Arrivals of passengers at a taxi stand form a Poisson process L with rate ?; passengers come singly...

Question

Arrivals of passengers at a taxi stand form a Poisson process L with rate ?; 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
X_t = L_t - M_t , t>=0,
a) The process X = (X _t) is a compound Poisson process that is, it has the form
X_t = Y_Nt
where Y₀ =0 and Y_n = Z₁ + Z₂ +...+ Z_n 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 {X_t = 3}, P {X_t = -2}, P {X_t = 0}, Interpret what these probabilities are.
c) Y = (Y_n) _{n ? N} is a Markov chain, what is its state space? Classify its states when ? > µ, and when
?
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 X_t has to be in the set D = {-3, -2, -1, 0, 1, 2, …}. Show that X still has the form
X_t= Y_Nt
but the Markov chain Y has transition probabilities different from those in the preceding problem.
a) Compute the probabilities
P_ij= P{Y _n+1 =j/Y_n = i} i,j ? D
b) Classify the states when ?

Compute the limiting probabilities p_j= lim_n-infinity P{X_n = j}.

d) What can you say about
lim_t-infinity P_i {X_t = j}.

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David · Accepted Answer

SOLUTION: 
a) Xt = Lt – Mt 
Now need to show that we can write the process X= (Xt) as YNt where 
Y0=0 and Yn= Z1+Z2+...+Zn where Zi ‘s are iid and Nt is a poisson 
process. 
Define Nt= Lt+Mt 
Now we know that the sum of 2 independent Poisson processes with 
parameters a and b  is a Poisson process with parameter (a+b). 
Since L and M are 2 independent Poisson process with parameters λ and 
μ respectivelty, N=L+M is a Poisson process with parameter λ+μ. 
Let an event   occur if either a taxi arrives or a passenger arrives. Hence 
the occurrences of these events  is the Poisson process N with 
parameter λ+μ. And at any time t, Nt ~ Poi( (λ+μ)t). This is the 
characterisation of N. 
Now define Zi= 1 if a passenger arrives during the ith event. 
And Zi=-1 if a taxi arrives during the ith event. 
This is the characterisation of the Zi’s. 
Then YNt= Z1+Z2+...+ZNt=Lt-Mt=Xt 
Now Zi are clearly identical. 
Since the two processes L and M are independent and the inter-arrival 
times are iid exponential, the Zi’s are also independent. This is because 
of the memoryless property of the exponential distribution. Given  a taxi 
arrives or a passenger arrives the time taken for the next taxi or 
passenger to arrive is independant of the previous occurrences . Hence 
the Zi’s are independant. 
Hence the Zi’s are iid and each Zi is independant of Nt,

Arrivals of passengers at a taxi stand form a Poisson process L with rate ?; passengers come singly and they wait patiently for their turn until a taxi shows up. Taxis arrive empty to the same stand...

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