We consider a queueing system with two servers. The customers arrive according to a Poisson process with rate λ = 1, and the system capacity is equal to four customers. The service times are independent random variables having an exponential distribution. Each server is able to serve two customers at a time. If a server attends to only one customer, he does so at rate 𝜇 = 2, whereas the service rate is equal to 1 when two customers are served at the same time.

Indication, If two customers are served together, then they will leave the system at the same time. Moreover, if there are two customers in the system, then one of the servers may be free.

(a) Write the balance equations of the system.

(b) Let T* be the total time that a given customer entering the system will spend in it.

(i) Calculate, in terms of the limiting probabilities, and supposing that no customers arrive during the service period of the customer in question, the distribution function of T*,

(ii) Under the same assumption as in (i), does the random variable T* have the memoryless property? Justify.