  In this paper, the dominant factor affecting the performance of active
intelligent reflecting surface (IRS) aided wireless communication networks in
Rayleigh fading channel, namely the average signal-to-noise ratio (SNR)
$\gamma_0$ at IRS, is studied. Making use of the weak law of large numbers, its
simple asymptotic expression is derived as the number $N$ of IRS elements goes
to medium-scale and large-scale. When $N$ tends to large-scale, the asymptotic
received SNR at user is proved to be a linear increasing function of a product
of $\gamma_0$ and $N$. Subsequently, when the BS transmit power is fixed, there
exists an optimal limited reflective power at IRS. At this point, more IRS
reflect power will degrade the SNR performance. Additionally, under the total
power sum constraint of the BS transmit power and the power reflected by the
IRS, an optimal power allocation (PA) strategy is derived and shown to achieve
0.83 bit rate gain over equal PA. Finally, an IRS with finite phase shifters
being taken into account, generates phase quantization errors, and further
leads to a degradation of receive performance. The corresponding closed-form
performance loss expressions for user's asymptotic SNR, achievable rate (AR),
and bit error rate (BER) are derived for active IRS. Numerical simulation
results show that a 3-bit discrete phase shifter is required to achieve a
trivial performance loss for a large-scale active IRS.
