A company has 3 staff cars for use by its executives. Executives come to the car pool according to a Poisson process with rate of 2/hour. If there is a car available then the executive drives off and returns an exponentially distributed amount of time later with mean of 2.5 hours. If there is not a car available then an executive will wait only if there is no more than 2 others already waiting, otherwise they leave and call a cab. Required:
a. Develop a model for this system. Define states and events and draw the transition rate diagram.
b. Construct rate balance equations and determine a vector.
c. Assume that the operating cost of a car is $20/hour while in use and assume that an average cab fare is $35. For this system, what is the total operating plus cab fare cost (per hour) in the long-run?
d. Assume there is a surplus car in another location that could be brought over and used. Would it be cost effective to do so?
e. Assume there are 3 cars again. Now assume that the waiting executives are only willing to wait a certain amount of time before giving up and calling a cab. Time studies indicate that this amount of time is exponentially distributed and equal to 20 minutes. For this system, what is the total operating plus cab fare cost (per hour) in the long-run?