the current service of a non-priority unit is complete. (2) Each customer under priority and non-priority units service provided by a single server on a first come - first served basis. The service time for both priority and non-priority units follows general (arbitrary) distributions with distribution functions B i (s) and the density functions b i (s), i = 1, 2. (3) Let µ i (x)dx be the conditional probability of completion of the priority and non-priority units service during the interval (x, x dx], given that the elapsed service time is x, so that µ i (x) = b i(x) 1 B i (x), and, therefore, b i (s) = µ i (s)e s µ i (x)dx. (4) We further assume that as soon as the service of the last priority unit present in the system is completed, the server has the option to take a vacation of random length with probability θ, in which case the vacation starts immediately or else with probability

4 64 G. Ayyappan and P. Thamizhselvi (1 θ) he may decide to continue serving the non-priority units present in the system, if any. In the later case, if there is no non-priority unit present in the system, the server remains idle in the system waiting for the new units to arrive. (5) The vacation time follow general (arbitrary) distribution with distribution function V (s) and the density function v(s). Let γ(x)dx be the conditional probability of a completion of a vacation during the interval (x, x dx] given that the elapsed vacation time is x, so that γ(x) = v(x) 1 V (x), and, therefore, v(s) = γ(s)e s γ(x)dx. (6) If the server is busy or on vacation, an arriving non-priority customer either joins the queue with probability b or balks (does not join the queue) with probability (1 b). 3. Definitions and Notations We define the following notations: (1) P m,n(x, (1) t) = Probability that at time t, the server is active providing service and there are m (m ) priority units and n (n ) non-priority units in the queue excluding the one priority unit in service with elapsed service time for this customer is x. Accordingly, m,n(t) = m,n(x, t)dx denotes the probability that at time t there are m (m ) priority units and n (n ) nonpriority units in the queue excluding one priority unit in service without regard to the elapsed service time x of a priority unit. (2) V m,n (x, t) = Probability that at time t, the server is on vacation with elapsed vacation time x and there are m(m ) priority units and n (n ) non-priority units in the queue. Accordingly, V m,n (t) = V m,n (x, t)dx denotes the probability that at time t there are m (m ) priority units and n (n ) nonpriority units in the queue, without regard to the elapsed vacation time x. (3) P m,n(x, (2) t) = Probability that at time t, the server is active providing service and there are m (m ) priority units in the queue and n (n ) non-priority units in the queue excluding the one non-priority unit in service with elapsed service time for this customer is x. Accordingly, m,n(t) = m,n(x, t)dx

5 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 65 denotes the probability that at time t there are m (m ) priority units in the queue and n (n ) non-priority units in the queue excluding one non-priority unit in service without regard to the elapsed service time x of a non-priority unit. (4) Q(t) = Probability that at time t, there are no priority and non-priority customers in the system and the server is idle but available in the system. 4. Equations Governing the System The Kolmogorov forward equations to govern the model: t m,n(x, t) x m,n(x, t) = (λ 1 λ 2 µ 1 (x)) m,n(x, t) λ 1 λ 2 b C i m i,n (x, t) C j m,n j (x, t) λ 2(1 b) m,n(x, t); m 1, n 1, (1) t m,(x, t) x m,(x, t) = (λ 1 λ 2 µ 1 (x)) m,(x, t) λ 1 C i m i, (x, t) λ 2 (1 b)p m,(x, (1) t); m 1, n =, (2) t P,n(x, (1) t) x P,n(x, (1) t) = (λ 1 λ 2 µ 1 (x))p,n(x, (1) t) λ 2 b C j,n j (x, t) λ 2 (1 b)p,n(x, (1) t); m =, n 1, (3) t, (x, t) x, (x, t) = (λ 1 λ 2 µ 1 (x)), (x, t) λ 2 (1 b), (x, t); t V m,n(x, t) x V m,n(x, t) = (λ 1 λ 2 γ(x))v m,n (x, t) λ 1 λ 2 b m, n =, (4) C i V m i,n (x, t) C j V m,n j (x, t) λ 2 (1 b)v m,n (x, t); m 1, n 1, (5) t V m,(x, t) x V m,(x, t) = (λ 1 λ 2 γ(x))v m, (x, t) λ 1 C i V m i, (x, t) λ 2 (1 b)v m, (x, t); m 1, n =, (6) t V,n(x, t) x V,n(x, t) = (λ 1 λ 2 γ(x))v,n (x, t) λ 2 b C j V,n j (x, t) λ 2 (1 b)v,n (x, t); m =, n 1, (7) t V,(x, t) x V,(x, t) = (λ 1 λ 2 γ(x))v, (x, t) λ 2 (1 b)v, (x, t); m, n =, (8)

6 66 G. Ayyappan and P. Thamizhselvi t m,n(x, t) x m,n(x, t) = (λ 1 λ 2 µ 2 (x)) m,n(x, t) λ 1 λ 2 b C i m i,n (x, t) C j m,n j (x, t) λ 2(1 b) m,n(x, t); m 1, n 1, (9) t m,(x, t) x m,(x, t) = (λ 1 λ 2 µ 2 (x)) m,(x, t) λ 1 C i m i, (x, t) λ 2 (1 b)p m,(x, (2) t); m 1, n =, (1) t P,n(x, (2) t) x P,n(x, (2) t) = (λ 1 λ 2 µ 2 (x))p,n(x, (2) t) λ 2 b C j,n j (x, t) λ 2 (1 b)p,n(x, (2) t); m =, n 1, (11) t, (x, t) x, (x, t) = (λ 1 λ 2 µ 2 (x)), (x, t) λ 2 (1 b), (x, t); d dt Q(t) = (λ 1 λ 2 )Q(t) (1 θ), (x, t)µ 1 (x)dx m, n =, (12), (x, t)µ 2 (x)dx V, (x, t)γ(x)dx. (13) The above set of equations are to be solved under the following boundary conditions at x =. m,n(, t) = m,(, t) = λ 1 C m1 Q(t),n(, t) =, (, t) = λ 1 C 1 Q(t) V,n (, t) = θ m1,n(x, t)µ 1 (x)dx 1,n(x, t)µ 1 (x)dx m1,(x, t)µ 1 (x)dx 1, (x, t)µ 1 (x)dx m1,n(x, t)µ 2 (x)dx V m1,n (x, t)γ(x)dx; m 1, n 1, (14) m1,(x, t)µ 2 (x)dx V m1, (x, t)γ(x)dx; m 1, n =, (15) 1,n(x, t)µ 2 (x)dx 1, (x, t)µ 2 (x)dx V 1,n (x, t)γ(x)dx; m =, n 1, (16) V 1, (x, t)γ(x)dx; m, n =, (17),n(x, t)µ 1 (x)dx, n, (18)

