# Peak load reduction for distributed backup scheduling

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Peak load reduction for distributed backup scheduling Peter van de Ven joint work with Angela Schörgendorfer (Google) and Bo Zhang (IBM Research)

2

3 My research interests are in the design and analysis of large (distributed) stochastic systems: - performance analysis; scheduling; stochastic optimization and control

4 Part I: Peak load reduction for distributed backup scheduling

5 load 8am 6pm Load is highly varying throughout the day - network costs are non-linear in the hourly load - aim to reduce network costs by shifting peak load

6 load backups 8am 6pm - the load is comprised of different types of traffic - we focus on the backup traffic

7 load backups 8am 6pm Users (workstations) aim to do daily backups - the backup transmitted across the network to a central server - backup traffic can be shifted because of delay-tolerance

8 load load backups backups 8am 6pm 8am 6pm current target Backups are initiated by the users, based on local information only - backup policy is characterized by hourly backup probabilities However - users are not always connected - the server may have some capacity constraint

9 load load backups backups 8am 6pm 8am 6pm current target Question 1: How much traffic can the network handle for given backup probabilities? Question 2: How to choose the backup probabilities to minimize the network costs while ensuring regular backups?

10 Model outline

11 N users network server Consider N users that generate data (bits) over time - user backup to a central server - access to server over a data network

12 A 1(t) B(t) i A 2(t) A 3(t) A 4(t) A 5(t) network server - time is slotted - user i generates A i (t) bits in slot t - data is added to the backlog B i (t)

13 A 1(t) B(t) i A 2(t) A 3(t) A 4(t) A 5(t) network C i (t) server - users can only do a backup when connected to the network - connectivity is represented by C i (t) {0, 1}, i = 1,..., N

14 A 1(t) B(t) i σ i (t) A 2(t) A 3(t) A 4(t) A 5(t) network C i (t) server Connected users may decide to do a backup (σ i (t) = 1) or not (σ i (t) = 0)

15 A 1(t) B(t) i σ i (t) A 2(t) A 3(t) A 4(t) A 5(t) network C i (t) server Connected users may decide to do a backup (σ i (t) = 1) or not (σ i (t) = 0) The backlog evolution and offered backup load are given by B i (t + 1) = (A i (t) + B i (t))(1 C i (t)σ i (t)), N H(t) = (A i (t) + B i (t))c i (t)σ i (t) i=1

16 User type & Backup probability Consider the number of days since backup W i (t) {0, 1,... } (user type): W i (t + 1) = W i (t) ( 1 σ i (t)c i (t) ) + 1 {η(t+1)=0}, with η(t) := t mod T the hour of the day

17 User type & Backup probability Consider the number of days since backup W i (t) {0, 1,... } (user type): W i (t + 1) = W i (t) ( 1 σ i (t)c i (t) ) + 1 {η(t+1)=0}, with η(t) := t mod T the hour of the day The backup decision is random and may depend on W i (t) and the hour of the day u(t) {0,..., T 1}: { 1 w.p. ν(u(t), Wi (t)) σ i (t) = 0 otherwise

18 Capacity & network cost minimization

19 Let W(t) = (W 1 (t),..., W N (t)) and consider the Markov process U := {(η(t), W(t)), t = 0, 1,... } For fixed ν, we can analyze U to obtain insights into the backup system

20 The stationary distribution π of the process U satisfies where π(u, w) = π(0, 1)k 1 (u, w), u = 0,..., T 1; w 1, π(u, 0) = π(0, 1)k 2 (u), u = 1,..., T 1, w 1 k 1 (u, w) := m(0, u 1, w) m(0, T 1, y), k 2 (u) := u v=1 y=1 and with normalizing constant [ π(0, 1) = y=1 k 1 (v 1, y)c(v 1)ν(v 1, y), T 1 w=1 u=0 T 1 k 1 (u, w) + u=1 k 2 (u)] 1.

21 Let u m(u, u, w) := (1 c(v)ν(v, w)), u, u = 0,..., T 1; w = 0, 1,..., v=u and define g(u, w) := 1 m(u, T 1, w)m(0, u 1, w + 1), u, u = 0,..., T 1; w = 0, 1,..., Proposition If α(u) := lim w w( g(u, w) w 1) exists and α(u) (0, ) for all u {0, 1,..., T 1}, then U is positive recurrent.

