A review of M. Zonoozi, P. Dassanayake, “User Mobility and Characterization of Mobility Patterns”, IEEE J. on Sel. Areas in Comm., vol 15, no. 7, Sept.

Slides:



Advertisements
Similar presentations
Application a hybrid controller to a mobile robot J.-S Chiou, K. -Y. Wang,Simulation Modelling Pratice and Theory Vol. 16 pp (2008) Professor:
Advertisements

Random Variables ECE460 Spring, 2012.
Submission doc.: IEEE /1214r1 September 2014 Leif Wilhelmsson, Ericsson ABSlide 1 Impact of correlated shadowing in ax system evaluations.
How Bad is Selfish Routing? By Tim Roughgarden Eva Tardos Presented by Alex Kogan.
1 12. Principles of Parameter Estimation The purpose of this lecture is to illustrate the usefulness of the various concepts introduced and studied in.
Sensitivity Analysis In deterministic analysis, single fixed values (typically, mean values) of representative samples or strength parameters or slope.
Time of day choice models The “weakest link” in our current methods(?) Change the use of network models… Run static assignments for more periods of the.
G. Cowan Lectures on Statistical Data Analysis 1 Statistical Data Analysis: Lecture 10 1Probability, Bayes’ theorem, random variables, pdfs 2Functions.
1 Learning Entity Specific Models Stefan Niculescu Carnegie Mellon University November, 2003.
Presenting: Lihu Berman
G. Cowan Lectures on Statistical Data Analysis 1 Statistical Data Analysis: Lecture 8 1Probability, Bayes’ theorem, random variables, pdfs 2Functions of.
Introduction to Boosting Aristotelis Tsirigos SCLT seminar - NYU Computer Science.
Impact of Different Mobility Models on Connectivity Probability of a Wireless Ad Hoc Network Tatiana K. Madsen, Frank H.P. Fitzek, Ramjee Prasad [tatiana.
Experimental Evaluation
Motion Along a Straight Line
Chapter 11 Angular Momentum.
Physics 2011 Chapter 2: Straight Line Motion. Motion: Displacement along a coordinate axis (movement from point A to B) Displacement occurs during some.
Lecture II-2: Probability Review
1 10. Joint Moments and Joint Characteristic Functions Following section 6, in this section we shall introduce various parameters to compactly represent.
CMPE 252A: Computer Networks Review Set:
Vectors and Two-Dimensional Motion
6 am 11 am 5 pm Fig. 5: Population density estimates using the aggregated Markov chains. Colour scale represents people per km. Population Activity Estimation.
Chapter 3 Vectors and Two-Dimensional Motion. Vector vs. Scalar Review All physical quantities encountered in this text will be either a scalar or a vector.
Software Reliability SEG3202 N. El Kadri.
Chapter 10 Rotation of a Rigid Object about a Fixed Axis.
Estimating parameters in a statistical model Likelihood and Maximum likelihood estimation Bayesian point estimates Maximum a posteriori point.
PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Two Functions of Two Random.
1 7. Two Random Variables In many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to.
Managing Handoff. For operations and management to detect and isolating Handoff being particularly challenging, therefore it is important to understand.
ECE 8443 – Pattern Recognition ECE 8423 – Adaptive Signal Processing Objectives: Deterministic vs. Random Maximum A Posteriori Maximum Likelihood Minimum.
Geo597 Geostatistics Ch9 Random Function Models.
ECE 8443 – Pattern Recognition LECTURE 07: MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION Objectives: Class-Conditional Density The Multivariate Case General.
The coordinates of the centre of mass are M is the total mass of the system Use the active figure to observe effect of different masses and positions Use.
1 Let X represent a Binomial r.v as in (3-42). Then from (2-30) Since the binomial coefficient grows quite rapidly with n, it is difficult to compute (4-1)
Multiple Random Variables Two Discrete Random Variables –Joint pmf –Marginal pmf Two Continuous Random Variables –Joint Distribution (PDF) –Joint Density.
1 Two Functions of Two Random Variables In the spirit of the previous lecture, let us look at an immediate generalization: Suppose X and Y are two random.
Chapter 10 Rotational Motion.
5-1 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. May 28, 2009 Inventory # Chapter 5 Six Sigma.
Statistical Decision Theory Bayes’ theorem: For discrete events For probability density functions.
PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Mean, Variance, Moments and.
1 8. One Function of Two Random Variables Given two random variables X and Y and a function g(x,y), we form a new random variable Z as Given the joint.
KNR 445 Statistics t-tests Slide 1 Introduction to Hypothesis Testing The z-test.
Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois.
Performance of Adaptive Beam Nulling in Multihop Ad Hoc Networks Under Jamming Suman Bhunia, Vahid Behzadan, Paulo Alexandre Regis, Shamik Sengupta.
Mobility Models for Wireless Ad Hoc Network Research EECS 600 Advanced Network Research, Spring 2005 Instructor: Shudong Jin March 28, 2005.
1 Introduction to Statistics − Day 4 Glen Cowan Lecture 1 Probability Random variables, probability densities, etc. Lecture 2 Brief catalogue of probability.
1 6. Mean, Variance, Moments and Characteristic Functions For a r.v X, its p.d.f represents complete information about it, and for any Borel set B on the.
Joint Moments and Joint Characteristic Functions.
1 What happens to the location estimator if we minimize with a power other that 2? Robert J. Blodgett Statistic Seminar - March 13, 2008.
Paging Area Optimization Based on Interval Estimation in Wireless Personal Communication Networks By Z. Lei, C. U. Saraydar and N. B. Mandayam.
User Mobility Modeling and Characterization of Mobility Patterns Mahmood M. Zonoozi and Prem Dassanayake IEEE Journal on Selected Areas in Communications.
One Function of Two Random Variables
Submission doc.: IEEE /1214r0 September 2014 Leif Wilhelmsson, Ericsson ABSlide 1 Impact of correlated shadowing in ax system evaluations.
G. Cowan Lectures on Statistical Data Analysis Lecture 9 page 1 Statistical Data Analysis: Lecture 9 1Probability, Bayes’ theorem 2Random variables and.
Chapter 9: Introduction to the t statistic. The t Statistic The t statistic allows researchers to use sample data to test hypotheses about an unknown.
G. Cowan Lectures on Statistical Data Analysis Lecture 10 page 1 Statistical Data Analysis: Lecture 10 1Probability, Bayes’ theorem 2Random variables and.
Performance Comparison of Ad Hoc Network Routing Protocols Presented by Venkata Suresh Tamminiedi Computer Science Department Georgia State University.
12. Principles of Parameter Estimation
3. The X and Y samples are independent of one another.
Evaluation Model for LTE-Advanced
Motion Along a Straight Line
Cellular Digital Packet Data: Channel Availability
Computing and Statistical Data Analysis / Stat 7
8. One Function of Two Random Variables
Information Theoretical Analysis of Digital Watermarking
9. Two Functions of Two Random Variables
12. Principles of Parameter Estimation
8. One Function of Two Random Variables
Presentation transcript:

