Quantum-Amplified M/G/1/K Simulation: A Comparator-Controlled Framework for Arbitrary Service Distributions

TL;DR

This paper introduces the first quantum circuit for simulating finite-capacity M/G/1/K queues with arbitrary service distributions, achieving high fidelity and significant speed-up over classical methods.

Contribution

It presents a novel quantum simulation framework for non-Markovian queues with buffer constraints, enabling efficient performance analysis of complex service systems.

Findings

Fidelity above 0.99 with 4 qubits

Waiting-time errors within 3% in high load regimes

Order of magnitude reduction in estimation errors near capacity

Abstract

Finite-capacity single-server queues with general service-time distributions form the backbone of numerous real-world systems, yet classical simulation of performance metrics such as blocking probabilities and delay becomes computationally prohibitive as service variability or required precision increases. This work presents the first coherent quantum circuit for simulating an M/G/1/K queue under arbitrary service-time laws. The circuit encodes the service distribution through a logarithmic-depth ladder of $R_y$ rotations and enforces buffer constraints via a comparator-controlled phase gate, while preserving the quadratic speed-up of amplitude amplification. Grover iterations center on estimating the expected number of customers in the system, yielding provable $O(\sqrt{N})$ variance reduction and closed-form confidence bounds, where $N$ denotes the number of shots. Empirical evaluations on IBM quantum simulators across four service distributions and three traffic intensities demonstrate fidelity above 0.99 with four qubits and above 0.76 with ten qubits, with Jensen-Shannon divergence below 0.11. Waiting-time estimation errors decrease by an order of magnitude as system load approaches capacity and remain within 3% in high-traffic regimes using registers of up to 63 qubits. These results establish the first end-to-end quantum simulation framework for finite-buffer, non-Markovian queueing systems and provide a concrete foundation for quantum-accelerated performance analysis in service-oriented architectures.