A screening approach to nonparametric inference from the M/G/1 workload

We address a long-standing open problem posed by Hansen and Pitts (2006) on nonparametric inference for the service-time distribution in an M/G/1 workload model. We consider an M/G/1 queue with unknown arrival rate $\lambda >0$ and service-time distribution $B(\cdot )$, without assuming stability or stationarity. A statistician observes the workload process at discrete times $t=0,1,\dots ,n$ and aims to estimate $B(w)$ at a fixed point $w>0$. We propose an estimator $B_n(w)$ based solely on the observed workload trajectory. The construction relies on a screening mechanism that extracts conditionally i.i.d. compound Poisson increments from the workload process, thereby reducing the dependent-data problem to a Laplace-transform inversion framework. Under mild regularity assumptions on $B(\cdot )$, i.e., continuous differentiability on $[0,\mathrm{\infty })$, twice differentiability at $w$, and a finite second moment, we establish the bound $$\mathbb{E}\mid B_n(w)-B(w)\mid =\mathcal{O}\text{\hspace{0.17em}⁣}\left(\frac{\log n}{\sqrt{n}}\right),n\to \mathrm{\infty }\mathrm{.}$$This provides the first solution to the Hansen-Pitts problem achieving a parametric $L^1$-risk rate (up to a logarithmic factor), without requiring stationarity, stability, or knowledge of the arrival rate.