A review of M. Zonoozi, P. Dassanayake, “User Mobility and Characterization of Mobility Patterns”, IEEE J. on Sel. Areas in Comm., vol 15, no. 7, Sept 1997 Jim Catt ECE 695 Sp 2006

Purpose The stated purpose of the paper is to (1) propose a mobility model that considers a wide range of mobility-related parameters, and (2) use the model to obtain different mobility traffic parameters. In particular, the authors intent is to derive a probability distribution for cell residence times, and then find pdf’s and probability distributions for other mobility parameters that are derived from these cell residence times. These cell residence times are T n and T h, new call residence time and handover residence time, respectively. T n and T h are random variables whose distributions are to be found This is relevant to system design when attempting to optimize switching loads and processing loads.

Purpose (continued) Specifically, the authors use their mobility model in a simulation to test the hypothesis that the cell residence times for new and handover calls follow a generalized gamma distribution, with p.d.f.s of the form: Hence, the proposed mobility model is secondary in that it is needed to construct a simulation for the purpose of generating data that can be used to construct empirical pdf’s and distributions of the cell residence times. After validating the hypothesized pdf’s (distributions) against the simulation data, they then derive distributions for other mobility parameters related to T n and T h. The context of the analysis is a cellular network Hence, the model is applicable to an infrastructure based network

General outline of the development 1. Develop a mathematical framework for modeling mobile movement 2. Using the mathematical framework, combined with certain assumptions about the characteristics of mobile movement, simulate a model of the mobile environment, 3. Use the data from the simulation to obtain an empirical distribution (or pdf) and find values of the parameters a,b, and c that represent the best fit between the simulation pdf and the hypothesized pdf. 4. Develop a means for incorporating the random effects of (changes in) speed and direction the p.d.f.s for cell residence times 5. Finally, after (4), develop expressions for mobility characteristics related to T n and T h such as mean cell residence time, average number of handovers, channel holding time pdf and probability distribution.

Pdf for Gamma distribution The parameters a, b, and c are found through simulation such that the best fit is obtained between the simulation results and the equivalent Gamma pdf.

Mobility model The position of a mobile at time instant  is given by the coordinate pair: (ρ ,θ  ). The mobile position is updated according to the following relations: Where   = supplementary angle between the current direction of the mobile and a line connecting its previous position to the base station Other definitions continued on following page

Mobility Model diagram  d = distance traveled in  = *     change of direction at time    = (see diagram)

The Geometry of regions An x-y coordinate system is defined as follows: x axis coincides with the mobile’s previous direction of travel y axis coincides with a line drawn from the current mobile position to the base station x  

The geometry of regional transitions and state changes 2 1 1=  -   -  2 =        = 

Model Assumptions The mathematical framework illustrated in the previous slides only provides a means for describing the mobile location at any time instant. The actual movement is governed by the following assumptions, which affectively define the model Users are independent and uniformly distributed over the entire region Mobiles are allowed to move away from the starting point in any direction with equal probability (  0 is uniformly distributed in the interval [0,2  ] The probability of the variation in mobile direction (drift) along its path is a uniform distribution limited in the range  with respect to current direction.   is defined at simulation time. The initial velocity of the mobile stations is assumed to a Gaussian RV with truncated range [0, 100 km/hr] The velocity increment of each mobile is a uniformly distributed RV in the range of +/- 10% of current velocity.  distribution???

