Reliability Decay Curve (RDC) Overview

Updated 5 July 2026

Reliability Decay Curve (RDC) is a family of models representing reliability as a function of horizon length through first-passage probabilities, duration buckets, or error exponents.
It integrates methods from absorbing Markov chain analysis for LLM agents, duration-based success evaluation, and survival analysis in degradation studies to capture performance decline.
The framework emphasizes rigorous diagnostic tests like AIC and KS, ensuring robust model fit and clear distinction from similarly abbreviated terms in other fields.

Reliability Decay Curve (RDC) denotes a horizon-indexed representation of reliability whose exact mathematical meaning depends on the research context. In recent arXiv literature, the term has been formalized most explicitly in LLM-agent evaluation, where it appears either as a first-passage success distribution over execution steps or as repeated-attempt success aggregated by task-duration bucket. Related reliability, degradation, and communication-theory literatures use closely analogous objects: survival functions induced by threshold crossing, cumulative failure curves, operational-reliability functions under degrading capacity, or asymptotic error-exponent curves. Taken together, these sources suggest that RDC is best understood as a family of reliability-as-a-function-of-horizon constructions rather than a single cross-domain invariant metric (Tran-Truong et al., 27 Apr 2026, Khanal et al., 31 Mar 2026, Li et al., 2024).

1. Scope and terminological range

The published meanings of RDC in the supplied literature are not identical. In "Measuring the Unmeasurable: Markov Chain Reliability for LLM Agents" (Tran-Truong et al., 27 Apr 2026), RDC is exactly the $d$ -step reliability of an absorbing discrete-time Markov chain, namely a cumulative first-passage success probability over execution horizon. In "Beyond pass@1: A Reliability Science Framework for Long-Horizon LLM Agents" (Khanal et al., 31 Mar 2026), RDC is defined as a duration-bucketed map $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ , where $d$ indexes short, medium, long, and very long tasks. In "Reliability Function of Classical-Quantum Channels" (Li et al., 2024), the reliability function $E(\mathcal N,R)$ is presented as the paper’s version of an RDC: a curve describing the exponential decay rate of optimal decoding error as blocklength increases at fixed communication rate.

Context	RDC object	Index variable
Absorbing-DTMC agent traces	$\Pr[\tau_\oplus \le d \mid X_0=s_0]$	execution horizon $d$
Long-horizon agent benchmark	$\mathrm{pass}^k(\mathcal M,d)$	duration bucket $d$
Classical-quantum coding	$E(\mathcal N,R)$	coding rate $R$

A further terminological caution is necessary. In structural biology and NMR, RDC commonly means residual dipolar coupling, not Reliability Decay Curve; "Integrating NOE and RDC using sum-of-squares relaxation for protein structure determination" (Khoo et al., 2016) uses the acronym in that older sense. Any encyclopedia treatment therefore has to distinguish the reliability-science usage from the NMR usage explicitly.

2. First-passage RDC in absorbing Markov models of LLM agents

The most mathematically explicit recent definition appears in "Measuring the Unmeasurable: Markov Chain Reliability for LLM Agents" (Tran-Truong et al., 27 Apr 2026). The paper fits agent traces to an absorbing DTMC

$d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 0

with transient state set $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 1, transient-to-transient matrix $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 2, transient-to-success absorption probabilities $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 3, transient-to-failure absorption probabilities $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 4, and initial state $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 5 or initial distribution $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 6. In that setup, reliability is defined by the first hitting time of the success absorber: $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 7 The paper states explicitly that “the RDC is exactly the $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 8-step reliability $d \mapsto \mathrm{pass}^k(\mathcal M,d)$ 9.” It also states that RDC is a cumulative first-passage success probability, not a survival curve, not a failure-hazard curve, and not a complementary CDF.

The closed-form expression follows classical absorbing-chain reliability theory: $d$ 0 with asymptotic reliability

$d$ 1

The first-passage structure is explicit in

$d$ 2

and the exact first-passage mass is

$d$ 3

The paper further derives

$d$ 4

so monotone increase is automatic, and

$d$ 5

which supports shape analysis. Under the paper’s sufficient stochastic-monotonicity conditions, global concavity follows; conversely, it states that a convex window suggests a barrier, such as waiting for a tool response before progress becomes possible.

