Exit Time Research Articles

Variational Principles for Asymptotic Variance of General Markov Processes

A variational formula for the asymptotic variance of general Markov processes is obtained. As application, we get an upper bound of the mean exit time of reversible Markov processes, and some comparison theorems between the reversible and non-reversible diffusion processes.

Dynamic trading with uncertain exit time and transaction costs in a general Markov market

This paper investigates a dynamic trading problem with transaction cost and uncertain exit time in a general Markov market, where the mean vector and covariance matrix of returns depend on the states of the stochastic market, and the market state is regime switching in a time varying state set. Following the framework proposed by Gârleanu and Pedersen (2013), the investor maximizes his or her multi-period mean–variance utility, net of quadratic transaction costs capturing the linear price impact where trades lead to temporary linear changes in prices. The explicit expression for the optimal strategy is derived by using matrix theory technique and dynamic programming approach. Finally, numerical examples are provided to study the effects of transition cost and exit probability on the wealth process, the trading strategy, turnover rate and the total transaction cost.

A hybrid particle swarm optimization and simulated annealing algorithm for the job shop scheduling problem with transport resources

This work addresses a variant of the job shop scheduling problem in which jobs need to be transported to the machines processing their operations by a limited number of vehicles. Given that vehicles must deliver the jobs to the machines for processing and that machines need to finish processing the jobs before they can be transported, machine scheduling and vehicle scheduling are intertwined. A coordinated approach that solves these interrelated problems simultaneously improves the overall performance of the manufacturing system. In the current competitive business environment, and integrated approach is imperative as it boosts cost savings and on-time deliveries. Hence, the job shop scheduling problem with transport resources (JSPT) requires scheduling production operations and transport tasks simultaneously. The JSPT is studied considering the minimization of two alternative performance metrics, namely: makespan and exit time. Optimal solutions are found by a mixed integer linear programming (MILP) model. However, since integrated production and transportation scheduling is very complex, the MILP model can only handle small-sized problem instances. To find good quality solutions in reasonable computation times, we propose a hybrid particle swarm optimization and simulated annealing algorithm (PSOSA). Furthermore, we derive a fast lower bounding procedure that can be used to evaluate the performance of the heuristic solutions for larger instances. Extensive computational experiments are conducted on 73 benchmark instances, for each of the two performance metrics, to assess the efficacy and efficiency of the proposed PSOSA algorithm. These experiments show that the PSOSA outperforms state-of-the-art solution approaches and is very robust.

DOC, grazers, and resilience of phytoplankton to enrichment

AbstractPhytoplankton blooms often follow nutrient enrichment. Differences among lakes in light‐absorbing dissolved organic carbon (DOC) may shift bloom thresholds to higher nutrient loads and thereby increase resilience of lakes to enrichment. To explore this idea, we measured resilience to experimental enrichment of two lakes with contrasting DOC concentrations. We compared bloom thresholds in both lakes using a model of phytoplankton response to DOC and nutrients, a dynamic time series indicator of resilience, and two empirical measures of stochastic resilience, mean exit time and median survival time. For the dynamic indicator and ecosystem model the lake with higher DOC was more resilient to enrichment. However, the distributions overlapped for stochastic indicators of resilience of the two lakes. These analyses show that DOC interacts with mixing depth and zooplankton biomass to affect resilience. Strong contrasts in DOC and many observations are needed to discern effects of DOC on resilience to enrichment.

Extended Relevance Vector Machine-Based Remaining Useful Life Prediction for DC-Link Capacitor in High-Speed Train.

Remaining useful life (RUL) prediction is a reliable tool for the health management of components. The main concern of RUL prediction is how to accurately predict the RUL under uncertainties. In order to enhance the prediction accuracy under uncertain conditions, the relevance vector machine (RVM) is extended into the probability manifold to compensate for the weakness caused by evidence approximation of the RVM. First, tendency features are selected based on the batch samples. Then, a dynamic multistep regression model is built for well describing the influence of uncertainties. Furthermore, the degradation tendency is estimated to monitor degradation status continuously. As poorly estimated hyperparameters of RVM may result in low prediction accuracy, the established RVM model is extended to the probabilistic manifold for estimating the degradation tendency exactly. The RUL is then prognosticated by the first hitting time (FHT) method based on the estimated degradation tendency. The proposed schemes are illustrated by a case study, which investigated the capacitors' performance degradation in traction systems of high-speed trains.

