Exploiting partial information in queueing systems

We often try to draw inferences from partial observations of queueing systems in real-life situations. For example, if we observe many customer arrivals, we may presume that the system is crowded and many customers are served. Unfortunately, such an intuitive statement is not necessarily valid. We provide sufficient conditions under which the intuition can be justified, and investigate related properties of queueing systems. We also study a way to exploit the partial information in a quantitative manner for simple queueing systems. One numerical result is rather counterintuitive. Specifically, the number of customers in the system at time t given that the cumulative number of departures is a certain constant is not necessarily stochastically increasing in t for a simple M/M/1 system with finite capacity.