For a function $f:\mathbb R \rightarrow \mathbb R$, define $f(s,t) = f(t) - f(s)$. A two-sided Brownian motion $B$ is defined by $$ B(t) = \begin{cases} B^{(1)}(t), &t \ge 0 \\ B^{(2)}(-t) &t < 0, \end{cases} $$ where $B^{(1)}$ and $B^{(2)}$ are independent Brownian motions on $[0,\infty)$.
Now, Let $B_1$ and $B_2$ be two independent two-sided Brownian motions, let $\lambda$ be a positive constant, and let $$ q(t) = \sup_{-\infty < s \le t}\{B_1(s,t) + B_2(s,t) - \lambda(t - s)\}. $$
Then $q(t) \ge 0$ for all $t$, and it is not too hard to show that there exist random exceptional times such that $q(t) = 0$. For example, if $s < 0$ is a maximizer for $q(0)$, then $q(s) = 0$.
Now, let $B_1,B_2,B_3$ be independent two-sided Brownian motions, and set
$$q^{(3)}(t) = \sup_{-\infty < s \le u \le t}\{B_1(t,s) + B_2(s,u) + B_3(u,t) - \lambda(t - s)\}$$.
Again, $q^{(3)}(t) \ge 0$ for all $t$. Do there exist random times when $q^{(3)}(t) = 0$?
This problem can be thought of simply as a problem for Brownian motions, but it has an interpretation in the language of queuing theory. The increments $B_1(s,t)$ and $B_2(s,t)$ denote the number of arrivals and amount of service in the interval $(s,t]$, and $q(t)$ denotes the length of the queue at time $t$. These quantities are not integer-valued and are interpreted as scaling limits of queues under heavy traffic conditions. The times when $q(t) = 0$ denote the times when the queue in empty. One can iterate this queue so that the departures from the first queue become the arrivals for the next queue. In this sense, the process $q^{(3)}(t)$ has the interpretation as the sum of the queue-lengths for two queues, and $q^{(3)}(t) = 0$ means that both queues are simultaneously empty at time $t$. This particular formulation is from O'Connell and Yor, "Brownian Analogues of Burke's Theorem," (2001), but the original ideas come from the earlier work of Harrison (1985) and Harrison and Williams (1990,1992).