# A “finite” taxi stand has room for 2 customers to wait and 1 taxi to wait. Suppose customers… 1 answer below »

A “finite” taxi stand has room for 2 customers to wait and 1 taxi to wait. Suppose customers arrive as a Poisson process with rate ? and if the stand is full, an arriving customer simply leaves. Taxis arrive as a Poisson process (independent from the arrival process of the customers) with rate 2?. Each taxi takes a single customer. If there is 1 taxi waiting, an arriving taxi simply leaves. Note that any time it is not possible for both customers and taxis to be waiting. Let X(t) take values in {-1, 0, 1, 2} where negative values indicate the number of taxis waiting and positive values indicate the number of people waiting.

(a) (8 points) Sketch a state transition diagram showing rates for this birth and death process, X(t).

(b) (8 points) Write the Kolmogorov’s Forward Equations for Pij (t) when i, j = -1, 0, 1, 2.

(c) (9 points) Find the limiting probabilities, P-1, P0, P1, and P2.