Optimal location management for two-tier PCS networks

Optimal location management for two-tier PCS networks

Yang Xiao*

Computer Science Division, The University of Memphis, 373 Dunn Hall, Memphis, TN 38152, USA

Received 14 March 2002; revised 5 January 2003; accepted 26 January 2003

Abstract

One of the most important issues in Personal Communication Service (PCS) networks is location management, which keeps track of the

Mobile Terminals (MTs) moving from place to place. In this paper, we analytically derive cost functions of location updates and paging for a

dynamic movement-based location management scheme for PCS networks with two-tier mobility databases. We prove analytically that there

is a unique optimal movement threshold that minimizes the total cost of Home Location Register location updates, Visitor Location Register

location updates, and paging, per call arrival. An effective algorithm is proposed to find the optimal movement threshold. Furthermore, we

propose a hybrid location management scheme, in which when the call-to-mobility ratio is larger than a threshold, the optimal dynamic

movement-based scheme is adopted. Otherwise, the static location update is adopted. The Newton approximation method is adopted to find

this threshold. Our study shows that the proposed hybrid scheme outperforms both the dynamic movement-based scheme and the static

location update scheme.

q 2003 Elsevier Science B.V. All rights reserved.

Keywords: Location management; Home Location Register; Visitor Location Register; Analytical model

1. Introduction

The ANSI-41 [1] and Global System for Mobile

communication (GSM) Mobile Application Part (MAP)

[2] have been standardized to support mobility management

in 2G wireless cellular networks. Both ANSI-41 and MAP

use a two-tier system of home and visited location

databases. The Home Location Register (HLR) database is

used to record mobile users’ information, whereas the

Visitor Location Register (VLR) database is used to record

mobile users’ temporary location information. The service

area is partitioned into several Location Areas (LAs). Each

LA is associated with a VLR, and it consists many cells. In

each cell, there is a Base Station (BS) and many MTs. BSs in

an LA are connected to a Mobile Switching Center (MSC).

All MSCs are finally connected to Public Switched

Telephone Network (PSTN). Location update and paging

are two kinds of operations in location management, and

they involve a significant amount of cost. Location update is

a procedure to keep the track of an MT’s location, and

paging is a procedure to search for the MT by sending

polling signals to the cells in a Paging Area (PA).

In a static location update scheme for two-tier Personal

Communication Service (PCS) networks, the HLR location

update and VLR location update are performed when an MT

enters an LA, and the PA is the same as the LA. Therefore,

the PA size is fixed. When an MT moves back and forth

between two LAs near the boundary, excessive location

updates happen. To avoid such problems, dynamic location

update schemes are proposed. There are basically three

kinds of dynamic location update schemes [3]: movement-

based location update, distance-based location update, and

time-based location update. In the distance-based location

update scheme, the VLR location update is performed when

the distance between the current cell and the last cell where

the VLR location update is performed reaches a threshold d

in terms of number of cells. In the time-based location

update scheme, the VLR location update is performed in

each d units of time.

The movement-based location update scheme is the most

practical one among the three kinds of dynamic location

update schemes [7]. In a movement-based scheme, a VLR

location update is performed either when an MT crosses an

LA boundary or when the MT completes d movements

between cells, where d is the movement threshold. It is

reasonable that a VLR location update occurs when the MT

crosses an LA boundary since at this moment, an HLR

0140-3664/03/$ - see front matter q 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0140-3664(03)00059-8

Computer Communications 26 (2003) 1047–1055

www.elsevier.com/locate/comcom

* Tel.: þ1-19-016-782-487; fax: þ1-19-016-782-480.

E-mail address: [email protected] (Y. Xiao).

http://www.elsevier.com/locate/comcom

location update occurs anyway. Therefore, when the MT

crosses an LA boundary, both HLR and VLR location

updates should occur. The PA is a subset of an LA

consisting of a number of cells. Particularly, a PA is the area

within both the LA, where the last VLR location update is

performed, and a circular area with the diameter d 2 1 and

with the center where the last VLR location update is

performed [4]. Therefore, the PA size is a variable.

Many performance studies have been proposed for

movement-based schemes [3,7–9]. However, their schemes

[3,7–9] only consider that a VLR location update occurs

when the MT completes d movements between cells, and

fails to consider the case that a VLR location update also

occurs when the MT crosses an LA boundary (therefore, the

PA size is a variable). The problem becomes much more

complex when both above cases are considered with a

variable size PA. The difficulty exists since it is very difficult,

if not impossible, to derive the number of cell boundary

crossings between two LA boundary crossings when the

residence time in an LA follows a general distribution. To

overcome this difficulty, a recent paper [4] proposed a

complex analysis with an assumption that the residence time

in an LA follows an exponential distribution. The analytic

model proposed in Ref. [4] is very complex, and the optimal

movement threshold had not been studied theoretically.

