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In this paper we investigate an M/M/$\infty$ queue whose parameters depend onan external random environment that we assume to be a semi-Markovian processwith finite state space. For this model we show a recursive formula that allowsto compute all the factorial moments for the number of customers in the systemin steady state. The used technique is based on the calculation of the rawmoments of the measure of a bidimensional random set. Finally the case when therandom environment has only two states is deeper analyzed. We obtain anexplicit formula to compute the above mentioned factorial moments when at leastone of the two states has sojourn time exponentially distributed.Comment: 17 pages, 2 figure

M/M/∞ queues in semi-Markovian random environment