Stationary Distribution Research Articles

Supervised ML for Solving the GI/GI/1 Queue

We apply supervised learning to a general problem in queueing theory: using a neural net, we develop a fast and accurate predictor of the stationary system-length distribution of a GI/GI/1 queue—a fundamental queueing model for which no analytical solutions are available. To this end, we must overcome three main challenges: (i) generating a large library of training instances that cover a wide range of arbitrary interarrival and service time distributions, (ii) labeling the training instances, and (iii) providing continuous arrival and service distributions as inputs to the neural net. To overcome (i), we develop an algorithm to sample phase-type interarrival and service time distributions with complex transition structures. We demonstrate that our distribution-generating algorithm indeed covers a wide range of possible positive-valued distributions. For (ii), we label our training instances via quasi-birth-and-death(QBD) that was used to approximate PH/PH/1 (with phase-type arrival and service process) as labels for the training data. For (iii), we find that using only the first five moments of both the interarrival and service times distribution as inputs is sufficient to train the neural net. Our empirical results show that our neural model can estimate the stationary behavior of the GI/GI/1—far exceeding other available methods in terms of both accuracy and runtimes. History: Ram Ramesh, Area Editor for Data Science and Machine Learning. Funding: O. Baron received financial support from the Natural Sciences and Engineering Research Council of Canada (NERC) [Grant 458051]. D. Krass received financial support from the NERC [Grant 458395]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0263 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0263 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

INTEGRATED OF WEB APPLICATION RSHINY FOR MARKOV CHAIN AND ITS APPLICATION TO THE DAILY CASES OF COVID-19 IN WEST SUMATERA

Discrete-time of Markov chains, starting now referred to as Markov chains, have been widely used by previous researchers in predicting the phenomenon. The predictions were made by manual calculations and using separate software, including Maple, Matlab, and Microsoft Excel. The analysis takes a relatively long time, especially in calculating the number of transitions from each state. This research built an integrated R script for the Markov chain based on the web application RShiny to quickly, easily, and accurately predict a phenomenon. The Markov chain integrated R script is built via command-command to predict the day-n distribution with the n-step distribution and long-term probability using a stationary distribution. The RShiny web application built is limited to state two and three. The integrated web application RShiny for the Markov chain is used to predict the daily cases of COVID-19 in West Sumatra. Based on the analysis carried out in predicting the daily cases of COVID-19 in West Sumatra from March 26, 2020, to October 20, 2020, for the next three days and in the long term, the results show that there is a 51.2% probability of an increase in COVID-19 cases, a 43% probability that cases will decrease, and 5.8% chance of stagnant cases

Controlling smoking: A smoking epidemic model with different smoking degrees in deterministic and stochastic environments

Engaging in smoking not only leads to substantial health risks but also imposes considerable financial burdens. To deepen our understanding of the mechanisms behind smoking transmission and to address the tobacco epidemic, we examined a five-dimensional smoking epidemic model that accounts for different degrees of smoking under both deterministic and stochastic conditions. In the deterministic case, we determine the basic reproduction number, analyze the stability of equilibria with and without smoking, and investigate the existence of saddle–node bifurcation. Our analysis reveals that the basic reproduction number cannot completely determine the existence of smoking, and the model possesses bistability, indicating its dynamic is susceptible to interference from environmental noises. In the stochastic case, we establish sufficient conditions for the ergodic stationary distribution and the elimination of smokers by constructing appropriate Lyapunov functions. Numerical simulations suggest that the effects of inevitable random fluctuations in the natural environment on controlling the smoking epidemic may be beneficial, harmful, or negligible, which are closely related to the noise intensities, initial smoking population sizes, and the effective exposure rate of smoking transmission (β). Given the uncontrollable nature of environmental random effects, effective smoking control strategies can be achieved by: (1) accurate monitoring of initial smoking population sizes, and (2) implementing effective measures to reduce β. Therefore, it is both effective and feasible to implement a complete set of strong MPOWER measures to control smoking prevalence.

The Boltzmann equation for plane Couette flow

In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at y=\pm L moving relative to each other with opposite velocities (\pm \alpha L,0,0) along the x -direction. Assuming that the stationary state takes the specific form of F(y,v_{x}-\alpha y,v_{y},v_{z}) with the x -component of the molecular velocity sheared linearly along the y -direction, such steady flow is governed by a boundary value problem for a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous nonmoving diffuse reflection boundary condition. In the case of the Maxwell molecule collisions, we establish the existence of spatially inhomogeneous nonequilibrium stationary solutions to the steady problem for any small enough shear rate \alpha>0 via an elaborate perturbation approach using Caflisch’s decomposition together with Guo’s L^{\infty}\cap L^{2} theory. The result indicates a polynomial tail at large velocities for the stationary distribution. Moreover, the large time asymptotic stability of the stationary solution with exponential convergence is also obtained and as a consequence the nonnegativity of the steady profile is justified.

