Stochastic modelling and analysis of some generalized queueing networks and their applications

Abstract

Queueing theory is a field dedicated to the modelling and analysis of queues or waiting lines. It provides mathematical tools to optimise various processes in service systems to enhance overall system performance. Most of the real-world service systems operate under time- varying conditions, such as fluctuating arrival and service processes. Analysing the transient behaviour of such time-varying queueing systems is significantly more challenging than steady-state analysis. This research focuses on the transient analysis of time-varying queues, which have practical applications in real-life service systems. The study investigates the transient distributional law that links the virtual workload to customer waiting times in a non- stationary general single-server queueing system. Additionally, a simulation study is conducted to validate the transient measure alongside other performance measures. The research also introduces a general algorithmic framework to derive transient performance measures in a k-station Markovian tandem network, supported by numerical studies that analyse the transient behaviour of these performance measures. Furthermore, a comparative study is presented on a Markovian non-stationary two-station tandem network with finite queue capacity, examining different blocking mechanisms. The study provides explicit expressions for transient performance measures under both BAS and BBS blocking mechanisms. Another significant contribution of this research is the exploration of time- varying approximations for performance measures in a feed-forward open queueing network comprising single-server queues with time-varying arrival rates. An algorithm is developed to compute time-varying approximations for performance measures in feed-forward open queueing networks of Gt /G/1 queues. The thesis concludes by emphasizing the critical role of time-varying queues in real-life service systems and offers actionable recommendations for future research directions.