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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 Tn and Th, new call residence time and handover residence time, respectively. Tn and Th 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 Tn and Th. 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 Tn and Th such as mean cell residence time, average number of handovers, channel holding time pdf and probability distribution. Pdf for Gamma distribution where a is the gamma function a x a 1 e x dx 0 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 1 2 1 = - - 1 + 2 + = 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 Vavg = 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 = KR R = R + R = KR For the joint case, R = K K R b now becomes 1.84 R (new call), 1.22 R (handover call) Direction change d0 + d1 > R R d0 for constant v. d1 Speed change For an initial velocity, 0, the mobile will require t0 ' = R/0 to reach the cell v1 t0 boundary. However, if ' the mobile increases v0 t1 R = 0* t0 0 1 speed to 1, then the edge of the cell t’0 t’1 boundary is reached sooner. Under an assumption of constant velocity, the effective cell radius decreases. 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 Pn = probability that a non-blocked new call will require at least 1 handover Let Ph = probability that a nonfailed handover will require at least one more handover before completion Let PFh = probability that a handover attempt fails Simplification of E(H) PH k PH 1 Ph 1 PFh k 1 E ( H ) P( H 1) k P H 1 Ph 1 PFh k 1 k Note : Ph 1 PFh 1 let Ph 1 PFh Simplification of E(H) E H PH 1 k k 1 k E ( H ) P( H 1) k k k E ( H ) E ( H ) (1 ) E ( H ) PH 1 k k 1 P( H 1) k k k k k PH 1 k k k 1 k k PH 1 k n n k Simplification of E(H) 1 (1 ) E ( H ) P ( H 1) 0 1 P( H 1) E ( H ) 1 2 but P ( H 1) Pn 1 PFh (1 ) so Pn 1 PFh Pn 1 PFh E(H ) (1 ) 1 Ph 1 PFh Average number of handovers So, how do we evaluate Pn, Ph, and PFh? Define the RVs: Tn = new call residence time Th = handover call residence time Tc = call hold time The probability function for call holding time (Tc) is borrowed from classical tele-traffic theory: FTc(t) = 1 – e - ct Where average call hold time = 1/c Pn = P(Tc > Tn) Finding Pn, Ph, anf PFh Assertion: Tn is influenced by user mobility, and has no influence on Tc, therefore Likewise, Ph is found from : Pn and Ph are found numerically from these expressions PFh 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 TN, channel holding time of a new call, as: TN = min(Tn, Tc) Define TH, the channel holding time of a handover call, as: TH = min( Th, Tc) In this case, Tc is residual call time Assertion: Tn and Th are dependent on the physical movement of the mobile, and do not influence total call duration or residual call time (Tc). Channel Holding time distribution Therefore, the distributions for TN and TH can be found from the distributions for Tn, Th and Tc, using the fundamental probability theorem: P A B P( A) P( B) P( A B) Channel Holding time distribution Though not explicitly stated, we assume that: ct FTc (t ) 1 e t FTh (t ) f Th ( ) d t FTn (t ) f Tn ( ) d Channel Holding time distribution The distribution of the channel holding time in a given cell is a weighted function of FTN(t) and FTH(t). Substituting for , FTN(t) and FTH(t), FTch(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.