Time Distribution from Zero up to Beginning: of the Final Stop of Semi-Markov Diffusion Process on Interval with Unattainable Boundaries

A one-dimensional process with continuous trajectories on a non-negative semiaxis is considered. The process has the Markov property with respect to the first exit time from any open interval (semi-Markov process). This process is called diffusion if the probability of its first exit from any symmetric neighborhood of its initial point across any boundary tends to 1/2 as the length of this neighborhood tends to zero. The time distribution from zero up to the beginning of the final interval of constancy is investigated. This distribution depends on semi-Markov transition generating functions of the process. Representation for Laplace transform of this distribution is obtained in an integral form. The integrand of this representation explains the meaning of quadratic terms of Taylor expansion of a semi-Markov transition generating function by powers of the diameter of the symmetric neighborhood of the initial point of the process. Namely, the trajectory of the process has no final interval of constancy if the coefficient of such a quadratic term is equal to zero.

Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes

AbstractWe analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All eight Laplace transforms can be described in terms of two so-called scale functions associated with the upward one-sided exit time and with the upward two-sided exit time. All other Laplace transforms can be obtained from the above scale functions by taking limits, derivatives, integrals, and combinations of these.

Nationwide analysis of proximity of American College of Surgeons–verified and state-designated trauma centers to the nearest highway exit and associated prehospital motor vehicle collision fatalities

BackgroundMotor vehicle collisions remain a leading cause of trauma-related deaths. We aim to investigate the relationship between the proximity of trauma centers to the nearest highway exit and prehospital motor vehicle collision fatalities at the county level nationwide. MethodsThis was a cross-sectional study evaluating the association between the distance of trauma centers to the nearest highway exit and prehospital motor vehicle collision fatalities between the years 2014 and 2019. Prehospital motor vehicle collision fatalities were obtained from National Highway Traffic Safety Administration. Mapping software was used to determine the distance of trauma center to the nearest highway exit and transport time. Linear regression analysis was performed. ResultsA total of 2,019 American College of Surgeons–verified and/or state-designated trauma centers were included (211 Level 1, 356 Level 2, 491 Level 3, and 961 Level 4 trauma centers). Prehospital motor vehicle collision fatalities were positively correlated with the distance of trauma center to the nearest highway exit for counties with trauma centers located ≤5 miles from the nearest highway exit (r = 0.328; P < .001). In the 612 counties with a 10% increase in prehospital motor vehicle collision fatalities from 2014 to 2019, prehospital motor vehicle collision fatalities were also positively correlated with distance to the nearest highway exit (r = 0.302; P < .001). The counties with more dispersed distributions of trauma centers were significantly associated with motor vehicle collision fatalities (Spearman’s rank coefficient = 0.456; 95% confidence interval, 0.163–0.675; P = .003). ConclusionShorter distances between trauma centers and the nearest highway exit are associated with fewer prehospital motor vehicle collision fatalities for counties with trauma centers ≤5 miles of the nearest highway exit. Further enhancement of existing highway infrastructure and standardization of emergency medical services transport protocols are needed to address the burden of prehospital motor vehicle collision fatalities in the United States.

The double barrier problem for Brownian motion with Poissonian resetting

Let a < 0 < b be two fixed points. We consider a diffusive particle in one space dimension whose dynamics combines continuous-time Brownian motion with resetting at random Poisson times. We study the double barrier problem regarding the probability that starting from 0 the Brownian particle escapes (a, b) at the upper barrier b and compare how resetting modifies the exit probabilities. We also study the distribution of the corresponding exit time. We show that the resetting activity may either increase or decrease the meantime to exit a region. A precise condition involving the golden ratio separates both cases. Optimal resetting rates that minimize the mean escape time are considered.

The Parisian and ultimate drawdowns of Lévy insurance models

In this paper, inspired by the ideas of Parisian ruin and ultimate bankruptcy, we introduce two new stopping times for the (general) drawdown process, namely, the Parisian drawdown and ultimate drawdown under the exponential implementation delays. We provide quantitative analysis of their distributional properties of interest through the generalized scale functions, whose properties are examined as well. We then discuss their relationships with the existing results on exit times and occupation times as the application of our main results. Another application in the fair market valuation of drawdown insurance is also presented and illustrated in a numerical example.

