  We consider the problem of online service with delay on a general metric
space, first presented by Azar, Ganesh, Ge and Panigrahi (STOC 2017). The best
known randomized algorithm for this problem, by Azar and Touitou (FOCS 2019),
is $O(\log^2 n)$-competitive, where $n$ is the number of points in the metric
space. This is also the best known result for the special case of online
service with deadlines, which is of independent interest.
  In this paper, we present $O(\log n)$-competitive deterministic algorithms
for online service with deadlines or delay, improving upon the results from
FOCS 2019. Furthermore, our algorithms are the first deterministic algorithms
for online service with deadlines or delay which apply to general metric spaces
and have sub-polynomial competitiveness.