7 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 67, (, t) = λ 2 C 1 Q(t) (1 θ),n(, t) = λ 2 C n1 Q(t) (1 θ),1 (x, t)µ 1 (x)dx,1 (x, t)µ 2 (x)dx V,1 (x, t)γ(x)dx; m, n =, (19),n1(x, t)µ 1 (x)dx,n1(x, t)µ 2 (x)dx V,n1 (x, t)γ(x)dx; m =, n 1. (2) We assume that initially there are no customers in the system and the server is idle. So the initial conditions are P m,n() (1) = P m,() (1) = P,n() (1) =, () =, P m,n() (2) = P m,() (2) = P,n() (2) =, () =, (21) V m,n () = V m, () = V,n () = V, () =, and Q() = 1. Next, we define the following probability generating functions: z1 m z2 n P m,n(x, (1) t) = (x, z 1, z 2 ), z1 m P m (1) (x, t) = (x, z 1 ), m= n= n= m= m= n= z n 2 n (x, t) = (x, z 2 ), z m 1 m (x, t) = (x, z 1 ), m= n= n= z1 m z2 n V m,n (x, t) = V (x, z 1, z 2 ), z1 m V m (x, t) = V (x, z 1 ), and m= m= z m 1 z n 2 m,n(x, t) = (x, z 1, z 2 ), z n 2 n (x, t) = (x, z 2 ), z2 n V n (x, t) = V (x, z 2 ), n= which are convergent inside the circle given by z 1 1, z 2 1, and define the Laplace transform of a function f(t) as f(s) = f(t)e st dt. x m,n(x, s) (s λ 1 λ 2 b µ 1 (x)) m,n(x, s) = λ 1 λ 2 b x m,(x, s) (s λ 1 λ 2 b µ 1 (x)) m,(x, s) = λ 1 C i m i,n(x, s) (22) C j m,n j(x, s); m 1, n 1, (23) C i m i,(x, s); m 1, (24)

8 68 G. Ayyappan and P. Thamizhselvi x,n(x, s) (s λ 1 λ 2 b µ 1 (x)),n(x, s) = λ 2 b C j,n j(x, s); n 1, (25) x,(x, s) (s λ 1 λ 2 b µ 1 (x)),(x, s) =, (26) x V m,n(x, s) (s λ 1 λ 2 b γ(x))v m,n (x, s) = λ 1 C i V m i,n (x, s) λ 2 b C j V m,n j (x, s); m 1, n 1, (27) x V m,(x, s) (s λ 1 λ 2 b γ(x))v m, (x, s) = λ 1 x V,n(x, s) (s λ 1 λ 2 b γ(x))v,n (x, s) = λ 2 b C i V m i, (x, s); m 1, (28) C j V,n j (x, s); n 1, (29) x V,(x, s) (s λ 1 λ 2 b γ(x))v, (x, s) =, (3) x m,n(x, s) (s λ 1 λ 2 b µ 2 (x)) m,n(x, s) = λ 1 C i m i,n(x, s) λ 2 b x m,(x, s) (s λ 1 λ 2 b µ 2 (x)) m,(x, s) = λ 1 x,n(x, s) (s λ 1 λ 2 b µ 2 (x)),n(x, s) = λ 2 b C j m,n j(x, s); m 1, n 1, (31) C i m i,(x, s); m 1, (32) C j,n j(x, s); n 1, (33) x,(x, s) (s λ 1 λ 2 b µ 2 (x)),(x, s) =, (34) (s λ 1 λ 2 )Q(s) 1 = (1 θ) m,n(, s) = m,(, s) = λ 1 C m1 Q(s) m1,n(x, s)µ 1 (x)dx,(x, s)µ 1 (x)dx m1,n(x, s)µ 2 (x)dx,(x, s)µ 2 (x)dx V, (x, s)γ(x)dx, (35) V m1,n (x, s)γ(x)dx; m 1, n 1, (36) m1,(x, s)µ 1 (x)dx m1,(x, s)µ 2 (x)dx V m1, (x, s)γ(x)dx; m 1, (37)

