  Electricity cost is a dominant and rapidly growing expense in data centers.
Unfortunately, much of the consumed energy is wasted because servers are idle
for extended periods of time. We study a capacity management problem that
dynamically right-sizes a data center, matching the number of active servers
with the varying demand for computing capacity. We resort to a data-center
optimization problem introduced by Lin, Wierman, Andrew and Thereska that, over
a time horizon, minimizes a combined objective function consisting of operating
cost, modeled by a sequence of convex functions, and server switching cost. All
prior work addresses a continuous setting in which the number of active
servers, at any time, may take a fractional value.
  In this paper, we investigate for the first time the discrete data-center
optimization problem where the number of active servers, at any time, must be
integer valued. Thereby we seek truly feasible solutions. First, we show that
the offline problem can be solved in polynomial time. Second, we study the
online problem and extend the algorithm {\em Lazy Capacity Provisioning\/}
(LCP) by Lin et al. to the discrete setting. We prove that LCP is
3-competitive. Moreover, we show that no deterministic online algorithm can
achieve a competitive ratio smaller than~3. In addition, we develop a
randomized online algorithm that is 2-competitive against an oblivious
adversary. Moreover, we prove that 2 is a lower bound for the competitive ratio
of randomized online algorithms, so our algorithm is optimal.
  Finally, we address the continuous setting and give a lower bound of~2 on the
best competitiveness of online algorithms. This matches an upper bound by
Bansal et al. We prove that all lower bounds still hold in a problem variant
with more restricted operating cost functions, introduced by Lin et al.
