# How to compute the expected number of events in the following conditional renewal process?

I have a stochastic point process with event times $$\{x_1, x_2, ...\}$$ and I want to compute the expected number of events $$n(T)$$ over the interval $$[0,T]$$. The point process is generated as follows:

1. The first event is distributed according to density $$f(x)$$ over $$x \in (0,\infty)$$.
2. Conditioned on event $$x_n$$:

(i) With probability $$\alpha$$, $$x_{n+1}$$ is distributed according to the truncated density $$f(x > x_n)$$. The truncared distribution is defined as $$\frac{f(x)}{1- F(x_n)}$$ for $$x \in (x_n, \infty)$$.

(ii) With probability $$1-\alpha$$, $$x_{n+1} = x_n + d_n$$ where $$d_n$$ is distributed according to $$f(x)$$, which is similar to a renewal process.

How can we compute the expected number of points over the interval $$[0,T]$$? I tried to write the following equation:

$$n(T) = \alpha \int\limits_{x_1 = 0}^T f(x_1) (n(T-x_1)+1) dx_1$$

$$+ (1-\alpha) \int\limits_{x_1 = 0}^T \hspace{2mm} \int\limits_{x_2 = x_1}^T f(x_1) \frac{f(x_2)}{1-F(x_1)} (n(T-x_2)+2) dx_1 dx_2$$

Is it possible to solve this equation or is there a better way to solve this problem? Also, if it helps, you may also use the rate definition of point processes, where each event arrives at a rate $$\lambda (x |x_{n})$$. Here, $$x_n$$ is the latest event.