arXiv subjects

Daren B. H. Cline

Publications and source records attributed to Daren B. H. Cline.

9 recordsLinked to original sources

arXiv2024

On tail inference in scale-free inhomogeneous random graphs

Both empirical and theoretical investigations of scale-free network models have found that large degrees in a network exert an outsized impact on its structure. However, the tools used to infer the tail behavior of degree distributions in scale-free networks often lack a strong theoretical foundation. In this paper, we introduce a new framework for analyzing the asymptotic distribution of estimators for degree tail indices in scale-free inhomogeneous random graphs. Our framework leverages the relationship between the large weights and large degrees of Norros-Reittu and Chung-Lu random graphs. In particular, we determine a rate for the number of nodes $k(n) \rightarrow \infty$ such that for all $i = 1, \dots, k(n)$, the node with the $i$-th largest weight will have the $i$-th largest degree with high probability. Such alignment of upper-order statistics is then employed to establish the asymptotic normality of three different tail index estimators based on the upper degrees. These results suggest potential applications of the framework to threshold selection and goodness-of-fit testing in scale-free networks, issues that have long challenged the network science community.

math.ST↗

arXiv2024

Emergence of Multivariate Extremes in Multilayer Inhomogeneous Random Graphs

In this paper, we propose a multilayer inhomogeneous random graph model (MIRG), whose layers may consist of both single-edge and multi-edge graphs. In the single layer case, it has been shown that the regular variation of the weight distribution underlying the inhomogeneous random graph implies the regular variation of the typical degree distribution. We extend this correspondence to the multilayer case by showing that the multivariate regular variation of the weight distribution implies the multivariate regular variation of the asymptotic degree distribution. Furthermore, in certain circumstances, the extremal dependence structure present in the weight distribution will be adopted by the asymptotic degree distribution. By considering the asymptotic degree distribution, a wider class of Chung-Lu and Norros-Reittu graphs may be incorporated into the MIRG layers. Additionally, we prove consistency of the Hill estimator when applied to degrees of the MIRG that have a tail index greater than 1. Simulation results indicate that, in practice, hidden regular variation may be consistently detected from an observed MIRG.

math.PR↗

arXiv2022

Large Deviation Probabilities for Sums of Random Variables with Heavy or Subexponential Tails

Let $S_n$ be the sum of independent random variables with distribution $F$. Under the assumption that $-\log(1-F(x))$ is slowly varying, conditions for $$ \lim_{n\to\infty}\sup_{s\ge t_n}\left|{P[S_n>s]\over n(1-F(s))}-1\right| =0 $$ are given. These conditions extend and strengthen a series of previous results. Additionally, a connection with subexponential distributions is demonstrated. That is, $F$ is subexponential if and only if the condition above holds for some $t_n$ and $$ \lim_{t\to\infty}{1-F(t+x)\over 1-F(t)} = 1 \quad\text{for each real $x$.}$$

math.PR↗

arXiv2020

Inbound Replenishment and Outbound Dispatch Decisions under Hybrid Shipment Consolidation Policies: An Analytical Model and Comparison

We consider a distribution warehouse where both the inbound inventory replenishment and outbound dispatch decisions are subject to fixed (as well as per-unit) transportation charges and demand is stochastic. In order to realize scale economies associated with transportation operations both on the outbound and inbound sides, dispatch schedules must be synchronized over time with inventory replenishment decisions. An immediate delivery policy on the outbound side does not make economic sense because outbound dispatch operations and costs will benefit from a temporal shipment consolidation policy. Our particular interest in this setting is the exact modeling and analysis of hybrid shipment consolidation policies, in comparison to the time- and quantity-based counterparts. Since shipment consolidation prolongs customer waiting and inventory holding, we investigate average delay penalty per order and average inventory per time unit as critical measures of performance of the distribution operation, along with the annual cost. By fixing the expected inbound replenishment and outbound dispatch frequencies, we compare these measures among the alternative ways of operation under hybrid, time-based, and quantity-based policies. This comparison then lends itself to an explicit analytical comparison of average costs under these three policies, without a need for solving the corresponding optimization problems.