A central claim of the same paper is metric unification. Under i.i.d. repeated trials,

$d$ 6

The paper emphasizes that these are “restrictions or transformations of the same first-passage distribution.” On that reading, RDC is not another repeated-trial denominator convention; it is the horizon-wise slice of the single-run success-time law.

The fitted chain is produced by a deterministic pipeline. A trace corpus

$d$ 7

is featurized by $d$ 8, clustered by agglomerative Ward linkage, and assigned a transient-state taxonomy. Transition probabilities are estimated by Laplace-smoothed MLE with $d$ 9: $E(\mathcal N,R)$ 0 with analogous normalization for $E(\mathcal N,R)$ 1 and $E(\mathcal N,R)$ 2, and mass conservation

$E(\mathcal N,R)$ 3

The paper is notable for making validity conditions explicit. It requires transience, $E(\mathcal N,R)$ 4, invertibility of $E(\mathcal N,R)$ 5, and i.i.d. replications only when interpreting $E(\mathcal N,R)$ 6 and $E(\mathcal N,R)$ 7. It also tests the first-order Markov and time-homogeneity assumptions instead of taking them for granted. The composite certificate combines an AIC order-selection test,

$E(\mathcal N,R)$ 8

with a two-sample KS comparison of the analytic conditional success first-passage CDF to the empirical success first-passage distribution. Acceptance requires both $E(\mathcal N,R)$ 9 and $\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 0.

Empirically, the paper reports strong held-out agreement on seven controlled MAST-style frameworks under a strict 50/50 fit/test split. Held-out empirical RDCs overlay the analytic curves with maximum

$\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 1

and median $\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 2; the two-sample KS test on the first-passage CDF accepts the fitted chain on $\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 3 frameworks, with minimum $\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 4; and per-entry $\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 5 posterior and bootstrap intervals agree to approximately $\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 6 at the median. In this formulation, RDC becomes an audited reliability object with explicit assumptions, fit diagnostics, and uncertainty rather than an isolated benchmark plot.

3. Duration-bucket RDC in long-horizon agent benchmarking

A different operationalization appears in "Beyond pass@1: A Reliability Science Framework for Long-Horizon LLM Agents" (Khanal et al., 31 Mar 2026). There, RDC is introduced to measure how success degrades as tasks become longer-horizon. The paper first defines

$\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 7

and

$\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 8

It then defines

$\Pr[\tau_\oplus \le d \mid X_0=s_0]$ 9

where $d$ 0 is a duration bucket based on estimated human completion time. In this framework, RDC is not a step-indexed first-passage CDF; it is a duration-conditioned repeated-attempt success curve averaged over tasks in each bucket.

The benchmark design is itself part of the definition. The paper evaluates 396 tasks, with 33 tasks per domain-duration cell, across 10 models, 2 scaffolds, and $d$ 1 repeated episodes, for 23,392 completed episodes out of 23,760 planned. The full-study results show a net aggregate decline in pass@1 from short to very long tasks: $d$ 2 The paper describes this as a 24.3 percentage-point drop from short to very long. It also reports substantial rank inversions: for example, GLM-4.5 Air is best at short horizon with $d$ 3 but falls to $d$ 4 at very long, while DeepSeek V3, Kimi K2.5, and MiniMax M2.5 remain near or above $d$ 5 even at very long duration.

A major result is domain stratification. Using the Graceful Degradation Score (GDS), the paper reports aggregate domain trajectories

$d$ 6

$d$ 7

$d$ 8

Software engineering therefore shows the steepest reliability decay, document processing is nearly flat, and web research is intermediate and non-monotone. The paper explicitly warns that raw aggregate RDCs can be non-monotonic because duration buckets mix domains and because human-time duration is an imperfect proxy for agent horizon complexity.

