  This paper derives approximate outage probability (OP) expressions for uplink
cell-free massive multiple-input-multiple-output (CF-mMIMO) systems with and
without pilot contamination. The system's access points (APs) are considered to
have imperfect channel state information (CSI). The
signal-to-interference-plus-noise ratio (SINR) of the CF-mMIMO system is
approximated via a Log-normal distribution using a two-step moment matching
method. OP and ergodic rate expressions are derived with the help of the
approximated Log-normal distribution. For the no-pilot contamination scenario,
an exact expression is first derived using conditional expectations in terms of
a multi-fold integral. Then, a novel dimension reduction method is used to
approximate it by the sum of single-variable integrations. Both the
approximations derived for the CF-mMIMO systems are also useful for single-cell
collocated massive MIMO (mMIMO) systems and lead to closed-form expression. The
derived expressions closely match the simulated numerical values for OP and
ergodic rate.
