= Solution
{c}
A <Poisson process> of rate $\lambda>0$ starts from zero, has independent stationary increments, has unit jumps, and satisfies $N_t-N_s\sim\operatorname{Poisson}(\lambda(t-s))$ for $t>s$. Its inter-arrival times are independent with the <exponential distribution> of rate $\lambda$: any finite joint density is $\prod_i\lambda e^{-\lambda x_i}$ on positive coordinates.
The second arrival is a sum of two such variables. Convolution gives
$$
f_{T_2}(t)=\int_0^t\lambda e^{-\lambda s}\lambda e^{-\lambda(t-s)}\,ds
=\boxed{\lambda^2t e^{-\lambda t}\quad(t>0).}
$$
This is a gamma distribution of shape two and rate $\lambda$. If $S$ is the first arrival of the independent rate-$\mu$ process, exponential independence gives
$$
\boxed{\mathbb P(S<T_1)=\int_0^\infty\mu e^{-\mu s}e^{-\lambda s}\,ds
=\frac{\mu}{\lambda+\mu}.}
$$
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