= Solution
Use independent colour marks, also independent of arrival times, as in the standard marked-process model. Let $K$ be the index of the first white arrival. Then $\mathbb P(K=k)=p(1-p)^{k-1}$ and $K$ is independent of the interarrival times. Thus $T=\sum_{j=1}^KX_j$. Conditional on $K$, <independence> makes its <moment-generating function> a power of $M_X$, and summing the <geometric series> gives
$$
M_T(\theta)=\sum_{k=1}^\infty p(1-p)^{k-1}\left(\frac\lambda{\lambda-\theta}\right)^k
=\boxed{\frac{p\lambda}{p\lambda-\theta}},\qquad\theta<p\lambda.
$$
This is the <exponential distribution> of rate $p\lambda$, so $T\sim\operatorname{Exp}(p\lambda)$. The calculation assumes $p>0$; if $p=0$, there is no white arrival and $T=+\infty$ almost surely.
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