Consider the following queueing system. Customers arrive in pairs according to a Poisson process with rate = 1 customer pair/min. There is one server...

Consider the following queueing system. Customers arrive in pairs according to a Poisson process with rate λ= 1 customer pair/min. There is one server and room for two customers to wait in line. Service times are exponential with mean 30 s. If there is not room for both arriving customers, they both leave.

(a) Describe the system in a rate diagram and find the stationary distribution.

(b) Now suppose that pairs may split up. If there is not room for both, then with probability 1 /2 they both leave and with probability 1/2 one stays and the other leaves.

(c) Do (a) again under these assumptions.

Answers:

(a) π0 = 4/10, π1 = 2/10, π2 = 3/10, π3 = 1/10

(b) π0 = 16/43, π1 = 8/43, π2 = 12/43, π3 = 7/43