Unbounded Rates Research Articles

Risk-sensitive zero-sum games for continuous-time jump processes with unbounded rates and Borel spaces

This paper studies two-person zero-sum risk-sensitive stochastic games for continuous time jump processes with the following features: (1) the payoff and transition rates are unbounded and time-dependent, (2) the state and action spaces are general Borel sets, (3) the discount factors can vary in time. Differing from those with the risk-sensitive parameter as a differential variable in previous works, we introduce a new type of Shapley equation (NSE) with the time as a differential variable. Under some technical assumptions, we derive a representation of a solution to the NSE, which in turn is used to establish the existence of a solution to the NSE by iterations and approximation. With the help of the NSE, we prove the existence of the value of the game and a saddle-point equilibrium depending on the time, which explicitly shows the nonstationarity of the risk-sensitive discounted equilibria. We illustrate our results by two examples with unbounded transition and payoff rates.

A model for the momentum flux parameters of the 1D Two-Fluid Model in vertical annular flows: Linear stability and numerical analysis

The 1D Two-Fluid model is based on an area average process of the time and phase averaged Two-Fluid conservation equations to render the model tractable for industrial scale problems. Due to the averaging processes, the loss of information renders the standard model ill-posed for certain configurations, i.e., short wavelengths disturbances are amplified at an unbounded rate and unphysical solutions are obtained. Closure relations play a key role in this problem, since they are required to close the 1D system and they reintroduce missing physical parameters that may stabilize the flow and render the model well-posed. Two formulations for the liquid momentum flux parameter (CL) for vertical annular flows are proposed based on local velocities distributions. Differential viscous Kelvin-Helmholtz and von Neumann stability analyses are performed to evaluate the proposed CL formulations and three dynamic pressure models. Results have shown that dynamic pressure closure models introduce a small additional amount of damping into the growth rate curves, without a significant change in the hyperbolicity of the system. On the other hand, the novel CL models can guarantee well posedness of the linear system by introducing a growth rate plateau, blocking the unbounded growth of instabilities. Numerical simulations were also performed, and numerical dispersion relations were extracted from the results showing good agreement against LST data, validating the methodology. The novel CLmodels are evaluated against a large experimental database from the literature, showing that the proposed models outperform the standard constant CL values for both pressure drop and liquid film thickness.

Optimal speed profile of a DVFS processor under soft deadlines

We consider a Dynamic Voltage and Frequency Scaling (DVFS) processor executing jobs with obsolescence deadlines: A job becomes obsolete and is removed from the system if it is not completed before its deadline. The objective is to design a dynamic speed policy for the processor that minimizes its average energy consumption plus an obsolescence cost per deadline miss. Under Poisson arrivals and exponentially distributed deadlines and job sizes, we show that this problem can be modeled as a continuous time Markov decision process (MDP) with unbounded state space and unbounded rates. While this MDP admits a continuous time optimality equation for its average cost, the standard uniformization approach is not applicable. Inspired by the scaling method introduced by Blok and Spieksma, we first define a family of truncated MDPs and we then show that the optimal speed profiles are increasing in the number of jobs in the system and are uniformly bounded by a constant. Finally, we show that these properties are inherited from the original (infinite) system. The proposed upper bound on the optimal speed profile is tight and is used to develop an extremely simple policy that accurately approximates the optimal average cost in heavy traffic conditions.

Optimal control of a queue under a quality-of-service constraint with bounded and unbounded rates

We consider a simple Markovian queue with Poisson arrivals and exponential service times for jobs. The controller chooses state-dependent service rates from an action space. The queue has a finite buffer, and when full, new jobs get rejected. The controller’s objective is to choose optimal service rates that meet a quality-of-service constraint. We solve this problem analytically and compute it numerically under two cases: When the action space is unbounded and when it is bounded.

Moderate deviations in a class of stable but nearly unstable processes

We consider a stable but nearly unstable autoregressive process of any order. The bridge between stability and instability is expressed by a time-varying companion matrix An with spectral radius ρ(An)<1 satisfying ρ(An)→1. In that framework, we establish a moderate deviation principle for the empirical covariance only relying on the elements of An through 1−ρ(An) and, as a by-product, we establish a moderate deviation principle for the OLS estimator when Γ, the renormalized asymptotic variance of the process, is invertible. Finally, when Γ is singular, we also provide a compromise in the form of a moderate deviation principle for a penalized version of the estimator. Our proofs essentially rely on truncations and deviations of mn–dependent sequences, with an unbounded rate (mn).

