Arrivals of passengers at a taxi stand form a Poisson process L with rate λ; passengers come singly and they wait patiently for their turn until a taxi shows up. Taxis arrive empty to the same stand according to Poisson process M with rate μ ; and each taxicab waits there until a ride shows up (if there were not only waiting passengers). Let
Xt = Lt - Mt , t>=0,
a) The process X = (X t) is a compound Poisson process that is, it has the form
Xt = YNt
where Y0 =0 and Yn = Z1 + Z2 +...+ Zn characterize the process N. Characterize the random variables Z1 , Z2 ,...; are they independent, what is their distribution? Is N independent of Y?
b) Compute (enough to write down an explicit expression) P {Xt = 3}, P {Xt = -2}, P {Xt = 0}, Interpret what these probabilities are.
c) Y = (Yn) n ? N is a Markov chain, what is its state space? Classify its states when λ > μ, and when λ < μ. Give intuitive justifications.
Continuation. In the preceding problem, we now modify the taxicab behavior, when a taxicab arrives to find 3 taxicabs there (and therefore no passengers) it leaves immediately. So, the number Xt has to be in the set D = {-3, -2, -1, 0, 1, 2, ...}. Show that X still has the form
Xt = YNt
but the Markov chain Y has transition probabilities different from those in the preceding problem.
a) Compute the probabilities
Pij = P{Y n+1 =j/Yn = i} i,j ? D
b) Classify the states when λ < μ.
Compute the limiting probabilities πj = limn-infinity P{Xn = j}.
d) What can you say about limt-infinity Pi {Xt = j}.