Probability in the Engineering and Informational Sciences


O. J.  Boxma a1, D.  Perry a2 and F. A.  van der Duyn Schouten a3
a1 Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, and, CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
a2 Department of Statistics, University of Haifa, 31999 Haifa, Israel
a3 CentER for Economic Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands, and CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands


This paper is devoted to the analysis of a fluid queue with a buffer content that varies linearly during periods that are governed by a three-state semi-Markov process. Two cases are being distinguished: (i) two upward slopes and one downward slope, and (ii) one upward slope and two downward slopes. In both cases, at least one of the period distributions is allowed to be completely general. We obtain exact results for the buffer content distribution, the busy period distribution, and the distribution of the maximal buffer content during a busy period. The results are obtained by establishing relations between the fluid queues and ordinary queues with instantaneous input and by using level crossing theory.