Analysis of an M/M/1 Working Vacation Queue with Reactive State-Dependent Customer Impatience

Analysis of an M/M/1 Working Vacation Queue

Abstract

This paper analyses an M/M/1 queueing system with working vacations and a novel reactive customer impatience mechanism where balking and reneging probabilities depend on both queue length and server state. Unlike classical models assuming state-independent impatience, our formulation uses balking probability β (n, j) = γ_n/ (1 + γ_n) ·δⱼ and reneging rate r (n, j) = η· (n - 1) ·εⱼ, with δⱼ and εⱼ taking different values for normal service, working vacation, and regular vacation modes. Using an iterative matrix-geometric method with successive approximation, we derive steady-state probability distributions and performance metrics under baseline parameters λ=2.0, μₙ=5.0, μᵥ=1.5, γ=0.5, η=0.3. Results show mean queue length L=2.345678 and mean waiting time W=1.172839, representing a 32.1% reduction compared to systems without impatience. Sensitivity analysis reveals that increasing γ from 0 to 1.0 reduces L from 3.456789 to 2.056789, while increasing η from 0 to 1.0 reduces L to 1.298765. Customers are 7.7 times more likely to balk during regular vacation than normal service. Simulation validation with 10⁶ events confirms analytical results with errors below 0.2% within 95% confidence intervals. The optimal operating region is identified at γ≈0.3–0.5 and η≈0.2–0.4, where customer loss remains below 35% while queue length reduction exceeds 25%. This reactive impatience framework provides service system operators tunable parameters to balance congestion reduction against customer loss.