2. Consider a maintenance facility for a taxi company that has 25 vehicles. Taxi's that require maintenance come to the facility and queue up in first-in-first-out order. Analysis has shown that an individual working taxi will require maintenance an exponentially distributed amount of time with an average of 100 hours. The facility has three repairmen who are effectively identical. The time to repair a taxi is exponentially distributed with an average of about 45 minutes. If a taxi arrives at the facility and there are 5 other taxis already at the facility, then the taxi will be sent off-site for repair at an average cost of $250. Each taxi at the facility incurs a lost opportunity cost of $65/hour, i.e. money if could have made if it were not in need of repair.

a) Develop a Markov model for this system, i.e., define the states of the system, list the possible events and draw the transition rate diagram that will allow you to answer the questions below. State any assumptions.

b)Give theAmatrix for this problem (prior to replacing the last row with all 1's).

c) Assuming you have solved for the steady state probabilities,in terms of the 's,what is the throughput rate of this system (repaired taxis/hour).

d)Assuming you have solved for the steady state probabilities,in terms of the 's,what is the expected cost/hour of off-site repairs?

e) Assuming you have solved for the steady state probabilities,in terms of the 's,what is the expected lost opportunity cost/hour of taxis tied up at the facility?