22 Lemma The expected backlog size of a type-w user at hour u: S(u, w) = where T 1 u =0 = p(u, u, w)e π [ B(t) η(t) = u, W (t) = w, η(τ (t)) = u ]. [ E π B(t) η(t) = u, W (t) = w, η(τ (t)) = u ] p(u, u, w) = u 1 v=u +1 T 1 v=u +1 w =0 w =0 a(v), w = 0, a(v) + (w 1)â + u 1 v=0 a(v), w 1, π(u,w ) π(u,0) c(u )ν(u, w )m(u + 1, u 1, 0), w = 0, π(u,w ) π(0,1) c(u )ν(u, w )m(u + 1, T 1, 0), w 1.

23 load B(t) L(t) 8am 6pm We consider the sum of H(t) (backup load) and L(t) (extraneous load) - denote H(t; ν) to reflect the impact of ν

24 Network costs log10(kb) The data network is associated with a load-dependent hourly cost function g : R R We are interested in the average per-day network costs (T = 24) G(ν) = T 1 u=0 E{g ( L(u) + H(u; ν) ) },

25 We approximate G(ν) G(ν) := T 1 u=0 g ( E [ L(u) ] + E π [ H(u; ν) ]), The E [ L(u) ] can be obtained from measurements. Exploiting user independence: E π [H(u; ν)] = NT π(u, w) ( S(u, w) + a(u) ) c(u)ν(u, w), w=0

26 We aim to minimize the network costs, while ensuring regular backups min ν G(ν) such that T w=w j π(0, w) γ j, j = 1, 2,.... This problem is non-convex.

27 Implementation

28 We consider an IBM location with N = 5397 users 25 L u c u u u load L(u) connectivity c(u)

29 w = 1 w = 2 w = 3 w = 4 w 5 We truncate the state space at w = 5 and compute the ν(u, w) - the probability is high at night and low during working hours - users that have not done a backup in a while are more likely to schedule during peak hours

30 Comparison to original scheduling table Europe team s schedule: = Our schedule: =

31 Comparison to original scheduling table - Performance days without backup total load

32 Constrained system β Consider a constraint on the total amount of backup traffic: B i (t + 1) = (A i (t) + B i (t))(1 C i (t)σ i (t) min{1, β/b(t)}) Users are no longer independent - stability conditions and proofs are more involved - use mean field limit to compute long-term average behavior

### LECTURE 4. Last time: Lecture outline

LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random

### How performance metrics depend on the traffic demand in large cellular networks

How performance metrics depend on the traffic demand in large cellular networks B. B laszczyszyn (Inria/ENS) and M. K. Karray (Orange) Based on joint works [1, 2, 3] with M. Jovanovic (Orange) Presented

### INTEGRATED OPTIMIZATION OF SAFETY STOCK

INTEGRATED OPTIMIZATION OF SAFETY STOCK AND TRANSPORTATION CAPACITY Horst Tempelmeier Department of Production Management University of Cologne Albertus-Magnus-Platz D-50932 Koeln, Germany http://www.spw.uni-koeln.de/

### ON THE K-WINNERS-TAKE-ALL NETWORK

634 ON THE K-WINNERS-TAKE-ALL NETWORK E. Majani Jet Propulsion Laboratory California Institute of Technology R. Erlanson, Y. Abu-Mostafa Department of Electrical Engineering California Institute of Technology

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### On the Traffic Capacity of Cellular Data Networks. 1 Introduction. T. Bonald 1,2, A. Proutière 1,2

On the Traffic Capacity of Cellular Data Networks T. Bonald 1,2, A. Proutière 1,2 1 France Telecom Division R&D, 38-40 rue du Général Leclerc, 92794 Issy-les-Moulineaux, France {thomas.bonald, alexandre.proutiere}@francetelecom.com

### Reinforcement Learning

Reinforcement Learning LU 2 - Markov Decision Problems and Dynamic Programming Dr. Martin Lauer AG Maschinelles Lernen und Natürlichsprachliche Systeme Albert-Ludwigs-Universität Freiburg martin.lauer@kit.edu

