Question: A mechanic is responsible for keeping two machines in working order. The time until a working machine breaks down is exponentially distributed with a mean of 12 h. The mechanic's repair time is exponentially distributed with a mean of 8 h.
a. Show that this queuing process is a birth-and-death process by defining the states, n = 0, 1, 2, 3, ...; specifying the state-dependent mean arrival and service rates, λn and µn for n = 0, 1, 2, 3, ...; and constructing the rate diagram. Also specify the criteria defining a birth-and-death process and make sure this process satisfies these criteria.
b. Specify the balance equations and use them to determine the steady-state probability distribution for finding n customers in the system, Pn, n = 0, 1, 2, 3,....
c. Use the definitions and Little's law to determine L, Lq, W, and Wq.
d. Determine the fraction of time that at least one machine is working.