eCommons Collection:http://hdl.handle.net/1813/39422015-04-19T10:04:39Z2015-04-19T10:04:39ZFunctional central limit theorem for negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flowsJung, PaulOwada, TakashiSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/392882015-04-07T05:05:24Z2015-04-06T00:00:00ZTitle: Functional central limit theorem for negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows
Authors: Jung, Paul; Owada, Takashi; Samorodnitsky, Gennady
Abstract: We prove a functional central limit theorem for
partial sums of symmetric stationary long range dependent heavy tailed
infinitely divisible processes with a certain type of negative
dependence. Previously only positive dependence could be treated. The
negative dependence involves cancellations of the Gaussian second
order. This
leads to new types of {limiting} processes involving stable random
measures, due to heavy tails, Mittag-Leffler processes, due to long
memory, and Brownian motions, due to the Gaussian second order
cancellations.2015-04-06T00:00:00ZMulti-Period Stock Allocation Via Robust OptimizationJackson, PeterMuckstadt, Johnhttp://hdl.handle.net/1813/392752015-03-26T05:08:09Z2015-03-25T00:00:00ZTitle: Multi-Period Stock Allocation Via Robust Optimization
Authors: Jackson, Peter; Muckstadt, John
Abstract: In this paper we re-visit a long-standing multi-echelon inventory al-location problem from a robust optimization perspective. We formulate the problem as a one warehouse, N-retailer, multi-period, stock allocation problem in which holding costs are identical at each location and
no stock is received from outside suppliers for the duration of the planning horizon. Stock may be transferred from the central warehouse to
the retailers instantaneously and without cost at the beginning of each
period for which the central warehouse still has stock on hand. No other
stock transfers are allowed. Under this set-up, the only motive for holding inventory at the central warehouse for allocation in future periods is
the so-called risk-pooling motive. The dynamic programming formulation
of this problem requires a state space too large for practical computation. Various approximation methods have been proposed for variants of
this problem. We apply robust optimization to this problem extending
the typical uncertainty set to capture the risk pooling phenomenon and
extending the inventory policy to allow for an adaptive, non-anticipatory
shipment policy. We show how to represent the uncertainty set compactly
so that it grows by no more than the square of the number of retailers.
The problem can be solved using Benders decomposition in the general
case. In the special case of no initial retailer inventories, two periods, and
identical retailers, a relaxed form of the problem admits a closed form
solution with surprising insights. Summarizing the experimental results
of the paper, we see both confirmation of the value of the robust optimization approach as well as managerial insights into the design and operation
of multi-echelon inventory systems.2015-03-25T00:00:00ZClimbing down Gaussian peaksAdler, RobertSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/390772015-01-29T06:02:16Z2015-01-28T00:00:00ZTitle: Climbing down Gaussian peaks
Authors: Adler, Robert; Samorodnitsky, Gennady
Abstract: How likely is the high level of a continuous Gaussian random field on
an Euclidean space to
have a ``hole'' of a certain dimension and depth? Questions of this
type are difficult, but in this paper we make progress on questions
shedding new light in existence of holes. How likely is the field to
be above a high level on one compact set (e.g. a sphere) and to be
below a fraction of that level on some other compact set, e.g. at the
center of the corresponding ball? How likely is the field to be below that
fraction of the level anywhere nside the ball? We work on the
level of large deviations.2015-01-28T00:00:00ZNumerical Validation of Fill Rate Estimation Methods for Two- and Three-Demand Class Rationing Policies with One-for-One Replenishment and General Lead Time DistributionsVicil, OguzhanJackson, Peterhttp://hdl.handle.net/1813/390192015-01-09T06:03:23Z2015-01-08T00:00:00ZTitle: Numerical Validation of Fill Rate Estimation Methods for Two- and Three-Demand Class Rationing Policies with One-for-One Replenishment and General Lead Time Distributions
Authors: Vicil, Oguzhan; Jackson, Peter
Abstract: In this report, we conduct numerical simulations of two- and three-demand class inventory threshold rationing systems under one-for-one replenishment policies. The performance metrics of interest are the fill rates of the high priority demand classes (the gold fill rate in the two demand
class system and the platinum and gold fill rates in the three-demand class system).
Our main interest is in the sensitivity of these fill rates to the form of the replenishment lead time probability distribution and the resulting quality of approximation methods used to estimate
these fill rates. We consider three approximation methods: what we call the single cycle approach attributed to Dekker et al and Deshpande et al, the embedded Markov chain approach of Fadigloglu and Bulut, and the continuous time Markov chain approach of Vicil and Jackson.