This framework complements RDC with three additional metrics: Variance Amplification Factor (VAF), Graceful Degradation Score (GDS), and Meltdown Onset Point (MOP). RDC captures the mean success trend across horizons; VAF captures how duration amplifies across-task variance; GDS measures partial progress when binary success fails; and MOP tracks trajectory-local behavioral collapse using sliding-window tool-call entropy. The paper further reports that memory scaffolds universally hurt long-horizon performance across all 10 models.

A methodological caveat is internal to the paper itself. RDC is formally defined over $d$ 9, but the main empirical tables and figure mostly report pass@1 versus duration with $\mathrm{pass}^k(\mathcal M,d)$ 0 repeated episodes. That definitional drift is acknowledged in the paper’s own discussion and is important when comparing this benchmark-style RDC to the first-passage RDC of absorbing-chain models.

Several adjacent literatures do not always use the exact label RDC, but they define objects that are effectively reliability-decay curves in the ordinary survival-theoretic sense. "Trajectory-Aware Reliability Modeling of Democratic Systems" (Zaytsev et al., 22 Apr 2026) treats democratic stability as a reliability problem over a multicomponent state vector

$\mathrm{pass}^k(\mathcal M,d)$ 1

with stable region $\mathrm{pass}^k(\mathcal M,d)$ 2, reliability $\mathrm{pass}^k(\mathcal M,d)$ 3, and failure indicator $\mathrm{pass}^k(\mathcal M,d)$ 4. The paper defines failure through threshold crossing within a fixed horizon $\mathrm{pass}^k(\mathcal M,d)$ 5. It does not explicitly plot a full RDC, but it states that failure risk is the probability that predicted trajectories cross degradation thresholds within the horizon. This suggests a conditional horizon-indexed survival curve

$\mathrm{pass}^k(\mathcal M,d)$ 6

or equivalently a cumulative threshold-crossing curve $\mathrm{pass}^k(\mathcal M,d)$ 7, as the closest reliability-decay interpretation.

Two degradation-modeling papers supply similarly direct pathways. "Reliability modeling and statistical analysis of accelerated degradation process with memory effects and unit-to-unit variability" (Chen et al., 2023) defines failure time by threshold crossing,

$\mathrm{pass}^k(\mathcal M,d)$ 8

and reliability by

$\mathrm{pass}^k(\mathcal M,d)$ 9

Its degradation model

$d$ 0

incorporates fractional Brownian motion memory effects and unit-to-unit variability, and the paper estimates $d$ 1 under normal operating conditions by Monte Carlo. "Modeling Multivariate Degradation Data with Dynamic Covariates Under a Bayesian Framework" (Lin et al., 7 Apr 2025) uses multiple degradation characteristics, dynamic covariates, correlated random effects, and posterior predictive simulation to estimate the induced failure-time distribution $d$ 2; in that setting, the natural RDC is the complement

$d$ 3

A third line of work makes the curve explicit but changes the physical meaning of reliability. "Reliability of coherent systems whose operating life is defined by the lifetime and power of the components" (Bayramoglu, 23 Jan 2025) studies $d$ 4-out-of- $d$ 5 systems in which both component survival and time-degrading power matter. With $d$ 6, where $d$ 7 is a time-decreasing stable function, the paper’s principal object is the operational reliability function

$d$ 8

That curve decays jointly because components fail and because the effective power threshold $d$ 9 becomes harder to satisfy over time.

Other papers support broader generalization. "A Degradation Performance Model With Mixed-type Covariates and Latent Heterogeneity" (Sun et al., 2021) does not define RDC explicitly, but it provides a degradation-to-reliability pathway under mixed scalar and functional covariates and latent unit effects. "Dynamic Model Agnostic Reliability Evaluation of Machine-Learning Methods Integrated in Instrumentation & Control Systems" (Chen et al., 2023) defines a non-time-based analog: LADDR uses

$E(\mathcal N,R)$ 0

with Mahalanobis-type distance $E(\mathcal N,R)$ 1 to map distance from training support to a continuously decaying reliability score. This suggests that some RDC-like constructions are indexed by exposure, distance, or cumulative stress rather than calendar time.

