  We study a variant of the $k$-server problem, the infinite server problem, in
which infinitely many servers reside initially at a particular point of the
metric space and serve a sequence of requests. In the framework of competitive
analysis, we show a surprisingly tight connection between this problem and the
$(h,k)$-server problem, in which an online algorithm with $k$ servers competes
against an offline algorithm with $h$ servers. Specifically, we show that the
infinite server problem has bounded competitive ratio if and only if the
$(h,k)$-server problem has bounded competitive ratio for some $k=O(h)$. We give
a lower bound of $3.146$ for the competitive ratio of the infinite server
problem, which implies the same lower bound for the $(h,k)$-server problem even
when $k/h \to \infty$ and holds also for the line metric; the previous known
bounds were 2.4 for general metric spaces and 2 for the line. For weighted
trees and layered graphs we obtain upper bounds, although they depend on the
depth. Of particular interest is the infinite server problem on the line, which
we show to be equivalent to the seemingly easier case in which all requests are
in a fixed bounded interval away from the original position of the servers.
This is a special case of a more general reduction from arbitrary metric spaces
to bounded subspaces. Unfortunately, classical approaches (double coverage and
generalizations, work function algorithm, balancing algorithms) fail even for
this special case.
