Quantum Estimation of Delay Tail Probabilities in Scheduling and Load Balancing

TL;DR

This paper introduces a quantum simulation framework for estimating delay tail probabilities in complex queueing systems, leveraging regenerative simulation and tail bounds to enable quantum estimation with manageable circuit complexity.

Contribution

It develops a novel quantum simulation approach for delay tail probabilities in infinite state space models using regenerative truncation and tail bounds.

Findings

Quantum algorithms reduce sample complexity for tail probability estimation.

Framework applies to models with infinite state spaces, avoiding mixing time issues.

Bounds on qubit and circuit complexity are provided for specific queueing models.

Abstract

Estimating delay tail probabilities in scheduling and load balancing systems is a critical but computationally prohibitive task due to the rarity of violation events. Quantum Amplitude Estimation (QAE) offers a generic quadratic reduction in sample complexity 1/sqrt(p) vs 1/p, but applying it to steady-state queueing networks in challenging: classical simulations involve unbounded state spaces and random regeneration cycles, whereas quantum circuits have fixed depth and finite registers. In this paper, we develop a framework for quantum simulation of delay tail probabilities based on truncated regenerative simulation. We show that regenerative rare-event estimators can be reformulated as deterministic, reversible functions of finite random seeds by truncating regeneration cycles. To control the resulting bias, we use Lyapunov drift and concentration arguments to derive exponential tail bounds on regeneration times. This allows the truncation horizon--and hence the quantum circuit depth--to be chosen such that the bias is provably negligible compared to the statistical error. The proposed framework enables quantum estimation in models with countably infinite state spaces, avoiding the challenge of determining the sufficient mixing time required for direct finite-horizon simulation. We provide bounds on qubit and circuit complexity for a GI-GI-1 queue, a wireless network under MaxWeight scheduling, and a multi-server system with Join-the-Shortest-Queue (JSQ) routing.