5. Error-exponent RDC in classical-quantum channel coding

In information theory, the relevant curve is not survival over time but error decay with blocklength. "Reliability Function of Classical-Quantum Channels" (Li et al., 2024) defines, for a cq channel $E(\mathcal N,R)$ 2 and rate $E(\mathcal N,R)$ 3,

$E(\mathcal N,R)$ 4

and the reliability function

$E(\mathcal N,R)$ 5

Operationally, if $E(\mathcal N,R)$ 6, the optimal average decoding error decays exponentially in blocklength $E(\mathcal N,R)$ 7, and larger $E(\mathcal N,R)$ 8 means steeper error decay. The paper explicitly interprets this reliability function as its version of an RDC: $E(\mathcal N,R)$ 9

The main result is a lower bound in terms of Petz Rényi information: $R$ 0 and, in the high-rate regime above a critical rate $R$ 1, an exact formula: $R$ 2 This determines the reliability function of general cq channels in that region and resolves Holevo’s conjectured lower-bound formula. In this literature, the RDC is therefore a rate-indexed exponent curve rather than a horizon-wise success probability.

The conceptual contrast with agent-evaluation RDCs is sharp. The DTMC-based RDC of LLM agents is a first-passage CDF over steps within one run; the duration-bucket RDC is a repeated-attempt success profile over task classes; the cq-channel reliability function is an asymptotic exponent describing how quickly optimal average error vanishes with blocklength. A plausible implication is that the term RDC is portable only at a high level of abstraction: in each case it denotes structured reliability decay, but the underlying measure, index set, and asymptotic regime differ.

6. Common confusions, methodological cautions, and acronym ambiguity

Several recurrent confusions appear across this literature. First, RDC is not a universal primitive. In one agent paper it is a first-passage success CDF over execution steps (Tran-Truong et al., 27 Apr 2026); in another it is a duration-bucketed repeated-attempt success curve (Khanal et al., 31 Mar 2026); in cq coding it is an error-exponent function of communication rate (Li et al., 2024). Taken together, these papers suggest that any use of RDC should specify the sample space, conditioning structure, and horizon variable explicitly.

Second, the absorbing-chain formulation requires a particularly strict semantic distinction. "Measuring the Unmeasurable" states that RDC is

$R$ 3

and explicitly warns against confusing this with the survival function

$R$ 4

In that paper, RDC is cumulative success-within- $R$ 5, not survival, not a failure-hazard curve, and not a complementary CDF (Tran-Truong et al., 27 Apr 2026).

Third, curve interpretation depends on model adequacy. In the DTMC setting, scalar metrics alone “do not identify the success-time distribution being estimated, test whether traces support that distribution, or quantify finite-trace uncertainty”; the paper therefore requires a composite certificate combining $R$ 6 and $R$ 7, and states that if AIC or KS rejects, first-passage interpretation should stop until the data are re-segmented or modeled more richly (Tran-Truong et al., 27 Apr 2026). In the long-horizon benchmark setting, duration-bucket RDCs can be non-monotonic because duration is defined by estimated human completion time and because domain mixing alters aggregate slopes (Khanal et al., 31 Mar 2026). In dynamic-covariate degradation models, the curve depends materially on threshold choice, future covariate modeling, and the assumed dependence structure across degradation characteristics (Lin et al., 7 Apr 2025).

Finally, the acronym itself is ambiguous. In protein-structure determination, RDC ordinarily means residual dipolar coupling, with measurement model

$R$ 8

and has no connection to reliability decay (Khoo et al., 2016). That ambiguity is well established and should be resolved from context rather than acronym alone.

In contemporary arXiv usage, the most influential reliability-science reading of RDC is the move from a single scalar benchmark number to a curve that preserves horizon information. Whether that curve is a first-passage success CDF, a duration-conditioned repeated-attempt success profile, a threshold-crossing survival function, or an asymptotic error exponent, its core role is the same: to expose how reliability changes as the effective horizon grows.