Assignment:
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.
Provide complete and step by step solution for the question and show calculations and use formulas.