Tail Decay Rate Research Articles

Simple and Explicit Bounds for Multiserver Queues with 11−ρ Scaling

We consider the first-come-first-serve (FCFS) [Formula: see text] queue and prove the first simple and explicit bounds that scale as [Formula: see text] under only the assumption that interarrival times have finite second moment, and service times have finite [Formula: see text] moment for some [Formula: see text]. Here, ρ denotes the corresponding traffic intensity. Conceptually, our results can be viewed as a multiserver analogue of Kingman’s bound. Our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay. The strength of our bounds (e.g., in the form of tail decay rate) is a function of how many moments of the service distribution are assumed finite. Our bounds scale gracefully, even when the number of servers grows large and the traffic intensity converges to unity simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale better than [Formula: see text] in certain asymptotic regimes. In these same asymptotic regimes, we also prove bounds for the tail of the steady-state number in service. Our main proofs proceed by explicitly analyzing the bounding process that arises in the stochastic comparison bounds of Gamarnik and Goldberg for multiserver queues. Along the way, we derive several novel results for suprema of random walks and pooled renewal processes, which may be of independent interest. We also prove several additional bounds using drift arguments (which have much smaller prefactors) and point out a conjecture that would imply further related bounds and generalizations. We also show that when all moments of the service distribution are finite and satisfy a mild growth rate assumption, our bounds can be strengthened to yield explicit tail estimates decaying as [Formula: see text], with [Formula: see text], depending on the growth rate of these moments. Funding: Financial support from the National Science Foundation [Grant 1333457] is gratefully acknowledged. Supplemental Material: The supplemental appendix is available at https://doi.org/10.1287/moor.2022.0131 .

Large Deviations for a Class of Multivariate Heavy-Tailed Risk Processes Used in Insurance and Finance

Modern risk modelling approaches deal with vectors of multiple components. The components could be, for example, returns of financial instruments or losses within an insurance portfolio concerning different lines of business. One of the main problems is to decide if there is any type of dependence between the components of the vector and, if so, what type of dependence structure should be used for accurate modelling. We study a class of heavy-tailed multivariate random vectors under a non-parametric shape constraint on the tail decay rate. This class contains, for instance, elliptical distributions whose tail is in the intermediate heavy-tailed regime, which includes Weibull and lognormal type tails. The study derives asymptotic approximations for tail events of random walks. Consequently, a full large deviations principle is obtained under, essentially, minimal assumptions. As an application, an optimisation method for a large class of Quota Share (QS) risk sharing schemes used in insurance and finance is obtained.

Upper tail for homomorphism counts in constrained sparse random graphs

AbstractConsider the upper tail probability that the homomorphism count of a fixed graph H within a large sparse random graph Gn exceeds its expected value by a fixed factor . Going beyond the Erdős–Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random dn‐regular graphs (provided H has a regular 2‐core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs (extending the known results for ), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erdős–Rényi graphs.

Analysis of the symmetric join the shortest orbit queue

This work introduces the join the shortest queue policy in the retrial setting. We consider a Markovian single server retrial system with two infinite capacity orbits. An arriving job finding the server busy, it is forwarded to the least loaded orbit. Otherwise, it is forwarded to an orbit randomly. Orbiting jobs of either type retry to access the server independently. We investigate the stability condition, the stationary tail decay rate, and obtain the equilibrium distribution by using the compensation method.

Markov modulated fluid network process: Tail asymptotics of the stationary distribution

We consider a Markov modulated fluid network with a finite number of stations. We are interested in the tail asymptotics behavior of the stationary distribution of its buffer content process. Using two different approaches, we derive upper and lower bounds for the stationary tail decay rate in various directions. Both approaches are based on a well-known time-evolution formula of a Markov process, so-called Dynkin’s formula, where a key ingredient is a suitable choice of test functions. Those results show how multidimensional tail asymptotics can be studied for the more than two-dimensional case, which is known as a hard problem.

