Robust Queueing for Single-Server Queues with Abandonment

Abstract: Single-server queues with customer abandonment arise in call centers and many service systems, but steady-state performance measures remain analytically intractable beyond Markovian assumptions. This paper develops Robust Queueing (RQ) approximations for the mean steady-state virtual waiting time (offered waiting time) in the GI/GI/1+GI model. The approach starts from a reverse-time supremum representation of the virtual waiting time as the reflection of an effective net-input process that accounts for abandonments. We approximate effective net-input increments by their mean plus a robustness parameter times their standard deviation. For the drift, we introduce a Poisson-surrogate compensator and show that the associated correction term is asymptotically negligible in the long-patience regime. For variability, we propose two implementable surrogates: (i) a deterministic time-change approximation that yields a first RQ algorithm, and (ii) a refined algorithm based on a heavy-traffic limit that produces a scale-dependent variance function capturing the variance-reduction effect of abandonment. The resulting steady-state approximation reduces to a one-dimensional fixed point solvable by bisection and takes as input the arrival index of dispersion for counts (IDC), the service-time squared coefficient of variation, and the patience-time distribution. We further show how to extend the method to queues in series by feeding an approximation of the upstream departure IDC into the downstream RQ algorithm. Extensive numerical experiments demonstrate that the refined RQ approximation is accurate across underload, critical loading, and overload, and remains robust relative to existing heavy-traffic and hazard-rate-scaling benchmarks.

• The paper develops two robust queueing fixed-point algorithms that estimate mean virtual waiting time using arrival variability, service variability, patience distributions, and a calibrated robustness parameter.
• The heavy-traffic analysis identifies underloaded, critically loaded, and overloaded regimes, showing that one calibration rule remains consistent across all three and approaches the classical value of √2 in deep underload.
• The refined method models abandonment-induced variance reduction and performs accurately across broad simulations, though it can show negative bias when patience is very short and the system nears a loss model.

Overview and motivation

This paper develops Robust Queueing (RQ) approximations for the mean steady-state virtual waiting time $E[Z(\infty)]$ in the $GI/GI/1{+}GI$ queue, where arrivals are renewal, service times are i.i.d. general, and patience times are i.i.d. general, so that customers abandon if service has not begun before their patience expires (2603.00982). The virtual waiting time (offered waiting time) is the natural performance metric in this setting: it underlies delay announcements and can be used to approximate the abandonment probability and effective queue length. Exact steady-state analysis of this model is available only in Markovian special cases; abandonment creates a nonlinear feedback loop between waiting times and which customers remain in queue, so general primitives require approximation.

The paper's methodological starting point is a reverse-time supremum representation of the virtual waiting time as the reflection of an effective net-input process that counts only work from customers who will eventually be served. RQ then replaces the stochastic increment inside the supremum by its mean plus a robustness parameter $b$ times its standard deviation, converting steady-state estimation into a one-dimensional fixed-point equation solvable by bisection. The inputs required are low-dimensional traffic descriptors: the arrival index of dispersion for counts (IDC), the service-time squared coefficient of variation (SCV), and the patience-time distribution.

Reverse-time representation and drift approximation

The paper first reviews RQ for the $G/GI/1$ model, where workload equals the running supremum of reverse-time net-input increments, and stresses that because the RQ surrogate depends on scale-dependent variability encoded by the index of dispersion for work (IDW), it is essential to apply the reverse-time representation before approximating increments—a distinction that does not arise in Brownian queues, where increments have stationary independent increments.

For the abandonment model, the key object is the effective arrival process counting only customers whose patience exceeds their offered waiting time. Under Poisson arrivals, predictable thinning gives an exact compensator $\lambda\int_0^t \bar F_\alpha(Z(u-))du$, yielding exact drift formulas. For renewal arrivals, the paper decomposes the drift into this Poisson surrogate plus a correction term $\delta_t(s)$ and proves that the expected correction vanishes in the iterated limit as the abandonment rate $\alpha\downarrow 0$ and then $t\to\infty$ (long-patience regime). This is the main assumption underlying the drift approximation: it is exact for Poisson arrivals and asymptotically justified for renewal arrivals only when patience is long relative to the load gap. In overload, the paper additionally establishes via Lemma 3.3-type results that $\alpha Z(t)\to F^{-1}((\rho-1)/\rho)$, the fluid equilibrium threshold.

