Search Results (18)

Traffic in large cities is increasing due to continuous urbanization, the construction of new housing complexes and the accompanying new street network. The growth of cities creates prerequisites for increasing the intensity of transport, pedestrian, and bicycle flows, especially during peak periods. To improve the conditions in which traffic flows, it is necessary to introduce an effective method for reducing delays that arise at intersections, especially those regulated by traffic light systems. One of the possible approaches to this is to coordinate the operation of traffic light systems. The main thing in this is to determine relatively accurate times for the movement of individual flows, for which adequate traffic models are needed. This article presents a model of the movement of transport flows when starting from the first intersection in a coordinated mode of operation of traffic light systems. This is of particular importance when determining the times of individual signals and, above all, has an impact on the moment for switching on the permitting signal at the next intersection. The presented model aims to provide an opportunity to determine accurate times of passage of vehicles through consecutive intersections that operate in a coordinated mode of traffic light systems. Full article

First-exit problems are studied for two-dimensional diffusion processes with jumps according to a Poisson process. The size of the jumps is distributed as an exponential random variable. We are interested in the random variable that denotes the first time that the sum of the two components of the process leaves a given interval. The function giving the probability that the process will leave the interval on its left-hand side satisfies a partial differential–integral equation. This equation is solved analytically in particular cases by making use of the method of similarity solutions. The problem of calculating the mean and the moment-generating function of the first-passage time random variable is also considered. The results obtained have applications in various fields, notably, financial mathematics and reliability theory. Full article

We analyze a discrete-time random walk on the vertices of an unbounded two-dimensional L-lattice. We determine the probability generating function, and we prove the independence of the coordinates. In particular, we find a relation of each component with a one-dimensional biased random walk with time changing. Therefore, the transition probabilities and the main moments of the random walk can be obtained. The asymptotic behavior of the process is studied, both in the classical sense and involving the large deviations theory. We investigate first-passage-time problems of the random walk through certain straight lines, and we determine the related probabilities in closed form and other features of interest. Finally, we develop a simulation approach to study the first-exit problem of the process thought ellipses. Full article

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone. Full article

Workflows orchestrate a collection of computing tasks to form a complex workflow logic. Different from the traditional monolithic workflow management systems, modern workflow systems often manifest high throughput, concurrency and scalability. As service-based systems, execution time monitoring is an important part of maintaining the performance for those systems. We developed a trace profiling approach that leverages quantitative verification (also known as probabilistic model checking) to analyse complex time metrics for workflow traces. The strength of probabilistic model checking lies in the ability of expressing various temporal properties for a stochastic system model and performing automated quantitative verification. We employ semi-Makrov chains (SMCs) as the formal model and consider the first passage times (FPT) measures in the SMCs. Our approach maintains simple mergeable data summaries of the workflow executions and computes the moment parameters for FPT efficiently. We describe an application of our approach to AWS Step Functions, a notable workflow web service. An empirical evaluation shows that our approach is efficient for computer high-order FPT moments for sizeable workflows in practice. It can compute up to the fourth moment for a large workflow model with 10,000 states within 70 s. Full article

A simple model is built to evaluate quantitatively the individual feeling of the passage of time using a sociophysics approach. Given an objective unit of time like the year, I introduce an individualized mirror-subjective counterpart, which is inversely proportional to the number of objective units of time already experienced by a person. An associated duration of time is then calculated. Past and future individual horizons are also defined together with a subjective speed of time. Furthermore, I rescale the subjective unit of time by activating additional clocks connected to ritualized socializations, which mark and shape the specific times of an individual throughout their life. The model shows that without any ritual socialization, an individual perceives their anticipated life as infinite via a “soft” infinity. The past horizon is also perceived at infinity but with a “hard” infinity. However, the price for the first ritualized socialization is to exit eternity in terms of the anticipated future with the simultaneous reward of experiencing a finite moment of infinity analogous to that related to birth. I then extend the model using a power law of the number of past objective units of time to mitigate the phenomenon of shrinking of time. The findings are sound and recover common feelings about the passage of time over a lifetime. In particular, the fact that time passes more quickly with aging with a concomitant slowing down of the speed of time. Full article

The quantum Wigner function and non-equilibrium equation for a microscopic particle in one spatial dimension ($1D$) subject to a potential and a heat bath at thermal equilibrium are considered by non-trivially extending a previous analysis. The non-equilibrium equation yields a general hierarchy for suitable non-equilibrium moments. A new non-trivial solution of the hierarchy combining the continued fractions and infinite series thereof is obtained and analyzed. In a short thermal wavelength regime (keeping quantum features adequate for chemical reactions), the hierarchy is approximated by a three-term one. For long times, in turn, the three-term hierarchy is replaced by a Smoluchovski equation. By extending that $1D$ analysis, a new model of the growth (polymerization) of a molecular chain (template or $te$) by binding an individual unit (an atom) and activation by a catalyst is developed in three spatial dimensions ($3D$). The atom, $te$, and catalyst move randomly as solutions in a fluid at rest in thermal equilibrium. Classical statistical mechanics describe the $te$ and catalyst approximately. Atoms and bindings are treated quantum-mechanically. A mixed non-equilibrium quantum–classical Wigner–Liouville function and dynamical equations for the atom and for the $te$ and catalyst, respectively, are employed. By integrating over the degrees of freedom of $te$ and with the catalyst assumed to be near equilibrium, an approximate Smoluchowski equation is obtained for the unit. The mean first passage time (MFPT) for the atom to become bound to the $te$, facilitated by the catalyst, is considered. The resulting MFPT is consistent with the Arrhenius formula for rate constants in chemical reactions. Full article

