CONTINOUS TIME MARKOV CHAINS:

Jack is the owner of a car-wash operation with two automatic car washing system. Jack estimates the costumers arrive randomly according to Poisson process with a mean rate of 1 every 1.5 minutes. However, if there is already a car waiting for the service, these potential costumers will balk (drive on to another wash-car place). The time that can be attributed to washing one car by a car washing system has exponential distribution with a mean of 3 minutes, and service time is exponentially distributed. The net profit is 6 TL for each washed car. Assume that this operation operates for 10 hours per day, and if there are any car in the system at the closing time, before closing it completes its service to those cars.

a) Give a continuous-time Markov chain (CTMC) .Define states. Then, compute the limiting probabilities for each state and find expected number of cars in the system in a given time.

b) Calculate net profit of this operation per day.