### Dynamic Assignment of Dedicated and Flexible Servers in Tandem Lines

Dynamic Assignment of Dedicated and Flexible Servers in Tandem Lines Sigrún Andradóttir and Hayriye Ayhan School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA 30332-0205,

### Perron vector Optimization applied to search engines

Perron vector Optimization applied to search engines Olivier Fercoq INRIA Saclay and CMAP Ecole Polytechnique May 18, 2011 Web page ranking The core of search engines Semantic rankings (keywords) Hyperlink

### Renewal Theory. (iv) For s < t, N(t) N(s) equals the number of events in (s, t].

Renewal Theory Def. A stochastic process {N(t), t 0} is said to be a counting process if N(t) represents the total number of events that have occurred up to time t. X 1, X 2,... times between the events

### Reinforcement Learning

Reinforcement Learning LU 2 - Markov Decision Problems and Dynamic Programming Dr. Joschka Bödecker AG Maschinelles Lernen und Natürlichsprachliche Systeme Albert-Ludwigs-Universität Freiburg jboedeck@informatik.uni-freiburg.de

### Brownian Motion and Stochastic Flow Systems. J.M Harrison

Brownian Motion and Stochastic Flow Systems 1 J.M Harrison Report written by Siva K. Gorantla I. INTRODUCTION Brownian motion is the seemingly random movement of particles suspended in a fluid or a mathematical

### Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

### Basic Multiplexing models. Computer Networks - Vassilis Tsaoussidis

Basic Multiplexing models? Supermarket?? Computer Networks - Vassilis Tsaoussidis Schedule Where does statistical multiplexing differ from TDM and FDM Why are buffers necessary - what is their tradeoff,

### Modeling and Performance Evaluation of Computer Systems Security Operation 1

Modeling and Performance Evaluation of Computer Systems Security Operation 1 D. Guster 2 St.Cloud State University 3 N.K. Krivulin 4 St.Petersburg State University 5 Abstract A model of computer system

### Random access protocols for channel access. Markov chains and their stability. Laurent Massoulié.

Random access protocols for channel access Markov chains and their stability laurent.massoulie@inria.fr Aloha: the first random access protocol for channel access [Abramson, Hawaii 70] Goal: allow machines

### The Exponential Distribution

21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding

### Bisimulation and Logical Preservation for Continuous-Time Markov Decision Processes

Bisimulation and Logical Preservation for Continuous-Time Markov Decision Processes Martin R. Neuhäußer 1,2 Joost-Pieter Katoen 1,2 1 RWTH Aachen University, Germany 2 University of Twente, The Netherlands

### IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS

IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)

### Coordinated Pricing and Inventory in A System with Minimum and Maximum Production Constraints

The 7th International Symposium on Operations Research and Its Applications (ISORA 08) Lijiang, China, October 31 Novemver 3, 2008 Copyright 2008 ORSC & APORC, pp. 160 165 Coordinated Pricing and Inventory

### 1 Limiting distribution for a Markov chain

Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

### Adaptive Threshold Policies for Multi-Channel Call Centers

Adaptive Threshold Policies for Multi-Channel Call Centers Benjamin Legros a Oualid Jouini a Ger Koole b a Laboratoire Génie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290 Châtenay-Malabry,

### Optimal Power Cost Management Using Stored Energy in Data Centers

1 Optimal Power Cost Management Using Stored Energy in Data Centers Rahul Urgaonkar, Bhuvan Urgaonkar, Michael J. Neely, and Anand Sivasubramaniam Dept. of EE-Systems, University of Southern California

### IN THIS PAPER, we study the delay and capacity trade-offs

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 15, NO. 5, OCTOBER 2007 981 Delay and Capacity Trade-Offs in Mobile Ad Hoc Networks: A Global Perspective Gaurav Sharma, Ravi Mazumdar, Fellow, IEEE, and Ness

### Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

### A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY

Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,

### Stochastic Inventory Control

Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the

### Spectral Measure of Large Random Toeplitz Matrices

Spectral Measure of Large Random Toeplitz Matrices Yongwhan Lim June 5, 2012 Definition (Toepliz Matrix) The symmetric Toeplitz matrix is defined to be [X i j ] where 1 i, j n; that is, X 0 X 1 X 2 X n

