Useful new results on the time to lose lock for a simple Markov tracking scheme

We study the time to lose lock of a simple and popular Markov tracking scheme wherein a signal or status indicator is monitored every T seconds. Loss of tracking is declared if the signal is not detected for s consecutive time units; otherwise, tracking is maintained. To date, there exists a closed form expression for the mean time to lose lock E[TLL] and also an expression involving a matrix inverse for the variance V[TLL], but apparently no other results on the distribution function of TLL are known. We obtain a closed form expression for the variance V[TLL] which is easy to evaluate, and also show that in many cases of interest the standard deviation of TLL is very nearly equal to the mean E[TLL]. As a result, the distribution function of TLL cannot be non-trivially bounded in terms of the mean and standard deviation. To overcome this problem, we derive general bounds on the distribution function of TLL which facilitates design of the tracking loop and also statistically characterizes the time to lose lock in a rigorous manner. Examples are given to demonstrate the usefulness and ease of applicability of the derived results