Quickest change detection in multiple on-off processes: Switching with memory

We consider the quickest detection of idle periods in multiple on-off processes. At each time, only one process can be observed, and the observations are random realizations drawn from two different distributions depending on the current state (on or off) of the chosen process. Switching back to a previously visited process is allowed, and measurements obtained during previous visits are taken into account in decision making. The objective is to catch an idle period in any of the on-off processes as quickly as possible subject to a constraint on the probability of mistaking a busy period for an idle one. Assuming geometrically distributed busy and idle times, we establish a Bayesian formulation of the problem within a decision-theoretic framework. Basic structures of the optimal decision rules are established. Based on these basic structures, we propose a low-complexity threshold policy for switching among processes and declaring idle periods. The near optimal performance of this threshold policy is demonstrated by a comparison with a genie-aided system which defines an upper bound on the optimal performance. This problem finds applications in spectrum opportunity detection in cognitive radio networks where a secondary user searches for idle channels in the spectrum.