Non-Persistent Retrial Queueing System with Two Types of Heterogeneous Service

Transcription

1 Global Journal of Theoretical and Applied Mathematics Sciences. ISSN Volume 1, Number 2 (211), pp Research India Publications Non-Persistent Retrial Queueing System with Two Types of Heterogeneous Service 1 J. Ebenesar Anna Bagyam and 2 Dr. K. Udayachandrika 1 Assistant Professor in Mathematics, SNS College of Technology, Coimbatore-64135, Tamilnadu, India ebenesar.j@gmail.com 2 Professor in Mathematics, Avinashilingam Deemed University for Women, Coimbatore Tamilnadu, India Abstract The steady state behaviour of an M/G/1 retrial queue with non-persistent customers is analyzed. Customers are allowed to balk or renege at particular time. If the server is busy at the time of arrival of primary customer, then with probability 1 α it leaves the system without service and with probability α it enters into an orbit. Similarly, if the server is occupied at the time of arrival of an orbital customer, with probability 1 β it leaves the system without service and with probability β it goes back to the orbit. The server provides two types of service and each customer may choose either type. Retrial time and service times of type 1 and type 2 are arbitrarily distributed. The steady state queue size distribution of number of customers in the retrial group and expected number of customers in the orbit are derived. Some performance measures are obtained and numerical illustration is presented. Keywords: Retrial Queue, Two Types of Service, Non-Persistent. Mathematics Subject Classification: 65K25, 9B22 Introduction Retrial queues have been widely used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems. Retrial queueing models are characterized by the feature that arriving calls which find a server busy, do not line up or leave the system immediately forever, but go to some virtual place called as orbit and try their luck again after some random

2 158 J. Ebenesar Anna Bagyam and Dr. K. Udayachandrika time. During the last two decades considerable attention has been paid to the analysis of queueing system with repeated calls. For comprehensive survey, see Artalejo [1, 2], Falin and Templeton [4], Artalejo and Gomez-Corral [3]. Many practical queueing systems especially those with balking and reneging have been widely applied to many real-life problems, such as the situations involving impatient telephone switchboard customers, the hospital emergency rooms handling critical patients and the inventory systems with storage of perishable goods [5]. In this paper, we consider a single server queueing model with impatient customers in which the server provides two types of service and each arriving customer has the option of choosing either type of service. Model Description Consider a single server retrial queueing system in which a primary customer arrives according to Poisson stream of rate λ. If an arriving customer finds the server idle, the customer enters the service immediately and leaves the system after service completion. If the server is busy, the customer enters the retrial queue with probability α or balks with probability 1 α. Customers in the orbit compete for access to service. Access from orbit to the server is arbitrarily distributed with distribution function A(x), density function a(x), and Laplace transform A (s). On retrial, an orbiting customer obtains service immediately if the server is idle, otherwise the customer will decide either to leave the system with probability 1 β or return to the orbit with probability β. The server provides two types of service, type 1 and type 2. Customers opt first type with probability p and the other with probability q (=1 p). The service time of a customer in type i (i = 1, 2) is generally distributed with distribution function B i (x), density function b i (x), Laplace transform B i (s) and first two moments μ i1 and μ i2. Let η(x), μ 1 (x) and μ 2 (x) be the conditional completion rates (at time x) respectively for repeated attempts, for type 1 service and type 2 service. The state of the system at time t can be described by the Markov process {N(t): t } = {J(t), X(t), ξ (t), ξ 1 (t), ξ 2 (t): t } where J(t) denotes the server state, 1 or 2 according as the server being idle, busy in type 1 service or busy in type 2 service. Let X(t) denote the number of customers in the retrial queue at time t. If J(t) = and X(t) >, then ξ (t) represents the elapsed retrial time, if J(t) = 1, ξ 1 (t) represents the elapsed service time of type 1 service, if J(t) = 2, ξ 2 (t) represents the elapsed service time of type 2 service. Stability Condition Theorem The necessary and sufficient condition for the system to be stable is α λ (p μ 11 + q μ 21 ) < 1 β + β A (λ)

3 Non-Persistent Retrial Queueing System 159 Proof Let S (k) be the generalized service time of the k th customer. Then {S (k) } are independently and identically distributed with expected value E(S k ) = p μ 11 + q μ 21. Let P(S) and P() be respectively the probabilities that the system is blocked and idle. E() be the expected idle time. Then P(S) = (k) E(S ) (k) E(S ) + E( ) and P() = E( ) (k) E(S ) + E( ) The arrival rate at the retrial queue is αλp(s). The exit rate from the retrial queue by entering service is A ( λ) P( ) E( ) The exit rate from the retrial queue by leaving the system when a primary customer arrives first at the server is (1 A P( ) (λ)) (1 β). The total exit rate from the retrial queue is A ( λ) P( ) E( ) + (1 A ( λ)) (1 β) P( ) E( ) = (1 β + βa (λ)) E( ) P( ). E( ) For stability, the arrival rate should be less than the exit rate. Thus αλp(s) < P( ) E( ) (1 β + βa (λ)) and hence αλ(p μ 11 + q μ 21 ) < (1 β + βa (λ)). Steady State Distribution For the process {N(t): t }, define the probabilities (t) = P{J(t) =, X(t) = } n (t, x) = P{J(t) =, X(t) = n, x ξ (t) < x + } for t, x and n 1 W 1, n (t, x) = P{J(t) = 1, X(t) = n, x ξ 1 (t) < x + } for t, x and n W 2, n (t, x) = P{J(t) = 2, X(t) = n, x ξ 2 (t) < x + } for t, x and n By supplementary variable technique the system of equations that governs the model are: d (t) dt = λ (t)+ W1, (t, x) μ 1 (x) + W2, (t, x) μ 2 (x) (1) n (t, x) = (λ + η(x)) n (t, x), n 1 (2)