9 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 69,n(, s) =,(, s) = λ 1 C 1 Q(s) V,n (, s) = θ 1,n(x, s)µ 1 (x)dx,(, s) = λ 2 C 1 Q(s) (1 θ),n(, s) = λ 2 C n1 Q(s) (1 θ) 1,(x, s)µ 1 (x)dx 1,n(x, s)µ 2 (x)dx 1,(x, s)µ 2 (x)dx V 1,n (x, s)γ(x)dx; n 1, (38) V 1, (x, s)γ(x)dx, (39),n(x, s)µ 1 (x)dx; n, (4),1(x, s)µ 1 (x)dx,1(x, s)µ 2 (x)dx V,1 (x, s)γ(x)dx; m, n =, (41),n1(x, s)µ 1 (x)dx,n1(x, s)µ 2 (x)dx V,n1 (x, s)γ(x)dx; m =, n 1. (42) Now we multiply equations (23), (25), (27), (29), (31) and (33) by z n 2 summing over n from 1 to, adding to equations (24), (26), (28), (3), (32) and (34) and using the generating function defined in equation (22), we get x m (x, s, z 2 ) (s λ 1 λ 2 b[1 C(z 2 )] µ 1 (x)) m (x, s, z 2 ) = λ 1 m C i m i,(x, s, z 2 ); m 1, n =, (43) x (x, s, z 2 ) (s λ 1 λ 2 b[1 C(z 2 )] µ 1 (x)) (x, s, z 2 ) =, (44) x V m(x, s, z 2 ) (s λ 1 λ 2 b[1 C(z 2 )] γ(x))v m (x, s, z 2 ) m = λ 1 C i V m i, (x, s, z 2 ); m 1, n =, (45) x V (x, s, z 2 ) (s λ 1 λ 2 b[1 C(z 2 )] γ(x))v (x, s, z 2 ) =, (46) x m (x, s, z 2 ) (s λ 1 λ 2 b[1 C(z 2 )] µ 2 (x)) m (x, s, z 2 ) = λ 1 m C i m i,(x, s, z 2 ); m 1, n =, (47) x (x, s, z 2 ) (s λ 1 λ 2 b[1 C(z 2 )] µ 2 (x)) (x, s, z 2 ) =. (48) Now we multiply equations (43), (45) and (47) by z1 m summing over m from 1 to, adding to equations (44), (46) and (48) and using the generating function defined in equation (22), we

10 7 G. Ayyappan and P. Thamizhselvi get x (x, s, z 1, z 2 ) (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )] µ 1 (x)) (x, s, z 1, z 2 ) =, (49) x V (x, s, z 1, z 2 ) (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )] γ(x))v (x, s, z 1, z 2 ) =, (5) x (x, s, z 1, z 2 ) (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )] µ 2 (x)) (x, s, z 1, z 2 ) =. For the boundary conditions, we multiply both sides of equations (36) and (37) by z1 m1 summing over m from 1 to, adding to equations z 1 (38) and z 1 (39) and use the the equation (22), we get z 1 n (, s, z 1 ) = n (x, s, z 1 )µ 1 (x)dx n (x, s, z 1 )µ 2 (x)dx z 1 (, s, z 1 ) = λ 1 C(z 1 )Q(s) (x, s, z 1 )µ 2 (x)dx,n(x, s)µ 1 (x)dx,n(x, s)µ 2 (x)dx (x, s, z 1 )µ 1 (x)dx,(x, s)µ 2 (x)dx V n (x, s, z 1 )γ(x)dx (51) V,n (x, s)γ(x)dx, (52),(x, s)µ 1 (x)dx V (x, s, z 1 )γ(x)dx V, (x, s)γ(x)dx. (53) Now multiply equations (52) and (42) by z2 n summing over n from 1 to, adding to equations (53) and z 2 (41) and using the equation (22), we get z 1 (, s, z 1, z 2 ) = λ 1 C(z 1 )Q(s) (x, s, z 2 )µ 1 (x)dx z 2 (, s, z 2 ) = λ 2 C(z 2 )Q(s) (1 θ) (1 θ),(x, s)µ 1 (x)dx (x, s, z 1, z 2 )µ 1 (x)dx (x, s, z 1, z 2 )µ 2 (x)dx V (x, s, z 1, z 2 )γ(x)dx (x, s, z 2 )µ 1 (x)dx (x, s, z 2 )µ 2 (x)dx V (x, s, z 2 )γ(x)dx (x, s, z 2 )µ 2 (x)dx V (x, s, z 2 )γ(x)dx, (54),(x, s)µ 2 (x)dx V, (x, s)γ(x)dx. (55)

11 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 71 Now multiply equation (4) by z n 2 summing over n from to, use the equation (22), we get V (, s, z 2 ) = θ Integrate equation (49) between to x, we obtain (x, s, z 2 )µ 1 (x)dx. (56) (x, s, z 1, z 2 ) = (, s, z 1, z 2 ) { (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])x e x µ 1 (t)dt}. (57) Again integrate (57) by parts with respect to x, and we get (s, z 1, z 2 ) = (, s, z 1, z 2 )[ 1 B 1(s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) ]. (58) (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) Multiply equation (57) by µ 1 (x) and integrate with respect to x, we get (x, s,z 1, z 2 )µ 1 (x)dx = (, s, z 1, z 2 )B 1 (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]). (59) Now equation (44) can be rewritten as, x (x, s, z 2 ) (s λ 1 λ 2 b[1 C(z 2 )] µ 1 (x)) (x, s, z 2 ) =, which on integration gives { (s λ (x, s, z 2 ) = 1 λ 2 b[1 C(z 2 )])x (, s, z 2 )e Again integrate by parts with respect to x x µ 1 (t)dt}. (6) (s, z 2 ) = (, s, z 2 )[ 1 B 1(s λ 1 λ 2 b[1 C(z 2 )]) ]. (61) (s λ 1 λ 2 b[1 C(z 2 )]) Now multiply (6) by µ 1 (x), and integrate over x, we get (x, s, z 2 )µ 1 (x)dx = (, s, z 2 )B 1 (s λ 1 λ 2 b[1 C(z 2 )]). (62)

