Optimal Dynamic Mobility Management for PCS Networks

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Optimal Dynamic Mobility Management for PCS
Networks

• Li, Kameda, and Li
• IEEE Transactions on Networking
• June 2000

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I. Introduction

• Mobile computing system
• The integration of mobile communications and
  computing
• Personal communication system (PCS)
• A new mobile computing system that enable users
  to economically transfer any form of information
  between any desired locations at any time
• Cells
• Base stations
• Mobile switching center(MSC)

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• Mobility management
• How to track the mobile users that move from
  place to place in PCS networks
• Two basic operations in mobility management
• Location update
• The process through which the system tracks the
  location of mobile users that are not in
  conversations
• The up-to-date location information of a mobile
  user is reported by the mobile users dynamically
• A location area may include one or more cells.
• paging
• When an incoming call arrives, the system
  searches for the mobile users by sending polling
  signal to cells in the location area.

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• Cost of location update and paging
• Wireless bandwidth and processing power at the
  mobile users, base stations, and MSCs
• A large location area will result in a decrease
  in the cost of location update and an increase in
  the cost of paging, and vice versa.
• To determine the size of the location area is a
  critical problem for minimizing the total cost of
  location update and paging.

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• Location update and paging schemes
• Static
• The location area size is fixed
• Problem excess location updates as a mobile
  user moves back and forth between two location
  areas
• Dynamic
• The location area size is determined dynamically
  according to the changes of mobility and calling
  patterns of mobile users

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• Three dynamic location update schemes
• Distance-based
• The location update is performed whenever the
  distance between the current cell for the mobile
  user and the last cell in which the last update
  was performed is d.
• The location area is an area in which the central
  cell is the last cell where the last update
  occurred surrounded by d rings of cells
• Movement-based ? (used in this paper)
• The location update is performed whenever a
  mobile user completes d movements between cells
  (the location update movement threshold)
• The location area is an area in which the central
  cell is the last cell where the last update
  occurred surrounded by d rings of cells
• Time-based
• The location update is performed every t units of
  time.
• The size of location area is calculated according
  the the mobility of the mobile user user.

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II. Modeling and System Description

• The probability density function of cell
  residence time is pm(t) which has
  Laplace-Stieltjes transform fm(s) and mean 1/?m
• When a mobile user leaves a cell, there is an
  equal probability that any one of the immediate
  neighboring cells is selected as the destination.
• The movement-based location update scheme is
  considered in this paper.
• A location update occurs when the number of
  boundary crossing since the last location update
  registration equals a threshold d.
• The center cell is the cell where the last
  location registration occurs.

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• Assume that incoming call arrivals to each mobile
  user follow a Poisson process with rate ?c
• As soon as a call for a mobile user arrives, the
  network initiates a paging process to locate the
  called mobile user.
• The paging area is the covering area within a
  distance d - 1 from the center cell.

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• Two popular cell configurations are studied in
  this paper
• Hexagonal cell configuration
• Mesh cell configuration (Ri the set of cells
  in the ith ring)

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• The size of each cell is determined based on the
  number of mobile channels available per cell and
  the channel allocation scheme used.
• The location tracking mechanism can be applied to
  both macrocell (several kilometers) and microcell
  (several hundreds of meters) environments.
• The size and shape of cells are indirectly
  reflected by the cell residence time.
• If the size of cells is small, the mean residence
  time will be relatively small, and vice versa.

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• Assume that homogeneous cells (of the same shape
  and the same size) are used.
• The distance is measured in terms of the number
  of rings such that the distance between a given
  center cell and the cells belonging to set Ri is
  i rings.
• The number of cells in ring i,
• Paging schemes
• Selective paging (Akyildix et al.)
• The paging is performed in one of the subarea
  only
• All cells in the location area are paged. ?

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III. Problem Formulation

• A. Cost of Location Update
• U the cost for performing a location update
• ?(j) the probability that there are j boundary
  crossings between call arrivals
• The expected location update cost per call
  arrival U ? 1 ?(d) ?(2d-1)
  2 ?(2d) ?(3d-1) 3
  ?(3d) ?(4d-1) ?

