AI 中文总结
该研究推导了完整$M/PH_2/1$族队列的闭式分布，通过BuTools明确其数值精度格局，证明$PH_2$是代数可处理性边界，为两阶段服务模型提供了现成参考。
AI 中文摘要
两阶段相位型（$PH_2$）队列在排队论中具有独特地位：其灵活性足以同时刻画低变异性和高变异性服务过程，同时结构足够清晰以实现完整的代数表征。尽管这类队列广泛应用于电信、医疗保健和制造业领域，但尚未有研究将其显式稳态分布和逗留时间分布整合到单一研究中并开展系统数值验证。我们推导了完整$M/PH_2/1$族（包括Erlang-2、次指数2、超指数2和Coxian-2）的队列长度分布$p_n = A_1 r_1^n + A_2 r_2^n$和逗留时间密度，所有系数均为系统参数的显式函数。通过与BuTools的全面验证，我们绘制了矩阵分析计算的数值精度格局：在极端参数范围（$\rho$高达0.999，$C_s^2$超过10000）下，队列长度评估可达到机器精度；而当高流量强度与高服务变异性共同作用时，通过矩阵指数计算的逗留时间评估会损失多达10位有效数字。这些符号表达式支持数值输出无法实现的分析操作：精确敏感性分析、闭式阈值优化和严格的尾概率保证。我们证明$PH_2$是代数可处理性的边界：即使是最简单的三阶段队列（$M/E_3/1$）也具有普遍负判别式，在任意流量强度下均会出现复根，因此其分布无法表示为实几何项的和；对于$k \geq 5$个阶段，阿贝尔-鲁菲尼定理完全排除了根式解的存在。本研究结果可作为两阶段服务模型的现成参考资料。
英文摘要
Two-phase phase-type ($PH_2$) service distributions are widely used in call center, healthcare and manufacturing models: they capture coefficients of variation both above and below unity while remaining parsimonious enough for reliable statistical fitting. We derive explicit closed-form queue-length distributions $p_n = A_1 r_1^n + A_2 r_2^n$ and sojourn-time densities $f_{W_s}(t) = c_1 e^{α_1 t} + c_2 e^{α_2 t}$ for the complete $M/PH_2/1$ family (Erlang-2, hypoexponential-2, hyperexponential-2 and Coxian-2), with every coefficient given as an explicit function of the system parameters and consolidated in ready-to-use reference tables. Because the results are functions rather than numerical values, they support analytical operations that numerical output cannot deliver directly: exact sensitivity derivatives, closed-form threshold optimization, capacity sizing by bisection on an exact cumulative distribution, and $O(1)$ tail-probability evaluation at arbitrary queue length. We further prove that $PH_2$ marks the boundary of algebraic tractability: the three-phase queue $M/E_3/1$ has a universally negative discriminant, forcing complex roots at every traffic intensity, so its distribution admits no representation as a sum of real geometric terms, and for $k \geq 5$ phases the Abel-Ruffini theorem precludes radical solutions. Validation against the matrix-analytic library BuTools confirms the derivations and characterizes where such computation degrades: queue-length evaluation holds machine precision throughout, while sojourn-time evaluation via the matrix exponential loses up to ten significant digits when high traffic intensity and high service variability act jointly. An application calibrated to published surgical time data for 46,322 cases yields an exact affine law for the sensitivity of overflow risk to case mix.
Comments28 pages, 12 figures, 8 tables. v2: revised title; new worked application (Section 7) calibrated to published surgical time data; closed-form results reorganized; expanded discussion of phase-level versus scalar-level closed forms