### CHAPTER 3 SECURITY CONSTRAINED OPTIMAL SHORT-TERM HYDROTHERMAL SCHEDULING

60 CHAPTER 3 SECURITY CONSTRAINED OPTIMAL SHORT-TERM HYDROTHERMAL SCHEDULING 3.1 INTRODUCTION Optimal short-term hydrothermal scheduling of power systems aims at determining optimal hydro and thermal generations

### STABILITY OF LU-KUMAR NETWORKS UNDER LONGEST-QUEUE AND LONGEST-DOMINATING-QUEUE SCHEDULING

Applied Probability Trust (28 December 2012) STABILITY OF LU-KUMAR NETWORKS UNDER LONGEST-QUEUE AND LONGEST-DOMINATING-QUEUE SCHEDULING RAMTIN PEDARSANI and JEAN WALRAND, University of California, Berkeley

1 2 3 4 5 6 90 180 360 600 900 W kg M stop start P M 1. 2. M 3. P 4. M M 1. 2. P 3. P 4. M M 1. 2. P 3. P 4. M 250 250 250 1. P 2. 3. P pm h M1 M2 min sec kg M 1. 2. 3. M M 1. 2. 3. M M M 1. 2. M 90

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### ALOHA Performs Delay-Optimum Power Control

ALOHA Performs Delay-Optimum Power Control Xinchen Zhang and Martin Haenggi Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556, USA {xzhang7,mhaenggi}@nd.edu Abstract As

### 6.231 Dynamic Programming and Stochastic Control Fall 2008

MIT OpenCourseWare http://ocw.mit.edu 6.231 Dynamic Programming and Stochastic Control Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 6.231

### Cloud for PC SOFT applications Pricing

Cloud for PC SOFT applications Pricing Date when pricing takes effect: 6 January 2016 10:51 Overview The Cloud for PC SOFT applications offers operating platforms for your WINDEV, WEBDEV or WINDEV Mobile

### http://www.guido.be/intranet/enqueteoverview/tabid/152/ctl/eresults...

1 van 70 20/03/2014 11:55 EnqueteDescription 2 van 70 20/03/2014 11:55 3 van 70 20/03/2014 11:55 4 van 70 20/03/2014 11:55 5 van 70 20/03/2014 11:55 6 van 70 20/03/2014 11:55 7 van 70 20/03/2014 11:55

### Assessment of robust capacity utilisation in railway networks

Assessment of robust capacity utilisation in railway networks Lars Wittrup Jensen 2015 Agenda 1) Introduction to WP 3.1 and PhD project 2) Model for measuring capacity consumption in railway networks a)

### Center for Mathematics and Computational Science (CWI) Phone: (+31)20-592-4135

Peter van de Ven Center for Mathematics and Computational Science (CWI) Phone: (+31)20-592-4135 Stochastics department Amsterdam, The Netherlands 2014-present Research interests ven@cwi.nl Applied probability;

### Hydrodynamic Limits of Randomized Load Balancing Networks

Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli

### Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

Single-Period Balancing of Pay Per-Click and Pay-Per-View Online Display Advertisements Changhyun Kwon Department of Industrial and Systems Engineering University at Buffalo, the State University of New

### Wireless Network Coding with partial Overhearing information

Wireless network coding with partial overhearing information Georgios S. Paschos Massachusetts Institute of Technology (MIT) Joint work with: C. Fragkiadakis University of Thessaly, Greece L. Georgiadis

### Reserve Valuation in Power Systems with High Penetration of Wind Power

Reserve Valuation in Power Systems with High Penetration of Wind Power Antonio J. Conejo Univ. Castilla La Mancha Spain 2011 1 La Mancha 2 3 4 5 7 Nuclear Power Plants 5 Nuclear Power Plants 1 Nuclear

### 3. Renewal Theory. Definition 3 of the Poisson process can be generalized: Let X 1, X 2,..., iidf(x) be non-negative interarrival times.