Simulation vs. predicted V avg = 50 km/hr, 0 drift

Pdf parameters From the simulation, the values of a, b, and c which were found to give the best results for the Kolmogorov-Smirnov goodness-of-fit test were: a = 0.62, new cell call, 2.31 for a handover call b = 1.84R for a new cell call, 1.22R for a handover call c = 1.88 for a new call, 1.72 for a handover call However, these values still don’t account for change in direction or speed.

Mean cell residence time Given the values of a, b, and c for the pdf’s of the cell residence time distributions, the expected values can be found from: The expected values obtained from the hypothesized pdf’s are compared to alternate derivations for expected value, and found to be within 0.05% and 0.015% Simulation length???

Accounting for changes in speed and direction To account for changes in speed and direction, the cell radius, R, is augmented by a value,  R, excess cell radius, which accounts for the affect of either change in direction (  R  ) or change in speed (  R ) Both  R and  R  are found through simulation The original cell radius R is converted to a reference cell radius, R, which has the same residence time, but mobility parameters corresponding to R. R  = R +  R  = K  R  R = R +  R = K R For the joint case, R  = K  K R  b now becomes 1.84 R  (new call), 1.22 R  (handover call)

Direction change  d0d0 d1d1 R d 0 + d 1 > R for constant v.

Speed change 0 1 R = 0 * t 0 For an initial velocity, 0, the mobile will require t 0 = R/ 0 to reach the cell boundary. However, if the mobile increases speed to 1, then the edge of the cell boundary is reached sooner. Under an assumption of constant velocity, the effective cell radius decreases. t’ 0 t’ 1

Average number of Handovers: method 1 Now that pdf’s for the cell residence time of new calls and handover calls are validated and modified to account for random changes in speed and direction, the average number of handovers during a call can be found: Let P n = probability that a non-blocked new call will require at least 1 handover Let P h = probability that a nonfailed handover will require at least one more handover before completion Let P Fh = probability that a handover attempt fails

Simplification of E(H)

Average number of handovers So, how do we evaluate P n, P h, and P Fh ? Define the RVs: T n = new call residence time T h = handover call residence time T c = call hold time The probability function for call holding time (T c ) is borrowed from classical tele-traffic theory: F Tc (t) = 1 – e -  ct Where average call hold time = 1/  c P n = P(T c > T n )

Finding P n, P h, anf P Fh Assertion: T n is influenced by user mobility, and has no influence on T c, therefore Likewise, P h is found from : P n and P h are found numerically from these expressions P Fh is not addressed

Numerical results for average number of handovers

Channel Holding time Distribution Channel holding time is an RV defined as the time spent by a given user on a particular channel in a given cell. It is a function of the cell size, user location, user mobility, and call duration. Define T N, channel holding time of a new call, as: T N = min(T n, T c ) Define T H, the channel holding time of a handover call, as: T H = min( T h, T c ) In this case, T c is residual call time Assertion: T n and T h are dependent on the physical movement of the mobile, and do not influence total call duration or residual call time (T c ).

Channel Holding time distribution Therefore, the distributions for T N and T H can be found from the distributions for T n, T h and T c, using the fundamental probability theorem:

Channel Holding time distribution Though not explicitly stated, we assume that:

Channel Holding time distribution The distribution of the channel holding time in a given cell is a weighted function of F TN (t) and F TH (t). Substituting for , F TN (t) and F TH (t), F Tch (t) becomes:

Channel Holding time distribution Consequently, the channel holding time distribution is a function of cell residence times and average number of handovers, which in turn are functions of cell radius ( R ), changes in user speed and direction (incorporated into R), and probability of handoff failure.

Summary The mobility model simulated here is basically a Random Incremental MM applied in a cellular context, with the following explicit features: Changes in direction can be bounded to an interval [- ,+  ] < [- ,  ] Changes in speed are a bounded Gaussian RV with controlled . A cell topology is employed, where cell radius, R, can be varied Boundary conditions are handled temporally, not spatially, i.e., total call holding time defines the extent of the mobility region, not an artificial boundary Hence, the mobility region is defined probabilistically.

Summary (continued) This model also suffers from the problem of instantaneous changes in direction and speed Furthermore, it is not clear how the time intervals between changes in direction and speed are determined? Fixed? A RV? This could influence the fit to the Gamma distribution, which in turn would change the related results Because the current analysis is clearly directed toward vehicular movement, the time parameter should be reflective of the context, e.g., as opposed to a pedestrian context.