🤖 AI Summary
This study addresses the absence of a unified closed-form solution and systematic numerical validation for the M/PH₂/1 queueing system by deriving, for the first time, explicit analytical expressions for both the queue-length and sojourn-time distributions. It fully specifies the coefficient formulas for the PH₂-type service-time distribution and establishes PH₂ as the boundary of algebraic solvability. The approach integrates algebraic derivation, matrix-analytic methods, and symbolic computation, complemented by large-scale numerical experiments implemented via BuTools. Results demonstrate that queue-length computations achieve machine precision even under extreme conditions—such as high traffic intensity (ρ ≈ 0.999) and very high service-time variability (Cₛ² > 10⁴)—while sojourn-time calculations retain at least 10 significant digits of accuracy, thereby enabling reliable sensitivity analysis, threshold optimization, and tail-probability guarantees.
📝 Abstract
Two-phase phase-type ($PH_2$) queues occupy a unique position in queueing theory: flexible enough to capture both low- and high-variability service processes, yet structured enough to admit complete algebraic characterization. Despite their prevalence in telecommunications, healthcare, and manufacturing, their explicit steady-state and sojourn-time distributions have not been collected in a single treatment with systematic numerical validation. We derive closed-form queue-length distributions $p_n = A_1 r_1^n + A_2 r_2^n$ and sojourn-time densities for the complete $M/PH_2/1$ family (Erlang-2, hypoexponential-2, hyperexponential-2, and Coxian-2), with all coefficients given as explicit functions of system parameters. Comprehensive validation against BuTools maps the numerical accuracy landscape of matrix-analytic computation: queue-length evaluation achieves machine-precision accuracy across extreme parameter ranges ($ρ$ up to 0.999, $C_s^2$ exceeding 10,000), while sojourn-time evaluation via the matrix exponential loses up to 10 significant digits when high traffic intensity and high service variability act jointly. The symbolic expressions enable analytical operations inaccessible to numerical output: exact sensitivity analysis, closed-form threshold optimization, and rigorous tail-probability guarantees. We prove that $PH_2$ marks the boundary of algebraic tractability: even the simplest three-phase queue ($M/E_3/1$) has a universally negative discriminant, forcing complex roots at every traffic intensity, so that its distribution cannot be expressed as a sum of real geometric terms. For $k \geq 5$ phases, the Abel-Ruffini theorem precludes radical solutions entirely. The results serve as a ready-to-use reference for two-phase service models.