2. Arrivals to a single server system only enter the system if they nd the server idle. Service times are exponential with rate a. Wet and dry arrival periods alternate. In a wet period, arrivals are according to a Poisson process with rate A1; in a dry period, arrivals are according to a Poisson process with rate A2. Wet periods last for an exponential time with rate 01 and dry periods last for an exponential time with rate (12. Let the state of the system be (1, j) if we are currently in a wet period and there are 3' in the system, 3' = 0.1; and let it be (2, j) if we are currently in a dry period and there are 3' in the system. (a) Give the transition diagram. (b) Give the balance equations. Do not solve the equations. In terms of the long run proportions P,,j,'i = 1,2, 3' = 0.1 nd (c) the average number of customers in the system; (d) the proportion of arrivals that do not enter the system. 3. Consider the previous problem, and let T1 be the time until the next arrival if we are currently in a wet period, and let T2 be the time if we are currently in a dry period. If A1 = 2, A2 = 1, a1 = 2, a2 = 4. nd E[T1]