Large Deviations Principle in the M/M/1 Queueing System

In this article, we study the principle of large deviations applied to the M/M/1 queueing model, focusing particularly on the sojourn time of patients in a hospital service. This stochastic model, which is based on Poisson arrivals and service times following an exponential distribution, offers both a relevant and simple illustration of the random dynamics of patient flow. Our aim is to analyze the asymptotic behavior of the average sojourn time and to quantify the probability of atypical delays that could significantly influence the efficiency of healthcare systems. From a theoretical point of view, we derive the associated rate function and establish explicit exponential bounds for the probability of prolonged stays. These results clearly illustrate the exponential decay of rare but essential events. Furthermore, we validate our theoretical results through numerical simulations, which demonstrate the coherence between analytical predictions and empirical assessments. This research contributes both to understanding the mathematical aspects of large deviations in queueing systems and to developing a practical model for hospital operations, providing a quantitative framework to forecast and manage cases of extreme congestion in healthcare services.