A Multiserver Tandem Queue with a Specialist Server Operating with a Vacation Strategy.

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]