An assembly line consists of two stations in tandem. Each station can hold only one item at a time. When an item is completed in station 1, it moves into station 2 if the latter is empty; otherwise it remains in station 1 until station 2 is free. Items arrive at station 1 according to a Poisson process with rate λ. However, an arriving item is accepted only if there is no other item in the station; otherwise it is lost from the system. The time that an item spends at station 1 is exponentially distributed with mean 1/μ1, and the time that it spends at station 2 is exponentially distributed with mean 1/μ2.

Let the state of the system be defined by (m, n), where m is the number of items in station 1 and n is the number of items in station 2.

a. Give the state-transition-rate diagram of the process.

b. Calculate the limiting state probabilities pmn.