It has been concluded that a single-server queuing system with exponentially distributed service and interarrival times can be modeled as a birth-and-death process with state-dependent mean service and arrival rates, μn and λn, respectively:

μn n n n n n =





=

=

 − =

0 0 1 2 3 3 0

for 0 1 2 3 otherwise for otherwise

, ,, ,, ,





a. Construct the corresponding rate diagram.

b. Calculate the stationary probabilities for finding exactly n customers in the system,

{Pn, n = 0, 1, 2, 3, …}.

c. Determine the expected number of customers in the queuing system, L, and in the queue, Lq. Also determine the expected time a customer spends in the system, W, in the queue, Wq, and in the service facility, Ws.

TABLE 6.5 Birth-and-Death Process for Exercise 6.7 State (n) Mean Birth Rate (λn) Mean Death Rate (μn)
0 2 0 1 2 2 2 2 4 3 0 4