Poisson Arrival Rate Research Articles

Stochastic Roll Dynamics of Smooth and Impacting Vessels in Random Waves

Ship instability is frequently associated with extreme roll motion. In this paper smooth and vibro-impact dynamics of vessels rolling under random wave excitations including nonlinearities is investigated. The vessel is represented by a single degree of freedom roll model. The random excitation due to waves and ice is modeled by a sum of random pulses with Poisson arrival rate. Stochastic P-bifurcation analysis is carried out by examining the variation with parameter of the joint probability density functions (jpdf) of the response computed numerically by the solution of the corresponding Fokker–Planck–Kolmogorov (FPK) equation. The finite element (FE), finite difference (FD), path integral (PI) and radial basis neural (RBN) network formulation methods are used in the solution of the FPK equation. The D-bifurcation analysis is analyzed by computing the largest Lyapunov exponent. The mean upcrossing rate is computed using Rice’s formula. The random wave excitation is also modeled as the response of a second order linear filter to white noise and the random response of the ship is investigated by solution of the corresponding FPK equation of the four dimensional Markov system by the FD method. Non-smooth coordinate transformations like the Zhuravlev and Ivanov transformations are used converting the discontinuous systems to equivalent smooth systems in the impact dynamics analysis. The adaptive time stepping method (ATSM) approach is used to accurately locate the discontinuity point and a Brownian tree approach is used to direct the integration along the correct Brownian path. Different models for nonlinear damping and the restoring moment are considered and their effects on ship stochastic response such as chaotic motion, unbounded rotational motion, impact modulation motion and ship capsizing are investigated.

Flattening the curve: Insights from queueing theory.

The worldwide outbreak of the coronavirus was first identified in 2019 in Wuhan, China. Since then, the disease has spread worldwide. As it is currently spreading in the United States, policy makers, public health officials and citizens are racing to understand the impact of this virus on the United States healthcare system. They fear a rapid influx of patients overwhelming the healthcare system and leading to unnecessary fatalities. Most countries and states in America have introduced mitigation strategies, such as using social distancing to decrease the rate of newly infected people. This is what is usually meant by flattening the curve. In this paper, we use queueing theoretic methods to analyze the time evolution of the number of people hospitalized due to the coronavirus. Given that the rate of new infections varies over time as the pandemic evolves, we model the number of coronavirus patients as a dynamical system based on the theory of infinite server queues with time inhomogeneous Poisson arrival rates. With this model we are able to quantify how flattening the curve affects the peak demand for hospital resources. This allows us to characterize how aggressive societal policy needs to be to avoid overwhelming the capacity of healthcare system. We also demonstrate how curve flattening impacts the elapsed lag between the times of the peak rate of hospitalizations and the peak demand for the hospital resources. Finally, we present empirical evidence from Italy and the United States that supports the insights from our model analysis.

Traffic Volume Estimate Based on Low Penetration Connected Vehicle Data at Signalized Intersections: A Bayesian Deduction Approach

The emergence of connected vehicle (CV) technologies has created new traffic control opportunities, among them, is the potential to estimate volume without approach lane detection. Rather than requiring the expense and effort to install and maintain detector systems, this new “detector-free” method permits traffic volume to be estimated from CV GPS trajectory data. Unfortunately, however, CV GPS methods are limited not only to locations where CV GPS data can be recorded, but also limited to time when CV GPS data is recorded. The goal of this research was to overcome these limitations and permit volume estimation to be accomplished under any location or condition, including low-penetration CV environments. The contributions made by this work are significant in two respects. First, it creates an improved queue-based method to estimate intersection approach volumes during each signal cycle with sparse CV data. Second, the research demonstrates the application of a Bayesian deduction method to approximate volume with no CV trajectory data. To accomplish this, traffic volumes are assumed to be time-dependent Poisson distributed throughout the day, and CV data were used to estimate CV volume and further set as prior to deduce the time-dependent Poisson arrival rate. To verify and evaluate the accuracy and effectiveness of this new method under a range of potential traffic conditions, a simulation case study and a NGSIM case study were implemented. Results of both case studies resulted in estimated-to-actual arrival rate average errors as low as 4.2 percent and volume estimation errors as low as 0.9 percent.

Asymptotic analysis of a storage allocation model with finite capacity: Marginal and conditional distributions

We consider a dynamic storage allocation model with a finite number of storage spaces. There are m primary spaces and R secondary spaces. Items arrive according to a Poisson process and an arriving item takes the lowest ranked available space. Each item occupies a storage space for an exponentially distributed amount of time. Letting N1 and N2 denote the numbers of occupied primary and secondary spaces, we examine the steady-state marginal distribution of N2 and derive various conditional limit laws for Prob[N1=k∣N2=r]. The joint process (N1,N2) behaves as a random walk in a lattice rectangle. We scale the storage capacities m and R to be large and study the problem asymptotically as the Poisson arrival rate λ becomes large. We previously derived detailed asymptotic results for the joint distribution Prob[N1=k,N2=r] and we use these results here to study the marginal and conditional distributions.

