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Higher-Order Approximations of Sojourn Times in M/G/1 Queues via Stein's Method

Bihan Chatterjee, Siva Theja Maguluri, Debankur Mukherjee

TL;DR

This work characterizes the pre-asymptotic behavior of the stationary sojourn time in an M/G/1 queue under heavy traffic by developing a higher-order Stein's method framework. The authors derive explicit $O(\varepsilon^k)$ bounds in the Zolotarev distance between the scaled sojourn time and an exponential limit when the first $k+1$ moments of the service-time distribution match those of the exponential distribution, using a generator-expansion approach and tight derivative bounds of the Stein solution. They establish a hierarchy of accuracy controlled by moment matching, with corollaries translating to Wasserstein bounds and moment convergence. This provides a systematic refinement of the classical exponential heavy-traffic approximation and has practical implications for more accurate performance evaluation in service systems, especially for modest orders of $k$ where a small number of moment-matches yields substantial gains.

Abstract

We study the stationary sojourn time distribution in an M/G/1 queue operating under heavy traffic. It is known that the sojourn time converges to an exponential distribution in the limit. Our focus is on obtaining pre-asymptotic, higher-order approximations that go beyond the classical exponential limit. Using Stein's method, we develop an approach based on higher-order expansions of the generator of the underlying Markov process. The key technical step is to represent higher-order derivatives in terms of lower-order ones and control the resulting error via derivative bounds of the Stein equation. Under suitable moment-matching conditions on the service distribution, we show that the approximation error decays as a high-order power of the slack parameter $\varepsilon=1-ρ$. Error bounds are established in the Zolotarev metric, which further imply bounds on the Wasserstein distance as well as the moments. Our results demonstrate that the accuracy of the exponential approximation can be systematically improved by matching progressively more moments of the service distribution.

Higher-Order Approximations of Sojourn Times in M/G/1 Queues via Stein's Method

TL;DR

This work characterizes the pre-asymptotic behavior of the stationary sojourn time in an M/G/1 queue under heavy traffic by developing a higher-order Stein's method framework. The authors derive explicit bounds in the Zolotarev distance between the scaled sojourn time and an exponential limit when the first moments of the service-time distribution match those of the exponential distribution, using a generator-expansion approach and tight derivative bounds of the Stein solution. They establish a hierarchy of accuracy controlled by moment matching, with corollaries translating to Wasserstein bounds and moment convergence. This provides a systematic refinement of the classical exponential heavy-traffic approximation and has practical implications for more accurate performance evaluation in service systems, especially for modest orders of where a small number of moment-matches yields substantial gains.

Abstract

We study the stationary sojourn time distribution in an M/G/1 queue operating under heavy traffic. It is known that the sojourn time converges to an exponential distribution in the limit. Our focus is on obtaining pre-asymptotic, higher-order approximations that go beyond the classical exponential limit. Using Stein's method, we develop an approach based on higher-order expansions of the generator of the underlying Markov process. The key technical step is to represent higher-order derivatives in terms of lower-order ones and control the resulting error via derivative bounds of the Stein equation. Under suitable moment-matching conditions on the service distribution, we show that the approximation error decays as a high-order power of the slack parameter . Error bounds are established in the Zolotarev metric, which further imply bounds on the Wasserstein distance as well as the moments. Our results demonstrate that the accuracy of the exponential approximation can be systematically improved by matching progressively more moments of the service distribution.
Paper Structure (13 sections, 10 theorems, 172 equations)

This paper contains 13 sections, 10 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Model Description and Preliminaries
  3. Main Results
  4. Proofs
  5. Proof for $k=2$
  6. Proof for general $k$
  7. Proof of Preliminary Lemmas
  8. Alternative approach for $k=2$
  9. Conclusion and Future Directions
  10. Acknowledgment
  11. Proof of Lemma \ref{['lemma general k']}
  12. Proof of Lemma \ref{['lemma:alt']}
  13. Proof of Lemma \ref{['lemmadistancebound']}

Key Result

Lemma 1

For probability laws $\mu,\sigma$ we have Moreover, if the first $k-1$ moments match, then where $L_k$ is a constant depending on $k$.

Theorems & Definitions (29)

  • Lemma 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • proof : Proof of Corollary \ref{['cor2']}
  • Corollary 2
  • proof : Proof of Corollary \ref{['cor3']}
  • Lemma 2
  • ...and 19 more