Stationary Coupled-Flow Approximation for Time-Varying Retrial Queues

Although stationary queueing models, such as Erlang-C and Erlang-A, are widely used for performance evaluation in service systems, they do not account for time dependence and retrials, which limits their applicability. Building on stationary $M/M/c+G$ formulas and point-wise stationary approximations (PSAs), this paper proposes the stationary coupled-flow (SCF) approximation. The method partitions the time horizon into intervals, models each as an $M/M/c+G$ in steady state, and iteratively determines the arrival rate, accounting for new customers and retrials. This yields an approximation of long-run performance metrics of retrial queues with stationary or time-varying arrivals. We prove that the SCF approximation converges to a finite limit for constant and cyclic arrival rates. Extensive numerical experiments demonstrate the high accuracy of the SCF approximation. Furthermore, SCF outperforms PSA, particularly during peak periods. Therefore, SCF provides a fast, transparent tool for planning and staffing in services with retrials. It remains reliable when the exact distribution of retrials is uncertain, enabling day-to-day scheduling and scenario testing under time-varying demand. We find that the retrial probability plays a stronger role than the retrial-time distribution. The impact of retrials in time-varying queues is driven by two effects: the state at the abandonment epoch and the cycle length-retrial time interaction. We posit that as retrial times increase, the first effect dampens congestion feedback, whereas the second effect can amplify or smooth fluctuations. Under sufficiently high variability in retrial times, the cycle length-retrial time interaction becomes negligible, leaving the system governed by the abandonment-epoch state.

History: Accepted by Shane Henderson, Area Editor for Simulation, Stochastic Models, & Stochastic Optimization.

Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grant 522848863].

Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2025.1626) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2025.1626). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/.