Security practices in retail properties in a developing economy: a logistic analysis of impacts on customers’ satisfaction and patronage

PurposeThe purpose of this study is to examine the impact of a soft facility practice in the organized retail sector. In specifics, it draws context from a security practice and assesses its effect on customers’ satisfaction and patronage in retail properties.Design/methodology/approachThe research method is quantitative. The study used a survey research design with the use of a structured questionnaire for data collection. The data were collected from the head of households who purchase items in the retail shops. It applied a logistic regression model to estimate customers’ satisfaction and the effect of the current security practice on patronage.FindingsContrary to expectation, descriptive analysis of data shows that respondents were satisfied with the security procedure with weighted means scores (3.62, 3.74, 3.78 and 3.66) above average for bag check at exits, reconciliation of receipts with purchase, the attitude of security personnel during exit checks and time taken during exit checks. With specific reference to bag checks at the exit, logit shows that 32% were neutral while 8% were dissatisfied with the security practice. Among the dissatisfied, logit shows an odds ratio of 0.059, which implies that they are likely to verbally express their dissatisfaction with the shop. On the other hand, the chances that they would not do this are more likely with an odds ratio of 162818201.343. Further, continuous patronage (Loyalty) is strongly less likely with an odds ratio of 1.250E-22. This was corroborated by a similar odd ratio of 4.068E-11 estimated for those that would take the exit option due to the bag’s check.Research limitations/implicationsThe limitation of this study is that samples were randomly drawn from an unknown population of customers. However, the study was guided by Cochran (1963) to select a valid representative sample and support the reliability of the research findings.Practical implicationsThe findings on satisfaction imply that the convenience and swiftness associated with shopping in a retail shop had been eroded by the current security facility practices which may lead to a reduction in the growth and retail sector turnover. While many dissatisfied customers would have exited if there are alternative shops with more customer-friendly security practices, the limited number of organized retail shops in the study area will prevent this from happening.Originality/valueLiterature on the management of facilities in real estate is quite vast; however, not much attention has been paid to the management of security in the retail sector particularly in Nigeria. This study is, therefore, novel, as it provides seminal evidence on this important topic and will serve as a reference for further research in Nigeria.

High quality timetables for Italian schools

This work introduces a complex variant of the timetabling problem, which is motivated by the case of Italian schools. The new requirements enforce to (i) provide teachers with the same idle times, (ii) avoid consecutive days with heavy workload, (iii) limit multiple daily lessons for each class, (iv) introduce shorter time units to differentiate entry and exit times. We present an integer programming model for this problem, which is denoted by Italian High School Timetabling Problem (IHSTP). However, requirements (i), (ii), (iii) and (iv) cannot be expressed according to the current XHSTT standard. Since the IHSTP model is very hard to solve by an off-the-shelf solver, we present a two-step optimization method: the first step optimally assigns teachers to lesson times and the second step assigns classes to teachers. An extensive experimentation is performed on the model by realistic and real instances from Italian schools, as well as benchmark instances from the literature. Finally, the experiments show that the method is effective in solving both this new problem and the simplified problem without the new requirements.

Evaluation of the start of surgical closure as a milestone for forecasting the time remaining to exit the operating room: A retrospective, observational cohort study

