A multiserver tandem queue with a specialist server operating with a vacation strategy

In the queueing system considered, service is provided at two stations, station 1 and station 2 operating in tandem. The station 1 is a multi-server station with $c$ identical servers working in parallel and station 2 is equipped with a single server called 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 to the station 1 according to a Markovian arrival process. An arriving customer directly enters 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 out system with the complimentary probability $1-p$. If the buffer is full, then all the customers coming out of the 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. Stability condition for this system is established and 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 represent the causality clinic and specialist server represents an expert giving consultation at the request of a threshold number of patients.