A review of M. Zonoozi_ P. Dassanayake_ “User Mobility and

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A review of M. Zonoozi_ P. Dassanayake_ “User Mobility and Powered By Docstoc
					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

   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
         General outline of the development

1.   Develop a mathematical framework for modeling mobile
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
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

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


       The geometry of regional transitions and state changes



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
       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
   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

                       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
   Let PFh = probability that a handover attempt fails
                   Simplification of E(H)

PH  k   PH  1  Ph  1  PFh 
                                          k 1

E ( H )  P( H  1)   k P H  1  Ph  1  PFh k 1

Note : Ph  1  PFh  1
let Ph  1  PFh   
             Simplification of E(H)

E H   PH  1   k   k 1

  E ( H )  P( H  1)   k  k

E ( H )  E ( H )  (1   )  E ( H ) 
 PH  1   k   k 1  P( H  1)   k  k
                k                           k

                             k
 PH  1   k     k  
                     k 1

             k           k    
                             
 PH  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
P ( H  1)  Pn  1  PFh   (1   )
        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
     FTh (t )    f
                          Th   ( ) d

     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

   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
       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

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