12 72 G. Ayyappan and P. Thamizhselvi Performing similar operations on equations (5) and (51), we get { (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])x V (x, s, z 1, z 2 ) = V (, s, z 1, z 2 )e x γ(t)dt}, V (s, z 1, z 2 ) = V (, s, z 1, z 2 )[ 1 V (s λ 1[1 C(z 1 )] λ 2 b[1 C(z 2 )]) ], (64) (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) V (x, s, z 1, z 2 )γ(x)dx = V (, s, z 1, z 2 )V (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]), (65) { (s λ 1 λ 2 b[1 C(z 2 )])x V (x, s, z 2 ) = V (, s, z 2 )e x (63) γ(t)dt}, (66) V (x, s, z 2 )γ(x)dx = V (, s, z 2 )V (s λ 1 λ 2 b[1 C(z 2 )]). (67) However, by its definition and (x, s, z 1, z 2 ) = (, s, z 1, z 2 ) V (, s, z 1, z 2 ) = V (, s, z 2 ), (68) { (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])x e x µ 2 (t)dt}, (69) (s, z 1, z 2 ) = (, s, z 1, z 2 )[ 1 B 2(s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) ], (7) (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) (x, s, z 1, z 2 )µ 2 (x)dx = (, s, z 1, z 2 )B 2 (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]), { (s λ (x, s, z 2 ) = 1 λ 2 b[1 C(z 2 )])x (, s, z 2 )e x (71) µ 2 (t)dt}, (72) (x, s, z 2 )µ 2 (x)dx = (, s, z 2 )B 2 (s λ 1 λ 2 b[1 C(z 2 )]). (73) However, by its definition Using equation (68) in (65), we get (, s, z 1, z 2 ) = (, s, z 2 ). (74) V (x, s, z 1, z 2 )γ(x)dx = V (, s, z 2 )V (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]). (75) By using (62) in (56), we get V (, s, z 2 ) = θ (, s, z 2 )B 1 (s λ 1 λ 2 b[1 C(z 2 )]). (76)

13 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 73 Now substitute equations (59), (62), (65), (67), (71) and (73) in (54), we get {z 1 B 1 (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])} (, s, z 1, z 2 ) = λ 1 C(z 1 )Q(s) (, s, z 2 )B 1 (s λ 1 λ 2 b[1 C(z 2 )]){1 θv (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) θv (s λ 1 λ 2 b[1 C(z 2 )])} (, s, z 2 ){B 2 (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) B 2 (s λ 1 λ 2 b[1 C(z 2 )])}. (77) Using equations (35), (62), (67), (73) in (55), we get (, s, z 2 ) = {1 (s λ 1 λ 2 b[1 C(z 2 )])Q(s)} {z 2 B 2 (s λ 1 λ 2 b[1 C(z 2 )])} (, s, z 2 )B 1 (s λ 1 λ 2 b[1 C(z 2 )]){1 θ θv (s λ 1 λ 2 b[1 C(z 2 )])}. (78) {z 2 B 2 (s λ 1 λ 2 b[1 C(z 2 )])} By applying Rouchy s theorem, {z 1 B 1 (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])} has one and only one zero inside the circle, z 1 = 1 for Re (s) >, z 2 1. Then equation (77) gives [ λ 1 C(g(z 2 ))Q(s) (, s, z 2 ){B 2 (s λ 1 [1 C(g(z 2 ))] λ 2 b[1 C(z 2 )]) B 2 (s λ 1 λ 2 b[1 C(z 2 )])} ] (, s, z 2 ) = [ B1 (s λ 1 λ 2 b[1 C(z 2 )]){1 θv (s λ 1 [1 C(g(z 2 )] λ 2 b[1 C(z 2 )]) θv (s λ 1 λ 2 b[1 C(z 2 )])} ] Substitute (79) in (78) and (77), we get [ {1 (s λ1 λ 2 b[1 C(z 2 )]))Q(s)}. (79) (, s, z 2 ) = {1 θv (s λ 1 [1 C(g(z 2 )] λ 2 b[1 C(z 2 )]) θv (s λ 1 λ 2 b[1 C(z 2 )])} {1 θ θv (s λ 1 λ 2 b[1 C(z 2 )])}λ 1 C(g(z 2 )Q(s) ] [ {z2 B 2 (s λ 1 λ 2 b[1 C(z 2 )])}{1 θv (s λ 1 [1 C(g(z 2 )], (8) λ 2 b[1 C(z 2 )]) θv (s λ 1 λ 2 b[1 C(z 2 )])} {B 2 (s λ 1 [1 C(g(z 2 ))] λ 2 b[1 C(z 2 )]) B 2 (s λ 1 λ 2 b [1 C(z 2 )])}{1 θ θv (s λ 1 λ 2 b[1 C(z 2 )])} ] [ {1 (s λ1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]))Q(s)} (, s, z 1, z 2 ) = {1 θv (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) θv (s λ 1 λ 2 b[1 C(z 2 )])} (, s, z 2 ){z 2 B 2 (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])} {1 θv (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) θv (s λ 1 λ 2 b[1 C(z 2 )])} ] [. (81) {z1 B 1 (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])} {1 θv (s λ 1 [1 C(g(z 2 )] λ 2 b[1 C(z 2 )]) θv (s λ 1 λ 2 b[1 C(z 2 )])} ]

