Probability distribution of spectral hole duration in cognitive networks

Operation of cognitive secondary networks is critically dependent on the activity patterns of primary users. In this paper, we investigate the probability distribution of spectral holes, assuming that active and idle periods of primary users are independent random variables (which need not be identically distributed). We consider black, white, and gray holes, which correspond to time intervals when all channels are busy, all channels are idle, and some channels are busy while others are idle, respectively. We show that the duration of black and white holes may be described using an exponential approximation which holds regardless of the actual probability distribution of channel active and idle times, as long as the number of channels is not too small. The time interval between successive black hole occurrences is shown to be exponentially distributed as well. We also analyze the behavior of gray holes and quantify their impact using a simple proxy measure.