Assignment:
Consider a post office with two counters denote by C1 and C2 with separate waiting queues. The service times are iid rvs with exponential distribution with parameter µ = 1/4 [customer/min]. Customers arrive at the post office according to a Poisson process with rate λ = 1/5 [customer/min] and, once entered the office, they join the shortest queue or, if the queues have equal length, they choose a queue at random. Furthermore, the last customer of a queue will move to the other queue (in zero time) whenever this action will advance its position. Modeling the system state at time t with the pair
w(t) = (x1(t), x2(t)), where x1(t) and x2(t) are the number of customers in front of the counters C1 and C2, respectively, develop the following points.
a) Prove that w(t) is a (two-dimensional) MC.
b) Write the probability flow balance equations.