MSE Tail Analysis for Remote State Estimation of Linear Systems Over Multiantenna Random Access Channels

In this article, we consider the Kalman filter (KF)-based state estimation in the multisensor networked control systems. The sensors communicate plant state observations to a remote state estimator over a multiantenna random access network, resulting in random fading in measurements and rank-deficient observation updates in KF. In contrast to the conventional literature where average state estimation mean-square-error (MSE) is adopted as the estimation performance measure, we analyze the state estimation MSE tail distribution which guarantees fine-grained characterizations of the MSE sample path. We developed various sufficient conditions for the well behaved tail distribution of the MSE. Specifically, we propose the observable and unobservable cone decomposition of the transformed coordinate system induced by the singular value decomposition of the measurement matrix, which enables us to overcome the limitation of MSE analysis with rank deficient measurement matrix. In the noiseless plant case, we establish the sufficient condition such that the state estimation MSE is a supermartingale in high SNR regime. Via martingale convergence theorems, we show that the state estimation MSE is almost surely stable, which results in outlier-free MSE sample path. In the noisy plant case, we show that the state estimation MSE is uniformly upper bounded by a Markov chain in high SNR regime. We establish a sufficient condition for the existence of the steady-state distribution of the Markov chain. Via analyzing the stochastic fixed point equation associated with the stationary distribution of the Markov chain, we provide the closed-form expression for the MSE tail probability. We further established a power law that provides closed-form characterizations of the tradeoff between the communication resource and the MSE tail decay rate, where more communication resource consumption leads to faster exponential decaying MSE tail. We also recover the results in the existing literature when applying our developed theory to the scalar systems.

Jointly Robust Prior for Gaussian Stochastic Process in Emulation, Calibration and Variable Selection

Gaussian stochastic process (GaSP) has been widely used in two fundamental problems in uncertainty quantification, namely the emulation and calibration of mathematical models. Some objective priors, such as the reference prior, are studied in the context of emulating (approximating) computationally expensive mathematical models. In this work, we introduce a new class of priors, called the jointly robust prior, for both the emulation and calibration. This prior is designed to maintain various advantages from the reference prior. In emulation, the jointly robust prior has an appropriate tail decay rate as the reference prior, and is computationally simpler than the reference prior in parameter estimation. Moreover, the marginal posterior mode estimation with the jointly robust prior can separate the influential and inert inputs in mathematical models, while the reference prior does not have this property. We establish the posterior propriety for a large class of priors in calibration, including the reference prior and jointly robust prior in general scenarios, but the jointly robust prior is preferred because the calibrated mathematical model typically predicts the reality well. The jointly robust prior is used as the default prior in two new R packages, called "RobustGaSP" and "RobustCalibration", available on CRAN for emulation and calibration, respectively.

Latency Analysis for Distributed Coded Storage Systems

Modern communication and computation systems often consist of large networks of unreliable nodes. Still, it is well known that such systems can provide aggregate reliability via redundancy. While duplication may increase the load on a system, it can lead to significant performance improvement when combined with the judicious management of extra system resources. Prime examples of this abstract paradigm include multi-path routing across communication networks, content access from multiple caches in delivery networks, and master/slave computations on compute clusters. Several recent contributions in the area establish bounds on the performance of redundant systems, characterizing the latency-redundancy tradeoff under specific load profiles. Following a similar line of research, this paper introduces new analytical bounds and approximation techniques for the latency-redundancy tradeoff for a range of system loads and a class of symmetric redundancy schemes, under the assumption of Poisson arrivals, exponential service-rates, and fork-join scheduling policy. The proposed approach can be employed to efficiently approximate the latency distribution of a queueing system at equilibrium. Various metrics can subsequently be derived for this system, including the mean and variance of the sojourn time, and the tail decay rate of the stationary distribution. This paper also establishes the stability region in terms of arrival rates for redundant systems with certain symmetries. Finally, it offers selection guidelines for design parameters to provide latency guarantees based on the proposed approximations. Findings are substantiated by numerical results.

