2007/09/10 by Takis Konstantopoulos, Konstantopoulos, Takis, Andreas E. Kyprianou +7
Decision Sciences · Mathematics · #60G10 #60G50 #60G51 #90B15 #FOS: Mathematics #Probability (math.PR) #Simulation Techniques and Applications #math.PR #msc:60G10 #msc:60G50 #msc:60G51 #msc:90B15
paper · pdf · doi:10.48550/arxiv.0709.1456
32 pages, 6 figures
arxiv created 2007/09/10 · openalex publication_date 2007/09/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a certain Markov process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is always singular with respect to the Lebesgue measure which in many applications is ``close'' to reality. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator) hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the Palm calculus coming from the point process point of view.