In this paper, we will consider both the above cases with a

variable size PA, and propose an easier and more

systematical analytic model for cost functions of location

updates and paging. Furthermore, we prove analytically that

there is a unique optimal movement threshold that minimizes

the total cost per call arrival for two-tier PCS networks. An

effective algorithm is proposed to find the optimal movement

threshold. In order to improve the performance, we propose a

hybrid location management scheme, in which when the call-

to-mobility ratio (CMR) is larger than a threshold, the

optimal dynamic movement-based scheme is adopted.

Otherwise, the static location update is used. The Newton

approximation method is adopted to find this threshold.

The rest of this paper is organized as follows. We derive

several Lemmas in Section 2, which will be used in our cost

analysis model extensively. Section 3 establishes an analytic

model to formulate cost functions, provides an optimality

analysis, proposes an effective algorithm to find the optimal

movement threshold, formulates the cost function for the

hybrid scheme, and proposes an algorithm to find the hybrid

scheme’s threshold. Performance study and comparisons are

studied in Section 4. We conclude our paper in Section 5.

2. Preparations

In this section, we provide four Lemmas that will be used

extensively to derive the analytical model proposed in the

later sections. Lemma 1 calculates the probability that one

event occurs in one Poisson process before one event occurs

in a second and independent Poisson process. Lemma 2

calculates the probability that exactly n events occur in one

Poisson process between two events in a second and

independent Poisson process. Lemma 3 provides some

useful formulas. Finally, Lemma 4 calculates the mean

number of events occurring in one Poisson process between

two events in a second and independent Poisson process.

Similar results for Lemma 2 were also reported in Ref. [5].

Here we prove them with an easier and different approach.

Note that results in this section are very general and can be

used in other applications.

In a Poisson arrival process with the rate l; the

interarrival time is an exponential process with the rate l

[6], vice versa. Therefore, a Poisson process for arrivals and

an exponential process for interarrival times will be used

interchangeable in this paper.

Lemma 1. Let {N1ðtÞ; t $ 0} and {N2ðtÞ; t $ 0} be two

independent Poisson processes with rate l1 and l2;

respectively. Let t1 and t2 denote the time of the first event

of the first process and the time of the first event of the

second process, respectively. The probability that one event

occurs in the first process before one event occurs in the

second process is given as follows:

Pðt1 , t2Þ ¼l1

l1 þ l2

ð1Þ

Proof. The interarrival times of the first and second Poisson

processes follow exponential distributions with rate l1 and

l2; respectively, and they are independent [6]. Let X1 and X2

denote the above interarrival times for the first and second

Poisson processes, respectively. We have

Pðt1 , t2Þ ¼ PðX1 , X2Þ ¼ð1

0PðX1 , X2lX1

¼ xÞl1 e2l1x dx ¼ð1

0Pðx , X2Þl1 e2l1x dx

¼ð1

0e2l2xl1 e2l1x dx ¼

l1

l1 þ l2

A



respectively. Let aðl1; l2; nÞ denote the probability that

exactly n events occur in the first process between two

events, which occur in the second process. We have,

aðl1;l2; nÞ ¼l2

l1 þ l2

l1

l1 þ l2

� �n

ð2Þ

Proof. The time period between two events in the second

process is the same as the time period between the time at

the beginning and the time when the first event occurs in the

second process due to memoryless property of a Poisson

process. Let tn1 and t2 denote the time of the nth event of

Y. Xiao / Computer Communications 26 (2003) 1047–10551048

the first process and the time of the first event of the second

process, respectively. Therefore,

aðl1;l2; nÞ ¼ Pðtn1 , t2 , tnþ1

1 Þ ð3Þ

aðl1;l2; nÞ ¼ Pðtn1 , t2 AND t2 , tnþ1

1 Þ

aðl1;l2; nÞ ¼ Pðtn1 , t2ÞPðt2 , tnþ1

1 Þ

by memoryless property

aðl1;l2; nÞ ¼ Pðtn1 , t2Þ

l2

l1 þ l2

;

by Lemma 1.