14 74 G. Ayyappan and P. Thamizhselvi Substitute equations (81), (79) and (8) in (58), (64) and (7), respectively, (s, z 1, z 2 ) = (, s, z 1, z 2 )[ 1 B 1(s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) ], (82) (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) V (s, z 1, z 2 ) = θ (, s, z 2 )B 1 (s λ 1 λ 2 b[1 C(z 2 )]) [ 1 V (s λ 1[1 C(z 1 )] λ 2 b[1 C(z 2 )]) ], (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) (83) (s, z 1, z 2 ) = (, s, z 2 )[ 1 B 2(s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) ]. (s λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) (84) Thus (s, z 1, z 2 ), V (s, z 1, z 2 ), and (s, z 1, z 2 ) are completely determined from equations (82) to (84). 5. Steady state Analysis: Limiting Behavior In this section, we derive the steady state probability distribution for our queueing model. By applying the well-known Tauberian property, lim sf(s) = lim f(t). s t In order to determine Q, we use the normalizing condition (1, 1) V (1, 1) (1, 1) Q = 1. The steady state probability for an priority queueing system with a single server serving two queues M [X 1] /G 1 /1 and M [X 2] /G 2 /1 with balking and optional server vacation based on exhaustive service of the priority units are given by where (, z 2 ) = (z 1, z 2 ) = (, z 1, z 2 )[ 1 B 1(λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) ], (85) (λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) V (z 1, z 2 ) = θp (, z 2 )B 1 (λ 1 λ 2 b[1 C(z 2 )]) [ 1 V (λ 1[1 C(z 1 )] λ 2 b[1 C(z 2 )]) ], (λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) (86) (z 1, z 2 ) = (, z 2 )[ 1 B 2(λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) ], (λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) (87) [{1 (λ 1 λ 2 b[1 C(z 2 )]))Q} {1 θv (λ 1 [1 C(g(z 2 )] λ 2 b[1 C(z 2 )]) θv (λ 1 λ 2 b[1 C(z 2 )])} {1 θ θv (λ 1 λ 2 b[1 C(z 2 )])}λ 1 C(g(z 2 )Q ] [ {z2 B 2 (λ 1 λ 2 b[1 C(z 2 )])}{1 θv (λ 1 [1 C(g(z 2 )] λ 2 b[1 C(z 2 )]) θv (λ 1 λ 2 b[1 C(z 2 )])} {B 2 (λ 1 [1 C(g(z 2 ))] λ 2 b[1 C(z 2 )]) B 2 (λ 1 λ 2 b[1 C(z 2 )])} {1 θ θv (λ 1 λ 2 b[1 C(z 2 )])} ], (88)

15 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 75 (, z 1, z 2 ) = [{1 (λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]))Q} {1 θv (λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) θv (λ 1 λ 2 b[1 C(z 2 )])} (, z 2 ) {z 2 B 2 (λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])} {1 θv (λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]) θv (λ 1 λ 2 b[1 C(z 2 )])} ] [ {z1 B 1 (λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )])}. (89) {1 θv (λ 1 [1 C(g(z 2 )] λ 2 b[1 C(z 2 )]) θv (λ 1 λ 2 b[1 C(z 2 )])} ] Let W q (z 1, z 2 ) be the probability generating function of the queue size irrespective of the state of the system. Then adding equations (85) to (87), we obtain where W q (z 1, z 2 ) = (z 1, z 2 ) V (z 1, z 2 ) (z 1, z 2 ), (9) W q (z 1, z 2 ) = N 1(z 1, z 2 ) D(z 1, z 2 ) N 2(z 1, z 2 ) D(z 1, z 2 ), (91) N 1 (z 1, z 2 ) = f 1 (z 1, z 2 )Q{1 θv (f 1 (z 1, z 2 )) θv (f 3 (z 2 ))}{1 B 1 (f 1 (z 1, z 2 ))} θf 4 (z 2 )Q{1 V (f 1 (z 1, z 2 ))}{z 1 B 1 (f 1 (z 1, z 2 ))}, (92) N 2 (z 1, z 2 ) = (, z 2 )[θ{b 2 (f 2 (z 2 )) B 2 (f 3 (z 2 ))}{1 V (f 1 (z 1, z 2 ))}{z 1 B 1 (f 1 (z 1, z 2 ) )} {z 2 B 2 (f 1 (z 1, z 2 ))}{1 θv (f 1 (z 1, z 2 )) θv (f 3 (z 2 ))}{1 B 1 (f 1 (z 1, z 2 ))} {z 1 B 1 (f 1 (z 1, z 2 ))}{1 B 2 (f 1 (z 1, z 2 ))}{1 θv (f 2 (z 2 )) θv (f 3 (z 2 ))}], (93) D(z 1, z 2 ) = {z 1 B 1 (f 1 (z 1, z 2 ))}{1 θv (f 2 (z 2 )) θv (f 3 (z 2 ))}f 1 (z 1, z 2 ) (94) and f 1 (z 1, z 2 ) = λ 1 [1 C(z 1 )] λ 2 b[1 C(z 2 )]), f 2 (z 2 ) = λ 1 [1 C(g(z 2 )] λ 2 b[1 C(z 2 )], f 3 (z 2 ) = λ 1 λ 2 b[1 C(z 2 )], f 4 (z 2 ) = λ 1 C(g(z 2 ). We see that for z 1 = 1, z 2 = 1, W q (z 1, z 2 ) is indeterminate of the form. Therefore, we apply L Hopitals rule and on simplification, we obtain the result of equation (92) where B 1 () = 1, B 2 () = 1, B 1() = E(B 1 ) is the mean service time of a priority customer, B 2() = E(B 2 ) is the mean service time of a non-priority customer, V () = 1, V () = E(V ) is the mean vacation time, C(1) = 1, g(1) = 1, C (1) = E(I) is the mean batch size of the arriving customer for both priority and non-priority units and g (1) = E(I 1 ) is the mean batch size of