3. Renewal Theory Definition 3 of the Poisson process can be generalized: Let X 1, X 2,..., iidf(x) be non-negative interarrival times. Set S n = n i=1 X i and N(t) = max {n : S n t}. Then {N(t)} is a

P r o f. d r. W. G u e d e n s L i c. M. R e y n d e r s D 2 Chemie o e l s t e l l i n g B e pal i n g v an d e c o n c e n t r at i e z u u r i n w i t t e w i j n d o o r u i t v o e r i n g v an e

### Analysing the impact of CDN based service delivery on traffic engineering

Analysing the impact of CDN based service delivery on traffic engineering Gerd Windisch Chair for Communication Networks Technische Universität Chemnitz Gerd Windisch - Chair for Communication Networks

### 15 Markov Chains: Limiting Probabilities

MARKOV CHAINS: LIMITING PROBABILITIES 67 Markov Chains: Limiting Probabilities Example Assume that the transition matrix is given by 7 2 P = 6 Recall that the n-step transition probabilities are given

### Cross-layer Scheduling and Resource Allocation in Wireless Communication Systems

Cross-layer Scheduling and Resource Allocation in Wireless Communication Systems Srikrishna Bhashyam Department of Electrical Engineering Indian Institute of Technology Madras 21 June 2011 Srikrishna Bhashyam

### Load Balancing and Switch Scheduling

EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load

### Change Management in Enterprise IT Systems: Process Modeling and Capacity-optimal Scheduling

Change Management in Enterprise IT Systems: Process Modeling and Capacity-optimal Scheduling Praveen K. Muthusamy, Koushik Kar, Sambit Sahu, Prashant Pradhan and Saswati Sarkar Rensselaer Polytechnic Institute

### Oxfordshire Fire and Rescue Service

Oxfh F Ru Sv F f v bk K Sg Tw Lk KS2 PHSE Czhp 1C, 3A, 3E & 5D NAME: Ffgh U h f u wh h mg! 1 2 3 4 5 6 7 8 9 10 11 12 13 b f g h j k m 14 15 16 17 18 19 20 21 22 23 24 25 26 p q u v w z z 26-19 18-13 4x5

### OPTIMAL CONTROL OF FLEXIBLE SERVERS IN TWO TANDEM QUEUES WITH OPERATING COSTS

Probability in the Engineering and Informational Sciences, 22, 2008, 107 131. Printed in the U.S.A. DOI: 10.1017/S0269964808000077 OPTIMAL CONTROL OF FLEXILE SERVERS IN TWO TANDEM QUEUES WITH OPERATING

### Lecture notes on Moral Hazard, i.e. the Hidden Action Principle-Agent Model

Lecture notes on Moral Hazard, i.e. the Hidden Action Principle-Agent Model Allan Collard-Wexler April 19, 2012 Co-Written with John Asker and Vasiliki Skreta 1 Reading for next week: Make Versus Buy in

### Time-Of-Use Pricing Policies for Offering Cloud Computing as a Service

Time-Of-Use Pricing Policies for Offering Cloud Computing as a Service Denis Saure, Anshul Sheopuri, Huiming Qu, Hani Jamjoom, Assaf Zeevi Email: dsaure@gsb.columbia.edu, {sheopuri, hqu, jamjoom}@us.ibm.com,

### Optimized Asynchronous Passive Multi-Channel Discovery of Beacon-Enabled Networks

t t Technische Universität Berlin Telecommunication Networks Group arxiv:1506.05255v1 [cs.ni] 17 Jun 2015 Optimized Asynchronous Passive Multi-Channel Discovery of Beacon-Enabled Networks Niels Karowski,

### The Feasibility of Supporting Large-Scale Live Streaming Applications with Dynamic Application End-Points

The Feasibility of Supporting Large-Scale Live Streaming Applications with Dynamic Application End-Points Kay Sripanidkulchai, Aditya Ganjam, Bruce Maggs, and Hui Zhang Instructor: Fabian Bustamante Presented

### QoS for Cloud Computing!

PIREGRID EFA53/08 QoS for Cloud Computing! PIREGRID THEMATIC DAY! May, 10thn 2011! University of Pau! Prof. Congduc Pham! http://www.univ-pau.fr/~cpham! Université de Pau, France! What is Quality of Service?!

