Unbiasedness of the first-derivative IPA estimator for all customers

Prove that, in a first-come, first-served single-server (G/G/1) queue where the first customer arrives at time 0 with no initial delay (W1 ≡ 0), only the first service time S1 depends on the parameter θ and S1 is continuous with respect to θ, and all interarrival and service times are mutually independent and drawn from continuous distributions, the infinitesimal perturbation analysis estimator for the first derivative is unbiased for every customer index i ≥ 1; that is, establish that E[W_i'] = dE[W_i]/dθ for all i ≥ 1, where W_i denotes the waiting time of the i-th customer and the derivative is with respect to θ.

Background

The paper studies unbiased estimation of higher-order derivatives of expected waiting times in a first-come, first-served single-server queue using sample-path-based techniques and Leibniz integral calculus. A central theme is the unbiasedness of derivative estimators derived via infinitesimal perturbation analysis (IPA) under assumptions on the system dynamics and input distributions.

In the appendix devoted to the special case where only the first service time depends on the parameter, the authors explicitly state a conjecture asserting the unbiasedness of the first-derivative IPA estimator for all customers under standard continuity and independence assumptions on interarrival and service times.

References

Conjecture Under Assumptions \ref{ass1}-\ref{ass3}, $$ E[W_i'] = \frac{dE[W_i]}{d\theta} ~\forall i \geq 1. $$

— Augmenting Automatic Differentiation for a Single-Server Queue via the Leibniz Integral Rule  (2604.02900 - Fu, 3 Apr 2026) in Appendix: θ only affects S1 ("Appendix: θ only affects S_1, i.e., S_i^{(n)} = 0 ∀ i>1, n>1"), final lines