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In this paper, we consider a single server production inventory model with server breakdown and customer impatience while server is down. We assume that customer arrivals are according to a Markovian Arrival Process. If the server is busy at the time of arrival of a customer, the customer joins a queue of infinite capacity. The server is subjected to breakdowns, which occur according to a Poisson process. The service time of customers follows independent and identical phase type distribution and one unit of the inventory is required for each service. Repair starts immediately after a breakdown and the time taken for repair follows phase type distribution. The customer whose service was interrupted waits for the remaining service while the server is under repair. When repair is completed, the service of the interrupted customer is resumed from the interrupted stage. When the server is under repair, loss of customers from the infinite queue due to impatience occurs at exponentially distributed intervals. Such impatient customers go to a buffer of finite capacity, provided waiting space is available, else such customers leave the system for ever. The production of each inventory occurs in k stages of production and production time of each stage follows independent phase type distributions. The production of inventory is according to (s, S) policy. Here the steady state analysis has been done. Various performance measures have been evaluated and we derived distributions of certain important performance characteristics. We analysed the model numerically and graphically. [ABSTRACT FROM AUTHOR]

In the considered queueing system, the service is provided at two stations, station 1 and station 2 operating in tandem. Station 1 is a multi-server station with c identical servers working in parallel, and station 2 is equipped with a single server called the specialist server. The service times of each of the servers at station 1 follow an exponential distribution. The specialist server has phase-type distributed service times. Customers arrive at station 1 according to a Markovian arrival process. An arriving customer enters directly into service at station 1 if at least one of the servers is idle, otherwise joins an infinite queue. After receiving service at station 1, customers either proceed to station 2 or can exit the system. There is a finite buffer between two stations. When the buffer is not full, a customer coming out of the station 1 joins the buffer with a probability p or leaves the system with the complementary probability 1 − p. If the buffer is full, then all the customers coming out of station 1 are lost forever. The server at the station 2 will be turned on only if the number of customers in the buffer reaches a threshold. Once the server is turned on, the service will be rendered until the buffer is emptied. A stability condition for this system is established and the stationary distribution is obtained using matrix analytic methods. Various performance measures are also calculated. Our model is motivated by a hospital situation where station 1 represents the causalty clinic and the specialist server represents an expert giving a consultation upon the request of a threshold number of patients. [ABSTRACT FROM AUTHOR]