Distribution Function Of Waiting Time Research Articles

The Poissonian Origin of Power Laws in Solar Flare Waiting Time Distributions

Abstract In this study we aim for a deeper understanding of the power-law slope, α, of waiting time distributions. Statistically independent events with linear behavior can be characterized by binomial, Gaussian, exponential, or Poissonian size distribution functions. In contrast, physical processes with nonlinear behavior exhibit spatiotemporal coherence (or memory) and “fat tails” in their size distributions that fit power-law-like functions, as a consequence of the time variability of the mean event rate, as demonstrated by means of Bayesian block decomposition in the work of Wheatland et al. In this study we conduct numerical simulations of waiting time distributions N(τ) in a large parameter space for various (polynomial, sinusoidal, Gaussian) event rate functions λ(t), parameterized with an exponent p that expresses the degree of the polynomial function λ(t) ∝ t p . We derive an analytical exact solution of the waiting time distribution function in terms of the incomplete gamma function, which is similar to a Pareto type II function and has a power-law slope of α = 2 + 1/p, in the asymptotic limit of large waiting times. Numerically simulated random distributions reproduce this theoretical prediction accurately. Numerical simulations in the nonlinear regime (p ≥ 2) predict power-law slopes in the range of 2.0 ≤ α ≤ 2.5. The self-organized criticality model yields a prediction of α = 2. Observations of solar flares and coronal mass ejections (over at least a half solar cycle) are found in the range of α obs ≈ 2.1–2.4. Deviations from strict power-law functions are expected due to the variability of the flare event rate λ(t), and deviations from theoretically predicted slope values α occur due to the Poissonian weighting bias of power-law fits.

Continuous time random walk model for non-uniform bed-load transport with heavy-tailed hop distances and waiting times

Bed-load transport along widely graded river-beds typically exhibits anomalous dynamics, whose efficient characterization may require parsimonious stochastic models with pre-defined statistics involving the waiting time and hop distance distributions for sediment particles. This study employs a continuous time random walk (CTRW) model to characterize bed-load particle motions on a widely graded gravel-bed with cluster microforms built in our lab. Flume experiments guide the selection of the Mittag-Leffler (M-L) function as the waiting time distribution function, and the Lévy α-stable density for the hop distance distribution function in the CTRW model. Monte Carlo simulations show that the resulting CTRW model can well capture the observed flume experimental data (with either a continuous or an instantaneous source) with coexisting super- and sub-dispersion behaviors in the bed-load transport process. Analyses further discover the dual impact of clusters on the dynamics of fine sediment particles. Some particles are more likely to be blocked or trapped by clusters, while others have a high probability to be accelerated by the flow accelerating belt between the clusters. Therefore, with proper statistical distributions and relevant parameters for sediment waiting times and hop distances, the CTRW model may efficiently capture the complex dynamics in sediment transport.

Generic Schemes for Single-Molecule Kinetics. 3: Self-Consistent Pathway Solutions for Nonrenewal Processes.

We recently developed a pathway analysis framework (paper 1) for describing single-molecule kinetics for renewal (i.e., memoryless) processes based on the decomposition of a kinetic scheme into generic structures. In our approach, waiting time distribution functions corresponding to such structures are expressed in terms of self-consistent pathway solutions and concatenated to form measurable probability distribution functions (PDFs), affording a simple way to decompose and recombine a network. Here, we extend this framework to nonrenewal processes, which involve correlations between events, and employ it to formulate waiting time PDFs, including the first-passage time PDF, for a general kinetic network model. Our technique does not require the assumption of Poissonian kinetics, permitting a more general kinetic description than the usual rate approach, with minimal topological restrictiveness. To demonstrate the usefulness of this technique, we provide explicit calculations for our general model, which we adapt to two generic schemes for single-enzyme turnover with conformational interconversion. For each generic scheme, wherein the intermediate state(s) need not undergo Poissonian decay, the functional dependence of the mean first-passage time on the concentration of an external substrate is analyzed. When conformational detailed balance is satisfied, the enzyme turnover rate (related to the mean first-passage time) reduces to the celebrated Michaelis-Menten functional form, consistent with our previous work involving a similar scheme with all rate processes, thereby establishing further generality to this intriguing result. Our framework affords a general and intuitive approach for evaluating measurable waiting time PDFs and their moments, making it a potentially useful kinetic tool for a wide variety of single-molecule processes.

