  We consider online coordinated precoding design for downlink wireless network
virtualization (WNV) in a multi-cell multiple-input multiple-output (MIMO)
network with imperfect channel state information (CSI). In our WNV framework,
an infrastructure provider (InP) owns each base station that is shared by
several service providers (SPs) oblivious of each other. The SPs design their
precoders as virtualization demands for user services, while the InP designs
the actual precoding solution to meet the service demands from the SPs. Our aim
is to minimize the long-term time-averaged expected precoding deviation over
MIMO fading channels, subject to both per-cell long-term and short-term
transmit power limits. We propose an online coordinated precoding algorithm for
virtualization, which provides a fully distributed semi-closed-form precoding
solution at each cell, based only on the current imperfect CSI without any CSI
exchange across cells. Taking into account the two-fold impact of imperfect CSI
on both the InP and the SPs, we show that our proposed algorithm is within an
$O(\delta)$ gap from the optimum over any time horizon, where $\delta$ is a CSI
inaccuracy indicator. Simulation results validate the performance of our
proposed algorithm under two commonly used precoding techniques in a typical
urban micro-cell network environment.
