ORIE Technical Reports
http://hdl.handle.net/1813/3942
20160208T20:54:30Z

Multivariate Subexponential Distributions and Their Applications
http://hdl.handle.net/1813/40837
Multivariate Subexponential Distributions and Their Applications
Samorodnitsky, Gennady; Sun, Julian
We propose a new definition of a multivariate subexponential
distribution. We compare this definition with the two existing
notions of multivariate subexponentiality, and compute the
asymptotic behaviour of the ruin probability in the context of an
insurance portfolio, when multivariate subexponentiality
holds. Previously such results were available only in the case of
multivariate regularly varying claims.
20150916T00:00:00Z

Ful lling Orders in a MultiEchelon Capacitated Online Retail System: PART TWO, realtime purchasing and ful llment decision making
http://hdl.handle.net/1813/40146
Ful lling Orders in a MultiEchelon Capacitated Online Retail System: PART TWO, realtime purchasing and ful llment decision making
Li, Juan; Muckstadt, John
When ful lling customer orders, online retailers must operate their multiwarehouse systems with great
care to ensure that these orders are satis ed in a timely and cost e ective manner. We worked closely with
a major online retailer to design an e ective and e cient ful llment system. This included establishing
policies and procedures for ordering, receiving, storing and shipping of goods. Internal warehousing and
transportation practices were addressed, and new approaches for managing inventories were established. In
this paper we focus on one type of inventory management problem faced by the company when making
daily purchasing and allocation decisions. These decisions are of two types, the positioning of inventories in
their multiechelon system and the detailed manner in which they use inventories to ful ll speci c customer
orders. After reviewing some of the key attributes of models that address the two types of decision problems,
we present a computationally tractable approach for solving these problems for a system that must ful ll
many hundreds of thousands of orders daily.
20150513T00:00:00Z

Asymptotic Normality of Degree Counts in a Preferential Attachment Model
http://hdl.handle.net/1813/39933
Asymptotic Normality of Degree Counts in a Preferential Attachment Model
Resnick, Sidney; Samorodnitsky, Gennady
Preferential attachment is a widely adopted paradigm for understanding
the dynamics of
social networks. Formal statistical inference,
for instance GLM techniques, and model
verification methods will require knowing test statistics are asymptotically
normal even though node or count based
network data is nothing like classical data from
independently replicated experiments. We therefore study asymptotic
normality of degree counts for a sequence of growing simple undirected
preferential attachment graphs. The methods of proof rely on
identifying martingales and then exploiting the martingale central
limit theorems.
20150428T00:00:00Z

Functional central limit theorem for negatively dependent heavytailed stationary infinitely divisible processes generated by conservative flows
http://hdl.handle.net/1813/39288
Functional central limit theorem for negatively dependent heavytailed stationary infinitely divisible processes generated by conservative flows
Jung, Paul; Owada, Takashi; Samorodnitsky, Gennady
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, MittagLeffler processes, due to long
memory, and Brownian motions, due to the Gaussian second order
cancellations.
20150406T00:00:00Z

MultiPeriod Stock Allocation Via Robust Optimization
http://hdl.handle.net/1813/39275
MultiPeriod Stock Allocation Via Robust Optimization
Jackson, Peter; Muckstadt, John
In this paper we revisit a longstanding multiechelon inventory allocation problem from a robust optimization perspective. We formulate the problem as a one warehouse, Nretailer, multiperiod, 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 setup, the only motive for holding inventory at the central warehouse for allocation in future periods is
the socalled riskpooling 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, nonanticipatory
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 multiechelon inventory systems.
20150325T00:00:00Z

Climbing down Gaussian peaks
http://hdl.handle.net/1813/39077
Climbing down Gaussian peaks
Adler, Robert; Samorodnitsky, Gennady
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.
20150128T00:00:00Z

Numerical Validation of Fill Rate Estimation Methods for Two and ThreeDemand Class Rationing Policies with OneforOne Replenishment and General Lead Time Distributions
http://hdl.handle.net/1813/39019
Numerical Validation of Fill Rate Estimation Methods for Two and ThreeDemand Class Rationing Policies with OneforOne Replenishment and General Lead Time Distributions
Vicil, Oguzhan; Jackson, Peter
In this report, we conduct numerical simulations of two and threedemand class inventory threshold rationing systems under oneforone 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 threedemand 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
threedemand 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.
20150108T00:00:00Z

Timechanged extremal process as a random sup measure
http://hdl.handle.net/1813/37941
Timechanged extremal process as a random sup measure
Lacaux, CĂ©line; Samorodnitsky, Gennady
A functional limit theorem for the partial maxima of a long memory
stable sequence produces a limiting process that can be described as a
betapower 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
maxincrements. This deceptively simple time change hides the much
more delicate structure of the resulting process as a selfaffine
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 selfsimilar Fr\'echet processes with stationary
maxincrements.
20141009T00:00:00Z

Tauberian Theory for Multivariate Regularly Varying Distributions with Application to Preferential Attachment Networks
http://hdl.handle.net/1813/36712
Tauberian Theory for Multivariate Regularly Varying Distributions with Application to Preferential Attachment Networks
Resnick, Sidney; Samorodnitsky, Gennady
AbelTauberian theorems relate power
law behavior of distributions and their transforms. We formulate and
prove a multivariate version for nonstandard regularly varying
measures on R_+^p and then apply it to
prove that the joint distribution of in and outdegree in a directed edge
preferential attachement model has jointly regularly varying
tails.
20140625T00:00:00Z

Nonstandard regular variation of the indegree and the outdegree in the preferential attachement model
http://hdl.handle.net/1813/36711
Nonstandard regular variation of the indegree and the outdegree in the preferential attachement model
Samorodnitsky, Gennady; Resnick, Sidney; Towsley, Don; Davis, Richard; Willis, Amy; Wan, Phyllis
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
indegree and
outdegree
has jointly regularly varying
tails.
Typically the marginal tails of the indegree distribution and the outdegree
distribution have different regular variation indices and so the joint
regular variation is nonstandard.
Only marginal regular variation has been
previously established for this distribution in the cases where the
marginal tail indices are different.
20140625T00:00:00Z