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Notwithstanding much effort, explicit analytical results for average message delays for asymmetric token rings are currently unavailable. We develop an excellent and relatively simple analytic approximation which readily demonstrates the effects of the many system parameters on delay-throughput performance. The approximation applies to any cyclically served network (including rings, polling networks, and unidirectional fiber optic broadcast networks) with independent time-slot arrivals and gated service. The approximation becomes exact for symmetric loading. An equation for the moment-generating function of the joint terminal service times is obtained via an imbedded Markov chain. Two matrix equations are then derived, one for the mean terminal service times of the queues and another for the joint second central moments. A two-dimensional difference equation for the second central moments is then obtained. This difference equation together with Kleinrock's conservation law is used to approximate the mean delay at each queue. The peak approximation error is typically a few percent for various loading distributions including heavy or highly asymmetric loading, even if the statistics for message arrivals, message lengths, or walk times vary among the queues. Actual delay-throughput results are presented for various terminal loadings. Also included is a procedure to upperbound the error of our approximation for any message and walk time statistics, given the terminal utilizations.