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Quantum Estimation of Delay Tail Probabilities in Scheduling and Load Balancing

R. Srikant

TL;DR

The paper develops a quantum-accelerated framework for estimating rare delay-tail probabilities in queueing networks by recasting regenerative simulation as a finite-depth, deterministic function of random seeds. It deploys a truncation horizon and Lyapunov-drift based exponential bounds to control truncation bias, enabling reliable Quantum Amplitude Estimation (QAE) with fixed-depth circuits. The framework yields explicit qubit and circuit-complexity bounds for canonical models—GI/GI/1 queues, wireless MaxWeight networks, and JSQ-based multi-server systems—demonstrating a quadratic speedup in sampling rare events while maintaining rigorous error control. The approach extends to continuous-state systems via clipping and Nummelin splitting, preserving finite quantum resources and providing a path toward practical quantum-assisted reliability certification in networking systems.

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.

Quantum Estimation of Delay Tail Probabilities in Scheduling and Load Balancing

TL;DR

The paper develops a quantum-accelerated framework for estimating rare delay-tail probabilities in queueing networks by recasting regenerative simulation as a finite-depth, deterministic function of random seeds. It deploys a truncation horizon and Lyapunov-drift based exponential bounds to control truncation bias, enabling reliable Quantum Amplitude Estimation (QAE) with fixed-depth circuits. The framework yields explicit qubit and circuit-complexity bounds for canonical models—GI/GI/1 queues, wireless MaxWeight networks, and JSQ-based multi-server systems—demonstrating a quadratic speedup in sampling rare events while maintaining rigorous error control. The approach extends to continuous-state systems via clipping and Nummelin splitting, preserving finite quantum resources and providing a path toward practical quantum-assisted reliability certification in networking systems.

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.
Paper Structure (35 sections, 10 theorems, 114 equations, 3 algorithms)

This paper contains 35 sections, 10 theorems, 114 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Challenge: Steady-State vs. Finite-Depth
  3. Related Work
  4. Organization of the Paper
  5. Waiting Time Tail Estimation in $GI/GI/1$ Queues
  6. Truncation and Quantum Implementation
  7. Qubit and circuit complexity
  8. Registers and qubit count.
  9. Gate and depth complexity per cycle.
  10. Overall quantum complexity.
  11. Quantifying the Error Due to Truncation
  12. Certifying a waiting time tail bound: Main result for the $GI/GI/1$ queue
  13. Latency-Tail Estimation in Wireless Networks
  14. A Tail Bound for the Regeneration time
  15. Quantum Implementation and Complexity.
  16. ...and 20 more sections

Key Result

Theorem 1

Fix a truncation horizon $M$. Let $\hat{\mu}_Q$ be an estimate of $E[Y]=E[R_M]/M$ returned by QAE such that, with probability at least $1-\delta_Q$, Then, with probability at least $1-\delta_Q$, the corresponding estimate $\widehat{E[R]} := M\,\hat{\mu}_Q$ satisfies where $\beta$ is given by eq:beta-def.

Theorems & Definitions (24)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Remark 3
  • Theorem 4: Exponential tail of regeneration time
  • proof
  • ...and 14 more