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• 1) Calculation of ?(j)

• tc call interval
• R0 cell at the last call
• tMi period staying in Ri
• tm interval between the last call and moving
  out of R0
• tc,i time between entering Ri and the next call

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• Let tMi (cell residence time) be an independent
  identically distribution (iid) random variable
  with
• a general distribution function Gm(tMi),
• density function gm(tMi),
• and the Laplace-Stieltjes Transform

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• Let fc(t) and rm(t) be the density function of tc
  (call interval) and tm, respectively.
• Let Etc 1/?c and EtMi 1/ ?m
• Assume that the incoming phone call is a Poisson
  process
• From the memoryless property of the exponential
  distribution, tc,i has the same exponential
  distribution as tc

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• for tm, from the random observer property
  Stochastic Process - S. Ross, we have
• The density function(Eq. 5)
• The Laplace-Stieltjes Transform(Eq. 6)

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• The probability ?(K) that user p moves across K
  cells between two phone calls is derived for K
  0 and K ? 1
• For K 0

• where ? ?c/?m, the call-to-mobility ratio
  (CMR)
• ? lt 1 ?c lt ?m 1/?c gt 1/?m tc gt tmtc call
  arrival timetm cell residence time

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• For K ? 1

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• From (7) to (11), we have

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• 2) Simplify the cost function Cu

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• Substituting (13) into (2), we have

Eq. 16
Eq. 15
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• Substituting (15) and (16) into (14), we have

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• B. Cost of Paging
• Assume that the cost of polling a cell is P
• The number of cells in a paging area with
  threshold value d
• The expected paging cost per call arrival

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• C. Total Cost per Call
• TC(d)
• The sum of the cost of location update, Cu, and
  the cost of paging , Cp

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IV. Minimizing the Total Cost

• The goal of the optimal movement-based location
  update scheme
• To find the optimal threshold, d, that minimize
  the total cost per call, TC(d)

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• Theorem 1
• The location update cost, Cu(d), is a decreasing
  and convex function with respect to the threshold
  d and the paging cost, Cp(d), is an increasing
  and convex function with respect to the threshold
  d.

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• Proof

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• Corollary 2 a direct result from Theorem 1
• Theorem 3
• The value of d is a unique solution to (21) if
  and only if the following relation holds.
• That is,

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V. Proposed Algorithm
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• Step 1
• Compares the values of Cu(d) and Cp(d) at d
  1
• where d gt 0, Cu(d) is decreasing and Cp(d) is
  increasing
• If Cu(1) ? Cp(1), the optimal threshold ð
  should be 1
• Step 2
• Determines the interval d,ds which consists of
  ð
• ð should not be too large (e.g. gt 20), thus we
  set ð 10
• Step 3
• Determines the the optimal threshold ð in the
  interval d,d1 by using binary search
• Step 4
• Determines the the optimal threshold ð

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VI. Numerical Examination

• Assume that the call residence time follows the
  Gamma distribution.
• Let the Laplace-Stieltjes Transform , fm(s), of
  the Gamma distribution with mean 1/?m and
  variance ? is
• We studies effects of optimal threshold ? by
  various parameters
• Update cost U, polling cost P, CMR
  (Call-to-Mobility Ratio) ? ?c/?m, and
  variance of the cell residence variance ?
• Program in C and run on SPARC-20 workstation

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• A. Effects of CMR, Update Cost, and Paging Cost

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• B.Effects of Cell Residence Time Variance

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VII.Conclusion Remarks

• This paper studies a dynamic mobility management
  scheme the movement-based location update
  scheme.
• An analytical model is applied to formulate the
  costs of location update and paging per call
  arrival.
• The problem of minimizing the total cost per call
  arrival is expressed as an optimization problem
  that finds the optimal threshold in the
  movement-based location update scheme
• An effective algorithm is proposed.