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The number of periods which have much influence on the initial decision is vital to a real world decision maker who desires to ``validate'' the usefulness of applying a stationary known distribution model to a nonstationary partial information problem. This question can be studied by analytically or computationally studying the convergence rate of successively longer finite horizon problems with one or more extreme sets of terminal conditions. White , Schweitzer , Odoni , Hastings , and Morton  have contributed analytically to this question for the undiscounted Markov decision problem. Recently Boyse  adapted some of these results to the discrete time undiscounted Markov renewal programming problem. Here a simpler extension is given, which allows all the earlier results to be used directly, providing somewhat stronger results.