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

=

= 



for otherwise 0 1 2 3 0

, , ,

n n n

=

− = 

3 01 2 3 0

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.