Dual-channel call center model with exchange of requests and repeat calls

Computer simulation of stationary distribution of states in a call center for a two-channel RQ system with exchange of requests is presented. Such systems are becoming increasingly relevant due to the parallel use of both human dispatchers and voice intelligent bots for servicing calls. The simulation is based on solving the system of stationary Chapman-Kolmogorov equations for the Markov process describing the RQ-system. The system of equations proposed in the work differs from previous models by the presence of an exchange of requests between service channels in accordance with customer preferences. It takes into account the possibility of random repeated calls within a given average time. Service time in channels and delay time of requests in orbits have exponential distribution laws. Requests in orbit have the property of impatience, i.e. leave the system after some random time. To find the stationary distribution of states in orbits, the iterative numerical Gauss-Seidel method is used, which ensures fast convergence of calculations. Each channel has its own orbit of requests. The accuracy of the solution is controlled by increasing the maximum number of requests in orbit until the result stabilizes. The model demonstrates the sensitivity of system performance to the asymmetry of customer preferences when changing channels. Numerical simulation was carried out for the call center of the housing management company "StroyTekhnika" in the city of Voronezh. Application flows and repeat call parameters were calculated based on data from the company website and social network analysis. Accounting for repeat calls reduces system throughput compared to the option of completely impatient customers who do not use repeat calls. The case of completely impatient customers describes the limit state of a service system. At the same time, redistributing calls in favor of a more productive channel improves the overall performance of the system. The results obtained show the feasibility of using high-performance multi-channel voice bots while simultaneously stimulating a shift in customer preferences in favor of intelligent automata.

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.

Examining error-contaminated, long-span bridge buffeting vibration by Stochastic Differential Equations

This paper summarizes the development of an analytically-based and numerically-implemented algorithm for the analysis of uncertainty propagation in aeroelastic systems. The term “uncertainty” is utilized to describe either simplifications in the modeling of the fluid–structure interaction mechanism or the effect of experimental errors, for example emerging during wind-tunnel tests. The algorithm is applied to study the dynamic response of long-span bridges. This study provides, by theoretical derivations, estimation of the stationary joint-probability distribution of the generalized bridge response, accounting for perturbations in the buffeting loads. The formulation includes physical states, fluid–structure interaction states and a random variable, which quantifies the errors in the span-wise correlation of the buffeting deck load. A 1200m span bridge is used for illustration of the numerical algorithm.

Reliably Controlling Massive Traffic between a Sensor Network End Internet of Things Device Environment and a Hub Using Transmission Control Protocol Mechanisms

The constant growth of Internet of Things traffic is ensured by the ongoing evolution of the hierarchy of all hardware links of sensor networks. At the same time, the implementation of the Edge computing ideology regulates the complexity of the “first-mile” section (from the sensors array to the peripheral server). Here, the authors suggest paying attention to the growing share of massive traffic from target sensors in the total traffic of the sensors array. This circumstance makes it expedient to introduce an additional link to the peripheral server for summarizing massive traffic from target sensors. The authors present a sensor network end IoT device (SNEIoTD), implemented grounded on a reliable and cheap Raspberry Pi computing platform, as such a link. The introduction of this SNEIoTD makes it possible to reduce the probability of information loss from the critical infrastructure of a smart city and increase the flexibility of controlling the massive traffic of the first mile. In this context, the urgent task is the reliable control of information transfer from the SNEIoTD environment to a hub, which the authors formalize based on Transmission Control Protocol (TCP). This article proposes a mathematical model of the interaction of the main mechanisms of the TCP in the form of a queuing system. As part of this model, a semi-Markov process of an information transfer with a unified speed is selected and its stationary distribution is analytically formalized. A computationally efficient information technology for determining the TCP Window Size is formulated, taking into account the interaction of TCP mechanisms in the process of massive traffic control. Using the example of TCP Westwood+ protocol modification, it is shown that the results of the application of information technology permit increases in the stability of data transfer under the circumstances of increasing Round-Trip Times.

Optimizing search processes with stochastic resetting on the pseudofractal scale-free web.