Performance of Multi-Cell Massive MIMO Systems With Interference Decoding

We consider a multi-cell massive MIMO system where a time-division duplex protocol is used to estimate the channel state information via uplink pilots. When maximum ratio combining (MRC) is used at the BSs, the re-use of pilots across cells causes the pilot contamination effect which yields interference components that do not vanish as the number of base-station (BS) antennas $M \rightarrow \infty$. When treating interference as noise (TIN), this phenomenon limits the performance of multi-cell massive MIMO systems. In this paper, we analyze more advanced schemes based on simultaneous unique decoding (SD) as well as simultaneous non-unique decoding (SND) of the interference that can provide unbounded rate as $M \rightarrow \infty$. We also establish a worst-case uncorrelated noise technique for multiple-access channels to derive achievable rate expressions for finite $M$. Furthermore, we study a much simpler subset of SND (called S-SND) which provides a lower bound to SND and achieves unbounded rate as $M \rightarrow \infty$, and also outperforms SD for finite $M$. For the special cases of two-cell and three-cell systems, using a maximum symmetric rate allocation policy we compare the performance of different interference decoding schemes with that of TIN. Finally, we numerically illustrate the improved performance of the proposed schemes.

Risk-sensitive continuous-time Markov decision processes with unbounded rates and Borel spaces

This paper considers the finite-horizon risk-sensitive optimality for continuous-time Markov decision processes, and focuses on the more general case that the transition rates are unbounded, cost/reward rates are allowed to be unbounded from below and from above, the policies can be history-dependent, and the state and action spaces are Borel ones. Under mild conditions imposed on the decision process’s primitive data, we establish the existence of a solution to the corresponding optimality equation (OE) by a so called approximation technique. Then, using the OE and the extension of Dynkin’s formula developed here, we prove the existence of an optimal Markov policy, and verify that the value function is the unique solution to the OE. Finally, we give an example to illustrate the difference between our conditions and those in the previous literature.

Material stability and instability in non-local continuum models for dense granular materials

A class of common and successful continuum models for steady, dense granular flows is based on the$\unicode[STIX]{x1D707}(I)$model for viscoplastic grain-inertial rheology. Recent work has shown that under certain conditions,$\unicode[STIX]{x1D707}(I)$-based models display a linear instability in which short-wavelength perturbations grow at an unbounded rate – i.e. a Hadamard instability. This observation indicates that$\unicode[STIX]{x1D707}(I)$models will predict strain localization arising due to material instability in dense granular materials; however, it also raises concerns regarding the robustness of numerical solutions obtained using these models. Several approaches to regularizing this instability have been suggested in the literature. Among these, it has been shown that the inclusion of higher-order velocity gradients into the constitutive equations can suppress the Hadamard instability, while not precluding the modelling of strain localization into diffuse shear bands. In our recent work (Henann &amp; Kamrin,Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 6730–6735), we have proposed a non-local model – called the non-local granular fluidity (NGF) model – which also involves higher-order flow gradients and has been shown to quantitatively describe a wide variety of steady, dense flows. In this work, we show that the NGF model also successfully regularizes the Hadamard instability of the$\unicode[STIX]{x1D707}(I)$model. We further apply the NGF model to the problem of strain localization in quasi-static plane-strain compression using nonlinear finite-element simulations in order to demonstrate that the model is capable of describing diffuse strain localization in a mesh-independent manner. Finally, we consider the linear stability of an alternative gradient–viscoplastic model (Bouzidet al.,Phys. Rev. Lett., vol. 111, 2013, 238301) and show that the inclusion of higher-order gradients does not guarantee the suppression of the Hadamard instability.

K competing queues with customer abandonment: optimality of a generalised c mu -rule by the Smoothed Rate Truncation method

We consider a K-competing queues system with the additional feature of customer abandonment. Without abandonment, it is optimal to allocate the server to a queue according to the c mu -rule. To derive a similar rule for the system with abandonment, we model the system as a continuous-time Markov decision process. Due to impatience, the Markov decision process has unbounded jump rates as a function of the state. Hence it is not uniformisable, and so far there has been no systematic direct way to analyse this. The Smoothed Rate Truncation principle is a technique designed to make an unbounded rate process uniformisable, while preserving the properties of interest. Together with theory securing continuity in the limit, this provides a framework to analyse unbounded rate Markov decision processes. With this approach, we have been able to find close-fitting conditions guaranteeing optimality of a strict priority rule.

