Impact of Poisson Control on Markovian Queue Performance

In this study, we investigate a class of Markovian queueing systems where a controller examines the queue at times dictated by a Poisson process. At each examination, the controller adjusts the server speed to the lesser of the current queue length or a predefined maximum speed, maintaining this adjusted speed until the next examination. This mechanism introduces a two-dimensional Markov process characterized by the queue length and server speed. We explore the behavior of this system under two distinct regimes of control timing: infinitely frequent and infinitely infrequent examinations. These regimes provide foundational insights, while the intermediate regime presents significant analytical challenges due to the complex interplay between arrival, service, and control processes. For the infinite maximum speed case ((s ̅=∞)), we derive the joint generating function of the steady-state process using functional equations and analyze the asymptotic behavior for both high and low control rates. For the finite maximum speed scenario ((s ̅<∞)), we utilize matrix geometric methods to characterize the steady-state probabilities and the joint generating function. Additionally, we consider two variant models where the controller observes but does not alter the server speed. These variants help elucidate the impact of observational frequency on queue dynamics. Our results are contextualized through numerical simulations that illustrate the theoretical findings and demonstrate the practical implications for optimizing system performance metrics such as queue length and server speed. This work extends the existing literature on workload-dependent service rates and Markov chains in random environments, offering new perspectives on efficiently managing queueing systems with Poisson-based control mechanisms.

Keywords:

Markovian queues , Variable service speeds , Poisson process , Queue management , Control rates