Functional sensitivity analysis of ruin probability in the classical risk models

Sensitivity analysis investigates how the change in the output of a computational model can be attributed to changes of its input parameters. Identifying the input parameters that propagate more uncertainty on the ruin probability associated with insurance risk models is a challenging problem. In this paper, we consider the classical risk model, where an epistemic-uncertainty veils the true values of the claim size distribution rate and the Poisson arrival rate. Based on the available data for calibrating the probability distributions that model gaps of knowledge on these rates, and using the Taylor-series expansion methodology, we obtain the ruin probability under polynomial form in uncertain rates as a computational model. Specifically, we get a new sensitivity estimate of the ruin probability with respect to uncertain parameters. We provide a coherent framework within which we can accurately characterize statistically the uncertain ruin probability. In addition, we use the Markov's inequality to estimate the risk incurred by working with uncertain ruin probability rather than that evaluated at fixed parameters. A series of numerical experiments are presented to illustrate the potential of the proposed approach.

Martingales and buffer overflow for the symmetric shortest queue model

A variant of the standard symmetric system of two parallel queues under the join-the-shortest-queue policy is introduced. Here, the shortest queue has service rate $$\mu _1$$ , while the longest queue has rate $$\mu _2$$ , where $$\mu _1 + \mu _2 = 1$$ . In the case of equality, both queues are served at rate 1 / 2. Each queue has capacity K, which may be finite or infinite, and the global Poisson arrival rate is $$\rho $$ . Couplings show that, as $$\mu _2$$ varies, the model is totally ordered, in terms of both total number N(t) of customers in the system and longest queue length. The two extreme cases $$\mu _2= 0$$ and $$\mu _2= 1$$ then provide simple stochastic bounds for N(t) for arbitrary $$\mu _2$$ . The ordering partially extends to the enlarged model in which, whenever the shortest queue is empty, the idle server at that queue helps—or on the contrary disturbs—the other server. The previous bounds remain valid. In the extended setup, different martingales are next built from the infinite capacity process. Those lead to simple explicit formulations of the means and Laplace transforms of the hitting time of saturation, for the process with finite K started from any initial state. Asymptotics are then derived, as K gets large. Using one particular mean time, the stationary blocking probability is explicitly obtained, extending the result by Dester et al. regarding the standard symmetric model. Finally, the joint distribution of the time and state of the system at the first queue-length equality is expressed as a function of the inverse of a simple explicit matrix.

A Bayesian Poisson–Gaussian Process Model for Popularity Learning in Edge-Caching Networks

Edge-caching is recognized as an efficient technique for future cellular networks to improve network capacity and user-perceived quality of experience. To enhance the performance of caching systems, designing an accurate content request prediction algorithm plays an important role. In this paper, we develop a flexible model, a Poisson regressor based on a Gaussian process (GP), for the content request distribution. The first important advantage of the proposed model is that it encourages the already existing or seen contents with similar features to be correlated in the feature space and therefore it acts as a regularizer for the estimation. Second, it allows predicting the popularities of newly-added or unseen contents, whose statistical data is not available in advance. In order to learn the model parameters that yield the Poisson arrival rates or alternatively the content popularities, we invoke the Bayesian approach that is robust against over-fitting. However, the resulting posterior distribution is analytically intractable to compute. To tackle this, we apply a Markov chain Monte Carlo (MCMC) method to approximate this distribution, which is also asymptotically exact. Nevertheless, the MCMC is computationally demanding, especially when the number of contents is large. Thus, we employ the variational Bayes (VB) method as an alternative low-complexity solution. More specifically, the VB method addresses the approximation of the posterior distribution through an optimization problem. Subsequently, we present a fast block-coordinate descent algorithm to solve this optimization problem. Finally, extensive simulation results both on synthetic and real-world datasets are provided to show the accuracy of our prediction algorithm and the cache hit ratio (CHR) gain compared to existing methods in this paper.

Joint routing and scheduling control in a two-class network with a flexible server

In this study we analyze a queueing model with a Gurvich structure. In such a network, the controller may route incoming jobs to different classes, but they are routed to the same server. This structure, although it falls into the general class of stochastic processing networks, is somewhat unconventional. We focus on a single-server two-class version of a Gurvich network in this paper. For a Poisson arrival stream and exponential service rates, we develop a Markov decision process representation of the system and prove structural results on optimal routing and scheduling controls. We show that the optimal policy uses $$c\mu $$ scheduling and switching curve routing. We also investigate the fluid model and perturbation expansions thereof, which are useful in deriving near-optimal policies in the original network.

