  Two-tier femtocell networks is an efficient communication architecture that
significantly improves throughput in indoor environments with low power
consumption. Traditionally, a femtocell network is usually configured to be
either completely open or completely closed in that its channels are either
made available to all users or used by its own users only. This may limit
network flexibility and performance. It is desirable for owners of femtocell
base stations if a femtocell can partially open its channels for external users
access. In such scenarios, spectrum and energy efficiency becomes a critical
issue in the design of femtocell network protocols and structure. In this
paper, we conduct performance analysis for two-tier femtocell networks with
partially open channels. In particular, we build a Markov chain to model the
channel access in the femtocell network and then derive the performance metrics
in terms of the blocking probabilities. Based on stationary state probabilities
derived by Markov chain models, spectrum and energy efficiency are modeled and
analyzed under different scenarios characterized by critical parameters,
including number of femtocells in a macrocell, average number of users, and
number of open channels in a femtocell. Numerical and Monte-Carlo (MC)
simulation results indicate that the number of open channels in a femtocell has
an adverse impact on the spectrum and energy efficiency of two-tier femtocell
networks. Results in this paper provide guidelines for trading off spectrum and
energy efficiency of two-tier femtocell networks by configuring different
numbers of open channels in a femtocell.
