Question: Consider the entrance of a highway with N = 5 tollbooths. Tollbooths are of two types, namely A and B. Tollbooths of type A are enabled for automatic fast payment by radio transmission, whereas those of type B are reserved for cash payment. Accordingly, vehicles are divided into type A and B depending on whether they are enabled or not for fast payment. Suppose that the time taken by a vehicle to cross a tollbooth is a rv, with exponential distribution of mean myA = 1 seconds and myB = 24 seconds for tollbooth of type A and B, respectively. In the rush hour, the vehicles arrive at the highway entrance according to a Poisson process with parameter λ = 0.1 vehicle/s. On average, 20% of the vehicles are of type A. Vehicles choose at random among the tollbooths of their own type, irrespective of the queues, lengths. Assuming that there is a single tollbooth of type A and four of type B, determine the steady-state values of the following metrics:
a) the mean number of vehicles of type A and type B waiting in queue to enter the highway;
b) the mean queueing time of any vehicle.
c) Finally, find the number of tollbooths of type A that minimizes the mean queueing time of any vehicle.