Hierarchical Space-Time Modeling of Asymptotically Independent Exceedances With an Application to Precipitation Data

The statistical modeling of space-time extremes in environmental applications is key to understanding complex dependence structures in original event data and to generating realistic scenarios for impact models. In this context of high-dimensional data, we propose a novel hierarchical model for high threshold exceedances defined over continuous space and time by embedding a space-time Gamma process convolution for the rate of an exponential variable, leading to asymptotic independence in space and time. Its physically motivated anisotropic dependence structure is based on geometric objects moving through space-time according to a velocity vector. We demonstrate that inference based on weighted pairwise likelihood is fast and accurate. The usefulness of our model is illustrated by an application to hourly precipitation data from a study region in Southern France, where it clearly improves on an alternative censored Gaussian space-time random field model. While classical limit models based on threshold-stability fail to appropriately capture relatively fast joint tail decay rates between asymptotic dependence and classical independence, strong empirical evidence from our application and other recent case studies motivates the use of more realistic asymptotic independence models such as ours. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Bridging asymptotic independence and dependence in spatial extremes using Gaussian scale mixtures

Gaussian scale mixtures are constructed as Gaussian processes with a random variance. They have non-Gaussian marginals and can exhibit asymptotic dependence unlike Gaussian processes, which are asymptotically independent except in the case of perfect dependence. In this paper, we study the extremal dependence properties of Gaussian scale mixtures and we unify and extend general results on their joint tail decay rates in both asymptotic dependence and independence cases. Motivated by the analysis of spatial extremes, we propose flexible yet parsimonious parametric copula models that smoothly interpolate from asymptotic dependence to independence and include the Gaussian dependence as a special case. We show how these new models can be fitted to high threshold exceedances using a censored likelihood approach, and we demonstrate that they provide valuable information about tail characteristics. In particular, by borrowing strength across locations, our parametric model-based approach can also be used to provide evidence for or against either asymptotic dependence class, hence complementing information given at an exploratory stage by the widely used nonparametric or parametric estimates of the χ and χ̄ coefficients. We demonstrate the capacity of our methodology by adequately capturing the extremal properties of wind speed data collected in the Pacific Northwest, US.

Limit theorems for point processes under geometric constraints (and topological crackle)

We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Čech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological “crackle;” the continued presence of spurious homology in samples of topological structures, despite increased sample size.

A Model-Free Tail Index and Its Return Predictability

This paper provides a novel framework -- the cumulant generating function (cgf) of the market risk on the positive half real line -- for studying the market risk which can be replicated by cross sections of index option prices in a model-free manner. Within this unifying framework, independent of the underlying price process, the VIX index measures the height of the cgf at one while the SVIX index proposed by Martin (2016) measures the convexity of the cgf over the interval [0, 2]. A tail index of the market risk, TIX, is proposed based on this framework which measures the tail decay rate of the distribution of market risk revealing the market perceptions of extreme risk. The innovation in TIX strongly predicts market returns both in and out of sample, with monthly R2 statistics of 3.33% and 6.15%, respectively, outperforming the popular return predictors in the literature.

Modeling asymptotically independent spatial extremes based on Laplace random fields

We tackle the modeling of threshold exceedances in asymptotically independent stochastic processes by constructions based on Laplace random fields. Defined as mixtures of Gaussian random fields with an exponential variable embedded for the variance, these processes possess useful asymptotic properties while remaining statistically convenient. Univariate Laplace distribution tails are part of the limiting generalized Pareto distributions for threshold exceedances. After normalizing marginal distributions in data, a standard Laplace field can be used to capture spatial dependence among extremes. Asymptotic properties of Laplace fields are explored and compared to the classical framework of asymptotic dependence. Multivariate joint tail decay rates are slower than for Gaussian fields with the same covariance structure; hence they provide more conservative estimates of very extreme joint risks while maintaining asymptotic independence. Statistical inference is illustrated on extreme wind gusts in the Netherlands where a comparison to the Gaussian dependence model shows a better goodness-of-fit. In this application we fit the well-adapted Weibull distribution, closely related to the Laplace distribution, as univariate tail model.

A superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov-modulated two-dimensional reflecting process

Markov modulation is versatile in generalization for making a simple stochastic model which is often analytically tractable to be more flexible in application. In this spirit, we modulate a two dimensional reflecting skip-free random walk in such a way that its state transitions in the boundary faces and interior of a nonnegative integer quadrant are controlled by Markov chains. This Markov modulated model is referred to as a 2d-QBD process according to Ozaw (2013). We are interested in the tail asymptotics of its stationary distribution, which has been well studied when there is no Markov modulation. Ozawa studied this tail asymptotics problem, but his answer is not analytically tractable. We think this is because Markov modulation is so free to change a model even if the state space for Markov modulation is finite. Thus, some structure, say, extra conditions, would be needed to make the Markov modulation analytically tractable while minimizing its limitation in application. The aim of this paper is to investigate such structure for the tail asymptotic problem. For this, we study the existence of a right subinvariant positive vector, called a superharmonic vector, of a nonnegative matrix with QBD block structure, where each block matrix is finite dimensional. We characterize this existence under a certain extra assumption. We apply this characterization to the 2d-QBD process, and derive the tail decay rates of its marginal stationary distribution in an arbitrary direction. This solves the tail decay rate problem for a two node generalized Jackson network, which has been open for many years.

