Task Success Rate (TSR) Metrics

• Task Success Rate (TSR) is defined as the ratio of successfully completed tasks to total tasks, providing a clear measure of system effectiveness.
• It serves as a critical metric in domains such as software testing, crowdsourcing, and robotics by embedding statistical models and domain-specific evaluation protocols.
• TSR optimization leverages methodologies like reinforcement learning, probabilistic estimators, and error analysis to enhance performance and reliability.

Task Success Rate (TSR) quantifies the probability or proportion of tasks completed successfully within a specified context, capturing the effectiveness and reliability of a system, model, agent, or procedural workflow. TSR is a central metric used across fields such as software engineering, machine learning, robotics, quantum computing, networking, and human-computer interaction, with precise definitions and evaluation protocols varying according to domain-specific requirements and task semantics.

1. Formal Definitions and Computation

TSR is generally formalized as the ratio of the number of successfully completed tasks to the total number of attempted or assigned tasks over a given period or experiment:

$$\text{TSR} = \frac{\text{Number of Successfully Completed Tasks}}{\text{Total Number of Tasks}}$$

Specific instantiations depend on application context:

• Crowdsourced and Freelance Work: TSR is the proportion of tasks with fully correct submissions, often verified by passing all programmatically defined test cases (Karim et al., 2016, Noever et al., 16 May 2025).
• Software Testing and Automation: TSR may refer to the proportion of test scenarios or test cases that complete successfully, or, in reduction frameworks, the fraction of faults detected by the reduced test suite relative to the original (Trad et al., 2018, Ruland et al., 2022).
• Dialog or Navigation Systems: TSR commonly denotes per-task or per-dialog success, e.g., answering correctly in a game scenario or reaching the correct target in navigation (Testoni et al., 2021, Zhao et al., 2023).
• Stochastic and Physical Systems: TSR is often the frequency or probability that a triggered event elicits a correct or expected response, e.g., an excitable laser firing following stimulation (Tiana-Alsina et al., 2020).
• Robotics and Manipulation: TSR is the probability (possibly estimated under uncertainty) that an intended manipulation action succeeds, given sensory input and task constraints (Naik et al., 16 Mar 2024, Kambara et al., 26 Dec 2024, Tang et al., 13 Feb 2025).

Mathematical expressions of TSR may involve integrals, summations, or probabilistic estimators, particularly in the presence of uncertainty or non-deterministic elements (see Section 4).

2. Domain-Specific Methodologies and Applications

The methodologies for evaluating and maximizing TSR diverge substantially across research domains:

• Machine Learning and Crowdsourcing: Predictive models (e.g., Random Forests, SVMs) are trained to identify participants and conditions most likely to yield successful outcomes. Filtering and ranking mechanisms then use the predicted probabilities to maximize overall TSR and minimize wasted effort (Karim et al., 2016).
• Reinforcement Learning Curriculum Design: Approaches such as Success Induced Task Prioritization (SITP) automatically prioritize tasks that induce the greatest learning, adaptively reweighing task sampling probabilities based on recent success rate changes (see formula):

$p_i = \frac{\exp(S_i)}{\sum_j \exp(S_j)}$

where $S_i$ reflects recent progress on task $i$ (Nesterova et al., 2022).

• Software Test Suite Reduction and Regression Testing: TSR is optimized by selectively retaining or excluding tests according to their historical fault detection ability, structural/state coverage, and potential relevance to future code changes (Trad et al., 2018, Ruland et al., 2022).
• Quantum Computing: The task success rate (e.g., probability that a quantum circuit yields the correct result) can be predicted using sophisticated circuit- and error-aware estimators such as Quantum Vulnerability Analysis (QVA), which outperform naive multiplicative models by explicitly tracing error propagation at each cycle (2207.14446).
• Networked or Distributed Systems: Closed-form models for TSR incorporate stochastic task arrivals, resource blocking (Erlang loss model), and network delays, with predictions derived via Laplace transforms and renewal theory (Qi et al., 23 Jul 2025).
• Vision-Language and Robotic Systems: TSR is influenced by multimodal scene understanding, task-specific planning, and output supervision. Designed modules—such as 2D-3D prompt synthesis and supervisory feedback—ensure that spatial reasoning, logical consistency, and safe operation are achieved, leading to measured high TSRs (Tang et al., 13 Feb 2025, Naik et al., 16 Mar 2024, Kambara et al., 26 Dec 2024).

3. Predictive Models, Measurement Protocols, and Evaluation

TSR is both a performance metric and a target for system optimization:

• Measurement Protocols: Tasks are labeled as successful according to rigorous criteria, often requiring automated or manual validation (e.g., code correctness via test cases, successful completion detected by sensors, accurate end-to-end dialog fulfiLLMent, precise navigation stops within a specified radius) (Noever et al., 16 May 2025, Zhao et al., 2023).
• Estimation Under Uncertainty: In robotic and quantum domains, TSR is estimated as an expectation or integral over distribution(s) of possible states/errors:

$$P(\text{task} | o) = \int P(\text{task} | o, \Delta) \cdot P(\Delta | o) d\Delta$$

where $P(\Delta|o)$ is the estimated error distribution and $P(\text{task}|o, \Delta)$ is determined via simulation or analytic models (Naik et al., 16 Mar 2024).

