Non-exponential Distributions Research Articles

Anomalous diffusion of synaptic vesicles and its influences on spontaneous and evoked neurotransmission.

Neurons in the central nervous system communicate with each other by activating billions of tiny synaptic boutons distributed along their fine axons. These presynaptic varicosities are very crowded environments, comprising hundreds of synaptic vesicles. Only a fraction of these vesicles can be recruited in a single release episode, either spontaneous or evoked by action potentials. Since the seminal work by Fatt and Katz, spontaneous release has been modelled as a memoryless process. Nevertheless, at central synapses, experimental evidence indicates more complex features, including non-exponential distributions of release intervals and power-law behaviour in their rate. To describe these features, we developed a probabilistic model of spontaneous release based on Brownian motion of synaptic vesicles in the presynaptic environment. To account for different diffusion regimes, we based our simulations on fractional Brownian motion. We show that this model can predict both deviation from the Poisson hypothesis and power-law features in experimental quantal release series, thus suggesting that the vesicular motion by diffusion could per se explain the emergence of these properties. We demonstrate the efficacy of our modelling approach using electrophysiological recordings at single synaptic boutons and ultrastructural data. When this approach was used to simulate evoked responses, we found that the replenishment of the readily releasable pool driven by Brownian motion of vesicles can reproduce the characteristic binomial release distributions seen experimentally. We believe that our modelling approach supports the idea that vesicle diffusion and readily releasable pool dynamics are crucial factors for the physiological functioning of neuronal communication. KEY POINTS: We developed a new probabilistic model of spontaneous and evoked vesicle fusion based on simple biophysical assumptions, including the motion of vesicles before they dock to the release site. We provide closed-form equations for the interval distribution of spontaneous releases in the special case of Brownian diffusion of vesicles, showing that a power-law heavy tail is generated. Fractional Brownian motion (fBm) was exploited to simulate anomalous vesicle diffusion, including directed and non-directed motion, by varying the Hurst exponent. We show that our model predicts non-linear features observed in experimental spontaneous quantal release series as well as ultrastructural data of synaptic vesicles spatial distribution. Evoked exocytosis based on a diffusion-replenished readily releasable pool might explain the emergence of power-law behaviour in neuronal activity.

Global dynamics of heterogeneous epidemic models with exponential and nonexponential latent period distributions

AbstractMany epidemic models assume an exponential distribution for the latent stage, but this may not accurately represent reality and could impact disease transmission predictions. Previous studies for short time scale models have shown that the choice of latency distribution affects estimates of the epidemic peak, time to peak, and infection eradication time, but has little effect on the final infection size. However, it is unclear if these conclusions hold for long time scale models. To address this, we investigate the impact of different latency distributions on disease dynamics in long‐term models, comparing them with short‐term models. We propose two susceptible‐exposed‐infected‐hospitalized‐recovered () models with multiple groups, using exponential and gamma distributions for latency. We derive the basic reproduction number () for both models and prove the global stability of the equilibrium points. We conduct numerical simulations and find that the gamma distribution may lead to larger epidemic peak sizes and longer peak times compared to the exponential distribution. However, the impact of latency distribution on estimating the peak and time to peak is smaller in long‐term models than in short‐term models. Additionally, the effect on the final total infected size is negligible regardless of the time scale. Therefore, when analyzing long‐term epidemic dynamics using heterogeneity models, the choice of latency distribution does not significantly affect the results. Assuming an exponential distribution for the latency is sufficient for simplifying the model and facilitating analysis. Our study provides valuable insights for selecting appropriate mathematical models in epidemiology.

