The double-space parking problem

A double-space parking problem is studied for a parking lot of size M accommodating both private cars and buses. Upon arrival, a private car is either admitted to the parking lot, occupying a single spot, or waits in line until a spot becomes available. An arriving bus occupies double spots and is admitted only if there are at least two free spots. It balks from the system otherwise. The inflow is governed by two independent Poisson streams, with rates lambda(C) for cars and lambda(B) for buses. The sojourn time of a car or a bus inside the parking lot is exponentially distributed with parameters mu(C) and mu(B), respectively. The problem is formulated as a QBD process and analyzed via matrix geometric methods. Various performance measures are calculated, including mean number of cars inside, and outside, the parking lot; mean number of buses in the system; and the probability that an arriving bus is blocked. The dichotomy whether to split the M-spot lot into two separate lots, one for cars, the other for buses, is studied and the optimal split is calculated. Numerical results are presented via graphs. Finally, it is shown that from the point of view of the parking lot owner, it is equivalent to either charge a fixed entrance fee or charge per-time unit of usage.