Excluded Volume Effect in General Distributions

In queuing systems, the dynamics of the excluded volume effect, which describes the movement of objects with a physical volume, are generally not considered. Recent studies have presented a theoretical formulation that incorporates the excluded volume effect to describe the delayed dynamics in M/M/1 queuing systems. We attempted to extend these results to the GI/G/1 queuing systems. Although the GI/G/1 queuing model is a stochastic process that cannot be completely described by a Markov chain, recent studies have indicated that it can be solved numerically using matrix geometry methods in discrete time under certain conditions. We propose a theoretical and analytical methodology in addition to approximation formulas for a discrete-time GI/G/1 queuing model that incorporates the excluded volume effect by utilizing the matrix-geometric method. The approximation methods were validated under single-server conditions for practical applications. Furthermore, using a theoretical perspective based on the aggregation method, we conducted numerical experiments on multiple servers to evaluate the performance of the discrete-time GI/PH/c queuing model. This approach adapted the excluded volume effect to the geometric matrix method. After testing it with a hyper-gamma distribution, we observed practical agreement, albeit to a limited extent.