Efficient derivation of optimal signal schedules for multimodal intersections

Tramways decrease time, cost, and environmental impact of urban transport, while requiring multimodal intersections where trams arriving with nominal periodic timetables may have right of way over road vehicles. Quantitative evaluation of stochastic models enables early exploration and online adaptation of design choices, identifying operational parameters that mitigate impact on road transport performance.We present an efficient analytical approach for offline scheduling of traffic signals at multimodal intersections among road traffic flows and tram lines with right of way, minimizing the maximum expected percentage of queued vehicles of each flow with respect to sequence and duration of phases. To this end, we compute the expected queue size over time of each vehicle flow through a compositional approach, decoupling analyses of tram and road traffic. On the one hand, we define microscopic models of tram traffic, capturing periodic tram departures, bounded delays, and travel times with general (i.e., non-Exponential) distribution with bounded support, open to represent arrival and travel processes estimated from operational data. On the other hand, we define macroscopic models of road transport flows as finite-capacity vacation queues, with general vacation times determined by the transient probability that the intersection is available for vehicles, efficiently evaluating the exact expected queue size over time. We show that the distribution of the expected queue size of each flow at multiples of the hyperperiod, resulting from temporization of nominal tram arrivals and vehicle traffic signals, reaches a steady state within few hyper-periods. Therefore, transient analysis starting from this steady-state distribution and lasting for the hyper-period duration turns out to be sufficient to characterize road transport behavior over time intervals of arbitrary duration.We implemented the proposed approach in the novel OMNIBUS Java library, and we compared against Simulation of Urban MObility (SUMO). Experimental results on case studies of real complexity with time-varying parameters show the approach effectiveness at identifying optimal traffic signal schedules, notably exploring in few minutes hundreds of schedules requiring tens of hours in SUMO.

Stochastic modeling of enhancing energy efficiency for HAP in HAP-LEO systems using Markov regenerative process

The sixth-generation (6G) communication systems heavily rely on space-air-terrestrial networks (SAT), which integrate high-altitude platforms (HAPs) and low Earth orbit (LEO) satellites. LEO satellites play a crucial role in ensuring global coverage and supporting a large number of users, while HAPs are strategically deployed to cover regions with high demand, providing a more reliable communication service to ground users. The principal aim of the proposed model is to explore techniques for energy conservation and efficient energy harvesting within the multi-layered system for HAP. In this energy harvesting scenario, it is assumed that HAPs will store the energy units generated through solar power. Moreover, the presented model incorporates two energy-saving states: standby and sleep states, during which HAPs consume minimal or zero energy. To analyze the described multi-layer system, a stochastic model is formulated using Markov regenerative process (MRGP), with consideration for the mean sojourn times of energy-saving states following a non-exponential distribution. The numerical results vividly demonstrate the significant contribution of energy-saving states in enhancing the energy efficiency of the HAP system. Also, trade-offs have been presented to analyze the impact of energy harvesting and energy saving in the proposed model. These findings are visually presented in graphs, and to validate the proposed model, simulation results have been carried out.

An Exponentiality Test of Fit Based on a Tail Characterization against Heavy and Light-Tailed Alternatives

Log-concavity and log-convexity play a key role in various scientific fields, especially in those where the distinction between exponential and non-exponential distributions is necessary for inferential purposes. In the present study, we introduce a testing procedure for the tail part of a distribution which can be used for the distinction between exponential and non-exponential distributions. The conspiracy and catastrophe principles are initially used to establish a characterization of (the tail part of) the exponential distribution, which is one of the main contributions of the present work, leading the way for the construction of the new test of fit. The proposed test and its implementation are thoroughly discussed, and an extended simulation study has been undertaken to clarify issues related to its implementation and explore the extent of its capabilities. A real data case is also investigated.

Poisson distributions in stochastic dynamics of gene expression: What events do they count?

