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We present a single server in which customers arrive in batches and the server provides service one by one. The server provides two heterogeneous service stages such that service time of both stages is different and mandatory to all arriving customers in such a way that, after the completion of first stage, the second stage should also be provided to the customers. The server may subject to random breakdowns with brake down rate λ and, after break down, it should be repaired but it has to wait for being repaired and such waiting time is called delay time. Both the delay time and repair time follow exponential distribution. Upon the completion of the second stage service, the server will go for vacation with probability p or stay back in the system probability 1-p, if any. 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.

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.

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In this model, we present a batch arrival non- Markovian queueingmodel with second optional service, subject to random break downs andBernoulli vacation. Batches arrive in Poisson stream with mean arrivalrate (> 0), such that all customers demand the rst `essential' ser-vice, wherein only some of them demand the second `optional' service.The service times of the both rst essential service and the second op-tional service are assumed to follow general (arbitrary) distribution withdistribution function B1(v) and B2(v) respectively. The server may un-dergo breakdowns which occur according to Poisson process with breakdown rate . Once the system encounter break downs it enters the re-pair process and the repair time is followed by exponential distributionwith repair rate . Also the sever may opt for a vacation accordingto Bernoulli schedule. The vacation time follows general (arbitrary)distribution with distribution function v(s). The time-dependent prob-ability generating functions have been obtained in terms of their Laplacetransforms and the corresponding steady state results have been derivedexplicitly. Also the mean queue length and the mean waiting time havebeen found explicitly.