Transient solution of an $M^{[X]} / G / 1$ queuing model with feedback, random breakdowns and Bernoulli schedule server vacation having general vacation time distribution

Abstract

This paper analyze an $M^{[X]} / G / 1$ queue with feedback, random server breakdowns and Bernoulli schedule server vacation with general(arbitrary) distribution. Customers arrive in batches with compound Poisson process and are served one by one with first come first served basis. Both the service time and vacation time follow general (arbitrary) distribution. After completion of a service the may go for a vacation with probability $\theta$ or continue staying in the system to serve a next customer, if any with probability $1-\theta$. With probability $\mathrm{p}$, the customer feedback to the tail of original queue for repeating the service until the service be successful. With probability $1-p=q$, the customer departs the system if service be successful. The system may breakdown at random following Poisson process, whereas the repair time follows exponential distribution. We obtain the time dependent probability generating function in terms of their Laplace transforms and the corresponding steady state results explicitly. Also we derive the system performance  measures like average number of customers in the queue and the average waiting time in closed form.