The Poisson distribution is the probability distribution of the number of independent events in a given period. Although the Poisson distribution appears ubiquitously in various stochastic dynamics of gene expression, both as time-dependent distributions and stationary distributions, underlying independent events that give rise to such distributions have not been clear, especially in the presence of the degradation of gene products, which is not a Poisson process. I show that the variable following the Poisson distribution is the number of independent events where biomolecules are created, destined to survive until the end of a given time duration. This new viewpoint enables me to rederive the Poisson distribution as a time-dependent probability distribution for molecule numbers in various monomolecular reaction models of stochastic gene dynamics. Additionally, it allows me to derive an analytic form of the time-dependent probability distribution for multispecies monomolecular reaction models with species whose lifetimes follow nonexponential distributions, which is the convolution of the Poisson distribution with the multinomial distribution. This distribution is then utilized for deriving a novel series expansion form of a time-dependent distribution for a model with a stochastic production rate.

Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries

In this paper we develop an encounter-based model of a run-and-tumble particle (RTP) confined to a finite interval with partially absorbing, sticky boundaries at both ends. We assume that the particle switches between two constant velocity states ±v at a rate α. Whenever the particle hits a boundary, it becomes stuck by pushing on the boundary until either a tumble event reverses the swimming direction or it is permanently absorbed. We formulate the absorption process by identifying the first passage time (FPT) for absorption with the event that the time A(t) spent attached to either wall up to time t (the occupation time) crosses some random threshold . Taking to be an exponential distribution, , we show that the joint probability density for particle position X(t) and velocity state satisfies a well-defined boundary value problem (BVP) with κ 0 representing a constant absorption rate. The solution of this BVP determines the so-called occupation time propagator, which is the joint probability density for the triplet . The propagator is then used to incorporate more general models of absorption based on non-exponential (non-Markovian) distributions . We illustrate the theory by calculating the mean FPT and splitting probabilities for absorption. We also show how our previous results for partially absorbing, non-sticky boundaries can be recovered in an appropriate limit. Absorption now depends on the number of collisions of the RTP with the boundary. Finally, we extend the theory by taking absorption to depend on the individual occupation times at the two ends.

Optimally Scheduling Heterogeneous Impatient Customers

Problem definition: We study scheduling multi-class impatient customers in parallel server queueing systems. At the time of arrival, customers are identified as being one of many classes, and the class represents the service and patience time distributions as well as cost characteristics. From the system’s perspective, customers of the same class at time of arrival get differentiated on their residual patience time as they wait in queue. We leverage this property and propose two novel and easy-to-implement multi-class scheduling policies. Academic/practical relevance: Scheduling multi-class impatient customers is an important and challenging topic, especially when customers’ patience times are nonexponential. In these contexts, even for customers of the same class, processing them under the first-come, first-served (FCFS) policy is suboptimal. This is because, at time of arrival, the system only knows the overall patience distribution from which a customer’s patience value is drawn, and as time elapses, the estimate of the customer’s residual patience time can be further updated. For nonexponential patience distributions, such an update indeed reveals additional information, and using this information to implement within-class prioritization can lead to additional benefits relative to the FCFS policy. Methodology: We use fluid approximations to analyze the multi-class scheduling problem with ideas borrowed from convex optimization. These approximations are known to perform well for large systems, and we use simulations to validate our proposed policies for small systems. Results: We propose a multi-class time-in-queue policy that prioritizes both across customer classes and within each class using a simple rule and further show that most of the gains of such a policy can be achieved by deviating from within-class FCFS for at most one customer class. In addition, for systems with exponential patience times, our policy reduces to a simple priority-based policy, which we prove is asymptotically optimal for Markovian systems with an optimality gap that does not grow with system scale. Managerial implications: Our work provides managers ways of improving quality of service to manage parallel server queueing systems. We propose easy-to-implement policies that perform well relative to reasonable benchmarks. Our work also adds to the academic literature on multi-class queueing systems by demonstrating the joint benefits of cross- and within-class prioritization. Funding: A. Bassamboo received financial support from the National Science Foundation [Grant CMMI 2006350]. C. (A.) Wu received financial support from the Hong Kong General Research Fund [Early Career Scheme, Project 26206419]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.1190 .