IntroductionForecasting when patients will exit the operating room (OR) is important for managing add-on cases, coordinating breaks and relief, anticipating postanesthesia care unit admissions, and estimating to-follow surgeons’ start times. If coefficients of variation (CVs) for the interval from closing to OR exit (OR exit interval) were small, a reasonable predictive approach would be to initialize a timer at the start of closing using the mean OR exit interval, then decrement the time remaining each successive minute. However, if the CV were ≥0.32 (typical for many surgical procedure times), this approach would be invalid because actual average times remaining would significantly exceed the predictions. Futility would be expected because the corresponding approach is inaccurate for case duration prediction. MethodsWe retrospectively studied all OR anesthetics (n = 44,254) at an adult academic hospital from October 29, 2017, through January 11, 2022. The primary surgeon, scheduled primary Current Procedural Terminology code, OR entry to closing, and OR exit intervals were retrieved electronically. The CVs of the OR exit intervals were calculated, categorized by: primary CPT and surgeon; primary CPT; American Society of Anesthesiologists CPT; and Clinical Classification Software category. We compared, pairwise, the standard deviations (SD) of the OR entry to closing interval to the OR exit interval, determined if most (>50%) of the CVs for the OR exit interval were ≥0.32, and compared CVs for the OR exit interval among categories. We analyzed categories that were performed ≥30 times. ResultsFor all categories, the standard deviations (SD) of the OR exit intervals were considerably smaller than the paired SDs of OR entry to closing (P < 0.0001). The smallest CVs were from combining primary CPT and surgeon (median 0.4, IQR 0.34 to 0.47, P < 0.0001 vs. other categories). However, using the 244 analyzed combinations of primary CPT and surgeon, the best classification category, 81.3% of the CV (99% CI 74.1% to 87.2%, P < 0.0001) were ≥0.32. In other words, at least 74% of combinations would have greater proportional variability than observed commonly for surgical times. ConclusionsA simple timer started when surgical closure begins, decrementing each subsequent minute, is an invalid approach to estimating the time remaining in cases once closure begins. Rather, previously published methods for predicting the time remaining in ongoing OR cases are appropriate, with time-zero set when closing begins. OR status boards would be updated continually with the revised exit times, calculated as the current time plus the time remaining.

Construction of Power Grid Stability Analysis System Based on Stochastic Process Theory

In recent years, the proportion of new energy power generation in the power supply has increased yearly. However, the random volatility generated by new energy generation poses a new challenge to the stability of the power system. In terms of stability analysis, the traditional methods based on the deterministic system can not fully describe the influence of random fluctuation. In this case, this paper introduces the power grid stability analysis method considering random influence. Based on the original model, this paper introduces the random fluctuation simulated by the Gaussian process, establishes the power grid model described by nonlinear stochastic differential equations, and gives the numerical solution format of the model with the help of the Euler method. Secondly, compared with the traditional method of quantifying the size of the escaping state of the system, this method considers the influence of the random variable on the stability of the system at the same time. Therefore, this paper establishes the stability analysis system of the power grid and further considers the influence of power grid topology and system parameters on system stability. By analyzing the Brass paradox, it is found that it is more reasonable to quantify the stability of the system with exit time. Finally, this paper uses the stability analysis system to study the stability of the high voltage power grid in Shandong Province and gives suggestions to improve the stability.

On an Approximation of Average Cost per Unit Time Impulse Control of Markov Processes

In this paper we consider impulse control of continuous time Markov processes with average cost per unit time functional. This problem is approximated using impulse control problems stopped at the first exit time from an increasing sequence of open sets. We find a solution to the Bellman equation corresponding to the original problem and show that stopped impulse control problems approximate optimal value of the cost functional.

Some explicit expressions for GBM with Markovian switching and parameter estimations

In this paper, we focus on the mean exit time and the scale function for the geometric Brownian motion with Markovian switching, in which the drift coefficients and the diffusion coefficients are associated with regime changes. The explicit expressions of mean exit time and scale function are obtained by solving the corresponding Poisson problem. Furthermore, we estimate parameters for the geometric Brownian motion with Markovian switching by composite likelihood and explore some properties of the estimates. Computer simulations are performed to illustrate our proposed algorithm, showing high accuracy of the estimators.

Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics.

Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g., the diffusion coefficient or the heat conductivity, which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated because two alternative formulations of the inhomogeneous transport equationsexist in the literature-the Smoluchowski and Fokker-Planck equations, which are also related to the so-called Ito-Stratonovich dilemma. Using the theory of statistical physics, we show that the two formulations, usually regarded as distinct models, are physically equivalent. From this result we develop efficient estimates for the transverse space-dependent diffusion coefficient in fluids near a phase boundary. Our method requires only measurements of escape probabilities and mean exit times of molecules leaving a narrow spatial region. We test our estimates in three case studies: (i) a Langevin model of a Büttikker-Landauer ratchet; atomistic molecular-dynamics simulations of liquid-water molecules in contact with (ii) vapor, and (iii) soap (surfactant) film which has promising applications in physical chemistry. Our analysis reveals that near the surfactant monolayer the mobility of water molecules is slowed down almost twice with respect to the bulk liquid. Moreover, the diffusion coefficient of water correlates with the transition from hydrophilic to hydrophobic parts of the film.

