M^x/G/1FEEDBACK QUEUE WITH THREE STAGE HETEROGENEOUS SERVICE, MULTIPLE ADAPTIVE VACATION AND CLOSED DOWN TIMES

We consider a batch arrival queueing system with three stage heterogeneous service provided by a single server with different (arbitrary) service time distributions. Each customer undergoes three stages of heterogeneous service. As soon as the completion of third stage of service, if the customer is dissatisfied with his service, he can immediately join the tail of the original queue. The vacation period has two heterogeneous phases. After service completion of a customer the server may take a phase one Bernoulli vacation. Further, after completion of phase one Bernoulli vacation the server may take phase two optional vacation. The vacation times are assumed to be general. In addition we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly. Also the mean number of customers in the queue and the system are also derived. Some particular cases and numerical results are discussed.


INTRODUCTION
During the last three or four decades, queueing models with vacations had been the subject of interest to queueing theorists of deep study because of their applicability and theoretical structures in real life situations such as manufacturing and production systems, computer and communication systems, service and distribution systems, etc. The M X /G/1 queue has been studied by numerous authors including Scholl and Kleinrock 1983, Gross and Harris, 1985, Doshi 1986, Kashyap and Chaudhry 1988, Shanthikumar 1988, Choi and Park 1990and Madan 2000, 2005. Krishnakumar et al., 2002 considered an M/G/1 retrial queue with additional phase of service. Madan and Anabosi 2003, have studied a single server queue with optional server vacations based on Bernoulli schedules and a single vacation policy. Madan and Choudhury 2005, have studied a single server queue with two phase of heterogeneous service under Bernoulli schedule and a general vacation time. Thangaraj and Vanitha 2010 have studied a single server M X /G/1 feedback queue with two types of service having general distribution. Levy and Yechiali 1976, Baba 1986, Keilson and Servi 1986, C.Gross and C.M. Harris 1985, Takagi 1992, Borthakur and Chaudhury 1997, Cramer 1989, and many others have studied vacation queues with different vacation policies. In some queueing systems with batch arrival there is a restriction such that not all batches are allowed to join the system at all time. This policy is named restricted admissibility.
Madan and Choudhury 2005 proposed anqueueing system with restricted admissibilty of arriving batches and Bernoulli schedule server vacation. In this paper, we consider a batch arrival queueing system with three stage heterogeneous service provided by a single server with different (arbitrary) service time distributions. Each customer undergoes three stage heterogeneous service. As soon as the completion of third stage of service, if the customer is dissatisfied with his service, he can immediately join the tail of the original queue as a feedback customer with probability p to repeat the service until it is successful or may depart the system with probability 1 − p if service happens to be successful. The vacation period has two heterogeneous phases. Further, after service completion of a customer the server may take phase one vacation with probability r or return back to the system with probability 1 − r for the next service. After the completion of phase one vacation the servermay take phase two optional vacation with probability θ or return back to the system with probability1 − θ. In addition we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. This paper is organized as follows. 2. Supplementary variable technique. The mathematical description of our model is given in section 3. Definitions and Equations governing the system are given in section 4. The time dependent solution have been obtained in section 5. and corresponding steady state results have been =1 =1 1 =0 derived explicitly in section 6. Mean queue size and mean system size are computed in section 7. Some particular cases are given in section 8. Conclusion are given in section 9 respectively. So if ( ) is the LST of the service time, the steady state queue size equations are given by 10 − 10 0 = 10 − 11 0 − 00 (0) ( ) Multiply Eq. (7) by 0 and Eq. (8) by ( ≥ 1) and take the summation from = 0 , we have The basic steady state equations are 00 − ∆ , + ∆ = 00 , 1 − ∆ 1 − 1 0 10 − ∆ , + ∆ = 10 , 1 − ∆ + 11 0, ∆ + 00 0, ∆ Using the LST in (9), we have 2.3. Queue Size Distribution 1 , − 1 , 0 = 1 , − −1 1 , 0 − Move the first coefficients ( , )of (1) -(3) to the 10 0 − 1 , − 00 0 left side and take the limit as ∆ → 0, we get (10) and therefore, m = e 0 μi , = 1,2,3, . ..
By substituting = 0 in the equation (13), the equation (14) becomes Moreover, after the completion of third stage of service, if the customer is dissatisfied with his service, he can immediately join the tail of the original queue as a feedback customer for receiving another service with probability . Otherwise the customer may depart forever from the system with probability 1 − .
e) As soon as the third stage of service is completed, the server may take phase one Bernoulli vacation with probability or may continue staying in the system with − 1 − − probability 1 − . After completion of phase one vacation the server may take phase two which represents the PGF of number of customers in queue in an arbitrary time epoch.