4 16 J. Ebenesar Anna Bagyam and Dr. K. Udayachandrika W 1, (t, x) = (αλ + μ 1 (x)) W 1, (t, x), (3) W 1, n (t, x) = (αλ + μ 1 (x)) W 1, n (t, x) + α λ W 1, n 1 (t, x), n 1 (4) W 2, (t, x) = (αλ + μ 2 (x)) W 2, (t, x) (5) W 2, n (t, x) = (αλ + μ 2 (x)) W 2, n (t, x) + α λ W 2, n 1 (t, x), n 1 (6) with the boundary conditions, n (t, ) = W1, n(t, x) W 1, (t, ) = λ p (t) + p W 1, n (t, ) = λ p β n + 1 t, x) μ 1 (x) + n t, x) 1 t, x) ( + p W μ 2 (x), n 1 (7) 2, n(t, x) ( η(x) + λ p (1 β) n + 1 t, x) 1 t, x) ( η(x) + λ p (1 β) ( (8) ( n 1 (9) W 2, (t, ) = λ q (t) + q W 2, n (t, ) = λq β n + 1 t, x) n t, x) 1 t, x) ( +q ( η(x) + λ q (1 β) n + 1 t, x) 1 t, x) ( η(x) + λ q (1 β) ( (1) ( (11) Assuming the stability condition αλ[p μ 11 + q μ 21 ] < 1 β + β A (λ), define the limiting probabilities n (x), W 1, n (x) and W 2, n (x) corresponding to n (t, x), W 1, n (t, x) and W 2, n (t, x). The steady state equations corresponding to (1) (11) are λ = W 1, ( x) μ 1 (x) + W 2, ( x) μ 2 (x) (12)

5 Non-Persistent Retrial Queueing System 161 d n (x) = (λ + η(x)) n (x), n 1 (13) d W 1, (x) = (α λ + μ 1 (x)) W 1, (x) (14) d W 1, n (x) = (α λ + μ 1 (x)) W 1, n (x) + α λ W 1, n 1 (x), n 1 (15) d W2,(x) = (α λ + μ 2 (x)) W 2, (x) (16) d W 2, n (x) = (α λ + μ 2 (x)) W 2, n (x) + α λ W 2, n 1 (x), n 1 (17) With boundary conditions n () = W 1, n( x) μ 1 (x) + W 1, (x) = λ p + p W 1, n (x) = λ p β n+ 1 x) n x) 1 x) W 2,n ( x) μ 2(x) n 1 (18) ( η(x) + λ p (1 β) ( + p n+ 1 x) 1 x) ( η(x) + λ p (1 β) ( (19) (, n 1 (2) W 2, (x) = λ q + q W 2, n (x) = λ q β ( η(x) + λ q (1 β) 1 x) ( +q n x) and the normalizing condition is n x) n= 1 + ( + n= W1,n (x) n+ 1 x) 1 x) ( η(x)+λq (1 β) + n= Define the probability generating functions (z, x) = = n 1 n x) ( z n ; W 1 (z, x) = n= ( (21) n+ 1 x) (, n 1 (22) W2,n (x) = 1 (23) W1,n (x) z n and W 2 (z, x) = n= W2,n (x) z n Multiplying equations (12) to (22) by suitable powers of z and summing the terms,

