A single and batch service queue with random breakdowns

We consider a single and batch service queue with Poisson arrivals and general service time distributions. In this model the server serves the customers either one at a time (manual service) or in batches (service by a machine). If the number of units in the queue is less than the control level c, then the server adopts the manual service (serves the units one by one). However, if the queue size is c or more, then the server adopts the batch service and serves the units in a batch of size c. Furthermore, the server may break down while providing either the single service or the batch service. Finally, a cost model is developed to determine the optimal value of c.