Multi-Class ICU Bed Reservation Under Bursty Arrivals: A Generalized Loss Model Framework with Fairness Optimization

The effective allocation of Intensive Care Unit (ICU) beds is critical for balancing timely access across patient groups of differing clinical urgency, particularly during demand surges. This paper extends the threshold-based loss-queuing framework to three patient classes, emergency, semi-urgent, and elective, governed by a two-tier coordinate reservation policy with thresholds (k1,k2). Under Poisson arrivals, we derive an exact product form steady-state distribution and closed-form blocking probabilities for all three classes; under Interrupted Poisson Process (IPP) arrivals, we construct a block-tridiagonal Markov chain over the full (i,j,l,ξ) state space and obtain dimensionally consistent blocking formulas via a lifted rate matrix. Fitted to the publicly available MIMIC-IV Medical ICU dataset, the IPP captures bursty emergency arrival patterns with mean and variance deviations below 0.1%. A comprehensive parameter sensitivity sweep over traffic utilization ρ∈[0.40,0.95] and burstiness index z∈[1.0,5.0] identifies three distinct operating regimes and yields policy recommendation charts for direct clinical use. An adaptive optimization framework selects (k1*,k2*) to minimize a weighted blocking loss subject to fairness constraints, achieving a Jain fairness index above 0.999 throughout the high-load region. Analytical predictions are validated against discrete-event simulation with a 100% pass rate at a ±1.5 percentage-point criterion. We further demonstrate that the blocking probabilities for all three classes are insensitive to the LoS distribution beyond its mean across six service-time distributions spanning coefficients of variation from 0.45 to 2.24, with a 100% pass rate across all 72 (class, distribution, setting) combinations, broadening the model’s applicability to diverse real-world scenarios. The findings provide actionable guidance for ICU managers in determining fair and efficient three-tier bed reservation thresholds.