16 76 G. Ayyappan and P. Thamizhselvi the arriving non-priority units during the busy period of priority units. N 1 (1, 1) = 2Q(λ 1 C (1) λ 2 bc (1)){B 1()(λ 1 C (1) λ 2 bc (1))}{1 θ θv (λ 1 )} 2V ()(λ 1 C (1) λ 2 bc (1)){1 B 1()(λ 1 C (1) λ 2 bc (1))}θQλ 1, (95) N 2 (1, 1) = (, 1)[2θ{1 B 2 (λ 1 )}V ()(λ 1 C (1) λ 2 bc (1)){1 B 1()(λ 1 C (1) λ 2 bc (1))} 2(λ 1 C (1) λ 2 bc (1)){1 θ θv (λ 1 )}{B 1() B 2()}], (96) D(1, 1) = 2{1 B 1()(λ 1 C (1) λ 2 bc (1))}(λ 1 C (1) λ 2 bc (1)){1 θ θv (λ 1 )} (97) and [ {1 θ θv (λ1 )}Q(λ 1 E(I)E(I 1 ) λ 2 be(i)) (, 1) = λ 1 QθE(V )(λ 1 E(I)E(I 1 ) λ 2 be(i))] [. (98) {1 θ θv (λ1 )}{1 E(B 2 )(λ 1 E(I)E(I 1 ) λ 2 be(i))} {1 B 2 (λ 1 )}{θe(v )(λ 1 E(I)E(I 1 ) λ 2 be(i)) θv (λ 1 )λ 2 be(i)} ] We shall use the normalizing condition W q (1, 1) Q = 1, we get [ {1 E(B1 )(λ 1 E(I) λ 2 be(i))}{1 θ θv (λ 1 )} ] Q = [ {1 θ θv (λ1 )} {1 θ θv (λ 1 )}{E(B 2 ) E(B 1 )} λ 1 θe(v ){1 E(B 1 )(λ 1 E(I) λ 2 be(i))} ] (, 1)θ{1 B 2 (λ 1 )}E(V ){1 E(B 1 )(λ 1 E(I) λ 2 be(i))} The utilization factor is given by. (99) ρ = 1 Q, (1) that is, [ {E(B1 )(λ 1 E(I) λ 2 be(i))}{1 θ θv (λ 1 )} ρ = {1 θ θv (λ 1 )}{E(B 2 ) E(B 1 )} λ 1 θe(v ){1 E(B 1 )(λ 1 E(I) λ 2 be(i))} ] (, 1)θ{1 B 2 (λ 1 )}E(V ){1 E(B 1 )(λ 1 E(I) λ 2 be(i))} [, (11) {1 θ θv (λ1 )} {1 θ θv (λ 1 )}{E(B 2 ) E(B 1 )} λ 1 θe(v ){1 E(B 1 )(λ 1 E(I) λ 2 be(i))} ] (, 1)θ{1 B 2 (λ 1 )}E(V ){1 E(B 1 )(λ 1 E(I) λ 2 be(i))} where ρ < 1 is the stability condition under which the steady state exists. Equation (1) gives the probability that the server is idle. Substituting equation (99) into (91), we have completely and explicitly determined W q (z 1, z 2 ), the probability generating function of the queue size.

17 AAM: Intern. J., Vol. 11, Issue 1 (June 216) The Average Queue Length The mean number of customer in priority queue under the steady state is L q1 = d dz 1 W q1 (z 1, 1) z1 =1 (12) and the mean number of customer in the non-priority queue under the steady state is then where L q2 = d dz 2 W q2 (1, z 2 ) z2 =1, (13) L q1 = D 1(1)N 1 (1) D 1 (1)N 1 (1) D 1(1)N 2 (1) D 1 (1)N 2 (1), 3(D 1(1)) 3(D 1(1)) (14) L q2 = D 2(1)n 1 (1) D 2 (1)n 1(1) D 2(1)n 2 (1) D 2 (1)n 2(1), 3(D 2(1)) 3(D 2(1)) (15) N 1 (1) = 2Qλ 1 E(I)E(B 1 )λ 1 E(I){1 θ θv (λ 1 )} 2Qλ 1 θe(v )λ 1 E(I) {1 E(B 1 )λ 1 E(I)}, N 1 (1) = Q{ 3λ 1 E(I[I 1])E(B 1 )λ 1 E(I) 3λ 1 E(I){E(B 2 1)(λ 1 E(I)) 2 E(B 1 )λ 1 E(I[I 1])}}{1 θ θv (λ 1 )} 6Q{λ 1 E(I)E(B 1 )λ 1 E(I)}θE(V )λ 1 E(I) 3θλ 1 Q{(E(V 2 )(λ 1 E(I)) 2 E(V )λ 1 E(I[I 1]))(1 E(B 1 )λ 1 E(I)) (E(V )λ 1 E(I))(E(B 2 1)(λ 1 E(I)) 2 E(B 1 )λ 1 E(I[I 1]))}, N 2 (1) = 2 (, 1)[θ{1 B 2 (λ 1 )}E(V )λ 1 E(I)(1 E(B 1 )λ 1 E(I)) E(B 2 )λ 1 E(I) {1 θ θv (λ 1 )}], N 2 (1) = (, 1)[ 3θ{1 B 2 (λ 1 )}(E(V 2 )(λ 1 E(I)) 2 E(V )λ 1 E(I[I 1])) (1 E(B 1 )λ 1 E(I)) 3θ{1 B 2 (λ 1 )}E(V )λ 1 E(I)(E(B 2 1)(λ 1 E(I)) 2 E(B 1 )λ 1 E(I[I 1])) 6(E(B 2 )λ 1 E(I))(E(B 1 )λ 1 E(I))θE(V )λ 1 E(I) 3[(E(B 2 2)(λ 1 E(I)) 2 E(B 2 )λ 1 E(I[I 1]))]{1 θ θv (λ 1 )}], D 1(1) = 2λ 1 E(I)[1 E(B 1 )λ 1 E(I)](1 θ θv (λ 1 )), D 1 (1) = [ 3λ 1 E(I[I 1])[1 E(B 1 )λ 1 E(I)] 3λ 1 E(I)[E(B 2 1)(λ 1 E(I)) 2 E(B 1 )λ 1 E(I[I 1])]](1 θ θv (λ 1 )), n 1(1) = 2Qλ 2 be(i)e(b 1 )λ 2 be(i){1 θ θv (λ 1 )} 2Qλ 1 θe(v )λ 2 be(i)e(b 1 )λ 2 be(i), n 1 (1) = Q{ 3λ 2 be(i[i 1])E(B 1 )λ 2 be(i) 3λ 2 be(i){e(b 2 1)(λ 2 be(i)) 2 E(B 1 )λ 2 E(I[I 1])}}{1 θ θv (λ 1 )} 6Q{λ 2 be(i)e(b 1 )λ 2 be(i)}θ{e(v )λ 2 be(i) V (λ 1 )λ 2 be(i)} 6θλ 1 E(I)E(I 1 )Q{E(V )λ 2 be(i)}{e(b 1 )λ 2 be(i)} 3θλ 1 Q{(E(V 2 )(λ 2 be(i)) 2 E(V )λ 2 be(i[i 1]))(E(B 1 )λ 2 be(i)) 3θλ 1 Q(E(V )λ 2 be(i))(e(b 2 1)(λ 2 be(i)) 2 E(B 1 )λ 2 be(i[i 1]))},