### Periodic Capacity Management under a Lead Time Performance Constraint

Periodic Capacity Management under a Lead Time Performance Constraint N.C. Buyukkaramikli 1,2 J.W.M. Bertrand 1 H.P.G. van Ooijen 1 1- TU/e IE&IS 2- EURANDOM INTRODUCTION Using Lead time to attract customers

### Dual-side Dynamic Controls for Cost Minimization in Mobile Cloud Computing Systems

Dual-side Dynamic Controls for Cost Minimization in Mobile Cloud Computing Systems Yeongin Kim, Jeongho Kwak and Song Chong Department of Electrical Engineering, KAIST E-mail: y.kim, h.kwak}@netsys.kaist.ac.kr,

### On the Operation and Value of Storage in Consumer Demand Response

On the Operation and Value of Storage in Consumer Demand Response Yunjian Xu and Lang Tong Abstract We study the optimal operation and economic value of energy storage operated by a consumer who faces

### The Ergodic Theorem and randomness

The Ergodic Theorem and randomness Peter Gács Department of Computer Science Boston University March 19, 2008 Peter Gács (Boston University) Ergodic theorem March 19, 2008 1 / 27 Introduction Introduction

### Optimization of Supply Chain Networks

Optimization of Supply Chain Networks M. Herty TU Kaiserslautern September 2006 (2006) 1 / 41 Contents 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal

### Binomial lattice model for stock prices

Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }

### An Available-to-Promise Production-Inventory. An ATP System with Pseudo Orders

An Available-to-Promise Production-Inventory System with Pseudo Orders joint work with Susan Xu University of Dayton Penn State University MSOM Conference, June 6, 2008 Outline Introduction 1 Introduction

### Notes from Week 1: Algorithms for sequential prediction

CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 22-26 Jan 2007 1 Introduction In this course we will be looking

### On the dynamic programming principle for static and dynamic inverse problems

On the dynamic programming principle for static and dynamic inverse problems Stefan Kindermann, Industrial Mathematics Institute Johannes Kepler University Linz, Austria joint work with Antonio Leitao,

### Single and Multi-Period Optimal Inventory Control Models with Risk-Averse Constraints

Single and Multi-Period Optimal Inventory Control Models with Risk-Averse Constraints Dali Zhang and Huifu Xu School of Mathematics University of Southampton Southampton, SO17 1BJ, UK Yue Wu School of

### Optimal Email routing in a Multi-Channel Call Center

Optimal Email routing in a Multi-Channel Call Center Benjamin Legros Oualid Jouini Ger Koole Ecole Centrale Paris, Laboratoire Génie Industriel, Grande Voie des Vignes, 99 Châtenay-Malabry, France VU University

### Graph Theory and Complex Networks: An Introduction. Chapter 08: Computer networks

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.20, steen@cs.vu.nl Chapter 08: Computer networks Version: March 3, 2011 2 / 53 Contents

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Coursework 3: Pseudo-Random Numbers and Monte Carlo Methods

Coursework 3: Pseudo-Random Numbers and Monte Carlo Methods The aim of this project is to explore random number generation and its use in simulation. It is, of course, impossible to generate a truly random

### Serial Agile Production Systems with Automation

OPERATIONS RESEARCH Vol. 53, No. 5, September October 25, pp. 852 866 issn 3-364X eissn 1526-5463 5 535 852 informs doi.1287/opre.5.226 25 INFORMS Serial Agile Production Systems with Automation Wallace

### Cooperative Multiple Access for Wireless Networks: Protocols Design and Stability Analysis

Cooperative Multiple Access for Wireless Networks: Protocols Design and Stability Analysis Ahmed K. Sadek, K. J. Ray Liu, and Anthony Ephremides Department of Electrical and Computer Engineering, and Institute

### DMD20 Satellite Control Channel (SCC)

DMD20 Satellite Control Channel (SCC) White Paper WP016 Rev. 1.0 October 2004 Radyne ComStream, Inc. 3138 E. Elwood St. Phoenix, AZ 85034 (602) 437-9620 Fax: (602) 437-4811 Introduction This white paper

### 6.6 Scheduling and Policing Mechanisms

02-068 C06 pp4 6/14/02 3:11 PM Page 572 572 CHAPTER 6 Multimedia Networking 6.6 Scheduling and Policing Mechanisms In the previous section, we identified the important underlying principles in providing

### Fluid Approximation of Smart Grid Systems: Optimal Control of Energy Storage Units

Fluid Approximation of Smart Grid Systems: Optimal Control of Energy Storage Units by Rasha Ibrahim Sakr, B.Sc. A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfillment

### Bindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12

Why eigenvalues? Week 8: Friday, Oct 12 I spend a lot of time thinking about eigenvalue problems. In part, this is because I look for problems that can be solved via eigenvalues. But I might have fewer

### Discrete Dynamic Optimization: Six Examples

Discrete Dynamic Optimization: Six Examples Dr. Tai-kuang Ho Associate Professor. Department of Quantitative Finance, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan 30013,

### M/M/1 and M/M/m Queueing Systems

M/M/ and M/M/m Queueing Systems M. Veeraraghavan; March 20, 2004. Preliminaries. Kendall s notation: G/G/n/k queue G: General - can be any distribution. First letter: Arrival process; M: memoryless - exponential

### Portfolio Using Queuing Theory

Modeling the Number of Insured Households in an Insurance Portfolio Using Queuing Theory Jean-Philippe Boucher and Guillaume Couture-Piché December 8, 2015 Quantact / Département de mathématiques, UQAM.

### Maximizing System Throughput Using Cooperative Sensing in Multi-Channel Cognitive Radio Networks

Maximizing System Throughput Using Cooperative Sensing in Multi-Channel Cognitive Radio Networks Shuang Li, Zizhan Zheng, Eylem Ekici and Ness Shroff Abstract In Cognitive Radio Networks (CRNs), unlicensed

### Dynamic Cost-Per-Action Mechanisms and Applications to Online Advertising

Dynamic Cost-Per-Action Mechanisms and Applications to Online Advertising Hamid Nazerzadeh Amin Saberi Rakesh Vohra July 13, 2007 Abstract We examine the problem of allocating a resource repeatedly over

### 1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

### A Note on the Ruin Probability in the Delayed Renewal Risk Model

Southeast Asian Bulletin of Mathematics 2004 28: 1 5 Southeast Asian Bulletin of Mathematics c SEAMS. 2004 A Note on the Ruin Probability in the Delayed Renewal Risk Model Chun Su Department of Statistics

### EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

### Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

### Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition

Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition Jack Gilbert Copyright (c) 2014 Jack Gilbert. Permission is granted to copy, distribute and/or modify this document under the terms

### Understanding the division property

Understanding the division property Christina Boura (joint work with Anne Canteaut) ASK 2015, October 1, 2015 1 / 36 Introduction In Eurocrypt 2015, Yosuke Todo introduces a new property, called the division

### Optimal Taxation in a Limited Commitment Economy

Optimal Taxation in a Limited Commitment Economy forthcoming in the Review of Economic Studies Yena Park University of Pennsylvania JOB MARKET PAPER Abstract This paper studies optimal Ramsey taxation

### On the Oscillation of Nonlinear Fractional Difference Equations

Mathematica Aeterna, Vol. 4, 2014, no. 1, 91-99 On the Oscillation of Nonlinear Fractional Difference Equations M. Reni Sagayaraj, A.George Maria Selvam Sacred Heart College, Tirupattur - 635 601, S.India

### Joint Optimization of Overlapping Phases in MapReduce

Joint Optimization of Overlapping Phases in MapReduce Minghong Lin, Li Zhang, Adam Wierman, Jian Tan Abstract MapReduce is a scalable parallel computing framework for big data processing. It exhibits multiple

### The Joint Distribution of Server State and Queue Length of M/M/1/1 Retrial Queue with Abandonment and Feedback

The Joint Distribution of Server State and Queue Length of M/M/1/1 Retrial Queue with Abandonment and Feedback Hamada Alshaer Université Pierre et Marie Curie - Lip 6 7515 Paris, France Hamada.alshaer@lip6.fr

### IN RECENT years, there have been significant efforts to develop

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 3, MARCH 2006 525 Effective Bandwidth of Multimedia Traffic in Packet Wireless CDMA Networks with LMMSE Receivers: A Cross-Layer Perspective Fei

### Worked examples Random Processes

Worked examples Random Processes Example 1 Consider patients coming to a doctor s office at random points in time. Let X n denote the time (in hrs) that the n th patient has to wait before being admitted

### Basics of information theory and information complexity

Basics of information theory and information complexity a tutorial Mark Braverman Princeton University June 1, 2013 1 Part I: Information theory Information theory, in its modern format was introduced