Waiting Time Distribution Research Articles

Statistical properties and cosmological applications of fast radio bursts

Abstract Fast radio burst (FRB) is a type of extragalactic radio signal characterized by millisecond duration, extremely high brightness temperature, and large dispersion measure. It remains a mystery in the universe. Advancements in instrumentation have led to the discovery of 816 FRB sources and 7622 bursts from 67 repeating FRBs*. This field is undergoing rapid development, rapidly advancing our understanding of the physics of FRBs as new observational data accumulates. The accumulation of data has also promoted our exploration of our universe. In this review, we summarize the statistical analysis and cosmological applications using large samples of FRBs, including the energy functions, the waiting time distributions of repeating FRBs, probe of “missing baryons” and circumgalactic medium in the universe, measurements of cosmological parameters, exploration of the epoch of reionization history, and study of the gravitational lensing of FRBs.

Waiting time representation of discrete distributions

We discuss a representation of any probability distribution on the set of non-negative integers as a waiting time distribution in a sequence of independent Bernoulli trials. Several associated results are derived and illustrated by examples. Multivariate extensions are briefly treated as well.

First-passage and first-arrival problems in continuous-time random walks: Beyond the diffusion approximation.

Some exact solutions of the first-passage and first-arrival problems for the continuous-time random-walk model are obtained. On the basis of these exact solutions, the following has been revealed. First, for some jump-length distributions with a finite variance, the approximate solutions obtained in the diffusion approximation can differ significantly from the exact solutions. Second, for some waiting time distributions with a finite mean, the times of first passage and the times of first arrival can significantly depend on the ensemble under consideration. In particular, the mean first-passage time corresponding to the stationary ensemble can be significantly greater than the mean first-passage time corresponding to the nonaged ensemble. Third, for any continuous distribution of jump lengths, the probability of first arrival is zero for a point-like target. This last result is contrary to existing opinion, but it is consistent with the fact that a single point has a probability measure equal to zero in the probability space defined by a continuous distribution of jump lengths.

Inferring Kinetics and Entropy Production from Observable Transitions in Partially Accessible, Periodically Driven Markov Networks

For a network of discrete states with a periodically driven Markovian dynamics, we develop an inference scheme for an external observer who has access to some transitions. Based on waiting-time distributions between these transitions, the periodic probabilities of states connected by these observed transitions and their time-dependent transition rates can be inferred. Moreover, the smallest number of hidden transitions between accessible ones and some of their transition rates can be extracted. We prove and conjecture lower bounds on the total entropy production for such periodic stationary states. Even though our techniques are based on generalizations of known methods for steady states, we obtain original results for those as well.

A stochastic fluid model approach to the stationary distribution of the maximal priority process

We consider a two-class priority queue in which the priority of a customer increases linearly at some constant, class-dependent rate. We describe the related maximum priority process { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } , which is of interest since the stationary distribution of the maximum priority process at the times of the commencement of service gives information about the waiting time distributions of the two classes of customers. In the case where service times are exponential, we map a two-class maximum priority process { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } to a tandem fluid queue, and use this mapping in order to derive the stationary distribution of { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } . Further, we extended these results to the maximum priority process in which service times are phase-type distributed. We illustrate the theory with numerical examples.

Waiting Time Distribution of Giant Pulses from the Crab Pulsar Modeled with a Non-stationary Poisson Process

We study the waiting time distribution of giant pulses from the Crab pulsar observed with the Nanshan 26-m radio telescope at a center frequency of 1556 MHz with 512 MHz bandwidth. The observations were performed over a duration of 12.6 hours with 32 μs sampling interval. Our analysis has led to the detection of 2097 giant pulses above a threshold of 10 σ, with flux density > 100 Jy. The occurrence rate of giant pulses is characterized by a highly intermittent giant pulse productivity in short clusters with high rates, separated by relatively long quiescent intervals with low occurrence rates, especially for the giant pulses associated with the interpulse emission. The distribution of waiting times between two subsequent giant pulses displays a power-law tail that can be modeled with a non-stationary Poisson process, which indicates that giant pulses are independent and random events. Distinct waiting time distributions between giant pulses in the main pulse and interpulse phases are presented, which implies that the giant pulse emission mechanisms maybe different in the opposite magnetic poles. The ramification for our understanding of the radio emission mechanisms is discussed.

