analyze first-passage times

This paper addresses the challenge of efficiently and analytically modeling process execution time statistics from event logs. Methodologically, it introduces the first end-to-end analytical performance analysis framework based on semi-Markov processes: it directly infers execution time means and probability density functions (PDFs) from logs—bypassing simulation entirely. For discrete-time execution times, it employs exact convolution; for continuous-time cases, it approximates PDFs using Gaussian mixture models (GMMs), balancing accuracy, model compactness, and interpretability. Experiments show that the discrete-time approach achieves up to one order of magnitude speedup over simulation under small support sets, while GMM-based representation drastically reduces model size, enabling rapid what-if analysis. The core contribution is the first fully analytical, log-driven inference of semi-Markov performance models—eliminating reliance on traditional simulation-based approaches and establishing a new paradigm for scalable, interpretable process performance analysis.

This paper addresses the challenge of modeling bivariate first-hitting times under dependence and right-censoring—common in clinical dual-endpoint settings such as concurrent liver and kidney injury. We propose a joint first-hitting-time model that integrates a copula-based dependence structure with a compound Poisson process, enabling threshold-crossing analysis for correlated endpoints. Methodologically, we unify the characterization of dependence, censoring mechanisms, and compound Poisson first-hitting dynamics, and rigorously establish model identifiability. We develop a pseudo-likelihood estimation framework accommodating right-censoring and derive asymptotic theory showing root-n consistency and asymptotic normality of the estimators. Monte Carlo simulations confirm robust finite-sample performance. Applied to real-world mushroom poisoning data, our model successfully quantifies temporal dependence between hepatic and renal injury onset and significantly improves prognostic accuracy for multi-endpoint outcomes.

The latent Markov modeling community suffers from fragmented formalisms, inconsistent terminology, and disparate inference methods and software tools—severely hindering practical adoption. To address this, we propose a unified latent-variable Markovian framework for time-series and sequential data, systematically integrating paradigms including hidden Markov models, state-space models, and Markov-modulated Poisson processes under a recursive structural perspective. We introduce a modular (Lego-style) modeling language and develop the efficient R package *LaMa*, whose core implements numerically stable maximum-likelihood estimation in C++ with dynamic programming, forward–backward algorithms, and optimizations for state-dependent structures. This framework substantially lowers the modeling barrier, enabling rapid, robust, and reproducible parameter estimation across all supported models. Moreover, it provides a data-driven, practical roadmap for model selection—bridging theoretical flexibility with empirical usability.

Efficient likelihood computation remains challenging for generalized drift-diffusion models (GDDMs) when the drift rate varies dynamically with time-varying covariates (e.g., neural activity, visual fixation). To address this, we propose a staged, analytically driven fast inference algorithm. Our method extends the Cherkasov condition to GDDMs with time-varying boundaries and integrates stochastic differential equation analytical solutions, piecewise density propagation, and adaptive numerical integration to achieve high-precision likelihood estimation. Evaluated on canonical tasks—including the attentional drift-diffusion model (aDDM)—our approach matches Monte Carlo “gold-standard” accuracy while accelerating computation by one to two orders of magnitude. This work establishes the first likelihood computation framework for dynamic-covariate-coupled decision models that simultaneously ensures analytical tractability, computational efficiency, and broad applicability across GDDM variants.

Telecommunications engineering graduate students often lack foundational knowledge in probability theory and stochastic processes, hindering their mastery of teletraffic analysis. Method: This work develops a balanced theoretical–practical pedagogical framework centered on classical queueing models (e.g., M/M/1, M/G/1) and stochastic processes (e.g., Poisson processes, Markov chains, steady-state analysis), integrated with contemporary telecommunications use cases—including traffic modeling, resource allocation, and flow management—and reinforced through numerical simulation exercises. A structured background remediation module addresses prerequisite gaps. Contribution/Results: The resulting textbook has been adopted as a core course resource at multiple universities worldwide, demonstrably enhancing students’ practical competencies in performance modeling and optimization of communication systems.

This study addresses the challenge of quantifying optimal timing for proactive defense in single-attack scenarios by proposing a stochastic-process-based analytical framework, which introduces stochastic game theory into cybersecurity for the first time. By modeling defense as a continuous observation mechanism and integrating exponential distributions, Markovian Poisson arrival processes, Laplace–Carson transforms, and first-exit theory, the work explicitly derives the marginal distribution and conditional expectation of defensive actions. A joint detection function is constructed to precisely localize the attack instant. The approach not only enables visualization of defense density but also provides conditional expectations of observation times before and after an attack, facilitating dynamic calibration of low-latency proactive defense parameters according to threat intensity.

This work proposes a novel paradigm that unifies the entire statistical inference pipeline through a probabilistic language, aiming to coherently bridge observed data, inferential targets, and real-world decision-making. By treating probability and stochastic processes as a central “translation language,” the framework integrates tools from probability measures, likelihood theory, weak convergence, empirical processes, functional data analysis, M- and Z-estimation, kernel methods, and event-time processes into a common syntax. This synthesis connects classical theoretical foundations with modern data structures and practical applications. The approach is validated through historical and biomedical case studies, demonstrating its capacity to provide systematic modeling pathways for complex data while substantially enhancing inferential stability and predictive performance.

This work proposes a parameter estimation method based on the Expectation–Maximization (EM) algorithm for Markovian Arrival Process (MAP)-driven Quasi-Birth–Death (QBD) queueing systems, tailored to realistic scenarios where only coarse-grained data such as system utilization are available. Within a maximum likelihood framework, the approach infers sufficient statistics—including sojourn times, phase transitions, and service dynamics—underlying the hidden states directly from utilization time series. To the best of our knowledge, this is the first method capable of fully estimating MAP-QBD model parameters using solely utilization data. The study further introduces an innovative use of the Akaike Information Criterion (AIC) to automatically select the number of MAP phases, thereby mitigating overfitting. Experimental results demonstrate that the method accurately recovers both arrival and service parameters, offering a practical performance modeling tool for real-world systems lacking fine-grained event logs.

This study addresses the identification and estimation of equilibrium outcomes in synchronous games from interdependent stopping time data. The equilibrium is modeled as a system of mutually dependent first-passage times, wherein an individual’s incentive to stop intensifies as others cease activity, and payoffs are influenced by common shocks as well as both observed and unobserved heterogeneity. A key innovation lies in the use of spectrally negative Lévy processes to characterize common shocks, which facilitates nonparametric identification of the model. Building on this foundation, the authors develop a computationally tractable estimation framework that combines maximum simulated likelihood with simulated method of moments. Monte Carlo experiments demonstrate strong finite-sample performance, offering the first empirical tool for synchronous games that simultaneously ensures rigorous identification and computational feasibility.