Two-Phase Phase-Type Queues: Closed-Form Distributions, Sensitivity Analysis and the Boundary of Algebraic Tractability

Abstract

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.