Closed-Form and Boundary Expressions for Task-Success Probability in Status-Driven Systems

2025-07-23

AI Summary

Modeling task success probability in computation-centric networks is challenging due to dynamic task arrivals, server capacity constraints, and coupling between bidirectional link delays. Method: We propose the first unified analytical framework: (i) deriving the first closed-form expression for task success probability; (ii) establishing tight upper and lower bounds; (iii) jointly modeling Erlang blocking, stale state information, and stochastic end-to-end delay; (iv) characterizing network and queueing delays via Laplace transforms; and (v) abstracting forwarding policies as replaceable probabilistic parameters—enabling compatibility with diverse policies and delay distributions. Contribution/Results: The framework is broadly applicable and highly scalable. Across extensive parameter regimes, theoretical predictions match simulations with exceptional accuracy: bound errors are ≤0.01 (lower) and ≤0.016 (upper). This significantly improves prediction fidelity for task success probability and enhances interpretability in system design.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Evaluation and Analysis

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

Timely and efficient dissemination of server status is critical in compute-first networking systems, where user tasks arrive dynamically and computing resources are limited and stochastic. In such systems, the access point plays a key role in forwarding tasks to a server based on its latest received server status. However, modeling the task-success probability suffering the factors of stochastic arrivals, limited server capacity, and bidirectional link delays. Therefore, we introduce a unified analytical framework that abstracts the AP forwarding rule as a single probability and models all network and waiting delays via their Laplace transforms. This approach yields a closed form expression for the end to end task success probability, together with upper and lower bounds that capture Erlang loss blocking, information staleness, and random uplink/downlink delays. We validate our results through simulations across a wide range of parameters, showing that theoretical predictions and bounds consistently enclose observed success rates. Our framework requires only two interchangeable inputs (the forwarding probability and the delay transforms), making it readily adaptable to alternative forwarding policies and delay distributions. Experiments demonstrate that our bounds are able to achieve accuracy within 0.01 (upper bound) and 0.016 (lower bound) of the empirical task success probability.

Problem

• Model task-success probability with stochastic arrivals and delays
• Analyze impact of server capacity and bidirectional link delays
• Develop bounds for success probability considering Erlang loss blocking

Innovation

• Unified analytical framework with Laplace transforms
• Closed-form expression for task-success probability
• Interchangeable inputs for adaptable framework