Smart Insertion Strategies for Sustainable Operation of Shared Autonomous Vehicles

As shared autonomous vehicles (SAV) emerge as an economical and feasible mode of transportation in modern cities, effective optimization models are essential to simulate their service. Traditional optimization approaches, based on first-come-first-served principles, often result in sub-optimal outcomes and, more notably, can impact public transport (PT) operations by creating unnecessary competition. This study introduces four insertion strategies within the MATSim model of the Melbourne Metropolitan Area, addressing these challenges. Two strategies optimize SAV operations by considering overall network costs, and the other two make insertion decisions based on the available PT service in the network. The findings show that strategic insertions of the requests can significantly enhance SAV service quality by improving the vehicle load and decreasing vehicle and empty kilometers traveled per ride. The analysis indicates that these strategies are particularly effective for smaller fleet sizes, leading to an increased number of served rides and a more equitable distribution of wait times across the network, reflected in an improved Gini Index. The findings suggest that prioritization-based insertions significantly enhance service quality by prioritizing users with limited access to PT, ensuring that those with fewer PT options are served first, and encouraging a more integrated and sustainable urban transportation system.

Entropy production from waiting-time distributions for overdamped Langevin dynamics

For a Markovian dynamics on discrete states, the logarithmic ratio of waiting-time distributions between two successive, instantaneous transitions in forward and backward direction is a measure of time-irreversibility. It thus serves as an entropy estimator, which is exact in the case of a uni-cyclic network. We adopt this framework to overdamped Langevin dynamics, where such transitions have finite duration. By introducing milestones based on the observation of a particle at at least two milestones and an additional third event, we identify an entropy estimator that becomes exact for driven motion along a one-dimensional potential.

단수 휴가와 일량 제어정책을 갖는 이산시간 Geo/G/1 시스템의 대기시간 분석

In this paper, we derived the steady-state waiting time distribution of a discrete-time Geo/ G/1 queuing system under single vacation and D-policy. Customers such as WIP(work-in- process), parts, packets arrive at the system with Bernoulli process, and the server takes a single vacation with length of V when there are no more customers to service in the system. The restart condition is met when the total service time (workload, unfinished work) of customers waiting in the system at the end of vacation exceeds the workload threshold D. If the total amount of work in the system is less than or equal to D at the end of vacation, the server waits until the restart conditions are met and serves waiting customers. Otherwise, it starts to be busy as soon as the vacation ends. The average waiting time was also derived as a performance measure. Additionally, the computability of theoretical results was confirmed through a numerical example. The server control policy covered in this study can be applied in situations where it is desired to reduce operating costs per unit time due to high server restart costs (or energy consumption) in production and communication systems.

Non-Markovian gene expression

We study two non-Markovian gene-expression models in which protein production is a stochastic process with a fat-tailed nonexponential waiting time distribution (WTD). For both models, we find two distinct scaling regimes separated by an exponentially long time, proportional to the mean first passage time (MFPT) to a ground state (with zero proteins) of the dynamics, from which the system can only exit via a nonexponential reaction. At times shorter than the MFPT the dynamics are stationary and ergodic, entailing similarity across different realizations of the same process, with an increased Fano factor of the protein distribution, even when the WTD has a finite cutoff. Notably, at times longer than the MFPT the dynamics are nonstationary and nonergodic, entailing significant variability across different realizations. The MFPT to the ground state is shown to directly affect the average population sizes and we postulate that the transition to nonergodicity is universal in such non-Markovian models. Published by the American Physical Society 2024

