Transient and stationary distributions of queue content for fluid queues whose continuous input process has a density with respect to time are studied. Results are obtained via a method primarily introduced by Benes, when considering input processes increasing by jumps at random times. The general formulas are applied to the queue where the input density varies as a stationary Ornstein-Uhlenbeck process. Such fluid queue models are useful in the field of telecommunications for dimensioning studies of packet multiplexers which are offered a superposition of variable bit rate video sources.
Simonian, A., & Virtamo, J. (1991). Transient and stationary distributions for fluid queues and input processes with a density. SIAM Journal on Applied Mathematics, 51(6), 1732-1739. https://doi.org/10.1137/0151088