A semi-Markov queue with exponential service times
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In de Smit (1985) a general model for the single server semi-Markov queue was studied. Its solution was reduced to the Wiener-Hopf factorization of a matrix function. For the general case a formal factorization can be given in which the factors have probabilistic interpretations but cannot be calculated explicitly (see Arjas (1972)). Explicit factorizations can only be given in special cases. The present paper deals with the case in which customers arrive according to a Markov renewal process, while an arriving customer requires an exponential service time whose mean depends on the state of the Markov renewal process at his arrival. For this special case we shall obtain an explicit factorization. The model is formally described as follows.
KeywordsQueue Length Imaginary Axis Implicit Function Theorem Busy Period Interarrival Time
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