Palm calculus for a process with a stationary random measure and its application to fluid queues

The paper considers a process associated with a stationary random measure, which may have infinitely many jumps in a finite interval. Such a process is a generalization of a process with a stationary embedded point process, and is applicable to fluid queues. Here, fluid queue means that customers are modeled as a continuous flow. Such models naturally arise in the study of high speed digital communication networks. The paper first derives the rate conservation law for them, and then introduces a process indexed by the level of the accumulated input. This indexed process can be viewed as a continuous version of a customer characteristic of an ordinary queue, e.g., of the sojourn time. It is shown that the indexed process is stationary under a certain kind of Palm probability measure, called detailed Palm. By using this result, the sojourn time processes in fluid queues is considered. The continuous version of Little’s formula in the present framework is derived. A distributional relationship is given between the buffer content and the sojourn time in a fluid queue with a constant release rate.