Translocation of polymers through a wide-open conical pore

The transport of biomolecules across a cell membrane is an important phenomenon that plays a pivotal role in the functioning of biological cells. In this paper, we investigate such processes by modeling the translocation of polymers through a conical channel, directed from the wider opening to the narrow end of the conical channel. We use the molecular dynamics approach to study the problem. The effect of the different conical pore geometry and polymer lengths on translocation dynamics is determined from the behavior of the total translocation time, τ, and the waiting time distributions, w(s). The escape of polymer segments from the narrow end of the conical channel is tracked by studying the escape velocity profile (〈v i 〉). To demonstrate the asymmetric pore effects on the translocation dynamics, we compare the translocation process from both the terminals: the wider-opening and the narrow-end of the conical channel. We find striking differences in the translocation dynamics for both processes, which are in agreement with the experimental study. We have accounted for the effect of various rigidity, and increasing length of a polymer chain, on both types of processes. This computational study can be used to underline the translocation process from different conical pores.

Stochastic modeling of injection induced seismicity based on the continuous time random walk model

The spatiotemporal evolution of earthquakes induced by fluid injections into the subsurface can be erratic owing to the complexity of the physical process. To effectively mitigate the associated hazard and to draft appropriate regulatory strategies, a detailed understanding of how induced seismicity may evolve is needed. In this work, we build on the well-established continuous-time random walk (CTRW) theory to develop a purely stochastic framework that can delineate the essential characteristics of this process. We use data from the 2003 and 2012 hydraulic stimulations in the Cooper Basin geothermal field that induced thousands of microearthquakes to test and demonstrate the applicability of the model. Induced seismicity in the Cooper Basin shows all the characteristics of subdiffusion, as indicated by the fractional order power-law growth of the mean square displacement with time and broad waiting-time distributions with algebraic tails. We further use an appropriate master equation and the time-fractional diffusion equation to map the spatiotemporal evolution of seismicity. The results show good agreement between the model and the data regarding the peak earthquake concentration close to the two injection wells and the stretched exponential relaxation of seismicity with distance, suggesting that the CTRW model can be efficiently incorporated into induced seismicity forecasting.

Exact results for gene-expression models with general waiting-time distributions.

Complex molecular details of transcriptional regulation can be coarse-grained by assuming that reaction waiting times for promoter-state transitions, the mRNA synthesis, and the mRNA degradation follow general distributions. However, how such a generalized two-state model is analytically solved is a long-standing issue. Here we first present analytical formulas of burst-size distributions for this model. Then, we derive an iterative equation for the mRNA moment-generating function, by which mRNA raw and binomial moments of any order can be conveniently calculated. The analytical results obtained in the special cases of phase-type waiting-time distributions not only provide insights into the mechanisms of complex transcriptional regulations but also bring conveniences for experimental data-based statistical inferences.

Wait-time distributions for photoelectric detection of light

Wait-time distributions for photoelectric detection of light

The Propagation of Fast Radio Bursts in the Magnetosphere Shapes Their Waiting-time and Flux Distributions

The field of fast radio bursts (FRBs) has entered the age of fine characterization as observational results from different radio telescopes become more and more abundant. The large FRB sample is suitable for a statistical study. There is an interesting finding that the waiting-time distributions of very active repeating FRBs show a universal double-peaked feature, with left peaks lower than right ones. Assuming these two peaks are independent and initially comparable, we show that the observed asymmetric shape can be ascribed to the propagational effect in the magnetosphere. An FRB passing through the magnetized plasma will induce the circular motion of charged particles to form a current loop. This further leads to an induced magnetic field in the opposite direction with respect to the background field. As the effective field strength changes, the scattering absorption probability of the following FRB will be influenced. The absorption can be important under certain physical conditions and bursts with smaller time lags are easier to be absorbed. Also, there will be an imprint on the flux distribution as the scattering optical depth depends on burst luminosity as well.