Evaluations of heterogeneous epidemic models with exponential and non-exponential distributions for latent period: the Case of COVID-19.

Most of heterogeneous epidemic models assume exponentially distributed sojourn times in infectious states, which may not be practical in reality and could affect the dynamics of the epidemic. This paper investigates the potential discrepancies between exponential and non-exponential distribution models in analyzing the transmission patterns of infectious diseases and evaluating control measures. Two SEIHR models with multiple subgroups based on different assumptions for latency are established: Model Ⅰ assumes an exponential distribution of latency, while Model Ⅱ assumes a gamma distribution. To overcome the challenges associated with the high dimensionality of GDM, we derive the basic reproduction number ($ R_{0} $) of the model theoretically, and apply numerical simulations to evaluate the effect of different interventions on EDM and GDM. Our results show that considering a more realistic gamma distribution of latency can change the peak numbers of infected and the timescales of an epidemic, and GDM may underestimate the infection eradication time and overestimate the peak value compared to EDM. Additionally, the two models can produce inconsistent predictions in estimating the time to reach the peak. Our study contributes to a more accurate understanding of disease transmission patterns, which is crucial for effective disease control and prevention.

Evaluations of COVID-19 epidemic models with multiple susceptible compartments using exponential and non-exponential distribution for disease stages.

Mathematical models have wide applications in studying COVID-19 epidemic transmission dynamics, however, most mathematical models do not take into account the heterogeneity of susceptible populations and the non-exponential distribution infectious period. This paper attempts to investigate whether non-exponentially distributed infectious period can better characterize the transmission process in heterogeneous susceptible populations and how it impacts the control strategies. For this purpose, we establish two COVID-19 epidemic models with heterogeneous susceptible populations based on different assumptions for infectious period: the first one is an exponential distribution model (EDM), and the other one is a gamma distribution model (GDM); explicit formula of peak time of the EDM is presented via our analytical approach. By data fitting with the COVID-19 (Omicron) epidemic in Spain and Norway, it seems that Spain is more suitable for EDM while Norway is more suitable for GDM. Finally, we use EDM and GDM to evaluate the impaction of control strategies such as reduction of transmission rates, and increase of primary course rate (PCR) and booster dose rate (BDR).

Theory of tip structure–dependent microtubule catastrophes and damage-induced microtubule rescues

Microtubules are essential cytoskeletal polymers that exhibit stochastic switches between tubulin assembly and disassembly. Here, we examine possible mechanisms for these switches, called catastrophes and rescues. We formulate a four-state Monte Carlo model, explicitly considering two biochemical and two conformational states of tubulin, based on a recently conceived view of microtubule assembly with flared ends. The model predicts that high activation energy barriers for lateral tubulin interactions can cause lagging of curled protofilaments, leading to a ragged appearance of the growing tip. Changes in the extent of tip raggedness explain some important but poorly understood features of microtubule catastrophe: weak dependence on tubulin concentration and an increase in its probability over time, known as aging. The model predicts a vanishingly rare frequency of spontaneous rescue unless patches of guanosine triphosphate tubulin are artificially embedded into microtubule lattice. To test our model, we used invitro reconstitution, designed to minimize artifacts induced by microtubule interaction with nearby surfaces. Microtubules were assembled from seeds overhanging from microfabricated pedestals and thus well separated from the coverslip. This geometry reduced the rescue frequency and the incorporation of tubulins into the microtubule shaft compared with the conventional assay, producing data consistent with the model. Moreover, the rescue positions of microtubules nucleated from coverslip-immobilized seeds displayed a nonexponential distribution, confirming that coverslips can affect microtubule dynamics. Overall, our study establishes a unified theory accounting for microtubule assembly with flared ends, a tip structure-dependent catastrophe frequency, and a microtubule rescue frequency dependent on lattice damage and repair.

