# Inequalities between Arrival Averages and Time Averages in
Stochastic Processes Arising from Queueing Theory

A common scenario in stochastic models, especially in queueing theory, is
that an arrival counting process both observes and interacts with another
continuous-time stochastic process. In the case of Poisson arrivals, Wolff
[1982] recently proved that the proportion of arrivals finding the process
in some state is equal to the proportion of time it spends there, under a
lack of anticipation assumption. Inspired by Wolff's approach, we study in
this paper the related interesting question of when do we have inequalities
between these proportions. We establish two-sided inequalities under the
following three assumptions: (i) the interarrival-time distributions are of
type NBUE or NWUE, (ii) the process being observed have monotone sample
paths between arrival epochs, and (iii) the state of the process does not
depend on future jumps of the arrival process. These assumptions are
typically true in all standard queueing models and hence our results have
wide implications. Stochastic inequalities between limiting distributions of
interest, when they exist, also follow easily from our main result.