On the Distribution of Detection Delay for Quickest Spectrum Sensing

We derive closed-form expressions to approximate the distribution of detection delay for a time-invariant cumulative sum detector. In addition, we derive approximations for the distribution of detection delay for the general case, which can be applied to any independent and identically distributed channel. We also employ random walk and Brownian motion theory to derive approximate expressions. The validity of these expressions is verified using simulations. We also approximate the probability of missed detection and investigate the probability of long detection delays via analysis and simulations. Numerical results show that different signal-to-noise ratio and detection delay conditions require different approaches to achieve good approximations of the detection delay distribution. Results also illustrate that the remarkably simple Brownian motion approach provides the best approximation for longer delays and the type of fading channel has very little impact on long detection delays.