  We study the problem of online facility location with delay. In this problem,
a sequence of $n$ clients appear in the metric space, and they need to be
eventually connected to some open facility. The clients do not have to be
connected immediately, but such a choice comes with a penalty: each client
incurs a waiting cost (the difference between its arrival and connection time).
At any point in time, an algorithm may decide to open a facility and connect
any subset of clients to it. This is a well-studied problem both of its own,
and within the general class of network design problems with delays.
  Our main focus is on a new variant of this problem, where clients may be
connected also to an already open facility, but such action incurs an extra
cost: an algorithm pays for waiting of the facility (a cost incurred separately
for each such "late" connection). This is reminiscent of online matching with
delays, where both sides of the connection incur a waiting cost. We call this
variant two-sided delay to differentiate it from the previously studied
one-sided delay.
  We present an $O(1)$-competitive deterministic algorithm for the two-sided
delay variant. On the technical side, we study a greedy strategy, which grows
budgets with increasing waiting delays and opens facilities for subsets of
clients once sums of these budgets reach certain thresholds. Our technique is a
substantial extension of the approach used by Jain, Mahdian and Saberi [STOC
2002] for analyzing the performance of offline algorithms for facility
location.
  We then show how to transform our $O(1)$-competitive algorithm for the
two-sided delay variant to $O(\log n / \log \log n)$-competitive deterministic
algorithm for one-sided delays. We note that all previous online algorithms for
problems with delays in general metrics have at least logarithmic ratios.
