single-queuing-system

There is a single toll booth placed at the intersection of three roads and it serves cars arriving from these roads. The cars from the different roads arrive in accordance with a Poisson process with rates λ1, λ2 and λ3. Note that, for a Poisson process with an arrival rate of λ, the times between successive arrivals (interarrival times) are independent exponential random variables, with mean value of 1/ λ. Each car upon arrival, directly enters service at the toll booth if the toll booth is free and, if not, the car joins the queue. When the toll both finishes serving a car, the car leaves the system, and the next car in the queue, if there is any, enters service. The service times at the toll booth are assumed to be independent exponential random variables having mean 1/ μ.