math.OC↗

arXiv2020

Analytical Results on the Service Performance of Stochastic Clearing Systems

Stochastic clearing theory has wide spread applications in the context of supply chain and service operations management. Historical application domains include bulk service queues, inventory control, and transportation planning (e.g., vehicle dispatching and shipment consolidation). In this paper, motivated by a fundamental application in shipment consolidation, we revisit the notion of service performance for stochastic clearing system operation. More specifically, our goal is to evaluate and compare service performance of alternative operational policies for clearing decisions, as quantified by a measure of timely service referred as \emph{Average Order Delay} ($AOD$). All stochastic clearing systems are subject to service delay due to the inherent clearing practice, and $AOD$ can be thought of as a benchmark for evaluating timely service. Although stochastic clearing theory has a long history, existing literature on the analysis of $AOD$ as a service measure has several limitations. Hence, we extend the previous analysis by proposing a more general method for a generic analytical derivation of $AOD$ for any renewal-type clearing policy, including but not limited to alternative shipment consolidation policies in the previous literature. Our proposed method utilizes a new martingale point of view and lends itself for a generic analytical characterization of $AOD$, leading to a complete comparative analysis of alternative renewal-type clearing policies. Hence, we also close the gaps in literature on shipment consolidation via a complete set of analytically provable results regarding $AOD$ which were only illustrated through numerical tests previously.

math.OC↗

arXiv2019

Stochastic Clearing Systems with Multiple Input Processes

In this paper, we consider stochastic clearing systems with multiple drifted Brownian motion inputs. First, we propose an instantaneous rate policy, which is shown to be the optimal one among a large class of renewal type clearing policies in terms of average cost. Second, we propose a service measure about average weighted delay rate, and provide a unified method to calculate the service measure under different clearing policies. Moreover, we prove that under a fixed clearing frequency, the instantaneous rate policy outperforms a large class of clearing policies, and the instantaneous rate hybrid policy performs better than time-based policy, in terms of average weighted delay rate.

math.OC↗

arXiv2014

On Asymptotic Statistics for Geometric Routing Schemes in Wireless Ad-Hoc Networks

In this paper we present a methodology employing statistical analysis and stochastic geometry to study geometric routing schemes in wireless ad-hoc networks. In particular, we analyze the network layer performance of one such scheme, the random $\frac{1}{2}$disk routing scheme, which is a localized geometric routing scheme in which each node chooses the next relay randomly among the nodes within its transmission range and in the general direction of the destination. The techniques developed in this paper enable us to establish the asymptotic connectivity and the convergence results for the mean and variance of the routing path lengths generated by geometric routing schemes in random wireless networks. In particular, we approximate the progress of the routing path towards the destination by a Markov process and determine the sufficient conditions that ensure the asymptotic connectivity for both dense and large-scale ad-hoc networks deploying the random $\frac{1}{2}$disk routing scheme. Furthermore, using this Markov characterization, we show that the expected length (hop-count) of the path generated by the random $\frac{1}{2}$disk routing scheme normalized by the length of the path generated by the ideal direct-line routing, converges to $3π/4$ asymptotically. Moreover, we show that the variance-to-mean ratio of the routing path length converges to $9π^2/64-1$ asymptotically. Through simulation, we show that the aforementioned asymptotic statistics are in fact quite accurate even for finite granularity and size of the network.

cs.NI↗

arXiv2013

On the Random 1/2-Disk Routing Scheme in Wireless Ad Hoc Networks

Random 1/2-disk routing in wireless ad-hoc networks is a localized geometric routing scheme in which each node chooses the next relay randomly among the nodes within its transmission range and in the general direction of the destination. We introduce a notion of convergence for geometric routing schemes that not only considers the feasibility of packet delivery through possibly multi-hop relaying, but also requires the packet delivery to occur in a finite number of hops. We derive sufficient conditions that ensure the asymptotic \emph{convergence} of the random 1/2-disk routing scheme based on this convergence notion, and by modeling the packet distance evolution to the destination as a Markov process, we derive bounds on the expected number of hops that each packet traverses to reach its destination.

cs.NI↗

arXiv2005

Stability and the Lyapounov exponent of threshold AR-ARCH Models

The Lyapounov exponent and sharp conditions for geometric ergodicity are determined of a time series model with both a threshold autoregression term and threshold autoregressive conditional heteroscedastic (ARCH) errors. The conditions require studying or simulating the behavior of a bounded, ergodic Markov chain. The method of proof is based on a new approach, called the piggyback method, that exploits the relationship between the time series and the bounded chain. The piggyback method also provides a means for evaluating the Lyapounov exponent by simulation and provides a new perspective on moments, illuminating recent results for the distribution tails of GARCH models.

math.PR↗