Executive Summary : | Queueing theory is increasingly popular in various organizations, including flexible manufacturing and service systems, transportation systems, computer and telecommunication networks, and healthcare systems. To improve the performance and efficiency of these systems, detailed analytical investigation of waiting time and queue length distributions is crucial. The analysis of waiting time distribution is essential for satisfying customers and real-time video and audio traffics. This project proposal aims to present an efficient approach for evaluating the waiting time distribution for M/G/1, GI/M/1, and GI/G/1 queueing problems using the Laplace-Stieltjes transform (L.-S.T.) of differential equations of the number of customers in the system at arbitrary epoch. The work will be carried out based on the L.-S.T. of differential equations of the system length distribution at arbitrary epoch without evaluating the steady state distribution of numbers in the system. The proposed approach will derive the exact waiting time distribution directly from the L.-S.T. of differential equations of the system length distribution at arbitrary epoch without evaluating the steady state distribution of numbers in the system. Other methods such as the embedded Markov chain technique, semi-Markov process, and renewal theory can also be used to obtain relations among state probabilities at various time epochs. The aim is to study deep insights into and understanding of stochastic models of interest from a mathematical or applications perspective. The results are expected to contribute significantly to the queueing literature and be useful for further research and application in other areas. |