A statistical and analytical exploration of some generalized queueing systems with phase type distribution

Abstract

Queueing systems with vacation policies effectively model service systems where servers may be temporarily unavailable due to rest, maintenance, or operational interruptions. These manufacturing, models computer are widely systems, and applicable in service communication networks, industries. Incorporating vacation mechanisms enhances realism and allows for better performance analysis, making such models critical for understanding and improving real world service systems. This thesis presents a comprehensive analytical and inferential investigation of single server queueing models with phase-type service and vacation time distributions. A Geo/PH/1 queue with multiple adaptive vacations is analyzed in discrete time, while an M/PH/1 queue with random vacations is examined in continuous time. Phase-type distributions provide flexibility while retaining analytical tractability. Both classical and Bayesian inference approaches are developed for parameter estimation. Classical inference is performed using maximum likelihood estimation via the expectation–maximization algorithm, while Bayesian inference is carried out using Markov chain Monte Carlo techniques with Gibbs sampling. Sensitivity analysis is conducted to assess the robustness of Bayesian estimates. Numerical investigations based on simulated data, along with real data applications, validate the analytical results. Key performance measures, including system utilization, queue length, waiting time, and busy period, are evaluated to demonstrate the impact of vacation policies. The thesis concludes by emphasizing the importance of vacation queueing models in practical service systems and by providing meaningful insights and directions for future research. Keywords: Vacation queueing models, Phase-type distributions, Discrete and continuous time queues, Performance measures, Maximum likelihood estimation, Bayesian inference.