Modeling of occupant energy consumption behavior based on human dynamics theory: A case study of a government office building

The diversity and complexity of non-physical factors make it difficult to predict occupant behavior accurately. It's also the main reason for the great difference between actual and simulated energy consumption. In this paper, the human dynamics method is introduced to establish the occupant energy consumption behavior (OECB) models of multiple offices in a government office building, and the possible generation mechanism of the OECB is discussed to analyze its underlying reason for improvement of energy consumption system design. The results show the models of different single/multi-person office rooms can be well fitted by non-exponential distributions, such as truncated power-law distribution, stretched exponential distribution and log-normal distribution. Among them, the truncated power-law distribution model is the most suitable model for fitting the temporal statistical characteristics of OECB, and the power exponents of the models are close to 1. Furthermore, the generation mechanism of OECB in office rooms is well explained by Priority-based task queue model (PBTQM). The modeling results in this paper can be applied for occupancy classification with similar energy use habits, building performance simulation, energy consumption system design and energy conservation management.

Non-exponential tunneling ionization probability distribution as a function of different laser beam profiles

In this paper, we discussed the probability distribution of exponential and non-exponential tunneling ionization of atoms, taking into account that the tunneling is not instantaneous, but requires a very short time interval. Also, it was investigated how different laser beam profiles affect the probability distribution. These physical situations are analyzed for the valence electron of potassium atom exposed to a strong laser field in a wide range of intensities (1012 -1015 W/cm2). We use ADK theory formalism to compute probability distributions. The results demonstrate that the probability distribution in the non-exponential mode has a significantly lower value than in the exponential mode, calculated under the same conditions. We showed that various laser beam profiles on these probability distributions produce different tunneling time intervals.

Optimal switching paths of a non-Markovian model of gene bursty expression

Gene expression usually involves multiple biochemical reactions, resulting into the probability of waiting time between consecutive reaction steps is a non-exponential probability distribution, and creating molecular memory. Besides, some experiments indicate that burst of gene expression and positive feedback loops play important roles in gene expression and phenotypic variation. However, how molecular memory and the burst together affect metastable states in gene expression is still open, as far as our knowledge. In this paper, we build a generalized chemical master equation of gene bursty expression with a self-regulatory positive feedback. Using the Wentzel-Kramers-Brillouin (WKB) method, we study optimal paths of switching between bistable states of gene expression under the combination of molecular memory and the burst. Through theoretical analysis and numerical simulations, we find that the burst and molecular memory can change the metastable state structure of the mean-field system of our model, and that the burst can prolong optimal paths of switching from the high state of gene expression to the low state, and shorten optimal paths of switching from the low state to the high state, but molecular memory has the opposite effect on the optimal switching paths. This result is the further improvement on our previous work (Yin et al., 2021).

Production control problem with semi-Markov jump under stochastic demands and deteriorating inventories

We formulate a stochastic optimal production control problem of a manufacturing system with deteriorating items in the presence of random disturbances for a single machine multi-product. The dynamics of the inventory can be governed by an Itô stochastic differential equation. It is found in real market that the deteriorating inventory systems appears in several industrial sectors, such as food production, chemical and radioactive material, and pharmaceutical manufacturing. In the considered production system, the machine’s times to failure and times to repair are modeled as a semi-Markov process. The goal is to find the production rates in real-time to meet the random demands, through a minimization of the expected discounted cost of inventory holdings and shortages over a finite horizon. The optimality conditions are obtained in the form of time-dependent second-order Hamilton-Jacobi-Bellman (HJB) equations by using the dynamic programming and the stochastic calculus introduced by Itô. The model is formulated for n-items, and in particular, is illustrated with two items for some numerical data. The proposed model in the case of non-exponential distributions enables us to independently improve the coefficient of variation (CVup/down). Numerical methods are used to solve the optimality conditions and the structure of the control policies is confirmed by a sensitivity analysis.

