Transient solution of / G / 1 with Second Optional Service Subject to Reneging during Vacation and Breakdown time

In this paper, we present a batch arrival nonMarkovian queuing model with second optional service. Batches arrive in Poisson stream with mean arrival rate λ, such that all customers demand the first essential service, whereas only some of them demand the second „optional‟ service. We consider reneging to occur when the server is unavailable during the system breakdown or vacations periods. The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly.


INTRODUCTION
The research study on queuing systems with impatient customers has become an extensive and interesting area in queuing theory literature. In many real life situations, the arriving customers may be discouraged due to long queue, and decide not to join the queue and leave the system at once. This behavior of customers is referred as balking. Sometimes customers get impatient after joining the queue and leave the system without getting service. This behavior of customer recognizes as reneging. In the last few years, we see studies on queues with balking and reneging gaining significant importance. We see applications of queue with reneging in emergency services in hospitals dealing serious patients, communication systems, production and inventory system and many more. A queue with balking and reneging was initially studied by Haight (1957), Barrer (1957). Since then, extensive amount of work has been done on queuing systems related to impatient customers. Queues with balking and reneging have been studied by authors like Altman and Yechiali (2006), Ancker et al. (1963), Choudhury and Medhi (2011), in the last few years.
Vacation queuing models has been modeled effectively in various situations such as production, banking service, communication systems, and computer networks etc. Numerous authors are interested in studying queuing models with various vacation policies including single and multiple vacation policies. Batch arrival queue with server vacations was investigated by Yechiali (1975). An excellent comprehensive studies on vacation models can be found in Takagi (1991) and Doshi (1986) research papers. One of the classical vacation model in queuing literature is Bernoulli scheduled server vacation. Keilson and Servi(1987) introduced and studied vacation scheme with Bernoulli schedule discipline.
In this paper we consider queuing system such that the customers are arriving in batches according to Poisson stream. The server provides 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. We extend and develop this model by adding new assumptions reneging and system breakdowns.
Customers may renege (leave the queue after joining) during server breakdowns or during the time when the server takes vacation due to impatience. In real world, this is a very realistic assumption and often we come across such queuing situations.

MATHEMATICAL DESCRIPTION OF THE MODEL
The following assumptions are to be used describe the mathematical model of our study:  Customers arrive at the system in batches of variable size in a compound Poisson process and they are provided service one by one on a "first come first served" basis. Let be the first order probability that a batch of k customers arrives at the system during a short interval of time is the mean arrival rate of batches.
 There is a single server which provides the first essential service to all arriving customers. Let respectively be the distribution function and the density function of the first service times respectively.
 As soon as the first service of a customer is completed, then he may demand for the second service with probability r, or else he may decide to leave the system with probability 1-r in which case another customer at the head of the queue (if any) is taken up for his first essential service.
 The second service times as assumed to be general with the distribution function and the density function . Further, Let be the conditional probability density function of service completion during the interval ( , +d ] given that the elapsed service time is so that and therefore  As soon as the customer"s service is completed, the server may go for a vacation of random length V with probability or it may continue to serve the next customer with probability (1-p).
 On returning from vacation the server instantly starts serving the customer at the head of the queue, if any.
 The vacation time of the server follows general (arbitrary) distribution with distribution function V(s) and the density function v(s). Let be the condition probability of a completion of a vacation during the interval ( , +d ] given that the elapsed service time is so that and thus D e c e m b e r 11, 2 0 1 4  The system may break down at random and breakdowns are assumed to occur according to a Poisson stream with mean breakdown rate α > 0. Further the repair time follows general (arbitrary) distribution with distribution function R ) and the density function . Let β be the condition probability of a completion of a repair process so that and thus  In addition, customers arriving for service may become impatient and renege (leave the queue) after joining during vacations and breakdown times. Reneging is assumed to follow exponential distribution with parameter γ. Thus .Thus η is the probability that a customer can renege during a short interval of time (t, t +d ]  Various stochastic processes involved in the system are independent of each others.

DEFINITIONS AND EQUATIONS GOVERNING THE SYSTEM
probability that at time 't' the server is active providing ith service and there are 'n' customers in the queue including the one being served and the elapsed service time for this customer is . Consequently denotes the probability that at time't' there are 'n' customers in the queue excluding the one customer in service irrespective of the value of .
 probability that at time 't' the server is on vacation with elapsed vacation time , and there are 'n' customers in the waiting in the queue for service. Consequently = probability that at time 't' there are 'n' customers in the queue and the server is on vacation irrespective of the value of .  Probability that at time t, the server is inactive due to break down and the system is under repair while there are 'n' customers in the queue.  Probability that at time t, there are no customers in the system and the server is idle but available in the system The model is then, governed by the following set of differential-difference equations:  Now we multiply both sides of equation (4.11),(4.12),(4.13) and (4.14) by , sum over n from 1to ,yields. In order to determine completely, we have yet to determine the unknown which appears in the numerator of the right sides of equations (4.34), (4.35) and (4.36). For that purpose, we shall use the normalizing condition. D e c e m b e r 11, 2 0 1 4 where denote the steady state probabilities that the server is providing first stage of service, second stage of service and server under repair without regard to the number of customers in the queue. Now using equations (4.53), (4.54), (4.55) and (4.56) into the normalizing condition (4.52) and simplifying, we obtain (4.57) and hence, the utilization factor ρ of the system is given by (4.58) where ρ < 1 is the stability condition under which the steady states exits. Then if we substitute the values from (5.3), (5.4), (5.5) and (5.6) into (5.2) we obtain in the closed form. Further we find the mean system size L using Little"s formula. Thus we have where has been found by equation (5.2) and ρ is obtained from equation (4.58).

Conclusion
In this paper we have studied an /G/1 with Second Optional Service, reneging to occur when the server is unavailable during the system breakdown or vacations periods. The probability generating function of the number of customers in the queue is found using the supplementary variable technique. This model can be utilized in large scale manufacturing industries and communication networks.