Modeling of public transport waiting time indicator for the transport network of a large city

The need to develop and improve public passenger transport in major cities was noted. It was reflected that waiting time at bus stops is one of the factors that have a big impact on the passenger quality assessment of transport services. The results of an empirical study of the actual and anticipated waiting time at bus stops were given. It was noted that the reliability functions were used in the field of ride duration modeling, traffic restoration time after an accident, and length of making the decision to travel. The waiting time distribution functions using the lognormal function and the Weibull function were chosen. The results of modeling were objective, the dependent variables in it were the expected waiting time of passengers and the difference between the anticipated and the actual waiting time. The explanatory variables were sex, age, time period, purpose of the trip and the actual waiting time. The results of the research showed that the age, purpose of the trip and the time period influence the waiting time perception, prolong it and lead to its reassessment.

Generalized phase-space kinetic and diffusion equations for classical and dispersive transport

We formulate and solve a physically-based, phase space kinetic equation for transport in the presence of trapping. Trapping is incorporated through a waiting time distribution function. From the phase-space analysis, we obtain a generalized diffusion equation in configuration space. We analyse the impact of the waiting time distribution, and give examples that lead to dispersive or non-dispersive transport. With an appropriate choice of the waiting time distribution, our model is related to fractional diffusion in the sense that fractional equations can be obtained in the limit of long times. Finally, we demonstrate the application of this theory to disordered semiconductors.

Two-state theory of binned photon statistics for a large class of waiting time distributions and its application to quantum dot blinking.

A theoretical method is proposed for the calculation of the photon counting probability distribution during a bin time. Two-state fluorescence and steady excitation are assumed. A key feature is a kinetic scheme that allows for an extensive class of stochastic waiting time distribution functions, including power laws, expanded as a sum of weighted decaying exponentials. The solution is analytic in certain conditions, and an exact and simple expression is found for the integral contribution of "bright" and "dark" states. As an application for power law kinetics, theoretical results are compared with experimental intensity histograms from a number of blinking CdSe/ZnS quantum dots. The histograms are consistent with distributions of intensity states around a "bright" and a "dark" maximum. A gap of states is also revealed in the more-or-less flat inter-peak region. The slope and to some extent the flatness of the inter-peak feature are found to be sensitive to the power-law exponents. Possible models consistent with these findings are discussed, such as the combination of multiple charging and fluctuating non-radiative channels or the multiple recombination center model. A fitting of the latter to experiment provides constraints on the interaction parameter between the recombination centers. Further extensions and applications of the photon counting theory are also discussed.

Stretching of single poly-ubiquitin molecules revisited: Dynamic disorder in the non-exponential unfolding kinetics

A theoretical framework based on a generalized Langevin equation (GLE) with fractional Gaussian noise (fGn) and a power-law memory kernel is presented to describe the non-exponential kinetics of the unfolding of a single poly-ubiquitin molecule under a constant force [T.-L. Kuo, S. Garcia-Manyes, J. Li, I. Barel, H. Lu, B. J. Berne, M. Urbakh, J. Klafter, and J. M. Fernández, Proc. Natl. Acad. Sci. U.S.A. 107, 11336 (2010)]. Such a GLE-fGn strategy is made on the basis that the pulling coordinate variable x undergoes subdiffusion, usually resulting from conformational fluctuations, over a one-dimensional force-modified free-energy surface U(x, F). By using the Kramers' rate theory, we have obtained analytical formulae for the time-dependent rate coefficient k(t, F), the survival probability S(t, F) as well as the waiting time distribution function f(t, F) as functions of time t and force F. We find that our results can fit the experimental data of f(t, F) perfectly in the whole time range with a power-law exponent γ = 1/2, the characteristic of typical anomalous subdiffusion. In addition, the fitting of the survival probabilities for different forces facilitates us to reach rather reasonable estimations for intrinsic properties of the system, such as the free-energy barrier and the distance between the native conformation and the transition state conformation along the reaction coordinate, which are in good agreements with molecular dynamics simulations in the literatures. Although static disorder has been implicated in the original work of Kuo et al., our work suggests a sound and plausible alternative interpretation for the non-exponential kinetics in the stretching of poly-ubiquitin molecules, associated with dynamic disorder.