We study a reliability system subject to occasional random shocks of random magnitudes $W_0,W_1,W_2,\dots$ occurring at times ${\tau }_0,{\tau }_1,{\tau }_2,\dots$. Any such shock is harmless or critical dependent on $W_k\le H$ or $W_k>H$, given a fixed threshold H. It takes a total of N critical shocks to knock the system down. In addition, the system ages in accordance with a monotone increasing continuous function $\delta$, so that when $\delta \left(T\right)$ crosses some sustainability threshold D at time T, the system becomes essentially inoperational. However, it can still function for a while undetected. The most common way to do the checking is at one of the moments ${\tau }_1,{\tau }_2,\dots$ when the shocks are registered. Thus, if crossing of D by $\delta$ occurs at time $T\in \left({\tau }_k,{\tau }_{k+1}\right)$, only at time ${\tau }_{k+1}$, can one identify the system’s failure. The age-related failure is detected with some random delay. The objective is to predict when the system fails, through the Nth critical shock or by the observed aging moment, whichever of the two events comes first. We use and embellish tools of discrete and continuous operational calculus ($\mathcal{D}$-operator and Laplace–Carson transform), combined with first-passage time analysis of random walk processes, to arrive at fully explicit functionals of joint distributions for the observed lifetime of the system and cumulative damage to the system. We discuss various special cases and modifications including the assumption that D is random (and so is T). A number of examples and numerically drawn figures demonstrate the analytic tractability of the results. Full article

Let $Y\left(t\right)$ be a one-dimensional jump-diffusion process and $X\left(t\right)$ be defined by $\mathrm{d}X\left(t\right)=\rho \left[X\right(t),Y(t\left)\right]\mathrm{d}t$, where $\rho (·,·)$ is either a strictly positive or negative function. First-passage-time problems for the degenerate two-dimensional process $\left(X\right(t),Y(t\left)\right)$ are considered in the case when the process leaves the continuation region at the latest at the moment of the first jump, and the problem of optimally controlling the process is treated as well. A particular problem, in which $\rho \left[X\right(t),Y(t\left)\right]=Y\left(t\right)-X\left(t\right)$ and $Y\left(t\right)$ is a standard Brownian motion with jumps, is solved explicitly. Full article

We derive the explicit price of the perpetual American put option canceled at the last-passage time of the underlying above some fixed level. We assume that the asset process is governed by a geometric spectrally negative Lévy process. We show that the optimal exercise time is the first moment when the asset price process drops below an optimal threshold. We perform numerical analysis considering classical Black–Scholes models and the model where the logarithm of the asset price has additional exponential downward shocks. The proof is based on some martingale arguments and the fluctuation theory of Lévy processes. Full article

For the Wiener, Ornstein–Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the zero state. A detailed study of the asymptotic average of local time in the presence of an absorbing boundary is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions in the presence of an absorbing boundary in the zero state and between the first-passage time densities through zero for Wiener, Ornstein–Uhlenbeck, and Feller processes are proven. Moreover, some asymptotic results between the first-passage time densities through zero state are derived. Various numerical computations are performed to illustrate the role played by parameters. Full article

This article analyzes the behavior of a Brownian fluctuation process under a mixed strategic game setup. A variant of a compound Brownian motion has been newly proposed, which is called the Shifted Brownian Fluctuation Process to predict the turning points of a stochastic process. This compound process evolves until it reaches one step prior to the turning point. The Shifted Brownian Fluctuation Game has been constructed based on this new process to find the optimal moment of actions. Analytically tractable results are obtained by using the fluctuation theory and the mixed strategy game theory. The joint functional of the Shifted Brownian Fluctuation Process is targeted for transformation of the first passage time and its index. These results enable us to predict the moment of a turning point and the moment of actions to obtain the optimal payoffs of a game. This research adapts the theoretical framework to implement an autonomous trader for value assets including stocks and cybercurrencies. Full article

We consider the first-passage time problem for the Feller-type diffusion process, having infinitesimal drift $B_1(x,t)=\alpha \left(t\right)x+\beta \left(t\right)$ and infinitesimal variance $B_2(x,t)=2r\left(t\right)x$, defined in the space state $[0,+\infty )$, with $\alpha \left(t\right)\in \mathbb{R}$, $\beta \left(t\right)>0$, $r\left(t\right)>0$ continuous functions. For the time-homogeneous case, some relations between the first-passage time densities of the Feller process and of the Wiener and the Ornstein–Uhlenbeck processes are discussed. The asymptotic behavior of the first-passage time density through a time-dependent boundary is analyzed for an asymptotically constant boundary and for an asymptotically periodic boundary. Furthermore, when $\beta \left(t\right)=\xi r\left(t\right)$, with $\xi >0$, we discuss the asymptotic behavior of the first-passage density and we obtain some closed-form results for special time-varying boundaries. Full article

We consider the problem of the first passage time T of an inhomogeneous geometric Brownian motion through a constant threshold, for which only limited results are available in the literature. In the case of a strong positive drift, we get an approximation of the cumulants of T of any order using the algebra of formal power series applied to an asymptotic expansion of its Laplace transform. The interest in the cumulants is due to their connection with moments and the accounting of some statistical properties of the density of T like skewness and kurtosis. Some case studies coming from neuronal modeling with reversal potential and mean reversion models of financial markets show the goodness of the approximation of the first moment of T. However hints on the evaluation of higher order moments are also given, together with considerations on the numerical performance of the method. Full article

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance are included in the considered class. For these processes, the transition probability density function is explicitly determined. Moreover, simulation procedures are applied to the diffusion processes obtained starting from Wiener and Ornstein–Uhlenbeck processes. Specific examples in which the infinitesimal moments include periodic functions are discussed. Full article