First RQ algorithm and heavy-traffic calibration

The first algorithm treats the thinned input as a stationary renewal reward process evaluated at a deterministically rescaled time, giving a variance surrogate proportional to $\Lambda_t(s) I_w(\Lambda_t(s))/\mu^2$. The steady-state approximation reduces to a fixed point $GI/GI/1{+}GI$0 with $GI/GI/1{+}GI$1 nonincreasing, guaranteeing uniqueness and efficient bisection computation.

A central contribution is the heavy-traffic analysis. Under patience scaling $GI/GI/1{+}GI$2 and load scaling $GI/GI/1{+}GI$3, with the local behavior of $GI/GI/1{+}GI$4 at zero characterized by an integer $GI/GI/1{+}GI$5 (the order of the first nonzero derivative at the origin), the paper identifies a threshold $GI/GI/1{+}GI$6 and proves that both the RQ solution and the exact $GI/GI/1{+}GI$7 mean exhibit three distinct scalings:

Regime	Condition	Scaling of $GI/GI/1{+}GI$8
Underloaded	$GI/GI/1{+}GI$9, $b$0	$b$1 (abandonment negligible)
Critically loaded	$b$2	$b$3
Overloaded	$b$4, $b$5	$b$6

The critically-loaded limit constant depends on the patience distribution only through $b$7, recovering the reflected Ornstein–Uhlenbeck (ROU) limit when $b$8 and explaining why hazard-rate-type refinements are needed when $b$9. Matching the RQ heavy-traffic constant to the exact constant yields a calibration $G/GI/1$0 for the robustness parameter that is consistent across all three regimes; notably, the overloaded leading-order limit is independent of $G/GI/1$1 entirely, and the calibrated $G/GI/1$2 recovers the classical $G/GI/1$3 in the deep-underload limit. This means a single calibration rule covers underload, critical loading, and overload—an unusual property among approximations for queues with abandonment.

The same fixed point also yields approximations for secondary measures: abandonment probability $G/GI/1$4 via a mean-field substitution, mean waiting time of served customers via the Baccelli-style extension of Pollaczek–Khintchine, and effective queue length via Little's law.

Refined algorithm: variance reduction from abandonment

The first algorithm neglects the correlation between the thinning decision and the arrival process. The refined algorithm corrects this using a new heavy-traffic limit. The paper establishes joint convergence of the diffusion-scaled virtual waiting time, idle time, effective work input, and effective arrival count to a reflected diffusion with polynomial drift:

$G/GI/1$5

For $G/GI/1$6 this reduces to the classical ROU limit; for $G/GI/1$7 the polynomial-drift diffusion has an explicit stationary density of exponential-quadratic form. This regime differs from hazard-rate scaling in that the limit retains only the local derivative $G/GI/1$8 rather than the full patience distribution.

From this limit the paper defines a variance-reduction function $G/GI/1$9—the ratio of the limiting effective-input variance to the $\lambda\int_0^t \bar F_\alpha(Z(u-))du$0 Brownian benchmark—and proves it satisfies $\lambda\int_0^t \bar F_\alpha(Z(u-))du$1, is strictly decreasing in $\lambda\int_0^t \bar F_\alpha(Z(u-))du$2, tends to $\lambda\int_0^t \bar F_\alpha(Z(u-))du$3 as $\lambda\int_0^t \bar F_\alpha(Z(u-))du$4, and decreases strictly in the scaled load $\lambda\int_0^t \bar F_\alpha(Z(u-))du$5 with limits $\lambda\int_0^t \bar F_\alpha(Z(u-))du$6 and $\lambda\int_0^t \bar F_\alpha(Z(u-))du$7 as $\lambda\int_0^t \bar F_\alpha(Z(u-))du$8 and $\lambda\int_0^t \bar F_\alpha(Z(u-))du$9. These properties formalize the intuition that state-dependent abandonment feedback suppresses cumulative variability over longer horizons, more strongly under heavier load. Computationally, $\delta_t(s)$0 is obtained via a Clark–Ocone/Malliavin representation reduced to two one-dimensional parabolic PDEs solved offline per pair $\delta_t(s)$1; for the long-run constant $\delta_t(s)$2, an explicit Poisson-equation formula requiring only one-dimensional integration is provided.

