Markov chains and random walks in data communication receivers

In many data communication receivers up/down counters are used as a critical part of the processing to determine whether the symbol timing and/or carrier phase tracking phase-locked loops are in-lock or out-of-lock, and it is necessary to calculate the various probabilities for true and false indications of in-lock or out-of-lock. A random walk along a line (which is viewed as a Markov chain) is an exact model of an up/down counter. The random walk has N states, and in this application one end is a partially reflecting barrier, and the other end is an absorbing barrier or sink. Previously published analyses have focused on finding the average time to make a declaration and its variance. The author concentrates on finding the probabilities of making a true or a false declaration within a certain number of symbol intervals or within a certain length of time.