 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 $T={\sum }_{j=1}^K{X}_j$. Conditional on $K$, independence makes its moment-generating function a power of $M_X$, and summing the geometric series givesThis is the exponential distribution of rate $p\lambda$, so $T\sim \mathrm{Exp}(p\lambda )$. The calculation assumes $p>0$; if $p=0$, there is no white arrival and $T=+\infty$ almost surely.