Markovian queues with Poisson control

TL;DR

This paper analyzes Markovian queues controlled at Poisson times, focusing on the joint process of queue length and server speed, with special regimes and variants where the controller only observes.

Contribution

It introduces a novel queue control model with Poisson-based examination and studies its dynamics, including two regimes and variants with passive observation.

Findings

Analysis of two regimes with different time scales.

Insights into the complex dynamics in the intermediate regime.

Comparison of controlled and observer-only variants.

Abstract

We investigate Markovian queues that are examined by a controller at random times determined by a Poisson process. Upon examination, the controller sets the service speed to be equal to the minimum of the current number of customers in the queue and a certain maximum service speed; this service speed prevails until the next examination time. We study the resulting two-dimensional Markov process of queue length and server speed, in particular two regimes with time scale separation, specifically for infinitely frequent and infinitely long examination times. In the intermediate regime the analysis proves to be extremely challenging. To gain further insight into the model dynamics we then analyse two variants of the model in which the controller is just an observer and does not change the speed of the server.