• Task and Worker/Agent Recommendations: Predictive rankings and filtering (often with thresholding based on win probability, e.g., $p_{\text{winner}} > \frac{1}{3}$) optimize both human-in-the-loop and fully automated settings (Karim et al., 2016).

Metrics derived from TSR may include precision, recall, $F$-measure, area under the ROC curve, and more task-specific statistics such as mean percentile rank for dialog/navigational agents (Testoni et al., 2021).

4. Modeling, Uncertainty, and Theoretical Bounds

Advanced TSR modeling incorporates:

• Blocking and Queueing Effects: Analytical frameworks (e.g., in compute-first networking systems) use the Erlang–B loss formula for blocking probability:

$$B(\rho, C) = \frac{\rho^C / C!}{\sum_{k=0}^{C} \rho^k / k!}$$

and derive TSR as:

$$P_{\mathrm{succ}} = (1 - B(\rho, C)) \cdot \text{Delay Factors}$$

with delay factors captured by Laplace transforms of delay distributions (uplink, staleness, downlink) (Qi et al., 23 Jul 2025).

• Bounds: Upper and lower bounds on TSR are derived for different idealized or operational regimes (e.g., perfect information, negligible delay versus real-world staleness) and typically bracket observed success probabilities within 1–1.6% (Qi et al., 23 Jul 2025).
• Multimodal, Non-parametric Distributions: Robotic manipulation under pose uncertainty requires TSR to be computed by summing over discretized error grains and integrating simulation outcomes weighted by error likelihoods. This approach enables capture of complex error landscapes beyond simple Gaussians (Naik et al., 16 Mar 2024).

5. Practical Optimization and Impact Across Domains

Increasing TSR is a primary objective across a variety of settings:

• Crowdsourced and Freelance Work: Smart recommendation and risk prediction yield 3.5–4.6 person-days saved per task and significantly decrease wasted effort on non-succeeding tasks while maintaining high recall in identifying likely winners (Karim et al., 2016).
• Software Testing: Incorporation of substate profiling and composite test prioritization boosts defect detection without sacrificing suite reduction, thereby improving the TSR of regression workflows (Trad et al., 2018, Ruland et al., 2022).
• Quantum Compilation: Using structure- and propagation-aware CQV estimation, relative prediction error in program TSR is decreased up to 30-fold, guiding optimal compilation choices and error mitigation (2207.14446).
• Robotics and Planning: Fusing 3D spatial context into prompt synthesis and enforcing logical validation through SLMs raises the TSR for robotic task execution to 96.0%, with ablations showing necessity of both spatial grounding and supervision (Tang et al., 13 Feb 2025).
• Dialog and Navigation Agents: Approaches that focus only on TSR can lead to impoverished interaction quality; pairing TSR maximization with dialogue richness metrics (e.g., linguistic divergence) is necessary for robust and human-like systems (Testoni et al., 2021).

6. Limitations, Edge Cases, and Future Directions

• Metric-Task Alignment: Experiments indicate that model-level TSR does not always translate into real-world system-level success, especially in adversarial, temporal, or black-box transfer scenarios. For instance, in traffic sign recognition, new system-level metrics that model the effect of spatial memorization reveal much lower actual attack success than per-frame evaluations suggest (Wang et al., 15 Sep 2024).
• Interactive and Human-Centric Tasks: TSR as defined for automated or structured benchmarks may not capture the full complexity and iterative nature of human-robot or human-computer collaboration (Noever et al., 16 May 2025).
• Uncertainty Modeling: When outcome is conditioned on high-dimensional, uncertain, or multi-modal state estimates, effective TSR estimation requires simulation-based or probabilistic integration approaches, highlighting the need for scalable and expressive models (Naik et al., 16 Mar 2024, Kambara et al., 26 Dec 2024).
• Dynamically Shifting Environments: Forecasting TSR in settings subject to economic or social shocks calls for adaptive models leveraging real-time covariates (demand, supply) rather than static historical data, as shown in workforce reskilling program prediction (mean error 3.9%) (Hurwitz et al., 2021).

7. Representative Mathematical Expressions

Context/Domain	TSR Definition	Domain-Specific Formula or Notes
Software/Freelance	TSR = (Tasks Solved) / (Total Tasks) × 100%	Passes all test cases; mapped to task monetary value
RL Curriculum	TSR guides $\mathbf{p}_i = \exp(S_i) / \sum_j \exp(S_j)$	$S_i$: exponential moving average of SR changes
Quantum Computing	TSR = $1 - \text{CQV}$	CQV = cumulative quantum vulnerability
Stochastic Systems	$$P_{\text{succ}} = (1 - B(\rho, C)) \cdot G_Y(\Lambda) \cdot G_\gamma(\Lambda) \cdot G_\beta(\Lambda)$$	Laplace transforms, Erlang B, delays in CFN
Robotic Manipulation	$$P(\text{task}|o) = \sum_{n \in E} 1_E(n) \cdot P(n|o)$$	Non-parametric multimodal integration

LaTeX and algorithmic expressions are routine in TSR computation for expressing probability integrals, ranking thresholds, or statistical scoring rules (see cited papers for formal derivations).

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TSR is thus a unifying metric reflecting system or agent ability to achieve intended goals under operational constraints and uncertainty. Its definition, estimation, and maximization are foundational to empirical validation, design choices, and practical deployment of artificial, robotic, and human-in-the-loop systems across scientific and engineering disciplines.