  The Lonely Runner Conjecture asserts that if $n$ runners with distinct
constant speeds run on the unit circle $\mathbb{R}/\mathbb{Z}$ starting from
$0$ at time $0$, then each runner will at some time $t>0$ be lonely in the
sense that she/he will be separated by a distance at least $1/n$ from all the
others at time $t$. In investigating the size of $t$, we show that an upper
bound for $t$ in terms of a certain number of rounds (which, in the case where
the lonely runner is static, corresponds to the number of rounds of the slowest
non-static runner) is equivalent to a covering problem in dimension $n-2$. We
formulate a conjecture regarding this covering problem and prove it to be true
for $n=3,4,5,6$. Then, we use our method of proof to demonstrate that the
Lonely Runner Conjecture with Free Starting Positions is satisfied for $n=3,4$.
Finally, we show that the so-called gap of loneliness in one round (with
respect to the Lonely Runner Conjecture), where we have $m+1$ runners including
one static runner, is bounded from below by $1/(2m-1)$ for all integer $m\geq
2$.