18 78 G. Ayyappan and P. Thamizhselvi n 2(1) = 2 (, 1)[θ{1 B 2 (λ 1 )}E(V )λ 2 be(i)(e(b 1 )λ 2 be(i)) E(B 1 )λ 2 be(i) {1 θ θv (λ 1 )}], n 2 (1) = 6 (, 1)[θ{1 B 2 (λ 1 )}E(V )λ 2 be(i)(e(b 1 )λ 2 be(i)) (E(B 1 )λ 2 be(i)) {1 θ θv (λ 1 )}] 3 (, 1)[θ{1 B 2 (λ 1 )}(E(B 1 )λ 2 be(i)){(e(v 2 ) (λ 2 be(i)) 2 E(V )λ 2 be(i[i 1]))} θ{1 B 2 (λ 1 )}E(V )λ 2 be(i)(e(b 2 1) (λ 2 be(i)) 2 E(B 1 )λ 2 be(i[i 1])) (E(B 2 1)(λ 2 be(i)) 2 E(B 1 )λ 2 b E(I[I 1])){1 θ θv (λ 1 )} 2θE(B 1 )λ 2 be(i){e(v )λ 2 be(i) V (λ 1 )λ 2 be(i)}], D 2(1) = 2λ 2 be(i)[e(b 1 )λ 2 be(i)](1 θ θv (λ 1 )), D 2 (1) = [3λ 2 be(i[i 1])[E(B 1 )λ 2 be(i)](1 θ θv (λ 1 )) 3λ 2 be(i)(e(b 2 1) (λ 2 be(i)) 2 E(B 1 )λ 2 be(i[i 1]))](1 θ θv (λ 1 )) 6θλ 2 be(i)e(b 1 )λ 2 be(i) [E(V )(λ 1 E(I)E(I 1 ) λ 2 be(i)) V (λ 1 )λ 2 be(i)]. 7. The Average Waiting Time in the Queue Average Waiting time of a customer in the priority queue is W q1 = L q 1 λ 1. (16) Average Waiting time of a customer in the non-priority queue is W q2 = L q 2 λ 2. (17) where L q1 and L q2 have been found in equations (14) and (15). 8. Particular Cases Case 1: If there is no vacation, no balking, single arrival and the service time follows exponential ( λ 2 when θ =, b = 1, E(I) = 1, E(I1 ) = (µ λ 1 ), E[I(I 1)] = and E[I 1(I 1 1)] = 2λ 2 µ ). Then, (µ λ 1 ) 3 and Q = 1 E(B 1 )(λ 1 λ 2 ), ρ = E(B 1 )(λ 1 λ 2 ), L q1 = D 1(1)N 1 (1) D 1 (1)N 1 (1) 3(D 1(1)) 2 D 1(1)N 2 (1) D 1 (1)N 2 (1) 3(D 1(1)) 2 L q2 = D 2(1)n 1 (1) D 2 (1)n 1(1) D 2(1)n 2 (1) D 2 (1)n 2(1), 3(D 2(1)) 2 3(D 2(1)) 2

19 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 79 where N 1 (1) = 2Qλ 1 E(B 1 )λ 1, N 1 (1) = 3Qλ 1 E(B 2 1)(λ 1 ) 2, N 2 (1) = 2 (, 1)[E(B 2 )λ 1 ], N 2 (1) = 3 (, 1)[E(B 2 2)(λ 1 ) 2 ], D 1(1) = 2λ 1 [1 E(B 1 )λ 1 ], D 1 (1) = 3λ 1 E(B 2 1)(λ 1 ) 2, n 1(1) = 2Qλ 2 E(B 1 )λ 2, n 1 (1) = 3Qλ 2 E(B 2 1)(λ 2 ) 2, n 2(1) = 2 (, 1)[E(B 1 )λ 2 ], n 2 (1) = 6 (, 1)[E(B 1 )λ 2 ] 3 (, 1)[E(B 1 2)(λ 1 ) 2 ], D 2(1) = 2λ 2 [E(B 1 )λ 2 ], D 2 (1) = 3λ 2 E(B 2 1)(λ 2 ) 2. The above result coincides with the results of Gross and Harris (1985). Case 2: If there is no low priority queue, no vacation, single arrival, no balking and priority service, then also consider normal service and service time as follows exponentially ( when θ =, b = 1, E(I) = 1, E(I1 ) =, E[I(I 1)] =, λ 2 = and E(B 2 ) = ). Then, where Q = 1 E(B 1 )(λ 1 ), ρ = E(B 1 )(λ 1 ), L q1 = D 1(1)N 1 (1) D 1 (1)N 1 (1), 3(D 1(1)) 2 N 1 (1) = 2Qλ 1 E(B 1 )λ 1, N 1 (1) = 3Qλ 1 E(B 2 1)(λ 1 ) 2, D 1(1) = 2λ 1 [1 E(B 1 )λ 1 ] D 1 (1) = 3λ 1 E(B 2 1)(λ 1 ) 2. The above result coincides with the results of Gross and Harris (1985). 9. Numerical Results To numerically illustrate the results obtained in this work, we consider that the service time for priority and non-priority customers and the vacation time are exponentially distributed with rates µ 1, µ 2, and γ respectively.

