  This paper aims at semi-parametrically estimating the input process to a
L\'evy-driven queue by sampling the workload process at Poisson times. We
construct a method-of-moments based estimator for the L\'evy process'
characteristic exponent. This method exploits the known distribution of the
workload sampled at an exponential time, thus taking into account the
dependence between subsequent samples. Verifiable conditions for consistency
and asymptotic normality are provided, along with explicit expressions for the
asymptotic variance. The method requires an intermediate estimation step of
estimating a constant (related to both the input distribution and the sampling
rate); this constant also features in the asymptotic analysis. For subordinator
L\'evy input, a partial MLE is constructed for the intermediate step and we
show that it satisfies the consistency and asymptotic normality conditions. For
general spectrally-positive L\'evy input a biased estimator is proposed that
only uses workload observations above some threshold; the bias can be made
arbitrarily small by appropriately choosing the threshold.
