A Study on M x /G/1 Queuing System with Essential, Optional Service, Modified Vacation and Setup time

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1 A Study on M x /G/1 Queuing System with Essential, Optional Service, Modified Vacation and Setup time E. Ramesh Kumar 1, L. Poornima 2 1 Associate Professor, Department of Mathematics, CMS College of Science & Commerce, Coimbatore, Tamil Nadu 2 Department of Mathematics, CMS College of Science & Commerce, Coimbatore, Tamil Nadu Abstract:-A study on M x /G/1 queuing system where the arrival follows Poisson Process and the server provides service in two stages. The first stage service is essential and the second stage of service is optional. If the system is empty then the server moves for a vacation of random duration and after returning from vacation the server decides to take a second vacation which is optional. Such a customer behavior is considered in both busy time and vacation time of the system. The vacation time follows general (arbitrary) distribution. Before providing service to a new customer or a batch of customers that joins the system in the renewed busy period, the server enters into a random setup time process such that setup time follows exponential distribution. We discuss the transient behavior and the corresponding steady state results with the performance measures of the model. For this model, we obtain the time dependent solution and the corresponding steady state solutions. Also we derive the performance measure the mean queue size and the average waiting time explicitly. Keywords: Batch Arrival, Essential Service, Optional Service, Setup Time, Bernoulli Vacation. I I. INTRODUCTION n this paper we consider queuing system such that the customers are arriving in batches according to Poisson stream. The server provide a first essential service to all incoming customers and a second optional service will be provided to only some of them those who demand it. Both the essential and optional service times are assumed to follow general distribution. Queues with server vacations have emerged as an essential area of queuing theory and have been studied widely and fruitfully due to their various applications in communication system, manufacturing systems, textile processing industries, food industries, and chemical industries etc. In this models server s setup time corresponds to the preparatory work of the server before starting his service. Hur. S and Park. J [1] and Ke. J.C [2] is some of the authors who analyzed the N policy of M x /G/1 queuing models with server s setup time. The batch arrival queuing system with double threshold policy, setup time and vacation are analyzed by Lee. S.S [3] is among the most general queuing system with threshold policies. In everyday life there are queuing situations where all the arriving customers require the first essential service and some may require the second optional service provided by the same server. Madan K.C [4] has introduced the concept of second optional service, where the customers may depart from the system either with probability (1-r) or may immediately opt for second optional service with probability r. The classical single server vacation model was generalized by Seri. L.D and Finn S.G [5] by considering working vacation. Simple explicit formula for the mean, variance of the number of customers in the system was provided. Arumuganathan. R and Jeyakumar S [6] considered M x /G/1 queuing system with multiple vacations, setup times and closedown times under N- policy. In many real cases, Almasi B and Roszik J and Sztrik. J [7] examined a single server retrial queue with finite number of homogenous sources of calls and a single removable server. Stability conditions are provided by Sherman. N.P and Kharoufeh. J.P [8] for an M/M/1 retrial queue with infinite capacity orbit. Alfa. A.S and Yang. X [9] studied a multiserver queuing system with identical unreliable server with phase type distributed service time. Madan [4] who first introduced the concept of second optional service while studying the time dependent as well as steady behavior of an M/G/1 queuing system with no waiting capacity, using supplementary variable technique. Supplementary variable technique was used to develop the time dependent probability generating function in terms of their Laplace transform for M/G/1 queue by Al-Jararha. J and Madan K.C [10]. It concentrates on such queuing system with setup time. In this paper probability generating function of the steady state queue size at an arbitrary time, expected queue length, expected busy period, expected idle period and numerical illustrations are presents. II. MATHEMATICAL DESCRIPTION OF THE QUEUING MODEL Let λdt be the first order probability of arrival of customers in batches in the system during a short period of time (t, t+dt). The single server provides the first essential service to all arriving customers. Let B 1 (v) and b 1 (v) be the distribution function and the density function of first service times respectively. There is a single server which provides service following a general (arbitrary) distribution with distribution function B i (v) and density function b (v). Let µ i (x) dx be the conditional probability density function of service completion of i th Page 72

2 service during the interval (x, x+dx) given that the elapsed time is x, so that, and There is a single server which provides setup time following a general (arbitrary) distribution with distribution function Y i (v) and density function y (v). Let ξ i (x) dx be the conditional probability density function of service completion of i th service during the interval (x, x+dx) given that the elapsed time is x, so that, and The vacation time of the server follows a general(arbitrary)distribution with distribution function V 1 (s) and the density function v 1 (s).let v 1k (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 ; k=1, 2, 3..M and Figure: 1. Schematic Representation of the Queuing Model 2.1 Notations λ The following notations are follows: = Bulk arrival = k th vacation in n th customer in the queue. = i th service in n th customer in the queue. P 0 (x) = probability at time x there are no customers in the system and the server is idle. Y n (x), v(x), s1(x), s 2 (x), r(x) and g(x) = Probability density function of setup time, V, S 1, S 2, R and G. = Laplace-Stieltjes transform of v(x), s 1 (x), s 2 (x), r(x) and g(x) III. EQUATIONS GOVERNING THE SYSTEM We let, probability that at time t the server is active providing i th service and there are n (n 1) customers in the queue including the one being served and the elapsed service time for this customers is x. Consequently denotes the probability that at time t there are n customers in the queue excluding the one customer in i th irrespective of the value of x. service Probability that at time t the server is on k th vacation with elapsed vacation time x, and there are n (n 0) customers waiting in the queue for service. Consequently denotes the probability that at time t there are n customers in the queue and the server is on k th vacation irrespective of the value of x. Probability that at time t the server is on k th vacation with setup time x, and there are n (n 0) customers waiting in the queue for service. Consequently denotes the probability that at time t there are n customers in the queue and the server is on k th vacation irrespective of the value of x. (t) = Probability that at time t there are no customers in the system and the server is idle but available in the system. 3.1 Steady State System Size Distribution: The following equations are obtained for the queuing system, Page 73

