  With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a
fractional Brownian motion $A(\cdot)$ with negative drift, this paper studies
the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$. We
define two metrics that measure the distance between the (complementary)
distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M>\cdot)$. Our
main result states that both metrics roughly decay as $\exp(-\vartheta
t^{2-2H})$, where $\vartheta$ is the decay rate corresponding to the tail
distribution of the busy period in an fBm-driven queue, which was computed
recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs
extensively rely on application of the well-known large deviations theorem for
Gaussian processes. We also show that the identified relation between the decay
of the convergence metrics and busy-period asymptotics holds in other settings
as well, most notably when G\"artner--Ellis-type conditions are fulfilled.
