Reference no: EM13581872

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}.