  Small cell networks promise good quality of service (QoS) even for cell edge
users, however pose challenges to cater to the high-speed users. The major
difficulty being that of frequent handovers and the corresponding handover
losses, which significantly depend upon the speed of the user. It was shown
previously that the optimal cell size increases with speed. Thus, in scenarios
with diverse users (speeds spanning over large ranges), it would be inefficient
to serve all users using common cell radius and it is practically infeasible to
design different cell sizes for different speeds. Alternatively, we propose to
allocate power to a user based on its speed, e.g., higher power virtually
increases the cell size. We solve well known Hamiltonian Jacobi equations under
certain assumptions to obtain a power law, optimal for load factor and busy
probability, for any given average power constraint and cell size. The optimal
power control turns out to be linear in speed. We build a system level
simulator for small cell network, using elaborate Monte-Carlo simulations, and
show that the performance of the system improves significantly with linear
power law. The power law is tested even for the cases, for which the system
does not satisfy the assumptions required by the theory. For example, the
linear power law has significant improvement in comparison with the 'equal
power' system, even in presence of time varying and random interference. We
observe good improvement in almost all cases with improvements up to 89\% for
certain configurations.
