Ambient backscatter communication (ABC) is a promising method of reducing energy consumption in wireless communications. Previous works on signal detection for ABC often assume that the ambient source signal is always present during backscattering. However, this may not be the case due to the random traffic of the ambient source and the asynchronous operation between source and tag. In this work, the effect of source signal traffic on the detection performance is studied for ABC systems. Firstly, the performances of the existing detectors are analyzed in the presence of source traffic. Both random arrival and random departure are considered. The exponential and uniform traffic models are used. Their bit error rate expressions are obtained by taking advantage of different approximation methods. Then, new detectors taking into account the random traffic models are derived by weighting the samples exponentially or linearly with their arrival times. Numerical results show that the random source traffic could cause large performance degradation to the existing detectors, leading to error floors at small signal-to-noise ratios (SNRs). In particular, the exponential departure causes the largest performance degradation, followed by the uniform arrival and departure. Numerical results also show that the new detectors could have significant performance gains over the conventional detectors in the presence of source traffic. In some case, the gain could be over 3 dB in SNR, and it increases with the sample size and traffic parameters. However, this gain could become negative for large SNRs and small sample sizes due to the use of heuristic detection thresholds.