3 The steady state equations are, The Laplace-Stieltjes transforms of P n (1) (x), P n (2) (x), (x), V n (k) (x) and y n (x) are defined as: Page 74

4 (15) Taking Laplace-Stieltjes transform on both sides, we get [ ] [ ] θ [ ] [ ] 3.2. The system size distribution: ( θ) (θ) ( θ) (θ) ( θ) (θ) ( θ) (θ) Page 75

5 After simplification we get, ( ( ) ) [ ] ( ) ( ( ) ) 0 ( ) 1 a ( ( ) ) [ ( ) ( ( ) ) [ ] ( ) ] ( ( ) ) [ ] ( ) Let us define the following, Put θ=0, we get the PGF of a queue size P (z) at an arbitrary time epoch as { ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ). ( ) ( )/ ( ) } IV. THE STEADY STATE ANALYSIS Since P (z) satisfies the steady state condition P (1) =1. Since P (z) is analytic within and on the unit circle. The Numerator must vanish at the point of circles the equations can be solved by any suitable numerical technique. V. PERFORMANCE MEASURES Some important performance measures using the PGF of the queue size P (z) of (29) are derived Expected Busy Period Let T be the residence time that the server is rendering service or under second optional service. Therefore T=S with probability Π and T=S+G with probability 1-Π. The expected length of busy period is given by E (B) = E (B/J=0) P (J=0) +E (B/J=1) P (J=1) = *, - +, -, 5.2. Expected Idle Period: (30) Let I be the Idle Period random variable, then the expected idle period is, - (31) Where is the probability that there are n customers arriving during a vacation. The expected length of idle period due to multiple vacation E (I) is given by E (I) = E (I/U=0) P (U=0) +E (I/U=1) P (U=1) = E (V) P (U=0) + [E (V) +E (I)] P (U=1). Page 76

6 E (I) = Expected Queue Size: where, P (U=0) =, Let L q denote the mean number of customers in the queue under the steady state, then form. Mean waiting time of a customer could be found, as follows (37) By using little s Formula. L q = Since this formula gives, then we write P (z) = Where N (z) and D (z) are the numerator and denominator of the right hand side of equation (29) respectively, then we use, - Where {, - } { ( ), - ( ), - ( ( ), -, - ( ), ( -, - ( ) ( ( ) (, - } VI. NUMERICAL ILLUSTRATION: The unknown probabilities of the queue size distribution are computed using numerical techniques. Using Mat lab, the zeros of the function ( ) ( ) ( )are obtained and simultaneous equations are solved. A Numerical example is analyzed with the following assumptions: Table 1: Threshold value vs. performance measures with μ=7. λ E(Q) E(B) E(I) E(W) Where E (V 2 ) is the second moment of the vacation time and we substitute the values of from equations (31), (32), (33) and (34) in to (30) equation we obtain L q in a closed Figure 2 : Arrival rate vs Expected length of queue and expected waiting time Table 2: Threshold value vs. performance measures with μ=8. Page 77

7 λ E(Q) E(B) E(I) E(W) VII. CONCLUSION This paper analyze the steady state solution of batch arrival single server with second optional service such that first essential service for all incoming customer whereas few of them require a second optional service and multi optional vacation. The queue with working vacation may be applicable in modeling of many practical situations related to computers, communications and productions systems, etc., wherein the server works at different service rates rather than completely stopping the service during a vacation. REFERENCES [1]. Hur. S and Park. J (1999). The effect of different arrival rates on N-policy of M/G/1 with server setup. Applied Mathematical modeling [2]. Ke. J.C. Operating characteristics analysis of M x /G /1 system with variant vacation policy and balking, Applied Mathematical Modeling, Vol.31, (2007), [3]. Lee. H.W, Lee. S.S and Park. J.O (1994). Analysis of the M x /G/1 queue with N policy and multiple threshold control with early setup. International Journal of system science. [4]. Madan. K.C (2000). An M/G/1 queue with second optional service. Queuing systems. [5]. Seri. L.D and Finn. S.G. M/M/1 queues with working vacations (M/M/1/WV) Performance Evaluation, Vol.50, (2002), [6]. Arumuganathan. R and Jeyakumar. S. Steady state analysis of bulk queue with multiple vacations, setup times with N-policy and closedown times. Applied Mathematical Modeling. Vol.29, (2005) [7]. Almasi. B, Roszik. J.and Sztrik. J. Homogeneous finite source retrial queues with server subject to breakdowns and repair, Mathematical and Computer Modeling, Vol.42, (2005), [8]. Sherman. N.P and Kharoufeh. J.P. An M / M / 1 retrial queue with unreliable server, Operations Research Letters, Vol.34, (2006), [9]. Yang. X and Alfa. A.S. A class of multi server queuing system with server failure, Computers & Industrial Engineering, Vol. 56, (2009). [10]. Al-Jararha. J and Madan. K.C. An M/G/1 queue with second optional service with general service time distribution, Information Management Science, Vol. 14, (2003). Page 78