Combining the heavy-traffic variance function with a long-patience fixed-horizon limit yields the effective IDW approximation

$\delta_t(s)$3

a factorization separating intrinsic input variability from abandonment-induced feedback. The inner term is a convex combination of the original IDC and the Poisson IDC $\delta_t(s)$4, with weights set by the traffic intensity—an explicit expression of abandonment's regulating effect. Simulation comparisons for $\delta_t(s)$5 and $\delta_t(s)$6 models show the approximation tracks simulated effective IDW closely even though it is derived under the long-patience limit. The refined steady-state fixed point again has a unique solution, and its heavy-traffic limits coincide with those of the first algorithm except in the critically-loaded case, where the limit now depends nontrivially on $\delta_t(s)$7; calibration of $\delta_t(s)$8 there must be done numerically against the exact $\delta_t(s)$9 or $\alpha\downarrow 0$0 constants.

Numerical performance

The experimental design spans $\alpha\downarrow 0$1 parameter combinations ($\alpha\downarrow 0$2 arrival rates crossing $\alpha\downarrow 0$3 on both sides, $\alpha\downarrow 0$4 mean patience times from $\alpha\downarrow 0$5 to $\alpha\downarrow 0$6), benchmarked against the Ward–Glynn ROU approximation, the hazard-rate-scaling approximation, and the Huang–Gurvich universal bounds-based approximation. Key findings:

• For $\alpha\downarrow 0$7 baselines, the refined RQ approximation is accurate across all regimes and stable across patience distributions, despite being calibrated only on $\alpha\downarrow 0$8 and $\alpha\downarrow 0$9 in the infinite-patience heavy-traffic limit. Errors fall to single-digit percentages once mean patience reaches roughly $t\to\infty$0–$t\to\infty$1 service times.
• Benchmarks degrade sharply outside their intended regimes: Ward–Glynn substantially overestimates in heavy overload with long patience; hazard-rate scaling can dramatically overestimate in extreme underload and overload for Erlang patience; Huang–Gurvich is accurate in overload but significantly overestimates under short patience near or below critical load.
• The hardest regime for all methods is short patience comparable to the mean service time, where the system approaches a loss model; here refined RQ exhibits a consistent negative bias.
• For non-Poisson renewal inputs (e.g., $t\to\infty$2, $t\to\infty$3), refined RQ remains stable while two-moment-based benchmarks deteriorate, illustrating the value of the full IDC function over rate-and-SCV descriptors away from heavy traffic.

Extension to tandem queues

Because the RQ formulations consume the arrival process only through its IDC function, the framework extends to non-renewal input. For a tandem system with an upstream stable $t\to\infty$4 queue feeding a downstream abandonment queue, the departure IDC is approximated by the convex-combination propagation scheme of Whitt's network analyzer line of work, with an explicit closed-form weight derived from RBM correlation structure. Numerical results for two tandem configurations show absolute relative errors below $t\to\infty$5 in nearly all instances with mean patience at least $t\to\infty$6, demonstrating that the approach handles endogenous non-renewal arrivals without modification to the core algorithm.

Limitations and open questions

Several limitations are conceded explicitly. The drift approximation relies on neglecting the martingale correction term, which is justified asymptotically only in the long-patience regime; accuracy for very short patience (the near-loss regime) degrades, with a consistent negative bias. The variance-reduction function requires solving parabolic PDEs offline, and no closed form exists in general. Calibration of $t\to\infty$7 in the critically-loaded regime is numerical and derived only from $t\to\infty$8 baseline models, so its optimality for other primitives is empirical rather than proven. The interchange-of-limits justification for the variance-function corollary assumes uniform integrability of scaled second moments, which is imposed rather than verified. The tandem extension inherits the accuracy of existing departure-IDC approximations without independent error guarantees. Open questions raised by the paper include extending the framework to larger networks with abandonment, developing data-driven procedures for estimating IDC/IDW inputs and calibrating $t\to\infty$9 adaptively, and providing theoretical guarantees for the non-renewal and network settings.

Conclusion

This paper extends stochastic robust queueing to single-server queues with abandonment by combining a reverse-time reflection representation of the offered waiting time with a Poisson-surrogate drift and two progressively refined variance surrogates, the second grounded in a new polynomial-drift heavy-traffic limit that quantifies abandonment-induced variance reduction. The resulting approximations reduce to one-dimensional fixed points solvable by bisection, carry a single calibration consistent across underloaded, critically loaded, and overloaded regimes, and numerically outperform or match specialized heavy-traffic benchmarks across a wide parameter grid, including non-renewal inputs in tandem configurations.