Large Deviations and Applications

This study examines rare event behavior using the principle of large deviations, offering a more nuanced probabilistic framework than the law of large numbers and the central limit theorem. Although the classical results characterize typical fluctuations around the mean, large deviation theory is concerned with the probabilities of atypical events and their asymptotic decay rates. This method is applied to two models originating from applied probability. The first pertains to the sojourn time within an M/M/1 queueing system, characterized by exponentially distributed service times that are crucial for assessing congestion and system performance. The second model deals with the count of erroneous seconds seen in telecommunications systems, which can be modeled by a Poisson distribution. We investigate the empirical mean for both contexts and produce theoretical large deviation estimates that are grounded in the respective log-Laplace transforms. To compare empirical probabilities with their theoretical counterparts, numerical simulations are conducted. The outcomes demonstrate a distinct exponential reduction of deviation probabilities with the growth of sample size, which aligns closely with large deviation predictions. The findings demonstrate the importance of large deviation principles for quantifying rare yet critical events and show their applicability in analyzing the performance and reliability of queueing and communication systems.