Recently I was stumped by the calculation of the probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where $A_i \sim \text{exp}(\lambda), S_i \sim \text{exp}(\mu), L \sim \text{exp}(\lambda)$ are mutually independent, $\lambda \neq \mu$ are two positive integers, and $m$ is an integer parameter.

**My attempt:**

I failed at the very start while calculating the density function of $\sum_{i=1}^{m} (A_i + S_i)$: $$\sum_{i=1}^{m} (A_i + S_i) = \sum_{i=1}^{m}{A_i} + \sum_{i=1}^{m}{S_i} = \big( A \sim \Gamma(m, \lambda) \big) + \big( S \sim \Gamma(m, \mu) \big)$$ Considering the two Gamma random variable $A$ and $S$, their convolution is: $$p_{A+S}(a) = p_{A} \ast p_{S} (a) = \int_{0}^{a} f_{A}(a-y) f_{S}(y) dy \\ = \int_{0}^{a} \frac{\lambda e^{-\lambda (a-y)} (\lambda (a-y))^{m-1}}{\Gamma(m)} \frac{\mu e^{-\mu y} (\mu y)^{n-1}}{\Gamma(n)} dy \\ = e^{-\lambda a} \frac{\lambda^m \mu^n}{\Gamma(m) \Gamma(n)} \int_{0}^{a} e^{(\lambda - \mu) y} (a-y)^{m-1} y^{n-1} dy$$

**And my failure:**

I was not able to go any further with the integral. And you can image what a messy calculation we will run into when we are going to take the second step involving $\sum_{i=1}^{m} (A_i + S_i) \le L$.

**My problems:**

Generally, my problem is how to make the calculation tractable. Specifically,

- Has the probability been studied in research papers?

People interested in its background in "alternating renewal process" can refer to the following part of the post.- What is the density function of $\big( A \sim \Gamma(m, \lambda) \big) + \big( S \sim \Gamma(m, \mu) \big)$ given that $m$ is an integer and $\lambda \neq \mu$ are positive integers?

If there is no nice closed form, can we make some good approximation?

Related post: at Math.SE (without proper answers yet).- For the whole calculation, is it feasible to obtain numerical solution using mathematical softwares?

Are there any off-the-shelf libraries for this purpose?

**For people who want to know the background of the probability in "renewal process" (specifically, in "alternating renewal process"):**

Background:Consider an alternating renewal process, in which a system can be in one of two states: on or off. Whenever it is off, it takes a time $A_i \sim \text{exp}(\lambda)$ before turning on. Whenever it becomes on, it remains on for a time $S_i \sim \text{exp}(\mu)$. Initially the system is off. A cycle $c_i$ consists of the $i$th off-state and the $i$th on-state. All $A_i$ and $S_i$ are mutually independent and $\lambda \neq \mu$ are two positive integers.

Problem:Given a time period $L \sim \text{exp}(\lambda)$, what is the probability that exactly $m$ cycles occur in $L$?

Related post: at MO: its origin; many thanks to @user137846.

Are there any other perspectives leading to a different formula from the one mentioned above?