We confirm the superiority of the embedded Markov chain approach for the case of constant lead times but we find that the fill rates are relatively insensitive to the form of the lead time distribution and both latter approaches, the embedded Markov chain approach and the continuous time Markov chain approach, perform well over wide ranges of lead time variability. For the
three-demand class system, we demonstrate that it is possible to achieve highly differentiated fill rates by demand class and show that these fill rates can be estimated with high accuracy using the continuous time Markov chain approach, provided the fill rate of the lowest priority demand class (the silver fill rate) is not too low.2015-01-08T00:00:00ZTime-changed extremal process as a random sup measureLacaux, CélineSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/379412014-10-10T05:03:36Z2014-10-09T00:00:00ZTitle: Time-changed extremal process as a random sup measure
Authors: Lacaux, Céline; Samorodnitsky, Gennady
Abstract: A functional limit theorem for the partial maxima of a long memory
stable sequence produces a limiting process that can be described as a
beta-power time change in the classical Fr\'echet
extremal process, for beta in a subinterval of the unit
interval. Any such power time change in the extremal process
for 0<beta<1 produces a process with stationary
max-increments. This deceptively simple time change hides the much
more delicate structure of the resulting process as a self-affine
random sup measure. We uncover this structure and show that in a
certain range of the parameters this random measure arises as a limit
of the partial maxima of the same long memory stable sequence, but in
a different space. These results open a way to construct a whole new
class of self-similar Fr\'echet processes with stationary
max-increments.2014-10-09T00:00:00ZTauberian Theory for Multivariate Regularly Varying Distributions with Application to Preferential Attachment NetworksResnick, SidneySamorodnitsky, Gennadyhttp://hdl.handle.net/1813/367122014-06-26T05:04:32Z2014-06-25T00:00:00ZTitle: Tauberian Theory for Multivariate Regularly Varying Distributions with Application to Preferential Attachment Networks
Authors: Resnick, Sidney; Samorodnitsky, Gennady
Abstract: Abel-Tauberian theorems relate power
law behavior of distributions and their transforms. We formulate and
prove a multivariate version for non-standard regularly varying
measures on R_+^p and then apply it to
prove that the joint distribution of in- and out-degree in a directed edge
preferential attachement model has jointly regularly varying
tails.2014-06-25T00:00:00ZNonstandard regular variation of the in-degree and the out-degree in the preferential attachement modelSamorodnitsky, GennadyResnick, SidneyTowsley, DonDavis, RichardWillis, AmyWan, Phyllishttp://hdl.handle.net/1813/367112014-06-26T05:04:31Z2014-06-25T00:00:00ZTitle: Nonstandard regular variation of the in-degree and the out-degree in the preferential attachement model
Authors: Samorodnitsky, Gennady; Resnick, Sidney; Towsley, Don; Davis, Richard; Willis, Amy; Wan, Phyllis
Abstract: For the directed edge preferential attachment network growth model
studied by Bollobas et al. (2003) and
Krapivsky and Redner (2001), we prove that the joint distribution of
in-degree and
out-degree
has jointly regularly varying
tails.
Typically the marginal tails of the in-degree distribution and the out-degree
distribution have different regular variation indices and so the joint
regular variation is non-standard.
Only marginal regular variation has been
previously established for this distribution in the cases where the
marginal tail indices are different.2014-06-25T00:00:00ZGeneral inverse problems for regular variationDamek, EwaMikosch, ThomasRosinski, JanSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/344112013-10-03T05:03:23Z2013-10-02T00:00:00ZTitle: General inverse problems for regular variation
Authors: Damek, Ewa; Mikosch, Thomas; Rosinski, Jan; Samorodnitsky, Gennady
Abstract: Regular variation of distributional tails is known to be preserved by
various linear transformations of some random structures.
An inverse problem for regular
variation aims at understanding whether the regular variation of a
transformed random object is caused by regular variation of components
of the original random structure. In this paper we build up on previous
work and derive results in the multivariate case and in
situations where regular variation is
not restricted to one particular direction or quadrant.2013-10-02T00:00:00ZStock Optimization in Emergency Resupply Networks under Stuttering Poisson DemandChen, JJackson, P.L.Muckstadt, Jhttp://hdl.handle.net/1813/331862013-04-23T05:03:17Z2013-04-01T00:00:00ZTitle: Stock Optimization in Emergency Resupply Networks under Stuttering Poisson Demand
Authors: Chen, J; Jackson, P.L.; Muckstadt, J
Abstract: We consider a network in which field stocking locations (FSLs) manage multiple parts according
to an (S-1,S) policy. Demand processes for the parts are assumed to be independent
stuttering Poisson processes. Regular replenishments to an FSL occur from a regional stocking
location (RSL) that has an unlimited supply of each part type. Demand in excess
of supply at an FSL is routed to an emergency stocking location (ESL), which also employs
an (S-1,S) policy to manage its inventory. Demand in excess of supply at the ESL is backordered.
Lead time from the ESL to each FSL is assumed to be negligible compared to the
RSL-ESL resupply time. In companion papers we have shown how to approximate the joint
probability distributions of units on hand, units in regular resupply, and units in emergency
resupply. In this paper, we focus on the problem of determining the stock levels at the FSLs
and ESL across all part numbers that minimize backorder, and emergency resupply costs
subject to an inventory investment budget constraint. The problem is shown to be a nonconvex integer programming problem, and we explore a collection of heuristics for solving
the optimization problem.2013-04-01T00:00:00ZCalculation of ruin probabilities for a dense class of heavy tailed distributionsBladt, MogensNielsen, Bo FriisSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/315402013-03-06T06:01:31Z2013-03-05T00:00:00ZTitle: Calculation of ruin probabilities for a dense class of heavy tailed distributions
Authors: Bladt, Mogens; Nielsen, Bo Friis; Samorodnitsky, Gennady
Abstract: In this paper we propose a
class of infinite--dimensional phase--type distributions with
finitely many parameters as models for heavy tailed distributions.
The class of finite--dimensional distributions is dense in the class
of distributions on the positive reals and may hence approximate any
such distribution.
We
prove that formulas from renewal theory, and with a particular
attention to ruin probabilities, which are true for common
phase--type distributions also hold true for the
infinite--dimensional case. We provide algorithms for
calculating functionals of interest such as the renewal density and
the ruin probability. It might be of interest to approximate a given
heavy--tailed distribution of some other type by a distribution from
the class of infinite--dimensional phase--type distributions and to
this end we provide a calibration procedure which works for the
approximation of distributions with a slowly varying tail. An
example from risk theory, comparing ruin probabilities for a
classical risk process with Pareto distributed claim sizes, is
presented and exact known ruin probabilities for the Pareto case are
compared to the ones obtained by approximating by an
infinite--dimensional hyper--exponential distribution.2013-03-05T00:00:00Z