Nonparametric regression of state occupation, entry, exit, and waiting times with multistate right‐censored data

We construct nonparametric regression estimators of a number of temporal functions in a multistate system based on a continuous univariate baseline covariate. These estimators include state occupation probabilities, state entry, exit, and waiting (sojourn) time distribution functions of a general progressive (e.g., acyclic) multistate model. We subject the data to right censoring, and the censoring mechanism is explainable by observable covariates that could be time dependent. The resulting estimators are valid even if the multistate process is non-Markov. We study the performance of the estimators in two simulation settings. We establish large sample consistency of these estimators. We illustrate our estimators using a data set on bone marrow transplant recipients.

Stochastic hybrid simulation

This is a summary of the author’s PhD thesis supervised by Bart Scheers and Antoine Van de Capelle and defended on 20 November 2009 at the Royal Military Academy, Brussels. The thesis (https://lirias.kuleuven.be/handle/123456789/246952) is written in English and is available from the author upon request. This work deals with an extension to the hybrid simulation paradigm, i.e. the combination of event-driven simulation and analytical modelling, applied to packet telecommunication networks. In order to speed up the simulation only a small part of all packets, the foreground traffic, is processed in an event-driven way. On each arrival of a foreground packet, the waiting time of the packet is sampled from the virtual waiting time distribution function of the combined foreground and background traffic. This distribution function is stochastically modelled by the exact large deviations asymptotic of the virtual waiting time in a many sources regime. This novel methodology is not only valid for wired point-to-point queueing networks having a fixed transmission capacity, but it can also be applied to queueing networks for which the transmission capacity varies with the traffic load of all the elements in the network. The results obtained by the stochastic hybrid simulator are compared to full-blown event-driven simulations. An important reduction in simulation run-time is gained without sacrificing accuracy.

A necessary condition for the emergence of diffusion instability in media with nonclassical diffusion

A necessary condition is derived for the emergence of diffusion instability in media in which diffusion does not obey classical Fick laws. The equations derived by Yadav and Horsthemke [Phys. Rev. E 74, 066118 (2006)] using the continuous-time random walk model are employed as equations simulating reaction—diffusion processes. The waiting-time distribution function is represented by the sum of a finite number of exponents. It is shown that passage to the diffusion limit in the time variable is an incorrect operation if it is used to analyze diffusion instability in media with a distribution function that differs from the Poisson distribution function.

Toward a predictive model of Ca2+ puffs

We investigate the key characteristics of Ca(2+) puffs in deterministic and stochastic frameworks that all incorporate the cellular morphology of IP(3) receptor channel clusters. In the first step, we numerically study the Ca(2+) liberation in a three-dimensional representation of a cluster environment with reaction-diffusion dynamics in both the cytosol and the lumen. These simulations reveal that Ca(2+) concentrations at a releasing cluster range from 80 to 170 microM and equilibrate almost instantaneously on the time scale of the release duration. These highly elevated Ca(2+) concentrations eliminate Ca(2+) oscillations in a deterministic model of an IP(3)R channel cluster at physiological parameter values as revealed by a linear stability analysis. The reason lies in the saturation of all feedback processes in the IP(3)R gating dynamics, so that only fluctuations can restore experimentally observed Ca(2+) oscillations. In this spirit, we derive master equations that allow us to analytically quantify the onset of Ca(2+) puffs and hence the stochastic time scale of intracellular Ca(2+) dynamics. Moving up the spatial scale, we suggest to formulate cellular dynamics in terms of waiting time distribution functions. This approach prevents the state space explosion that is typical for the description of cellular dynamics based on channel states and still contains information on molecular fluctuations. We illustrate this method by studying global Ca(2+) oscillations.