Collaborative data-driven reliability analysis of multi-state fault trees

Fault tree modeling and failure analysis of systems that are equipped with sensors and meters are becoming more automated and less human-dependent. For a single system to benefit from its own collected data, it will need to wait for a long time to collect sufficient data to build representative models to increase its reliability. Therefore, if multiple systems with similar functionalities cooperate, the resolution of the collected data will increase. This leads to extracting fault trees with higher accuracy in failure detection and prediction. In this paper, we present an extended approach for collaborative Data-Driven Fault Tree Analysis (DDFTA) of a system which extracts repairable fault trees from time series data streaming from multiple systems/machines sharing similar functionalities. Results are analyzed to estimate the system’s reliability measures and investigate the effect of number of machines cooperating in data collection. Our method is not limited to binary (two states) components, nor to exponential distributions. Results show that applying collaborative data analytics significantly increases the accuracy of data-driven fault tree analysis, specifically for systems following nonexponential distributions.

Retrial BMAP/PH/N Queueing System with a Threshold-Dependent Inter-Retrial Time Distribution

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

Second-virial theory for shape-persistent living polymers templated by disks.

Living polymers composed of noncovalently bonded building blocks with weak backbone flexibility may self-assemble into thermoresponsive lyotropic liquid crystals. We demonstrate that the reversible polymer assembly and phase behavior can be controlled by the addition of (nonadsorbing) rigid colloidal disks which act as an entropic reorienting "template" onto the supramolecular polymers. Using a particle-based second-virial theory that correlates the various entropies associated with the polymers and disks, we demonstrate that small fractions of discotic additives promote the formation of a polymer nematic phase. At larger disk concentrations, however, the phase is disrupted by collective disk alignment in favor of a discotic nematic fluid in which the polymers are dispersed antinematically. We show that the antinematic arrangement of the polymers generates a nonexponential molecular-weight distribution and stimulates the formation of oligomeric species. At sufficient concentrations the disks facilitate a liquid-liquid phase separation which can be brought into simultaneously coexistence with the two fractionated nematic phases, providing evidence for a four-fluid coexistence in reversible shape-dissimilar hard-core mixtures without cohesive interparticle forces. We stipulate the conditions under which such a phenomenon could be found in experiment.

Orientational memory of active particles in multistate non-Markovian processes.

The orientational memory of particles can serve as an effective measure of diffusivity, spreading, and search efficiency in complex stochastic processes. We develop a theoretical framework to describe the decay of directional correlations in a generic class of stochastic active processes consisting of distinct states of motion characterized by their persistence and switching probabilities between the states. For exponentially distributed sojourn times, the orientation autocorrelation is analytically derived and the characteristic times of its crossovers are obtained in terms of the persistence of each state and the switching probabilities. We show how nonexponential sojourn-time distributions of interest, such as Gaussian and power-law distributions, can result from history-dependent transitions between the states. The relaxation behavior of the correlation function in such non-Markovian processes is governed by the history dependence of the switching probabilities and cannot be solely determined by the mean sojourn times of the states.

Memory shapes microbial populations.

Correct decision making is fundamental for all living organisms to thrive under environmental changes. The patterns of environmental variation and the quality of available information define the most favourable strategy among multiple options, from randomly adopting a phenotypic state to sensing and reacting to environmental cues. Cellular memory-the ability to track and condition the time to switch to a different phenotypic state-can help withstand environmental fluctuations. How does memory manifest itself in unicellular organisms? We describe the population-wide consequences of phenotypic memory in microbes through a combination of deterministic modelling and stochastic simulations. Moving beyond binary switching models, our work highlights the need to consider a broader range of switching behaviours when describing microbial adaptive strategies. We show that memory in individual cells generates patterns at the population level coherent with overshoots and non-exponential lag times distributions experimentally observed in phenotypically heterogeneous populations. We emphasise the implications of our work in understanding antibiotic tolerance and, in general, bacterial survival under fluctuating environments.