Consider a single-server queueing system where some potential customers balk (refuse to enter the system) and some customers who enter the system later get impatient and renege (leave without being served). Potential customers arrive according to a Poisson process with a mean rate of 4 per hour. An arriving potential customer who finds n customers already there will balk with the following probabilities:
0, if n 0, 1 2 , if n 1, P{balkn already there}
3 4 , if n 2, 1, if n 3.
Service times have an exponential distribution with a mean of 1 hour.
A customer already in service never reneges, but the customers in the queue may renege. In particular, the remaining time that the customer at the front of the queue is willing to wait in the queue before reneging has an exponential distribution with a mean of 1 hour. For a customer in the second position in the queue, the time that she or he is willing to wait in this position before reneging has an exponential distribution with a mean of 1 2 hour.

(a) Construct the rate diagram for this queueing system.

(b) Obtain the steady-state distribution of the number of customers in the system.

(c) Find the expected fraction of arriving potential customers who are lost due to balking.

(d) Find Lq and L.