Continuous Time Markov Chain - Immigration-Birth Process

An immigration-birth process with arrival rate *λ* and birth rate *β* may be described by the probability statement

*P*(*X*(*t +**δt*) *= x +* 1*| X*(*t*) *= x*)=(*λ* +*βx*)*δt* + *0*(*δt*)

Suppose that at time 0 the size of a population growing according to the above rule is 4. An ‘event’ is said to have occurred when the population size increases by one (and it is immaterial whether this increase is due to a random arrival or to a birth).

Write down an expression for the expected waiting time until the *n*th event (that is until the population size reaches *n* + 4).

Since X(0) = 4,

should the birthrate be *β* or 4*β*? I am confused here.

I am leaning towards 4*β*, so am I correct that the expected waiting time should be:

1/(*λ* + 4*β*) + 1/(*λ* + 8*β*) + 1/(*λ* + *12β*) + ... + 1/(*λ* + 4n*β*)

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__If the above is correct__, for the p.g.f. of a simple birth process with X(0) = 2 and birthrate β, should it be:

Π(s, t) = [(se^(-βt))/(1-s(1-e^(-βt)))]^2

*or*

Π(s, t) = [(se^(-2βt))/(1-s(1-e^(-2βt)))]^2

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Using the former (which I think is correct), is the following workings to find the probabilities at X(t) = 3 correct?

Π(3, t) = [(3e^(-βt))/(1-3(1-e^(-βt)))]^2

= (9e^(-2βt))/(-2+3e^(-βt))^2

= (9e^(-2βt))/(4-12e^(-βt)+9e^(-2βt))

Re: Continuous Time Markov Chain - Immigration-Birth Process

Any help will be appreciated. (Worried)