Bayesian predictive inference and threshold decisions for a discrete-time Geo/DP H/1 vacation queue under system-size observations

This paper develops a Bayesian predictive and decision-oriented framework for a discretetime Geo/DP H/1 queue with discrete phase-type service and vacation times under a single-vacation policy, observed only through a dependent system-size trajectory. A fixedorder, fully supported, labelled DPH working model and finite truncation yield a hidden Markov representation. Within this family, the parameter-to-latent-kernel mapping is injective; this result does not establish identifiability from the observed process alone. Posterior inference combines state-and-event augmentation, forward-filtering backwardsampling, conjugate parameter proposals, and an exact Metropolis–Hastings correction for the parameter-dependent stationary initial distribution. Posterior uncertainty is propagated to multi-step system-size distributions, exceedance probabilities, predictive quantiles, and threshold decisions under asymmetric loss. The numerical study combines repeated-sample experiments across three traffic levels and two observation lengths with truncation and prior sensitivity, phase-order and representation comparisons, controlled misspecification, posterior predictive checks, and plug-in and dependence-ignoring benchmarks. Longer observation records generally improve recovery and prediction, although the mean vacation duration remains comparatively difficult to estimate. Predictive and decision performance remain stable within the evaluated local perturbations, whereas ignoring temporal dependence substantially degrades both. The results support the framework as a coherent and auditable basis for prediction and cost-sensitive decision support from partially observed queueing dynamics.