20 8 G. Ayyappan and P. Thamizhselvi We base our numerical example on the result found in equations (99), (11), (14), (15), (118) and (119). For this purpose in Table I, we choose the following arbitrary values λ 2 θ =.3, b =.3, µ 1 = 9, µ 2 = 9, E(I) = 1, E[I(I 1)] =, E(I 1 ) = (µ λ 1 ), E[I 1 (I 1 1)] = 2λ 2µ (µ λ 1 ) 3, γ = 6 and λ 2 = 2. While λ 1 varies from.1 to 1. such that the stability condition is satisfied. Table I: Effect of λ 1 on various queue characteristics λ1 Q ρ L q1 L q2 W q1 W q It clearly shows that as long as increasing the arrival rate of priority units the servers idle time decreases while the utilisation factor, average queue length and waiting time of a queue for priority and non-priority units are all increases. In Table II, we choose the following arbitrary values θ =.3, b =.3, µ 1 = 9, µ 2 = 9, λ 2 E(I) = 1, E[I(I 1)] =, E(I 1 ) = (µ λ 1 ), E[I 1(I 1 1)] = 2λ 2µ (µ λ 1 ), γ = 6 and λ 3 1 = 2. Also λ 2 varies from.1 to 1. such that the stability condition is satisfied. It clearly shows that as long as increasing the arrival rate of non-priority units the servers idle time decreases while the utilisation factor, average queue lenth for both priority and non-priority unit and waiting time of a priority queue are all increases and waiting time of a non-priority units is decreases. Table II: Effect of λ 2 on various queue characteristics λ2 Q ρ L q1 L q2 W q1 W q Conclusion In this paper we studied a priority queueing system with balking and optional server vacation based on exhaustive service of the priority units. The server provides two types of service, namely priority and non-priority under non-preemptive priority rule. We derived the probability generating

21 AAM: Intern. J., Vol. 11, Issue 1 (June 216) 81 functions of the number of customers in the priority and non-priority units are found by using the supplementary variable technique, average queue size, the average waiting time for the priority and non-priority units and numerical results are also obtained. The above model finds potential application for guaranteeing different layers service for different customers. The Internet Protocol (IP) is not only successful in data communications, but IP emerges as the convergence layer for all forms of communication including voice, video or multimedia in general. The packet switched internet is taking over the circuit switched telephone network. The packet switched paradigm has great advantages in flexibility but it can give the same Quality of Service (QoS) as circuit switching. Therefore, all over the world great effort is put into improving the QoS packet switched systems. Priorities are a key instrument in giving each communication flow the QoS that it asks (and pays) for or Asynchronous Transfer Mode (ATM) was an attempt by telephone companies to design a network architecture that could efficiently transport both voice and data. Acknowledgment: The authors are thankful to the anonymous referees for their valuable comments and suggestions for the improvement of the paper. REFERENCES Ayyappan, G. and Muthu Ganapathi Subramanian, A. (29). Single server Retrial queueing system with Non-pre-emptive priority service and Single vacation-exhaustive service type, Pacific Asian Journal of Mathematics, Vol. 3, N. 1 2, pp Chae, K. C., Lee, H. W. and Ahn, C. W. (21). An arrival time approach to M/G/1 type queues with generalised vacations, Queueing Systems, Vol. 38, pp Chaudhry, M. L. and Templeton, J.G.C. (1983). A First Course In Bulk Queues. John Wiley & and Sons, New York. Cobham, A. (1954). Priority assignment in waiting line problems, Operations Research, Vol. 2, pp Doshi, B. T. (1986). Queueing systems with vacation-a survey, Queueing Systems, Vol. 1, No. 1, pp Gross, D. and Harris, C.M. (1985). Fundamentals of Queueing Theory, 2nd Edition. Wiley, New York. Haghighi, A. M. and Mishev, D. (26). A parallel priority queueing system with finite buffers, Journal of Parallel and Distributed Computing, Vol. 66, pp Haghighi, A. M. and Mishev, D. (213). Stochastic Three-stage Hiring Model as a Tandem Queueing Process with Bulk Arrivals and Erlang Phase-Type Selection: M K /M (k,k) /1 M Y /E r /1, Int. J. Mathematics in Operational Research, Vol. 5, No. 5, pp Hawkes, A.G. (1956). Time dependent solution of a priority queue with bulk arrivals, Operations Research, Vol. 34, pp Jain, M. and Charu, B. (28). Bulk Arrival Retrial Queue with Unreliable Server and Priority

22 82 G. Ayyappan and P. Thamizhselvi Subscribers, International Journal of Operations Research, Vol. 5, No. 4, pp Jau-Chauan, K. and Fu-Min, C. (29). Modified vacation policy for M/G/1 retrial queue with balking and feedback, Computers & Industrial Engineering, Vol. 57, pp Jinbio, W. and Zhaotong, L. (213). A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule, Computers & Industrial Engineering, Vol. 64, pp Kleinrock, L. (1976). Queueing system, Volume 2: Computer Applications, John Wiley & and Sons, New York. Madan, K.C. (211). A Non-Preemptive Priority Queueing System with a Single Server Serving Two Queues M/G/ and M/D/1 with Optonal Server Vacations Based on Exhaustive Service of the Priority Units, Applied Mathematics, Vol. 2, pp Takagi, H. (199). Time-dependent analysis of an M/G/1 vacation models with exhaustive service, Queueing Systems, Vol. 6, No. 1, pp Thangaraj, V. and Vanitha, S. (21). An M/G/1 queue with two-stage heterogeneous service compulsory server vacation and random breakdowns, Int. J. Contemp. Math. Sciences, Vol. 5, No. 7, pp