Customers arrive to a super market according to a Poisson process with intensity V = ½ per minute. The supermarket has two counters, that use a common queue. Counter 1 is always occupied. Counter 2 is opened when 3 or more customers are in the queue, and will be closed when the counter becomes idle (no customer is served at counter 2). The service time of a customer has an exponential distribution with mean 1/W = 1 minute.
a) Draw the transition diagram for this queueing system. Describe the states, transitions, and transition rates. Hint: define the states (i, j) with i the number of customers, j the number of counters in use.
b) Give the equilibrium equations.
You do not have to solve the equilibrium equations in b). The following questions must be answered in terms of the arrival intensities V, the average service time 1/W, and the equilibrium probabilities P(i,j).
c) Give the average number of customers in the queue.
d) Give the average waiting time per customer.
e) How many counters are open on average?
f) Which percentage of time all counters are occupied?
g) What is the fraction of time counter 2 is occupied?
h) Determine the average length of a period during which counter 1 is not occupied.