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Optimal Dynamic Mobility Management for PCS Networks Li, Kameda, and Li IEEE Transactions on Networking June 2000 1 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) 2 • 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. 3 • 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. 4 • 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 5 • 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 Ö (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. 6 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/lm • 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. 7 • Assume that incoming call arrivals to each mobile user follow a Poisson process with rate lc • 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. 8 • Two popular cell configurations are studied in this paper: – Hexagonal cell configuration – Mesh cell configuration (Ri = the set of cells in the ith ring) 9 • 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. 10 • 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. Ö 11 III. Problem Formulation • A. Cost of Location Update – U = the cost for performing a location update – a(j) = the probability that there are j boundary crossings between call arrivals – The expected location update cost per call arrival: d=1 d=2 d=3 =U{ 1 * [a(d) +…+ a(2d-1)] 1 2,3 3,4,5 + 2 * [a(2d) +…+ a(3d-1)] 2 4,5 6,7,8 + 3 * [a(3d) +…+ a(4d-1)] 3 6,7 9,10,11 +… } … from 0 … from 0 … from 0 1: to 1 1: to 2,3 1: to 3,4,5 2: to 2 2: to 4,5 2: to 6,7,8 3: to 3 3: to 6,7 3: to 9,10,11 12 • 1) Calculation of a(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 13 • 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： 14 • Let fc(t) and rm(t) be the density function of tc (call interval) and tm, respectively. • Let E[tc] =1/lc and E[tMi] = 1/ lm • 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 15 • for tm, from the random observer property [Stochastic Process - S. Ross], we have – The density function: (Eq. 5) – The Laplace-Stieltjes Transform: (Eq. 6) 16 • The probability a(K) that user p moves across K cell’s between two phone calls is derived for K = 0 and K ³ 1: • where q = lc/lm, – For K = 0: the call-to- mobility ratio (CMR) • q < 1: lc < lm 1/lc > 1/lm tc > tm tc call arrival time tm cell residence time 17 – For K ³ 1: Eq. 11 Eq. 9 Eq. 10 18 • From (7) to (11), we have 19 • 2) Simplify the cost function Cu 20 • Substituting (13) into (2), we have Eq. 15 Eq. 16 21 22 • Substituting (15) and (16) into (14), we have 23 • 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: 24 • C. Total Cost per Call – TC(d) – The sum of the cost of location update, Cu, and the cost of paging , Cp: 25 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) 26 • 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. 27 – Proof: 28 29 • 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, 30 V. Proposed Algorithm 31 • Step 1 – Compares the values of –Cu’(d) and Cp’(d) at d = 1 – where d > 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,d+s] which consists of ð – ð should not be too large (e.g. > 20), thus we set ð = 10 • Step 3 – Determines the the optimal threshold ð in the interval [d,d+1] by using binary search • Step 4 – Determines the the optimal threshold ð 32 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/lm and variance n is • We studies effects of optimal threshold d by various parameters: – Update cost U, polling cost P, CMR (Call-to-Mobility Ratio) q = lc/lm, and variance of the cell residence variance n • Program in C and run on SPARC-20 workstation 33 • A. Effects of CMR, Update Cost, and Paging Cost 34 35 • B.Effects of Cell Residence Time Variance 36 37 38 39 40 41 42 43 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. 44

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