The pseudofractal scale-free web (PSFW) is a well-known model for a scale-free network with small-world characteristics. Understanding the dynamic properties of this network can provide valuable insights into dynamic processes occurring in general scale-free and small-world networks. In this study we investigate search processes using discrete-time random walks on the PSFW to reveal the impact of the resetting position on optimizing search efficiency, as measured by the mean first-passage time (MFPT). At each step the walker has two options: with a probability of 1-γ, it moves to one of the neighboring sites, and with a probability of γ, it resets to the predefined resetting position. We explore various choices for the resetting position, present rigorous results for the MFPT to a given node of the network, determine the optimal resetting probability γ^{*} where the MFPT reaches its minimum, and evaluate the ratio of the minimum for MFPT to the MFPT without resetting for each case. Results show that, in large PSFWs, both the degree of the resetting position and the distance between the target and the resetting position significantly affect the search efficiency. A higher degree of the resetting position leads to a slower convergence of the walker to the target, while a greater distance between the target and the resetting position also results in a slower convergence. Additionally, we observe that resetting to a vertex randomly selected from the stationary distribution can significantly expedite the process of the walker reaching the target. The findings presented in this study shed light on optimizing stochastic search processes on large networks, offering valuable insights into improving search efficiency in real-world applications, where the target node's location is unknown.

Performance analysis and ANFIS computing of an unreliable Markovian feedback queueing model under a hybrid vacation policy

This study covers the stationary analysis of an unreliable Markovian queueing system with feedback based on a hybrid vacation policy. Customers arrive at the station at random using a Poisson process. Service time is distributed exponentially. The necessary and sufficient requirements for system stability have been demonstrated. Many performance metrics of this queueing system, such as the stationary queue length distribution, have been obtained from the steady-state probability distribution of the queueing model and the steady-state queue length using the matrix geometric method. To establish the model’s performance indicators, a few formulas have also been developed. The Adaptive Neuro-Fuzzy Inference System (ANFIS) was used to develop rules from the input datasets and generate different performance indices. Finally, the impact of system parameters and cost analysis was studied using numerical examples.

A Review of Birth-Death and Other Markovian Discrete-Time Queues

In this review article, we consider discrete-time birth-death processes and their applications to discrete-time queues. To make the analysis simpler to follow, we focus on transform-free methods and consider instances of non-birth-death Markovian discrete-time systems. We present a number of results within one discrete-time framework that parallels the treatment of continuous time models. This approach has two advantages; first, it unifies the treatment of several discrete-time models in one framework, and second, it parallels to the extent possible the treatment of continuous time models. This allows us to draw parallels and contrasts between the discrete and continuous time queues. Specifically, we focus on birth-death applications to the single server discrete-time model with Bernoulli arrivals and geometric service times and provide the reader with a simple rigorous detailed analysis that covers all five scheduling rules considered in the literature, with attention to stationary distributions at slot edges, slot centers, and prearrival epochs. We also cover the waiting time distributions. Moreover, we cover three Markovian models that fit the global balance equations. Our approach provides interesting insights into the behavior of discrete-time queues. The article is intended for those who are familiar with queueing theory basics and would like a simple, yet rigorous introductory treatment to discrete-time queues.

Conditional control of measurement-driven machines

The cycle model of a finite-time measurement-driven machine is proposed by repeatedly performing the quantum measurement, interaction of an atom with the single-mode field, and thermalization with heat reservoirs. The trajectories of the evolution of the atom reach a stationary probability distribution after sufficiently many iterations of the cycle, from which the average changes of the internal energy and the entropy of the atom are calculated. By taking the total work input as the sum of the average works during the measurement, unitary evolution, and information erasure processes, the energy conversion efficiencies of the machine as a refrigerator and a heat engine are determined. Results show that the cooperation of the quantum measurement and the unitary operation always turns the state of the atom into a desired state in every trajectory, yielding a high energy conversion efficiency. The investigation on the proposed model has potentially significant applications where the thermal manipulation at the quantum level is required.

Dynamical complexity driven by water temperature in a size-dependent phytoplankton–zooplankton model with environmental variability

In this paper, we propose a stochastic phytoplankton–zooplankton (PZ) system modeling the growth dynamic of phytoplankton, where the death rate of zooplankton and the growth rate of phytoplankton are considered to be dependent on water temperature, and random environmental perturbation is characterized by mean-reverting Ornstein–Uhlenbeck process. Mathematical theoretical work mainly studies the stability, persistence, and Hopf bifurcation of deterministic PZ model, as well as analyzes the stochastic extinction, persistence in mean, and ergodic stationary distribution of stochastic PZ model, which provides a theoretical basis for numerical simulations. Numerical simulation work mainly reveals that water temperature plays a crucial role in influencing the dynamic of size-dependent phytoplankton–zooplankton interactions, whether in deterministic or random environments. An important conclusion of this study is that the increase of water temperature is able to induce bubbling effect, however, the small phytoplankton cell size can weaken this bubbling effect, and the large phytoplankton cell size can strengthen this bubbling effect, which supports the experimental conclusion that body size plays a key role in mitigating the negative impact of warming on predator–prey interactions. Further, it is found that the plankton with large body size has the advantages of stable and sustainable survival in the cold environment, while the plankton with small body size has advantages of stable and sustainable survival in the warm environment. Therefore, it can be inferred that plankton will evolve into large body size in the cold environment, but they will evolve into small body size in the warm environment.