Generic Schemes for Single-Molecule Kinetics. 1: Self-Consistent Pathway Solutions for Renewal Processes

In this paper, we discuss a strategy for reducing a complex single molecule kinetic process to a set of generic structures (motifs) that are building blocks for a general kinetic scheme. In general, these motifs have complex kinetics (i.e., waiting time distribution functions) which are composed of fundamental kinetic steps. (1) First, we treat four different experimental single molecule measurements within both the usual kinetic framework (i.e., using the rate matrix) and the waiting time distribution function framework. The two frameworks are then shown to be equivalent and can be formulated on the basis of the first passage time distribution function of monitored single molecule events. (2) Second, to calculate this basic quantity, we decompose a complex kinetic scheme with the help of two kinetic motifs, sequential and branching, and derive self-consistent equations by convoluting waiting time distributions and first passage time distribution(s) along the reaction pathway(s). (3) As examples, two experimental systems, a chain reaction model with a special case of enzymatic reactions and a general kinetic model for fluorescence emission, are analyzed on the basis of a generic scheme composed of a monitored link, controlled link, and unknown link, each representing a possible subscheme associated with a complex waiting time distribution function. As a result, single molecule measurements of the generic scheme retain the same functional form when a kinetic link is altered within a subscheme, and different measurements can be classified and analyzed within the same framework. (4) Finally, to explore the physical reasons for nonexponential waiting time distribution, we use the example of blinking phenomena to discuss several scenarios of dynamic and static disorder and their implications for observed memory effects. The self-consistent pathway formalism is presented in this paper for renewal processes and will be generalized to nonrenewal processes with memory effects in a future publication.

Reaction-diffusion wave fronts on comblike structures

From the Hamilton-Jacobi formalism, an explicit expression for the speed of wave front propagation along the backbone of comblike structures is obtained. This expression, through the waiting-time distribution function, takes into account the number of sites and their distribution in the secondary branches. Our theoretical results are supported by numerical simulations of the reaction random-walk process on the structure. Finally, a more complex situation such as the Peano basin structure is also considered, both theoretically and numerically, exhibiting a good agreement too.

Relaxation in heterogeneous systems: A rare events dominated phenomenon

Abstract We have derived a general two-power-law relaxation function for heterogeneous materials using the maximum entropy principle for nonextensive systems. The power law exponents of the relaxation function are simply related to a global fractal parameter α and for large time to the entropy nonextensivity parameter q. For intermediate times the relaxation follows a stretched exponential behavior. The asymptotic power law behaviors both in the time and the frequency domains coincide with those of the Weron generalized dielectric function derived in the stochastic theory from an extension of the Levy central limit theorem. These results are in full agreement with the Jonscher universality principle and trace the origin of the large t power law universality (with system dependent exponent α and q ) to the scaling behavior of the extreme value distribution function of the effective macroscopic waiting time and the fluctuation of the number of relaxing entities.

Waiting time distribution for the M∕M∕m queue

A novel method is presented for the calculation of the waiting time distribution function for the M/M/m queue. It is shown that the conditional waiting time obeys an Erlang distribution with rate mμ, where μ is the service rate of a server. An explicit closed form solution is obtained by means of the probability density function of the Erlang distribution. The derivation of the result proved to be very simple. The significance of Khintchine's method and its close relation to the proposed method is pointed out. It is also shown that the waiting time distribution can be obtained from Takacs's waiting time distribution for the G/M/m queue as a special case. This reveals some insight into the significance of Takacs's more general, but rather complex, result.

An analysis of multiple-class vacation queues with individual thresholds

This study examines a multiple-class M/G/1 queue with multiple thresholds. Each class of customer arrives according to an independent Poisson process, and each class has its own threshold. The idle server is reactivated as soon as any one of the thresholds is reached. Both FCFS and non-preemptive priority cases are considered. The Laplace–Stieltjes transform of the waiting time distribution function and the mean waiting time for each class of customers are derived.