MATHEMATICAL DESCRIPTION OF THE MODEL
We assume the following to describe the equueing model of our study. a) Customers arrive at the system in batches of variable size in a compound Poisson process and they are provided one by one service on a first come -first served basis. optional vacation with probability or return back to the system with probability 1 − On returning from vacation the server starts instantly serving the customer a1t5t he head of the queue, if any. f) The server's vacation time follows a general (arbitrary) distribution with distribution function and density function . Let be the conditional probability of a completion of a vacation during the interval ( , + ] given that the elapsed vacation time is , so that = , = 1,2, … and therefore 1− t = e − 0 , = 1,2, …. g) The restricted admissibility of batches in which not all batches are allowed to join the 0 − 0 0 1 − system at all times. Let 0 ≤ ≤ 1 and 0 ≤ ≤ 1 be the probability that an arriving batch will be allowed to join the system during the period of server's nonvacation period and vacation period respectively. h) Various stochastic processes involved in the system are assumed to be independent of each other.

DEFINITIONS AND EQUATIONS GOVERNING
probability that at time t there are customers in the queue and the server is under phase two vacation irrespective of the value of .
Q(t) = Probability that at time t, there are no customers in the queue and the server is idle but available in the system. The model is then, governed by the following set of differential-difference equations: t, the server is active providing first stage of service , + , + + 1 , and there are ( ≥ 0 ) customers in the queue = 1 − 1 , excluding the one being served and the elapsed service time for this customer is . Consequently denotes the probability that at time t there are customers in the queue excluding one customer in the first stage of service irrespective of the value of .
probability that at time t there are customers in the queue excluding one customer in the second stage of service irrespective of the value of .

Theorem
The steady state probabilities for anM X /G/1 feedback (9q0u)eue with three stage 2 , heterogeneous service, feedback, Bernoulli vacation and optional server vacation with restricted ( (1 − ( ) ) 1 − admissibility are given by (1) 1 , (2) 1 , (3) 1 , (1) 1 (2) 1 are the where DR is given by equation (89). Thus steady state probabilities that the server is providing , and first stage of service, second stage of service, third stage of service, server under phase one and server , are completely determined from equations (90) to (94) which completes the proof of the theorem

THE STEADY STATE RESULTS
In this section, we shall derive the steady state probability distribution for our queueing model. To define the steady probabilities we suppress the argument t wherever it appears in the under phase two vacation, server under idle respectively without regard to the number of customers in the system. Proof: Multiplying both sides of equations (90) to (94) by s, taking limit as → 0, applying property (95) and simplifying, we obtain wheredr is given by equation (102). Therefore adding to equation (109),equating to 1 and simplifying, we get Q= 1 − (110) and hence the utilization factor of the system is given by + Let ( ) denote the probability generating function 1 2 3 + of the queue size irrespective of the state of the system. Then adding equations (103) to (107) we obtain where < 1 is the stability condition under which the steady state exists. Equation (110) gives the probability that the server is idle. Substituting from (110) into (108), we have completely and explicitly determined ,the probability generating function of the queue size.

THE MEAN QUEUE SIZE AND THE MEAN SYSTEM SIZE
Let denote the mean number of customers in the queue under the steady state. Then Further, we find the mean system size using Little's formula. Thus we have are numerator and denominator of the right = + 117 hand side of (93) respectively. Then we use where has been found by equation (112) and is = lim obtained from equation (111).