Risk-Sensitive Discounted Continuous-Time Markov Decision Processes with Unbounded Rates

This paper attempts to study the risk-sensitive discounted continuous-time Markov decision processes with unbounded transition and cost rates. Different from the case of bounded transition/cost rat...

Risk-sensitive average continuous-time Markov decision processes with unbounded rates

In this paper we study the risk-sensitive average cost criterion for continuous-time Markov decision processes in the class of all randomized Markov policies. The state space is a denumerable set, and the cost and transition rates are allowed to be unbounded. Under the suitable conditions, we establish the optimality equation of the auxiliary risk-sensitive first passage optimization problem and obtain the properties of the corresponding optimal value function. Then by a technique of constructing the appropriate approximating sequences of the cost and transition rates and employing the results on the auxiliary optimization problem, we show the existence of a solution to the risk-sensitive average optimality inequality and develop a new approach called the risk-sensitive average optimality inequality approach to prove the existence of an optimal deterministic stationary policy. Furthermore, we give some sufficient conditions for the verification of the simultaneous Doeblin condition, use a controlled birth and death system to illustrate our conditions and provide an example for which the risk-sensitive average optimality strict inequality occurs.

Optimization of the Superstable Linear Stochastic System Applied to the Model with Extremely Impatient Agents

We consider the problem of stochastic linear regulator over an infinite time horizon with superexponentially stable matrix in the equation of state dynamics. The form of the optimal control based on the criterion taking into account the information about the parameters of disturbances and the matrix stability rate was determined. The results obtained were used to analyze the model of a system with extremely impatient agents where the objective functional includes discounting by the asymptotically unbounded rate.

Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion

Abstract. Stochastic processes exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm), with the spectral slope at high frequencies being associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matérn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matérn process and its relationship to fBm. An algorithm for the simulation of the Matérn process in O(NlogN) operations is given. Unlike fBm, the Matérn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.

Lower Current Large Deviations for Zero-Range Processes on a Ring

We study lower large deviations for the current of totally asymmetric zero-range processes on a ring with concave current-density relation. We use an approach by Jensen and Varadhan which has previously been applied to exclusion processes, to realize current fluctuations by travelling wave density profiles corresponding to non-entropic weak solutions of the hyperbolic scaling limit of the process. We further establish a dynamic transition, where large deviations of the current below a certain value are no longer typically attained by non-entropic weak solutions, but by condensed profiles, where a non-zero fraction of all the particles accumulates on a single fixed lattice site. This leads to a general characterization of the rate function, which is illustrated by providing detailed results for four generic examples of jump rates, including constant rates, decreasing rates, unbounded sublinear rates and asymptotically linear rates. Our results on the dynamic transition are supported by numerical simulations using a cloning algorithm.

Point Process with Last-arrival-time Dependent Intensity and 1-Dimensional Incompressible Fluid System with Evaporation

We consider an infinite system of quasilinear first-order partial differential equations, generalized to contain spacial integration, which describes an incompressible fluid mixture of infinite components in a line segment whose motion is driven by unbounded and space-time dependent evaporation rates. We prove unique existence of the solution to the initial-boundary value problem, with conservation-of-fluid condition at the boundary. The proof uses a map on the space of collection of characteristics, and a representation based on a non-Markovian point process with last-arrival-time dependent intensity.

Countable state Markov decision processes with unbounded jump rates and discounted cost: optimality equation and approximations

This paper considers Markov decision processes (MDPs) with unbounded rates, as a function of state. We are especially interested in studying structural properties of optimal policies and the value function. A common method to derive such properties is by value iteration applied to the uniformised MDP. However, due to the unboundedness of the rates, uniformisation is not possible, and so value iteration cannot be applied in the way we need. To circumvent this, one can perturb the MDP. Then we need two results for the perturbed sequence of MDPs: 1. there exists a unique solution to the discounted cost optimality equation for each perturbation as well as for the original MDP; 2. if the perturbed sequence of MDPs converges in a suitable manner then the associated optimal policies and the value function should converge as well. We can model both the MDP and perturbed MDPs as a collection of parametrised Markov processes. Then both of the results above are essentially implied by certain continuity properties of the process as a function of the parameter. In this paper we deduce tight verifiable conditions that imply the necessary continuity properties. The most important of these conditions are drift conditions that are strongly related to nonexplosiveness.

Countable state Markov decision processes with unbounded jump rates and discounted cost: optimality equation and approximations

This paper considers Markov decision processes (MDPs) with unbounded rates, as a function of state. We are especially interested in studying structural properties of optimal policies and the value function. A common method to derive such properties is by value iteration applied to the uniformised MDP. However, due to the unboundedness of the rates, uniformisation is not possible, and so value iteration cannot be applied in the way we need. To circumvent this, one can perturb the MDP. Then we need two results for the perturbed sequence of MDPs: 1. there exists a unique solution to the discounted cost optimality equation for each perturbation as well as for the original MDP; 2. if the perturbed sequence of MDPs converges in a suitable manner then the associated optimal policies and the value function should converge as well. We can model both the MDP and perturbed MDPs as a collection of parametrised Markov processes. Then both of the results above are essentially implied by certain continuity properties of the process as a function of the parameter. In this paper we deduce tight verifiable conditions that imply the necessary continuity properties. The most important of these conditions are drift conditions that are strongly related to nonexplosiveness.

A practical guide to text mining with topic extraction

Text analytics continue to proliferate as mass volumes of unstructured but highly useful data are generated at unbounded rates. Vector space models for text data—in which documents are represented by rows and words by columns—provide a translation of this unstructured data into a format that may be analyzed with statistical and machine learning techniques. This approach gives excellent results in revealing common themes, clustering documents, clustering words, and in translating unstructured text fields (such as an open‐ended survey response) to usable input variables for predictive modeling. After discussing the collection and processing of text, we explore properties and transformations of the document‐term matrix (DTM). We show how the singular value decomposition may be used to drastically reduce the size of the document space while also setting the stage for automatic topic extraction, courtesy of the varimax rotation. This latent semantic analysis (LSA) approach produces factors that are compatible with graphical exploration and advanced analytics. We also explore Latent Dirichlet Allocation for topic analysis. We reference published R packages to implement the methods and conclude with a summary of other popular open‐source and commercial software packages. WIREs Comput Stat 2015, 7:326–340. doi: 10.1002/wics.1361This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences &gt; Clustering and Classification Statistical Learning and Exploratory Methods of the Data Sciences &gt; Pattern Recognition Statistical Learning and Exploratory Methods of the Data Sciences &gt; Text Mining

Solvability of Age-Structured Epidemiological Models with Intracohort Transmission

The standard version of the epidemiological model with continuous age structure (Iannelli, Mathematical Theory of Age-Structured Population Dynamics, 1995) consists of the linear McKendrick model for the evolution of a disease-free population, coupled with one of the classical (SIS, SIR, etc.) models for the spread of the disease. A natural functional space in which the linear McKendrick model is well posed is the space of integrable functions. However, in the so-called intracohort models, the disease term contains pointwise products of the unknown functions; that is, of the age-specific densities of susceptibles, infectives and other classes (if applicable) which render the standard semilinear perturbation technique of proving the well-posedness of the full model not applicable in that space. This is due to the fact that the product of two integrable functions need not be integrable. Therefore, most works on the well-posedness of such problems have been done under additional assumption that the disease-free population is stable and the equilibrium has been reached (Busenberg et al. SIAM J Math Anal 22(4):1065–1080, 1991, Prus, J Math Biol 11:65–84, 1981). This allowed for showing, after some algebraic manipulations, that the order interval [0, 1] in the space of integrable functions was invariant under the action of the linear McKendrick semigroup and thus the usual iteration technique could be applied in this interval, yielding a bounded solution. An additional advantage of adopting the stability assumption was that it eliminated from the model the death rate which, in all realistic cases, is unbounded and creates serious technical difficulties. The aim of this note is to show that a careful modification of the standard Picard iteration procedure allows for proving the same result without the stability assumption. Since in such a case we are not able to suppress the unbounded death rate, we will briefly present the necessary linear results in a way which simplifies and unifies some classical results existing in the literature (Inaba, Math Popul Stud 1:49–77, 1988, Prus, J Math Biol 11:65–84, 1981, Webb, Theory of Nonlinear Age Dependent Population Dynamics, 1985).

Study of liquid-mediated adhesion between 3D rough surfaces: A spectral approach

In this work, a three-dimensional model for liquid-mediated adhesion between two rough surfaces is presented. The approach is based on a spectral (multi-scale) representation of compressive rough surface deformation along with the capillary equations governing the tensile deformation. An iterative numerical algorithm is designed to solve the equations of elasticity and capillarity simultaneously. It is shown that, under certain conditions, a contact instability occurs leading to unbounded rates of change of tensile force, average gap and wetted radius. The effects of liquid volume, liquid surface tension, surface topography, nominal contact area, and external load on the stability of contact interface are studied. Key dimensionless ratios are identified that govern the equilibrium state and onset of instability.