Dissipation and fluctuation for a randomly kicked particle: Normal and anomalous diffusion

A stochastic model is considered which describes the motion of a one-dimensional classical particle kicked at random by heat bath particles. The effect of the distribution of the time intervals between the collision events on the motion of the test particle is studied. It is shown that the general pattern strongly depends on the long time behavior of the waiting time distribution function. The Brownian or normal diffusive type of motion is obtained when the distribution function has finite moments. If, however, all these moments diverge the diffusion has an anomalous character. The relaxation is characterized by power law dependences, while the mean square displacement is domin ated by the ballistic contribution. The second, superdiffusive term can be related to the momentum relaxation by a relation differing from the conventional Einstein relation. It is also important to emphasize that the information on the initial state of the test particle is essential for the long time behavior of the mean square displacement for the proper formulation of the dissipation-fluctuation relation.

Efficient Computation of Probabilities of Events Described by Order Statistics and Applications to Queue Inference

This paper derives recursive algorithms for efficiently computing event probabilities related to order statistics and applies the results in a queue inferencing setting. Consider a set of N i.i.d. random variables in [0, 1]. When the experimental values of the random variables are arranged in ascending order from smallest to largest, one has the order statistics of the set of random variables. Both a forward and a backward recursive O(N3) algorithm are developed for computing the probability that the order statistics vector lies in a given N-rectangle. The new algorithms have applicability in inferring the statistical behavior of Poisson arrival queues, given only the start and stop times of service of all N customers served in a period of continuous congestion. The queue inference results extend the theory of the “Queue Inference Engine” (QIE), originally developed by Larson in 1990 (Larson, R. C. 1990. The queue inference engine: Deducing queue statistics from transactional data. Management Sci. 36(5) 586–601, and 1991. The queue inference engine: Addendum. Management Sci. 37(6) 1062.). The methodology is extended to a third O(N3) algorithm, employing both forward and backward recursion, that computes the conditional average probability (averaged over all N customers in a congestion period) that the in-queue wait is less than t minutes, given the departure time data and assuming first come, first served service. To our knowledge, this result is the first O(N3) exact algorithm for computing points on the in-queue waiting time distribution function, conditioned on the start and stop time data. The paper concludes with an extension to the computation of certain correlations of in-queue waiting times. Illustrative computational results are included throughout. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Nonparametric Estimation of the Stationary Waiting Time Distribution Function for the $GI/G/1$ Queue

The $GI/G/1$ queueing model is regarded as a functional that maps the service and interarrival time distribution functions onto the stationary waiting time distribution function. By considering the output of the functional when it is applied to nonparametric estimators of the input distribution functions, we obtain a nonparametric estimator of the stationary waiting time distribution function. Using appropriate continuity and differentiability properties of the functional, we show that statistical properties of the input estimators carry over to corresponding properties for the stationary waiting time distribution function estimator.

A new model for polymer melts and concentrated solutions

A new mesoscopic model is presented for polymer melts and concentrated solutions. It is a single Kramers chain model in which elementary motions of the Orwoll–Stockmayer type are allowed. However, for this model, the bead jumps are no longer given by a Markovian probability, but rather are described by ‘‘a waiting time distribution function.’’ Such a distribution is supposed to occur when the chain is ‘‘frozen’’ in space until a ‘‘gap’’ in the solution or melt meets with the bead or chain segment. The time a bead must wait to jump is given by a distribution function with a single adjustable parameter β, which describes the long-time behavior of the distribution: ∼1/t1+β . We find that the model predicts non-Fickian diffusion in agreement with experimental data and Fickian diffusion for longer times which scales with chain length as 1/N2/α−1, where α is a function of β. For β=1.3, D∼1/N2.28. The autocorrelation of the end-to-end vector of the chain is a stretched-exponential form with a time constant which scales as the length of the chain to the 3.3 power for β=1.3.