Tail asymptotics of the stationary distribution for a two-node generalized Jackson network

Tail asymptotics of the stationary distribution for a two-node generalized Jackson network

Intermediate behavior of Kerr tails

The numerical investigation of wave propagation in the asymptotic domain of Kerr spacetime has only recently been possible thanks to the construction of suitable hyperboloidal coordinates. The asymptotics revealed an apparent puzzle in the decay rates of scalar fields: the late-time rates seemed to depend on whether finite distance observers are in the strong field domain or far away from the rotating black hole, an apparent phenomenon dubbed ‘splitting.’ We discuss far-field ‘splitting’ in the full field and near-horizon ‘splitting’ in certain projected modes using horizon-penetrating, hyperboloidal coordinates. For either case we propose an explanation to the cause of the ‘splitting’ behavior, and we determine uniquely decay rates that previous studies found to be ambiguous or immeasurable. The far-field ‘splitting’ is explained by competition between projected modes. The near-horizon ‘splitting’ is due to excitation of lower multipole modes that back excite the multipole mode for which ‘splitting’ is observed. In both cases ‘splitting’ is an intermediate effect, such that asymptotically in time strong field rates are valid at all finite distances. At any finite time, however, there are three domains with different decay rates whose boundaries move outwards during evolution. We then propose a formula for the decay rate of tails that takes into account the inter-mode excitation effect that we study.

Asymptotic analysis of simultaneous damages in spatial Boolean models

A notion of the positive spatial association is introduced in this paper to analyze spatial dependence of Boolean models with the focus on estimating the long-range spatial dependence. The explicit tail estimates for probabilities of simultaneous damage to two distant spatial regions are obtained using the regular variation method, and the long-range spatial covariance for the Boolean models with heavy-tailed grains is shown to decay at the power-law rate that is smaller than the tail decay rate of grains. Examples and applications to spatial reliability modeling are also discussed.

Stationary distribution of a multi-server vacation queue with constant impatient times

This paper analyzes a multi-server queueing system with multiple vacations in which the vacation duration of each server is exponentially distributed. When all the servers are on vacation, customers are impatient if their waiting times exceed a constant value. To derive the stationary distribution of the system, we employ the matrix analytic method. Specifically, our queueing model is represented as an M/G/1-type Markov chain, and the stationary distribution is simply obtained by using its transition structures. Furthermore, we explicitly derive the tail decay rate of the stationary distribution, and demonstrate that the decay rate is not always equal to that for the queueing model without vacations.

Tail asymptotics of the stationary distribution for M/M-JSQ with k parallel queues (abstract only)

We consider a parallel queueing model which has k identical servers. Assume that customers arrive from outside according to a Poisson process and join the shortest queue. Their service times have an i.i.d . exponential distribution, which is referred to as an M/MJSQ with k parallel queues. We are interested in the asymptotic behavior of the stationary distribution for the shortest queue length of this model, provided the stability is assumed. For this stationary distribution, it can be guessed conjectured that the tail decay rate is given by the k-th power of the traffic intensity of the corresponding M/M/k queue with a single waiting line. We prove this fact by obtaining the exactly geometric asymptotics. For this, we use two formulations. One is a quasi-birth-and-death (QBD for short) process which is typically used, and the other is a reflecting random walk on the boundary of the k + 1-dimensional orthant which is a key for our proof.

Conjectures on tail asymptotics of the marginal stationary distribution for a multidimensional SRBM

We are concerned with the stationary distribution of a d-dimensional semimartingale reflecting Brownian motion on a nonnegative orthant, provided it is stable, and conjecture about the tail decay rate of its marginal distribution in an arbitrary direction. Due to recent studies, the conjecture is true for d=2. We show its validity for the skew symmetric case for a general d.