A non-cooperative foundation for the continuous Raiffa solution

This paper provides a non-cooperative foundation for (asymmetric generalizations of) the continuous Raiffa solution. Specifically, we consider a continuous-time variation of the classic Ståhl–Rubinstein bargaining model, in which there is a finite deadline that ends the negotiations, and in which each player’s opportunity to make proposals is governed by a player-specific Poisson process, in that the rejecter of a proposal becomes proposer at the first next arrival of her process. Under the assumption that future payoffs are not discounted, it is shown that the expected payoffs players realize in subgame perfect equilibrium converge to the continuous Raiffa solution outcome as the deadline tends to infinity. The weights reflecting the asymmetries among the players correspond to the Poisson arrival rates of their respective proposal processes.

Decoding Performance in Low-Power Wide Area Networks With Packet Collisions

This paper proposes an analytical framework for the prediction of decoding error probabilities in heterogeneous wireless environments, where transmissions from various radio nodes with distinct Poisson-arrival rates and packet lengths populate the channel. Random channel access without feedback is assumed, where partial packet collisions can lead to the loss of packets. The analysis is based on the modeling of the collision length distribution between competing nodes. With recent results from information theory in the finite block length regime, we provide bounds on achievable decoding error probabilities for a given interference scenario. The new framework enables jointly considering inter- and intra-system interference, which is an important aspect in unlicensed radio bands. The framework’s applicability to optimize system designs is demonstrated for a typical low-power wide area network scenario. We study the tradeoff between reliability and code rate for a point-to-point link and present achievable throughput regions. The analysis reveals the superior performance of coded time-hopping spread spectrum systems. They reach very low error probability even under strong interference for wide ranges of practically relevant load regions.

Two rapid test methods for assessing transmission time reliability of a multiple-channel network with Poisson arrival and service rates

To shorten the duration of the network transmission time reliability test, two rapid methods are proposed for a multiple-channel network with Poisson arrival and service rates. For a network whose service rate is given and can be increased in practice, a rapid test can be conducted on a similar network with higher arrival and service rates. The test conditions are derived using the similarity theory, and the data collected in the test can be used to estimate the reliability of the original network. For a network whose service rate is unknown, a rapid test analogous to accelerated test can be applied, which only increases the arrival rate. The analytical expression of transmission time reliability is used to derive the accelerated model, and the data collected in the test can be used to extrapolate the network reliability under normal arrival rate. Our numerical study shows that the two proposed methods are efficient and the reliability estimates are quite accurate compared to the empirical estimate obtained from verification tests. In the rapid tests, the data collected in a given period of time increases along with the arrival rate, and the test duration can be shortened without affecting the sample size of packets.

A prediction-based dynamic file assignment strategy for parallel file systems

An analysis model of file assignment and access are generalized.A load prediction model in parallel file systems is proposed.A prediction-based dynamic file assignment strategy (PDFA) is proposed.We evaluate the effectiveness of the proposed algorithms. Nowadays, the rapid development of the internet calls for a high performance file system, and a lot of efforts have already been devoted to the issue of assigning nonpartitioned files in a parallel file system with the aim of pursuing a prompt response to requests. Yet most of the existing strategies still fail to bring about an optimal performance on system mean response time metrics, and new strategies which can achieve better performance in terms of mean response time become indispensable for parallel file systems. This paper, while addressing the issue of assigning nonpartitioned files in parallel file systems where the file accesses exhibit Poisson arrival rates and fixed service times, presents an on-line file assignment strategy, named prediction-based dynamic file assignment (PDFA), to minimize the mean response time among disks under different workload conditions, and a comparison of the PDFA with the well-known file assignment algorithms, such as HP and SOR. Comprehensive experimental results show that PDFA is able to improve the performance consistently in terms of mean response time among all algorithms for comparison.

Growth and Mitigation Policies With Uncertain Climate Damage

We analyze an endogenously growing economy in which production generates greenhouse gas emissions leading to global temperature increase. Global warming causes stochastic climate shocks, modeled by the Poisson process, which destroy part of the economy's capital stock. Part of the output may be devoted to emissions abatement and thus damages from climate shocks may be reduced. We solve the model in closed form and show that the optimal path is characterized by a constant growth rate of consumption and capital stock until a shock arrives, triggering a downward jump in both variables. The magnitude of the jump depends on the Poisson arrival rate, abatement efficiency, damage intensity, and the elasticity of intertemporal consumption substitution. Optimum mitigation policy consists of spending a constant share of output on abatement, which is an increasing function of the Poisson arrival rate, the economy's productivity, polluting intensity of output, and the intensity of environmental damage. Optimum growth and abatement react sharply to changes in the arrival rate and the damage intensity, suggesting more stringent climate policies for a realistic world with uncertainty compared to the certainty-equivalent case. We extend the baseline model by adding climate-induced fluctuations around the growth trend and stock pollution effects, showing the robustness of our results.

