We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we will show that previous laws (see ) can be generalized; furthermore, within this framework a new law can be derived as well, which gives higher-moment expressions for very general types of queueing systems; in particular, the laws hold for systems that allow customers to overtake one another. What's especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide closed-form expressions of all moments of the number of customers in the system in an M=G=1 preemptive-LCFS queue at a time t > 0, for any initial condition and for any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling). The phrase \closed-form" is used here to stress that the moments can be expressed in terms of probabilities that consist of convolutions of busy periods and residual busy periods, and so moment-matching methods can be used to generate very simple approximations of these quantities, as in . It is also worth noting that these results appear to be new for the M=M=1 queue as well (see , ), and so we use them to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see , ).
Fralix, B. H., & Riaño, G. (2010). A new look at transient versions of Little's law, and M/G/1 preemptive Last-Come-First-Served queues. Journal of Applied Probability, 47(2), 459-473. https://doi.org/10.1239/jap/1276784903