Mode-Independent Switching Stabilizing Control for Continuous-Time Linear Markovian Switching Systems

We present a switching control approach to studying the stabilization problem of linear Markovian switching systems. The switching signal in feedback loop is generated independently of the underlying switching signal. Moreover, the closed-loop switching signal can be accelerated to show the influence of its stationary distribution. As a result, the transition matrix of the closed-loop system approximates such an operator semigroup that its infinitesimal generator exactly is the combination of the closed-loop subsystem matrices with respect to this stationary distribution. Therefore, endowing this infinitesimal generator with certain stable structure may allow us to make the closed-loop system exponentially stable at the expense of fast switching speed.

Статистическая модель высоконаправленной цифровой антенной решетки пространственно-корреляционным методом обработки сигналов

The article presents a statistical model of a digital antenna array with a spatial correlation method of signal processing. The model is obtained on the basis of the Neumann-Pearson criterion for the conditions of a stationary Gaussian distribution of the output process in spatial correlation signal processing. This model makes it possible to numerically evaluate the probabilistic characteristics of radar signal detection for various orders of nonlinear processing under the influence of natural radio interference and can be recommended for the study and optimization of the construction of the antenna array aperture with spatial correlation signal processing, providing a given spatial noise immunity of radio engineering systems.

A subgeometric convergence formula for finite-level M/G/1-type Markov chains: via a block-decomposition-friendly solution to the Poisson equation of the deviation matrix

AbstractThe purpose of this study is to present a subgeometric convergence formula for the stationary distribution of the finite-level M/G/1-type Markov chain when taking its infinite-level limit, where the upper boundary level goes to infinity. This study is carried out using the fundamental deviation matrix, which is a block-decomposition-friendly solution to the Poisson equation of the deviation matrix. The fundamental deviation matrix provides a difference formula for the respective stationary distributions of the finite-level chain and the corresponding infinite-level chain. The difference formula plays a crucial role in the derivation of the main result of this paper, and the main result is used, for example, to derive an asymptotic formula for the loss probability in the MAP/GI/1/N queue.

Continuous‐time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

AbstractWe develop a new continuous‐time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events.

Threshold behavior and exponential ergodicity of an sir epidemic model: the impact of random jamming and hospital capacity.

This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease's persistence or eradication depends on the basic reproduction number [Formula: see text]. If [Formula: see text], the disease is eradicated with a probability of 1, while [Formula: see text] results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if [Formula: see text], the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required.

Randomized time Riemannian Manifold Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) algorithms, which combine numerical approximation of Hamiltonian dynamics on finite intervals with stochastic refreshment and Metropolis correction, are popular sampling schemes, but it is known that they may suffer from slow convergence in the continuous time limit. A recent paper of Bou-Rabee and Sanz-Serna (Ann Appl Prob, 27:2159-2194, 2017) demonstrated that this issue can be addressed by simply randomizing the duration parameter of the Hamiltonian paths. In this article, we use the same idea to enhance the sampling efficiency of a constrained version of HMC, with potential benefits in a variety of application settings. We demonstrate both the conservation of the stationary distribution and the ergodicity of the method. We also compare the performance of various schemes in numerical studies of model problems, including an application to high-dimensional covariance estimation.

Insights into unexpected relapse and recovery in HCV-infected patients by studying a stochastic within-host HCV model

The causes of viral rebound leading to disease relapse in some chronic hepatitis C virus-infected patients at the end of treatment with direct-acting antiviral drugs are still unclear, which poses a challenge to further improve the permanent cure rate of HCV-infected patients. In this paper, a stochastic within-host HCV model is studied in detail to gain insights into the phenomena of unexpected relapse and recovery in HCV-infected patients. Theoretically, the existence of an ergodic stationary distribution, which also reflects the long-term coexistence of uninfected hepatocytes and free viruses, is proved by constructing suitable stochastic Lyapunov functions at a threshold condition related to the basic reproduction number of the corresponding deterministic model. In addition, the sufficient conditions for the extinction of free viruses is also given, which implies that the viral load eventually drops to zero and HCV-infected patients are cured. In the numerical simulations, the phenomenon of unexpected relapse and recovery in HCV-infected patients is demonstrated by a few simple examples. By Monte Carlo method, under certain conditions, it is observed that the increase in noise intensities makes the phenomena of unexpected relapse and recovery in HCV-infected patients increase as well.