  A single server commences its service at time zero every day. A random number
of customers decide when to arrive to the system so as to minimize the waiting
time and tardiness costs. The costs are proportional to the waiting time and
the tardiness with rates $\alpha$ and $\beta$, respectively. Each customer's
optimal arrival time depends on the others' decisions, thus the resulting
strategy is a Nash equilibrium. This work considers the estimation of the ratio
$\displaystyle\theta\equiv \beta/(\alpha+\beta)$ from queue length data
observed daily at discrete time points, given that customers use a Nash
equilibrium arrival strategy. A method of moments estimator is constructed from
the equilibrium conditions. Remarkably, the method does not require estimation
of the Nash equilibrium arrival strategy itself, or even an accurate estimate
of its support. The estimator is strongly consistent and the estimation error
is asymptotically normal. Moreover, the asymptotic variance of the estimation
error as a function of the queue length covariance matrix (at sampling times)
is derived. The estimator performance is demonstrated through simulations, and
is shown to be robust to the number of sampling instants each day.
