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Knowledge vs. Experience: Asymptotic Limits of Impatience in Edge Tenants

Anthony Kiggundu, Bin Han, Hans D. Schotten

TL;DR

This work analyzes reneging and jockeying in a dual M/M/1 system under two information feeds: a closed-form Markovian estimator of residual waiting time and an online actor–critic policy. It establishes that with total-time patience and unequal service rates, the total waiting time grows linearly with backlog, causing abandonment to converge to 1 and the chance of a successful jockey to converge to 0, while showing robustness: if estimator error grows sublinearly, both feeds share the same asymptotic behavior. The authors validate the theory through simulations, revealing finite-backlog differences in delays, reneging, and transient jockeying that diminish as queues grow large, thereby clarifying when information value matters. The results offer practical guidance for lightweight telemetry and decision logic on edge/offload platforms, where the cost and freshness of information should be balanced against expected gains in finite regimes. Overall, the paper contributes a rigorous asymptotic robustness claim and a nuanced finite-regime comparison between analytic and learned information for congestion-management decisions in edge environments.

Abstract

We study how two information feeds, a closed-form Markov estimator of residual sojourn and an online trained actor-critic, affect reneging and jockeying in a dual M/M/1 system. Analytically, for unequal service rates and total-time patience, we show that total wait grows linearly so abandonment is inevitable and the probability of a successful jockey vanishes as the backlog approaches towards infinity. Furthermore, under a mild sub-linear error condition both information models yield the same asymptotic limits (robustness). We empirically validate these limits and quantify finite backlog differences. Our findings show that learned and analytic feeds produce different delays, reneging rates and transient jockeying behavior at practical sizes, but converge to the same asymptotic outcome implied by our theory. The results characterize when value-of-information matters (finite regimes) and when it does not (asymptotics), informing lightweight telemetry and decision-logic design for low-cost, jockeying-aware systems.

Knowledge vs. Experience: Asymptotic Limits of Impatience in Edge Tenants

TL;DR

This work analyzes reneging and jockeying in a dual M/M/1 system under two information feeds: a closed-form Markovian estimator of residual waiting time and an online actor–critic policy. It establishes that with total-time patience and unequal service rates, the total waiting time grows linearly with backlog, causing abandonment to converge to 1 and the chance of a successful jockey to converge to 0, while showing robustness: if estimator error grows sublinearly, both feeds share the same asymptotic behavior. The authors validate the theory through simulations, revealing finite-backlog differences in delays, reneging, and transient jockeying that diminish as queues grow large, thereby clarifying when information value matters. The results offer practical guidance for lightweight telemetry and decision logic on edge/offload platforms, where the cost and freshness of information should be balanced against expected gains in finite regimes. Overall, the paper contributes a rigorous asymptotic robustness claim and a nuanced finite-regime comparison between analytic and learned information for congestion-management decisions in edge environments.

Abstract

We study how two information feeds, a closed-form Markov estimator of residual sojourn and an online trained actor-critic, affect reneging and jockeying in a dual M/M/1 system. Analytically, for unequal service rates and total-time patience, we show that total wait grows linearly so abandonment is inevitable and the probability of a successful jockey vanishes as the backlog approaches towards infinity. Furthermore, under a mild sub-linear error condition both information models yield the same asymptotic limits (robustness). We empirically validate these limits and quantify finite backlog differences. Our findings show that learned and analytic feeds produce different delays, reneging rates and transient jockeying behavior at practical sizes, but converge to the same asymptotic outcome implied by our theory. The results characterize when value-of-information matters (finite regimes) and when it does not (asymptotics), informing lightweight telemetry and decision-logic design for low-cost, jockeying-aware systems.

Paper Structure

This paper contains 15 sections, 2 theorems, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contribution of this Paper
  4. System Modeling
  5. Setup and Assumptions
  6. Model of the Knowledge: Markovian Waiting time
  7. Reneging under Markovian knowledge
  8. Jockeying under Markovian knowledge
  9. Pure-death case: arrivals $\lambda_{tar}=0$ join ahead of jockey
  10. Birth-death case: $\lambda_{tar}>0$ join ahead of jockey
  11. Model of the Experience: Actor-Critic
  12. The Value Function and the Optimal Policy
  13. Simulation Results
  14. Conclusion and Outlook
  15. Acknowledgment

Key Result

Proposition 2.1

Let queues 1 and 2 be independent M/M/1 with service rates $\mu_{1},\mu_{2}$. Assume an arriving customer faces $n$ jobs in queue 1 and changing count of queued requests $m$ in queue 2, and draws a finite patience $T$ at entry which is consumed continuously (total time patience). If $\mu_{1}\neq \mu

Figures (3)

  • Figure 1: Under total-time patience, as queue lengths grow large the time already spent waiting in the original queue overwhelms any finite patience budget. Therefore almost every arrival abandons before being served, and the fraction of arrivals who actually switch and get served collapses to zero.
  • Figure 2: As the system grows in size, jockeying becomes negligible and reneging stabilizes at a high rate for both servers, regardless of whether decisions are guided by raw Markov state or NN-based information.
  • Figure 3: Loss metrics for the actor and critic networks decrease rapidly over episodes, indicating effective learning and convergence of the actor-critic algorithm.

Theorems & Definitions (5)

  • Proposition 2.1: Asymptotic behavior of jockeying and reneging under total time patience
  • Theorem 2.2: Robustness to information models — unequal rates
  • proof
  • proof
  • proof