Comparing Semi-Markov Processes

Abstract

Sufficient conditions are found for two semi-Markov processes to be stochastically ordered, i.e., for which two new semi-Markov processes can be constructed on a common probability space so that the new processes individually have the same distributions as the original processes and every sample path of the first new process lies below the corresponding sample path of the second new process. This ordering has recently been shown by Kamae, Krengel, and O’Brien (Kamae, T., V. Krengel, G. L. O’Brien. 1977. Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 899–912.) to be equivalent to the definition of stochastic order usually seen in the literature; see Veinott (Veinott, A. F. Jr. 1965. Optimal policy in a dynamic, single product, nonstationary inventory model with several demand classes. Oper. Res. 13 761–778.). The conditions are: (i) the initial distribution of the first process is stochastically smaller than that of the second; (ii) for each ordered triple of states j ≤ k ≤ 1 and (unordered) holding times s and t, the instantaneous transition rate of the first process from j to the set of states exceeding 1 given its holding time in j is s is not less than the corresponding transition rate of the second process where j and s are replaced by k and t respectively; and (iii) the dual of condition (ii) obtained by reversing the order of the states. The construction is accomplished for semi-Markov processes for which all subprobability transition rates are absolutely continuous with failure rates uniformly bounded over finite intervals by representing the two semi-Markov processes as compositions of discrete-time stochastic processes with a sequence of Poisson processes. This permits application of recent comparison results for discrete-time stochastic processes by O'Brien (O'Brien, G. L. 1975a. The comparison method for stochastic processes. Ann. Probab. 3 80–88; O'Brien, G. L. 1975b. Inequalities for queues with dependent interarrival and service times. J. Appl. Probab. 12 653–656.).