Estimating the probability that a queueing system violates a service-level agreement-e.g., that a customer's waiting time exceeds a threshold-is a rare-event estimation problem that, for most queueing models beyond the memoryless M/M/1 case, has no closed-form solution and must be estimated by classical Monte Carlo simulation. Quantum Amplitude Estimation (QAE) offers a proven quadratic reduction in the number of oracle calls needed to estimate such probabilities relative to classical sampling. We introduce Q-WAIT, an algorithm that discretizes a queue's stationary waiting-time or queue-length distribution, loads it into a quantum register via amplitude encoding, and applies Iterative Quantum Amplitude Estimation (IQAE) with an integer-comparator oracle to estimate SLA-violation probabilities. Unlike prior work applying QAE to Monte Carlo replacement in finance, we (i) apply the method to queueing performance metrics, including a non-Markovian G/G/1 queue with heavy-tailed service times built from a discrete-event simulation rather than a closed form, and (ii) benchmark against a classical baseline held to the same distribution-free statistical rigor as IQAE's own confidence interval, rather than an optimistic baseline that assumes the estimated probability in advance. On a simulator, we empirically recover the predicted query-complexity exponents (quantum:-1.43, classical:-2.00, versus theoretical-1 and-2), with a growing empirical speedup from 7× to 30× as target precision tightens, and show that the required oracle circuit's depth, two-qubit gate count, and ancilla overhead all scale linearly with the number of discretization qubits (R 2 = 1.000). We report these results, together with their methodological caveats and NISQera feasibility limits, as a basis for further work on quantum-accelerated queueing performance analysis.