PGF of a Poisson process using differential equations - telephone answering service Could really do with help on this particular problem. I am comfortable with finding PGFs for most process but have no idea where to start for this question which begins with difference equations. Thanks.
A telephone answering service receives calls whose frequency varies with time but independently of other calls perhaps with a daily pattern – more during the day than the night. The rate $\lambda(t)\geq 0$ becomes a function of the time $t.$ The probability that a call arrives in the small time interval $(t,t+\delta t)$ when $n$ calls have been received at time $t$ satisfies
$$p_{n+1}(t+\delta t) = p_{n-1}(t)(\lambda(t)\delta t+o(t\delta))+p_{n}(t)(1-\lambda(t)\delta t+o(\delta t))$$
with
$$p_o(t+\delta t) = (1-\lambda(t)\delta t+o(\delta t))p_o(t)$$
It is assumed that the probability of two or mote calls arriving in the time interval $(t,t+\delta t)$ is negligible. Find the set of differential-difference equations for $p_n(t)$ Hence show that the probability generating function for the process is given by $$G(s,t)= \exp\left[(s - 1)\int_0^t\lambda(u) \, du\right].$$
 A: To start with since we have been asked to generate differential difference equations, we should be looking at how derivate might come up here. We know by definition that
$$ \frac{dp_n}{dt} = lim_{\delta \rightarrow 0} \frac{p_n(t+\delta t)-p_n(t)}{\delta t}$$
Then substituting in what you know $p_n(t)$ and $p_n(t+\delta t)$ to be, we see that
$$ \frac{dp_n}{dt} = lim_{\delta \rightarrow 0} (p_{n-1}\lambda - p_n\lambda + O(\delta t))=p_{n-1}\lambda - p_n\lambda$$
And using the definition  for $p_0$,
$$ \frac{dp_0}{dt} = - p_0\lambda$$
This is our set of differential difference equations.
Then we have:
$$ G(s,t)=\sum\limits_{n=0}^{\infty} p_n(t)s^n$$
So that, assuming we can pass the derivative through the sum (which is true for $|S|<1$?):
$$
\frac{dG(s,t)}{dt} = \frac{d}{dt} \sum\limits_{n=0}^{\infty} p_n(t)s^n = \sum\limits_{n=0}^{\infty} \frac{dp_n(t)}{dt}s^n \\
= \sum\limits_{n=0}^{\infty} (p_{n-1}\lambda - p_n\lambda)s^n \\
$$
We then recognise the summand as being similar to that in the definition of G, so that we can write:
$$
\frac{dG(s,t)}{dt} = \lambda s G(s,t) - \lambda G(s,t) = \lambda (s-1) G(s,t)
$$
Then you can solve that (partial) differential equation in G using an integrating factor to give the desired result.