6 162 J. Ebenesar Anna Bagyam and Dr. K. Udayachandrika we obtain the following equations: (z, x) = [λ + η(x)] (z, x) (24) x W 1 (z, x) = [αλ (1 z) + μ 1 (x)] W 1 (z, x) (25) x W 2 (z, x) = [αλ (1 z) + μ 2 (x)] W 2 (z, x) (26) x (z, ) = p W 1 (z, ) =λp + z W 1 (z, x) μ 1 (x) + ( z, x) η(x) +λpβ W 2 (z, x) μ 2 (x) λ (27) λ p(1 β) ( z, x) + z q λ W 2 (z, ) = λq + z ( z, x) η(x) + λqβ ( z, x) + q(1 β) z ( z, x) (28) ( z, x) (29) The solutions of the partial differential equations (24) (26) are given by (z, x) = (z, ) e λx [1 A(x)] (3) W 1 (z, x) = W 1 (z, ) e α λ (1 z) x [1 B 1 (x)] (31) W 2 (z, x) = W 2 (z, ) e α λ (1 z) x [1 B 2 (x)] (32) Substituting (31) and (32) in (27) we get (z, ) = W 1 (z, ) B 1 (α λ (1 z)) + W 2 (z, ) B 2 (α λ (1 z)) λ (33) Using (3) (33) the equations (27), (28) and (29) yield W 1 (z, ) = λ p (1 z) [1 β + β A (λ)] / D(z) (34) W 2 (z, ) = λ q (1 z) [1 β + β A (λ)] / D(z) (35) (z, ) = λ z [1 p B 1 (α λ (1 z)) q B 2 (α λ (1 z))] / D(z) (36) where D(z) = [A (λ) + (1 A (λ)) (1 β + βz)] [pb 1 (αλ(1 z)) + q B 2 (αλ(1 z))] z (37) Define the partial generating function ψ(z) = function ψ(z, x). Then we have ψ (z, x), for any generating (z) = z [1 A (λ))] [1 pb 1 (αλ(1 z)) q B 2 (αλ(1 z))] / D(z) (38)

7 Non-Persistent Retrial Queueing System 163 W 1 (z) = p [1 β + β A (λ)] [1 B 1 (αλ(1 z))] / [αd(z)] (39) W 2 (z) = q [1 β + β A (λ)] [1 B 2 (αλ(1 z))] / [αd(z)] (4) Using the normalizing condition, the expression for is obtained as, = where, T 1 = 1 β + β A (λ) αλ [p μ 11 + q μ 21 ] T 2 = 1 β + β A (λ) + λ [p μ 11 + q μ 21 ] [1 β + (β α) A (λ)] T T 1 2 The probability generating function of the mean number of customer in the system is given by K(z) = + (z) + z W 1 (z) + zw 2 (z) = {[p B 1 (αλ(1 z)) + q B 2 (αλ(1 z))] [α A (λ) + α(1 A (λ)] (1 β + βz) αz(1 A (λ)) (1 β + β A (λ))] + z [1 β + (β α) A (λ)]} / [α D(z)] The probability generating function of the mean number of customers in the orbit is given by H(z) = + (z) + W 1 (z) + W 2 (z) = {[p B 1 (αλ(1 z)) + q B 2 (αλ(1 z))] [α A (λ) + α(1 A (λ)] (1 β + βz) αz(1 A (λ)) z (1 β + β A (λ))] + 1 β + β A (λ) α z A (λ)} / [α D(z)] Performance Measures In this section we derived the performance measures for the system under steady state The steady state probability that the server idle during the retrial time is (1) = αλ[1 - A (λ)] [p μ 11 + q μ 21 ] / T 2 The steady state probability that the server busy in type 1 service is W 1 (1) = λ p μ 11 [1 β + β A (λ)] / T 2 The steady state probability that the server busy in type 2 service is W 2 (1) = λ q μ 21 [1 β + β A (λ)] / T 2 The steady state probability that the server busy is W 1 (1) + W 2 (1) = λ [p μ 11 + q μ 21 ] [1 β + β A (λ)] / T 2 The mean number of customers in the system is N 2 N L s = K (1) = + 1 T3 2 T2 2 T1T 2

8 164 J. Ebenesar Anna Bagyam and Dr. K. Udayachandrika The mean number of customers in the orbit is L q = H (1) = L s λ [p μ 11 + q μ 21 ] [1 β + β A (λ)] / T 2 where N 1 = [1 β + β A (λ)] [1 + λ (p μ 11 + q μ 21 )] α λ A (λ) [p μ 11 + q μ 21 ] N 2 = λ 2 α [p μ 12 + q μ 22 ] [1 β + A (λ) (β α)] + 2λ [p μ 11 + q μ 21 ] [1 β + β A (λ) + α (1 β) (1 A (λ)] T 3 = 2 β λ [1 A (λ)] [p μ 11 + q μ 21 ] + λ 2 α [p μ 12 + q μ 22 ] Numerical Results Assume that the distributions of retrial times, and service times of type 1 and type 2 service are exponential with rate respectively η, μ 1 and μ 2. The combined effect of varying arrival rate and the retrial rate η on the performance measures (i) the mean number of customer in the system L s (ii) the steady state probability that the server is idle during the retrial time (1) and (iii) the steady state probability that the server busy in type 1 service W 1 (1) for fixed set of parameters (μ 1, μ 2, p, q, r) = (5, 5,.5,.5,.5,.5) are displayed as surfaces respectively in Figures (a) (c). References [1] Artalejo, J.R. (1999). Accessible bibliography on retrial queues, Mathematical and Computer Modeling, 3: 1-6. [2] Artalejo, J.R., (1999). A classified bibliography of research on retrial queues: progress in , Top 7: [3] Artalejo, J.R. and Gomez-Corral, A. (28). Retrial queueing systems A computational approach, Springer Verlag, Berlin. [4] Falin, G.I. and Templeton, J.G.C. (1997). Retrial queues, Chapman and Hall, London. [5] Robert, E. (1979). Reneging phenomenon of single channel queues, Mathematics of Operations Research, 4(2),