Consider a queueing system with 1 counter, to which groups of customers arrive according to a Poisson proces with intensity l. The size of a group is 1 with probability p and 2 with probability 1-p. Customers are served one by one. The service time has exponential distribution with mean m^{-1 }. Service times are mutually independent and independent of the arrival proces. The system may contain at most 3 customers. If the system is full upon arrival of a group, or if the system may contain only one additional customer upon arrival of a group of size 2, then all customers in the group are lost and will never return. Let Z(t) record the number of customers at time t.
(a) Explain why {Z(t), t³0}is a Markov proces and give the diagram of transitions and transition rates.
(b) Give the equilibrium equations (balance equations) for the stationary probabilities P_{n }, n=0,1,2,3.
(c) Compute these probabilities P_{n }, n=0,1,2,3.
The answers to the following questions may be provided in terms of the probabilities P_{n } (except for (h)).
(d) Give an expression for the average number of waiting customers.
(e) Give the departure rate and the rate at which customers enter the queue.
(f) Give an expression for the average waiting time of a customer.
(g) What is the fraction of time the counter is busy?
(h) What is the average length of an idle period?
(i) Determine from (g) and (h) the average length of a period the system is occupied (= at least 1 customer in the system).
(j) What is the rate at which groups of size 2 enter the queue?