The Supermarket Game

A supermarket game is considered with N FCFS queues with unit exponential service rate and global Poisson arrival rate Nλ. Upon arrival each customer chooses a number of queues to be sampled uniformly at random and joins the least loaded sampled queue. Customers are assumed to have cost for both waiting and sampling, and they want to minimize their own expected total cost. We study the supermarket game in a mean field model that corresponds to the limit as N converges to infinity in the sense that (i) for a fixed symmetric customer strategy, the joint equilibrium distribution of any fixed number of queues converges as N → ∞ to a product distribution determined by the mean field model and (ii) a Nash equilibrium for the mean field model is an ε-Nash equilibrium for the finite N model with N sufficiently large. It is shown that there always exists a Nash equilibrium for λ < 1 and the Nash equilibrium is unique with homogeneous waiting cost for [Formula: see text]. Furthermore, we find that the action of sampling more queues by some customers has a positive externality on the other customers in the mean field model, but can have a negative externality for finite N.

Markov modulated Poisson process models incorporating covariates for rainfall intensity

Time series of rainfall bucket tip times at the Beaufort Park station, Bracknell, in the UK are modelled by a class of Markov modulated Poisson processes (MMPP) which may be thought of as a generalization of the Poisson process. Our main focus in this paper is to investigate the effects of including covariate information into the MMPP model framework on statistical properties. In particular, we look at three types of time-varying covariates namely temperature, sea level pressure, and relative humidity that are thought to be affecting the rainfall arrival process. Maximum likelihood estimation is used to obtain the parameter estimates, and likelihood ratio tests are employed in model comparison. Simulated data from the fitted model are used to make statistical inferences about the accumulated rainfall in the discrete time interval. Variability of the daily Poisson arrival rates is studied.

MODELING QUEUEING SYSTEMS USING FUZZY ESTIMATORS

Queuing modeling is the mathematical approach for the analysis of waiting lines. The central problem in every queueing model is a trade-off decision: The manager must weigh the added cost of providing more rapid service against the inherent cost of waiting. In this paper, a new fuzzy approach to queueing modeling is presented in order to eliminate the disadvantages of point estimation and the relevant paradoxes when calculating the efficiency of a queue, in the case of unreliable data available. Both Poisson arrival rate and exponential service time are considered by using fuzzy estimators. The introduction of fuzzy estimators in performance measures of M/M/S queueing systems is presented to address the central estimation issue under uncertainty.

On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits

We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotesνof its capacity to the first queue and the remaining1−νto the second. We obtain exact integral representations for the joint probability distribution of the number of customers in this two-node network. Then we evaluate this distribution in various asymptotic limits, such as large numbers of customers in either/both of the queues, light traffic where arrivals are infrequent, and heavy traffic where the system is nearly unstable.

The supermarket game

A supermarket game is considered with N FCFS queues with unit exponential service rate and global Poisson arrival rate Nλ. Upon arrival each customer chooses a number of queues to be sampled unifor...

On semiparametric estimation of ruin probabilities in the classical risk model

The ruin probability of an insurance company is a central topic in risk theory. We consider the classical Poisson risk model when the claim size distribution and the Poisson arrival rate are unknown. Given a sample of inter-arrival times and corresponding claims, we propose a semiparametric estimator of the ruin probability. We establish properties of strong consistency and asymptotic normality of the estimator and study bootstrap confidence bands. Further, we present a simulation example in order to investigate the finite sample properties of the proposed estimator.

Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case

We consider two coupled queues with a generalized processor sharing service discipline. The second queue has a much smaller Poisson arrival rate than the first queue, while the customer service times are of comparable magnitude. The processor sharing server devotes most of its resources to the first queue, except when it is empty. The fraction of resources devoted to the second queue is small, of the same order as the ratio of the arrival rates. We assume that the primary queue is heavily loaded and that the secondary queue is critically loaded. If we let the small arrival rate to the secondary queue beO(ε), where0≤ε≪1, then in this asymptotic limit the number of customers in the first queue will be large, of orderO(ε-1), while that in the second queue will be somewhat smaller, of orderO(ε-1/2). We obtain a two-dimensional diffusion approximation for this model and explicitly solve for the joint steady state probability distribution of the numbers of customers in the two queues. This work complements that in (Morrison, 2010), which the second queue was assumed to be heavily or lightly loaded, leading to mean queue lengths that wereO(ε-1)orO(1), respectively.