A task such as the execution of a computer program or the transfer of a file on a communications link may fail and then needs to be restarted. Let the ideal task time be a constant $\ell$ and the actual task time $X$, a random variable. Tail asymptotics for $\mathbb{P}(X>x)$ is given under three different models: 1: a time-dependent failure rate $\mu(t)$ and a constant work rate $r(t)\equiv 1$; 2: Poisson failures and a time-dependent deterministic work rate $r(t)$; 3: as 2, but $r(t)$ is random and a function of a finite Markov process. Also results close to being necessary and sufficient are presented for $X$ to be finite a.s. The results complement those of Asmussen, Fiorini, Lipsky, Rolski & Sheahan [Math. Oper. Res. 33, 932--944, 2008] who took $r(t)\equiv 1$ and assumed the failure rate to be a function of the time elapsed since the last restart rather than wallclock time.

Keywords change of measure, computer reliability, fluid model, gaps, inhomogeneous Poisson process, Markov-modulation, Markov renewal theorem, tail asymptotics, time transformation