A note on [formula omitted] queueing system

This article investigates the discrete-time GeoX/G/1 queue with an early arrival system. The system length distributions at outside observer’s, post-departure and random epochs as well as the waiting time distribution of an arbitrary customer in a batch are derived. The system of difference equations developed using the supplementary variable technique allows us to directly compute the probability generating functions for these distributions. Some numerical results are provided in tabular form.

A minimal kinetic model for the interpretation of complex catalysis in single enzyme molecules.

Multi-exponential waiting-time distribution and randomness parameter greater than unity ascribe dynamic disorder in single-enzyme catalysis corroborated to the interplay of transforming conformers [English et al., Nat. Chem. Biol., 2006, 2, 87]. The associated multi-state model of enzymatic turnovers with statically heterogeneous catalytic rates misdescribes the non-linear uprising of the randomness parameter from unity in relation to the attributes of the fall-offs of the waiting-time distribution at different substrate concentrations. To resolve this crucial issue, we first employ a comprehensive stochastic reaction scenario and further rationalize and work out the minimal indispensable dynamic-disorder model that ensures the foregoing relationship upon comparison with the data. We elucidate that specific disregard for the transition rate coefficients in the multi-state model on account of the especially slow conformational transitions is the underlying reason for not achieving interrelation between the observables.

A Monte Carlo simulation of tracer diffusion in amorphous polymers.

Tracer diffusion in amorphous polymers is a sought-after quantity for a range of technological applications. In this regard, a quantitative description of the so-called decoupling from the reverse proportionality between viscosity and diffusion coefficient into a fractional one remains a challenge requiring a deeper insight. This work employs a Monte Carlo simulation framework in 3 dimensions to investigate the consequences of different scenarios for estimating this fractional exponent on the diffusion coefficient of tracers in polymers near glass transition. To this end, we adopted a continuous-time random walk model for tracer diffusion in the supercooled liquid state. The waiting time distribution of the diffusants was computed based on the rotational correlation times of the polymer. This proposed procedure is of particular interest because it brings the quantity of waiting time (and its statistics) in connection with the measurable observable of rotational times. In the framework of our simulations the aforementioned fractional exponent appears in the relation between the diffusant's waiting time and the rotational time of the diffusion medium. A limited comparison with experimental diffusivities from the literature revealed a reasonable agreement with a fractional exponent on the basis of the molar volumes of the diffusant and the monomeric unit. Finally, an analysis of time-averaged mean squared displacement pointed to normal Brownian dynamics for tracer diffusion in polymers above the glass transition temperature.

Scaling of earthquake waiting time distributions in northern Chile

SUMMARY In this study, we examine the scaling of earthquake waiting time distributions in Northern Chile. For this purpose, we analysed 7-yr high spatial resolution and low completeness magnitude IPOC seismic catalogue and 43-yr USGS catalogue. A unified earthquake moment–space–time scaling relation is empirically evaluated by calculating waiting times for different ranges of magnitude and epicentral area linear dimension and estimating the associated scaling coefficients, β analogue to b-value and γ, the correlation fractal dimension. We find a scaling function that can be characterized with three distinct regions, regions whose behaviour depends on whether seismicity is in the coastal area or at intermediate depth. Moreover, high resolution localizations from the IPOC catalogue allows us to further observe differences: the lower plane seismicity of the double seismic zone behaves just like that at intermediate depth, while the crustal upper plate and the upper plane from the double seismic zone seismicity behaves more like interface seismicity. Thus, the earthquake waiting time distribution primarily depends on whether seismicity is located in a crust-on-crust or in a crust–mantle system: having, respectively, high/low correlated behaviour in the short scale region, non-exponential/exponential decay in the transition middle region and, in all cases, long-term clustering with a slower than exponential decay in the long scale.

Some reflected autoregressive processes with dependencies

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered.