Title: Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching
Abstract We study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied. more »« less
We study ergodic properties of Markovian multiclass many-server queues that are uniform over scheduling policies and the size of the system. The system is heavily loaded in the Halfin–Whitt regime, and the scheduling policies are work conserving and preemptive. We provide a unified approach via a Lyapunov function method that establishes Foster–Lyapunov equations for both the limiting diffusion and the prelimit diffusion-scaled queuing processes simultaneously. We first study the limiting controlled diffusion and show that if the spare capacity (safety staffing) parameter is positive, the diffusion is exponentially ergodic uniformly over all stationary Markov controls, and the invariant probability measures have uniform exponential tails. This result is sharp because when there is no abandonment and the spare capacity parameter is negative, the controlled diffusion is transient under any Markov control. In addition, we show that if all the abandonment rates are positive, the invariant probability measures have sub-Gaussian tails regardless whether the spare capacity parameter is positive or negative. Using these results, we proceed to establish the corresponding ergodic properties for the diffusion-scaled queuing processes. In addition to providing a simpler proof of previous results in Gamarnik and Stolyar [Gamarnik D, Stolyar AL (2012) Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime: asymptotics of the stationary distribution. Queueing Systems 71(1–2):25–51], we extend these results to multiclass models with renewal arrival processes, albeit under the assumption that the mean residual life functions are bounded. For the Markovian model with Poisson arrivals, we obtain stronger results and show that the convergence to the stationary distribution is at an exponential rate uniformly over all work-conserving stationary Markov scheduling policies.
Arapostathis, Ari; Hmedi, Hassan; Pang, Guodong; Sandrić, Nikola(
, Stochastic Control, Optimization, and Applications, Yin G., Zhang Q. (eds). The IMA Volumes in Mathematics and its Applications)
We study the ergodic properties of a class of controlled stochastic differential
equations (SDEs) driven by a-stable processes which arise as the limiting
equations of multiclass queueing models in the Halfin–Whitt regime that have
heavy–tailed arrival processes. When the safety staffing parameter is positive, we
show that the SDEs are uniformly ergodic and enjoy a polynomial rate of convergence
to the invariant probability measure in total variation, which is uniform over
all stationary Markov controls resulting in a locally Lipschitz continuous drift. We
also derive a matching lower bound on the rate of convergence (under no abandonment).
On the other hand, when all abandonment rates are positive, we show that the
SDEs are exponentially ergodic uniformly over the above-mentioned class of controls.
Analogous results are obtained for Lévy–driven SDEs arising from multiclass
many-server queues under asymptotically negligible service interruptions. For these
equations, we show that the aforementioned ergodic properties are uniform over all
stationary Markov controls. We also extend a key functional central limit theorem
concerning diffusion approximations so as to make it applicable to the models studied
here.
Cheng, Hanke; Mark, Brian L.; Ephraim, Yariv; Chen, Chun-Hung(
, IEEE International Conference on Communications)
null
(Ed.)
In a wireless network with dynamic spectrum sharing, tracking temporal spectrum holes across a wide spectrum band is a challenging task. We consider a scenario in which
the spectrum is divided into a large number of bands or channels, each of which has the potential to provide dynamic spectrum access opportunities. The occupancy times of each band by primary users are generally non-exponentially distributed. We develop an approach to determine and parameterize a small selected subset of the bands with good spectrum access opportunities, using limited computational resources under noisy measurements. We model the noisy measurements of the received signal in each band as a bivariate Markov modulated Gaussian process, which can be viewed as a continuous-time bivariate Markov chain observed through Gaussian noise. The underlying
bivariate Markov process allows for the characterization of non-exponentially distributed state sojourn times. The proposed scheme combines an online expectation-maximization algorithm for parameter estimation with a computing budget allocation algorithm. Observation time is allocated across the bands to determine the subset of G out of G frequency bands with the largest mean idle times for dynamic spectrum access and at the same time to obtain accurate parameter estimates for this subset of bands. Our simulation results show that when channel holding times are non-exponential, the proposed scheme achieves a substantial improvement in the probability of correct selection of the best subset of bands compared to an approach based on a (univariate) Markov modulated Gaussian process model.
The traditional von Neumann architecture limits the increase in computing efficiency and results in massive power consumption in modern computers due to the separation of storage and processing units. The novel neuromorphic computation system, an in-memory computing architecture with low power consumption, is aimed to break the bottleneck and meet the needs of the next generation of artificial intelligence (AI) systems. Thus, it is urgent to find a memory technology to implement the neuromorphic computing nanosystem. Nowadays, the silicon-based flash memory dominates non-volatile memory market, however, it is facing challenging issues to achieve the requirements of future data storage device development due to the drawbacks, such as scaling issue, relatively slow operation speed, and high voltage for program/erase operations. The emerging resistive random-access memory (RRAM) has prompted extensive research as its simple two-terminal structure, including top electrode (TE) layer, bottom electrode (BE) layer, and an intermediate resistive switching (RS) layer. It can utilize a temporary and reversible dielectric breakdown to cause the RS phenomenon between the high resistance state (HRS) and the low resistance state (LRS). RRAM is expected to outperform conventional memory device with the advantages, notably its low-voltage operation, short programming time, great cyclic stability, and good scalability. Among the materials for RS layer, indium gallium zinc oxide (IGZO) has shown attractive prospects in abundance and high atomic diffusion property of oxygen atoms, transparency. Additionally, its electrical properties can be easily modulated by controlling the stoichiometric ratio of indium and gallium as well as oxygen potential in the sputter gas. Moreover, since the IGZO can be applied to both the thin-film transistor (TFT) channel and RS layer, it has a great potential for fully integrated transparent electronics application. In this work, we proposed amorphous transparent IGZO-based RRAMs and investigated switching behaviors of the memory cells prepared with different top electrodes. First, ITO was choosing to serve as both TE and BE to achieve high transmittance. A multi-target magnetron sputtering system was employed to deposit all three layers (TE, RS, BE layers) on glass substrate. I-V characteristics were evaluated by a semiconductor parameter analyzer, and the bipolar RS feature of our RRAM devices was demonstrated by typical butterfly curves. The optical transmission analysis was carried out via a UV-Vis spectrometer and the average transmittance was around 80% out of entire devices in the visible-light wavelength range, implying high transparency. We adjusted the oxygen partial pressure during the sputtering of IGZO to optimize the property because the oxygen vacancy concentration governs the RS performance. Electrode selection is crucial and can impact the performance of the whole device. Thus, Cu TE was chosen for our second type of device because the diffusion of Cu ions can be beneficial for the formation of the conductive filament (CF). A ~5 nm SiO 2 barrier layer was employed between TE and RS layers to confine the diffusion of Cu into the RS layer. At the same time, this SiO 2 inserting layer can provide an additional interfacial series resistance in the device to lower the off current, consequently, improve the on/off ratio and whole performance. Finally, an oxygen affinity metal Ti was selected as the TE for our third type of device because the concentration of the oxygen atoms can be shifted towards the Ti electrode, which provides an oxygengettering activity near the Ti metal. This process may in turn lead to the formation of a sub-stoichiometric region in the neighboring oxide that is believed to be the origin of better performance. In conclusion, the transparent amorphous IGZO-based RRAMs were established. To tune the property of RS layer, the sputtering conditions of RS were varied. To investigate the influence of TE selections on switching performance of RRAMs, we integrated a set of TE materials, and a barrier layer on IGZO-based RRAM and compared the switch characteristics. Our encouraging results clearly demonstrate that IGZO is a promising material in RRAM applications and breaking the bottleneck of current memory technologies.
Abstract. Satellite remote sensing provides a global view to processes on Earth that has unique benefits compared to making measurements on the ground, such as global coverage and enormous data volume. The typical downsides are spatial and temporal gaps and potentially low data quality. Meaningful statistical inference from such data requires overcoming these problems and developing efficient and robust computational tools.We design and implement a computationally efficient multi-scale Gaussian process (GP) software package, satGP, geared towards remote sensing applications. The software is able to handle problems of enormous sizes and to compute marginals and sample from the random field conditioning on at least hundreds of millions of observations. This is achieved by optimizing the computation by, e.g., randomization and splitting the problem into parallel local subproblems which aggressively discard uninformative data. We describe the mean function of the Gaussian process by approximating marginals of a Markov random field (MRF). Variability around the mean is modeled with a multi-scale covariance kernel, which consists of Matérn, exponential, and periodic components. We also demonstrate how winds can be used to inform covariances locally.The covariance kernel parameters are learned by calculating an approximate marginal maximum likelihood estimate, and the validity of both the multi-scale approach and the method used to learn the kernel parameters is verified in synthetic experiments. We apply these techniques to a moderate size ozone data set produced by an atmospheric chemistry model and to the very large number of observations retrieved from the Orbiting Carbon Observatory 2 (OCO-2) satellite. The satGP software is released under an open-source license.
Arapostathis, Ari, Pang, Guodong, and Zheng, Yi. Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching. Retrieved from https://par.nsf.gov/biblio/10282626. Advances in Applied Probability 53.1 Web. doi:10.1017/apr.2020.47.
Arapostathis, Ari, Pang, Guodong, & Zheng, Yi. Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching. Advances in Applied Probability, 53 (1). Retrieved from https://par.nsf.gov/biblio/10282626. https://doi.org/10.1017/apr.2020.47
Arapostathis, Ari, Pang, Guodong, and Zheng, Yi.
"Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching". Advances in Applied Probability 53 (1). Country unknown/Code not available. https://doi.org/10.1017/apr.2020.47.https://par.nsf.gov/biblio/10282626.
@article{osti_10282626,
place = {Country unknown/Code not available},
title = {Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching},
url = {https://par.nsf.gov/biblio/10282626},
DOI = {10.1017/apr.2020.47},
abstractNote = {Abstract We study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.},
journal = {Advances in Applied Probability},
volume = {53},
number = {1},
author = {Arapostathis, Ari and Pang, Guodong and Zheng, Yi},
editor = {null}
}
Warning: Leaving National Science Foundation Website
You are now leaving the National Science Foundation website to go to a non-government website.
Website:
NSF takes no responsibility for and exercises no control over the views expressed or the accuracy of
the information contained on this site. Also be aware that NSF's privacy policy does not apply to this site.