Influencing Discharge Efficiency: Addressing Interdisciplinary Communication, Transportation, and COVID-19 as Barriers.

At one tertiary, academic medical center, two general medicine units averaged 94% and 97% occupancy causing strain on patient throughput. This project was implemented at these two comparable general medicine units, totaling 64 beds. On each of these units, Pareto analyses on causal factors related to discharge order to exit time (DOTE) were performed. DOTE was defined as the period in minutes from when a provider orders a discharge to when the patient actually exits a room. Prime DOTE reduction opportunities were elicited that highlighted the need to address coordination of hospital discharge transportation; that is, arriving family members averaged 120 and 129 min for the two units, and medicars and ambulances averaged 122 and 156 min, which fell above the established 90-min overall strategic DOTE goal. Coordinating efficient discharges decreases the likelihood of hospital bottlenecking and improves patient satisfaction. The health care team is composed of physician and provider services, nursing, and case management, as well as the patient and family. Team-focused interventions aimed at reducing DOTE included leveraging interdisciplinary communication technology and messaging for efficiency and accuracy within the health care team and proactive scheduling of hospital discharge transportation arrival. Process objectives measured included percentage of the health care team educated and utilization of the discharge suite. Outcome objectives measured included median DOTE times, patient satisfaction, and emergency department boarding volume and times. Significantly, admissions for coronavirus disease-2019 (COVID-19) cases were also rapidly increasing early on during program implementation resulting in one of the two general medicine units to be designated for COVID-19 overflow. Using Lean methodology, the project design was formed based on the Institute for Healthcare Improvement's work on improving hospital-wide patient flow and the Agency for Healthcare Research and Quality's (AHRQ) IDEAL patient discharge framework to better achieve the well-known, triple aim. In response to COVID-19 demands, the Plan-Do-Study-Act process was warranted to be able to manage acute changes, using iterative processing. This program evaluation study assessed whether a communication training program that taught an interdisciplinary team of case managers, nurses, physicians, and related staff how to reduce DOTE was useful. The program had a material impact on the DOTE metric knowing that the hospital's ultimate strategic goal is to reduce DOTE to 90 min or less. A reduction in discharge time was documented when using weekly data from the hospital's discharge dashboard powered by the Maestro database. More specifically, nurses fully trained in the interdisciplinary communications program aimed to reduce DOTE had significantly lower DOTE outcomes on their discharges compared with untrained staff (i.e., average untrained = 127 min, average trained = 93 min). In addition, the fully trained nurses had 14% more of their discharges fall at or below the 90-min goal compared with untrained staff (i.e., untrained = 40%, trained = 54%). Supplemental research also suggested that the content of the communication training program was very relevant (e.g., empowering families to pick up the patients and using scheduling vs. will-call transportation strategies with patients lowered the DOTE metric). Corollary analyses showed that readmissions were also lowered, and patient satisfaction ratings increased. In addition, the interdisciplinary communications training program can benefit from being updated to include content on how COVID-19 issues adversely impact discharge times since significant relationships between various COVID-19 measures and higher discharge exit times were documented.

Market Exit and Minimax Regret

This paper shows how minimax regret sheds new light on an old economic topic, market-exit games. It focuses on wars of attrition, namely overcrowded duopoly markets where the strategic variable is the exit time. The only symmetric Nash equilibrium (NE) of the game studied is a mixed-strategy equilibrium that leads to a null expected payoff, i.e., the payoff a firm gets when it immediately exits the market. This result is not convincing, both from a behavioral and from a strategic viewpoint. The minimax regret approach that builds upon opposite regrets — exiting the market too late and exiting the market too early — is more convincing and ensures that both firms obtain a strictly positive expected payoff. Highlights Minimax regret behavior sheds new light on the optimal exit-times in overcrowded markets. Minimax regret behavior provides a new behavioral content to mixed strategies. In overcrowded duopolies, if the firms’ aim consists in minimizing regrets, then they get better payoffs than in the mixed-strategy NE.

A Collection of Results Relating the Geometry of Plane Domains and the Exit Time of Planar Brownian Motion

We prove a number of results relating exit times of planar Brownian motion with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.