Let $\boldsymbol{X}=\{X_1, X_2, ...\}$ denote the inter-arrival times of a renewal process, and the CDF $F(x)$ for $X_i$ is known. Assume that the first arrival is at time $t=0$.
For the arrivals within the time duration $(0, D]$, what is the probability that none of the inter-arrival time is larger than $\tau$? (Or all the inter-arrvial times are smaller than $\tau$?)
$D$ and $\tau$ are both constant, and $D>\tau$.
Let $N_t$ be the number of arrivals in $(0,D]$. The probability you look for is \begin{align} P(X_1\le \tau \cap \cdots \cap X_{N_D}\le \tau) & = \sum_{n\ge 0} P(X_1\le \tau \cap \cdots \cap X_{N_D}\le \tau | N_D=n) P(N_D=n)\\ & = \sum_{n\ge 0} P^n(X_1\le \tau) P(N_D=n) \end{align} I guess one needs to know more on $F(x)$ to go further.