Nondeterministic Polling Systems

A nondeterministic polling system is considered in which a single server serves a number of stations. The service discipline at each station is, consistently, either nonexhaustive, semiexhaustive, gated, or exhaustive. If the server polls a station i which uses either the nonexhaustive or the semiexhaustive service discipline, then the next station polled is station j with probability p_ij if there was service at station i. The service time at station i is a random variable which may depend on the station polled next. If no service is performed at station i, then the next station polled is station j with probability e_ij. The time to switch between stations i and j is a random variable which may depend on whether service was performed at station i or not. If the server polls a station i that follows either the exhaustive service discipline or the gated service discipline, then the next station polled is station j with probability p_ij regardless of whether there was service at station i or not. Cycle times and stability conditions are derived for this system, and Conservation Laws are obtained which express a weighted sum of the mean waiting times in terms of known data parameters. For systems with a mix of exhaustive and gated service stations, we show how the individual mean waiting times can be obtained.