At a private investigation agency, 2 detectives work. Each detective can work on at most 1 case at a time, and the two detectives never work on the same case. The agency receives 1 case offer per month on average. If both of the detectives are currently working on a case, the agency declines the case. Otherwise, they accept the case and assign it to an available detective. If both detectives are available, the case is assigned at random to one of them. A case lasts for 3 months on average.

a) Model the number of active cases at the agency with a continuous time Markov chain. What are the states? What assumptions do we need to make to ensure the Markov property holds? Calculate the generator. Pay attention to the rates!

b) What is the long-term average ratio of time when the younger detective is working on a case?

c) Right now, one of the detectives is working. What is the probability that he will finish the case before the agency receives a new case offer?