Excluded Volume Effect in GI/G/s Queues

This study developed an approximation framework for GI/G/s queues with excluded volumes. Building on a discrete-time GI/G/1 framework that embeds the totally asymmetric simple exclusion process (TASEP) via matrix-geometric and aggregation methods, we derived a Sakasegawa-type approximation with an effective service rate <inline-formula> <tex-math notation="LaTeX">$\mu ^{\prime }=\mu p_{e}$ </tex-math></inline-formula> capturing the exclusion-induced delay. A gamma-based moment approximation links the Yanagisawa M/M/1 system with excluded volume (E-Queue) and the Alfa model for GI/G/1 queues by matching the first two moments of geometric and gamma distributions, yielding an internal density <inline-formula> <tex-math notation="LaTeX">$\rho _{\mathrm {mean}}$ </tex-math></inline-formula> that is fed back into the Sakasegawa formula as a response coefficient in a two-stage correction scheme. For numerical validation, we used the Alfa model and an aggregated Alfa model for GI/PH/s queues as references, and evaluated it over ranges of squared coefficients of variation, utilization, server counts, and service-time distributions. Within the validated region (<inline-formula> <tex-math notation="LaTeX">$c_{a}^{2}\ge 1$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$c_{s}^{2}\le 1$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$s\le 4$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$h=0.1$ </tex-math></inline-formula>) (consistent with our previous study), the relative mean queue-length error is typically below 10%. Additional tests for exponential, Erlang, Weibull, Coxian-2, and static service times show full-range errors against the Alfa reference below 1% for light-tailed gamma-type cases and within a few percent for the more variable Coxian and Stacy cases, with mid-range traffic percentiles largely below 1%. We quantified the information loss from temporal discretization via Riemann sum approximations of gamma distributions and related it to the approximation error. Overall, the framework provides a data-assisted explicit approximation formula GI/G/s with a gamma-based two-stage correction and an error analysis that links the excluded volume.