Next, we will derive Pðtn1 , t2Þ:By Lemma 1, the probability

that the first event in the first process happens before that in

the second process is l1=ðl1 þ l2Þ: After the first event

happened in the first process, the probability that the second

event in the first process happens before the first event has

occurred in the second process is also l1=ðl1 þ l2Þ due to the

memoryless property of Poisson processes. Therefore,

Pðt21 , t2Þ ¼

l1

l1 þ l2

� �2

Similarly, due to the memoryless property of Poisson

processes, we can derive Pðt31 , t2Þ; Pðt4

1 , t2Þ; so on. By

using mathematical Induction, we can easily prove that for

any natural number n; we have

Pðtn1 , t2Þ ¼

l1

l1 þ l2

� �n

ð4Þ

Plug Eq. (1) into Eq. (3), we can get Eq. (2). A

Lemma 3. Let x be a real number such that 0 , x , 1: We

have,

X1i¼0

xi ¼1

1 2 xð5Þ

X1i¼0

ixi ¼x

ð1 2 xÞ2ð6Þ

bðl1;l2; dÞ ;X1m¼0

mXðmþ1Þd21

k¼md

aðl1;l2; kÞ

¼

l1

l1 þ l2

� �d

1 2l1

l1 þ l2

� �dð7Þ

X1j¼0

aðl1; l2; jÞ ¼ 1 ð8Þ

X1j¼0

jaðl1;l2; jÞ ¼l1

l2

ð9Þ

gðl1;l2; dÞ ;›bðl1;l2; dÞ

›d¼

l1

l1 þ l2

� �d

logl1

l1 þ l2

� �

1 2l1

l1 þ l2

� �d" #2

ð10Þ

fðl1;l2; dÞ ;›2bðl1;l2; dÞ

›2d

¼

1 þl1

l1 þ l2

� �d" #

l1

l1 þ l2

� �d

log2 l1

l1 þ l2

� �

1 2l1

l1 þ l2

� �d !3

. 0

ð11Þ

Proof.

X1i¼0

xi ¼ limk!1

Xk

i¼0

xi

!¼ lim

k!1

1 2 xkþ1

1 2 x

!¼

1

1 2 x

X1i¼0

ixi ¼X1i¼0

ði þ 1 2 1Þxi ¼X1i¼0

ði þ 1Þxi 2X1i¼0

xi

¼X1i¼0

ði þ 1Þxi 21

1 2 x¼X1i¼0

dðxiþ1Þ

dx2

1

1 2 x

¼

dX1i¼0

xiþ1

!

dx2

1

1 2 x¼

dx

1 2 x

� �dx

21

1 2 x

¼1

ð1 2 xÞ22

1

1 2 x¼

x

ð1 2 xÞ2

X1m¼0

mXðmþ1Þd21

k¼md

aðl1;l2; kÞ

¼X1m¼0

mXðmþ1Þd21

k¼md

l2

l1 þ l2

l1

l1 þ l2

� �k

¼l2

l1 þ l2

X1m¼0

m

l1

l1 þ l2

� �md

2l1

l1 þ l2

� �ðmþ1Þd

1 2l1

l1 þ l2

¼X1m¼0

ml1

l1 þ l2

� �md

2l1

l1 þ l2

� �ðmþ1Þd( )

¼X1m¼0

ml1

l1 þ l2

� �md

1 2l1

l1 þ l2

� �d( )

¼ 1 2l1

l1 þ l2

� �d( ) l1

l1 þ l2

� �d

1 2l1

l1 þ l2

� �d( )2

;

Y. Xiao / Computer Communications 26 (2003) 1047–1055 1049

by Eq. (6)

X1m¼0

mXðmþ1Þd21

k¼md

aðl1;l2; kÞ ¼

l1

l1 þ l2

� �d

1 2l1

l1 þ l2

� �d;

X1j¼0

aðl1;l2; jÞ ¼X1j¼0

l2

l1 þ l2

l1

l1 þ l2

� �j

¼l2

l1 þ l2

X1j¼0

l1

l1 þ l2

� �j

¼l2

l1 þ l2

1

1 2l1

l1 þ l2

¼ 1

X1j¼0

jaðl1;l2; jÞ ¼X1j¼0

jl2

l1 þl2

l1

l1 þl2

� �j

¼l2

l1 þl2

X1j¼0

jl1

l1 þl2

� �j

¼l2

l1 þl2

l1

l1 þl2

12l1

l1 þl2

� �2¼

l1

l2

fðl1;l2;dÞ¼›2bðl1;l2;dÞ

›2d¼

:

1þl1

l1þl2

� �d" #

l1

l1þl2

� �d

log2 l1

l1þl2

� �

12l1

l1þl2

� �d !3

.0

A



respectively. Let N denote the mean number of events

occurring in the first process between two events in the

second process. We have, N ¼ l1=l2:

Proof. Let X be a random variable that stands for the

number of events occurring in the first process between two

events in the second process. By Lemma 2 and the

relationship of an exponential process and a Poisson

process, we have,

PðX ¼ nÞ ¼l2

l2 þ l1

l1

l2 þ l1

� �n

N ¼X1n¼0

nPðX ¼ nÞ ¼X1n¼0

nl2

l2 þ l1

l1

l2 þ l1

� �n

¼l2

l2 þ l1

l1

l2 þ l1

1 2l1

l2 þ l1

� �2;

by Lemma 3

N ¼l1

l2

As one application of Lemma 4, we will consider that when

an MT trespasses a group of connected areas shown in Fig. 1,

and we derive the average number of area boundary

crossings, given the residence time distribution and a mobile

call arrival distribution.

Note that the areas are not necessary to be cells, and the

shapes of areas can be irregular. We draw with regular

shapes just for the demonstration purpose.

Let the residence time of an MT in an area and call

interarrival time follow exponential processes with rate h

and l; respectively. From Lemma 4, the mean number of

area boundary crossings between two call arrivals is N ¼

h=l: Intuitively, if treating the residence time as the diameter

›bðl1;l2; dÞ

›d¼

›

l1

l1 þ l2

� �d

1 2l1

l1 þ l2

� �d

0BBB@

1CCCA

›d

¼

1 2l1

l1 þ l2

� �d !

l1

l1 þ l2

� �d

logl1

l1 þ l2

� �2

l1

l1 þ l2

� �d

2l1

l1 þ l2

� �d

logl1

l1 þ l2

� � !

1 2l1

l1 þ l2

� �d" #2

¼

l1

l1 þ l2

� �d

logl1

l1 þ l2

� �

1 2l1

l1 þ l2

� �d" #2


of the area on average in time and the interarrival time of

two calls as the distance in time that it can travel, the number

of area boundary crossings should be the interarrival time

between two calls divided by the ‘diameter’ (the residence

time in the area), i.e. N ¼ ð1=lÞ=ð1=hÞ ¼ h=l: In other words,

Lemma 4 confirms our intuition.

3. Analytical modeling

3.1. Cost analysis

In this subsection, we will formulate cost functions of

HLR location update, VLR location update, and paging. We

assume that an MT’s residence times follow exponential

processes with mean 1=hLA and 1=hcell in an LA and a cell,

respectively. We expect 1=hLA . 1=hcell: We further assume

that call arrivals to each MT follow a Poisson process with

rate l:

Let CTotal; CHLR; CVLR; and CPaging denote the total

cost, the HLR location update cost, the VLR

location update cost, and the paging cost, respectively,

for each call arrival on average. Let RHLR and RVLR

denote the cost for each location update for the

HLR and the VLR, respectively. We expect RHLR .

RVLR: Let Rpoll denote the cost for polling a cell.

Note that RHLR and CHLR are different: RHLR is

the cost for each HLR location update, whereas CHLR

is the average cost of HLR location updates per call

arrival.

An HLR location update is performed when an MT first

enter an LA. Therefore, wherever an MT crosses the

boundary of an LA, an HLR location update action is

executed. By Lemma 4, the average number of HLR

location updates between two call arrivals is hLA=l:

Therefore, we have

CTotal ¼ CHLR þ CVLR þ CPaging ð12Þ

CHLR ¼ RHLR

hLA

lð13Þ

By Lemma 2, the probabilities that there are i LAs’, and k

cells’ boundary crossings between two call arrivals are

aðhLA; l; iÞ; and aðhcell; l; kÞ; respectively. Without loss of

generality, assume that the MT resides in the LA, LA0,

when the previous phone call arrived. Let NVLR denote the

average number of VLR location updates between two call

arrivals, and NVLR;i denote the average number of location

updates in VLRs with the movement-based location update

scheme when the MT receives the next phone call in the ith

LA, LAi; ði ¼ 0; 1; 2;…Þ: We have,

NVLR ¼X1i¼0

NVLR;iaðhLA; l; iÞ ð14Þ

CVLR ¼ RVLRNVLR ð15Þ

Note that the size of a PA is changed according to the

threshold d while the size of an LA is fixed. The probability

that there are k cell boundary crossings within LA0, and the

MT is still in LA0 when the next call arrives, is aðhcell;l; kÞ

by Lemma 2. The probability that there are k cell boundary

crossings within LA0 when the MT enters LA1 is

aðhcell;hLA; kÞ by Lemma 2. The probability that there are

k cell boundary crossings within LAm during the time period

within LAm; ðm ¼ 1;…; i 2 1Þ is aðhcell;hLA; kÞ by Lemma

2. The probability that there are k cell boundary crossings

after entering the last LAi until the next phone call arrival is

aðhcell;l; kÞ by Lemma 2. Therefore, NVLR;i will be

NVLR;i ¼ iX1m¼0

mXðmþ1Þd21

k¼md

aðhcell;hLA; kÞ þX1m¼0

�mXðmþ1Þd21

k¼md

aðhcell;l; kÞ

¼ ibðhcell;hLA; dÞ þ bðhcell;l; dÞ ð16Þ

Fig. 1. An MT crosses areas.

NVLR;i ¼

X1m¼0

mXðmþ1Þd21

k¼md

aðhcell;l; kÞ; for i ¼ 0

X1m¼0

mXðmþ1Þd21

k¼md

aðhcell;hLA; kÞ þ ði 2 1ÞX1m¼0

mXðmþ1Þd21

k¼md

aðhcell;hLA; kÞ þX1m¼0

mXðmþ1Þd21

k¼md

aðhcell;l; kÞ; for i . 0

8>>>>><>>>>>:


NVLR ¼X1i¼0

NVLR;iaðhLA; l; iÞ

¼X1i¼0

½ibðhcell;hLA; dÞ þ bðhcell;l; dÞ�aðhLA;l; iÞ

¼ bðhcell;hLA; dÞhLA

lþ bðhcell; l; dÞ ð17Þ

When an incoming call arrives, all the cells in the PA are

paged. The PA is the covering area within a distance d 2 1

from the center cell where the last VLR location update of

the MT is performed which belongs to the LA where the

center cell locates. The covering area within a distance d 2

1 from the center cell may cover more than one LA.

We assume that cells and LAs are two-dimensional.

However, our technique can be easily applied to a linear

shape system such as high way or railway. Let Ncell be the

number of cells in a PA. Assume that the PCS networks

have hexagonal configurations. We have

Ncell < 1 þXd21

m¼0

6m ¼ 3d2 2 3d þ 1

The right side of above equation is the number of cells with

up to d 2 1 rings. Since the number of cells in an LA is

much larger than 3d2 2 3d þ 1; we use 3d2 2 3d þ 1 to

approximate the number of cells in a PA. Therefore,

approximately

CPaging ¼ Rpollð3d2 2 3d þ 1Þ ð18Þ

To substitute Eqs. (13), (15), (17), and (18) into Eq. (12), we

obtain the total cost per call arrival for movement-based

location update scheme as follows:

CTotal ¼ RHLR

hLA

lþ Rpollð3d2 2 3d þ 1Þ

þ RVLR bðhcell;hLA; dÞhLA

lþ bðhcell; l; dÞ

� �ð19Þ

Please note that we consider the case that a VLR location

update occurs when the MT completes d movements

between cells, as well as the case that a VLR location

update occurs when the MT crosses an LA boundary.

Previous work [3,7–9] failed to consider the later case, and

therefore, made derivation easier.

Next, we will formulate the total cost for the static

location update scheme. Let Ncell be the number of cells in

an LA. Note that the PA is the same as the LA, and the PA

size is fixed. By Lemma 4, the average number of HLR

location updates and the average number of VLR location

updates between two call arrivals are hLA=l and hLA=l;

respectively. Let C0Total denote the total for the static location

update scheme. We can easily have

C0Total ¼ RHLR

hLA

lþ RVLR

hLA

lþ RpollNcell: ð20Þ

3.2. Optimization and algorithm

In this subsection, we will prove analytically that there is

a unique optimal movement threshold that minimizes the

total cost of HLR location updates, VLR location updates

and paging, per call arrival. We have the first and second

derivatives of Eq. (19) as follows:

›CTotal

›d¼ RVLR

hLA

lgðhcell;hLA; dÞ

þ RVLRgðhcell;l; dÞ þ Rpollð6d 2 3Þ ð21Þ

›2CTotal

›2d¼ RVLR

hLA

lgðhcell;hLA; dÞ

þ RVLRgðhcell;l; dÞ þ 6Rpoll . 0 ð22Þ

From Eq. (22), we know that CTotal is a convex function.

Therefore, the unique minimum point can be achieved at

›CTotal=›d ¼ 0:FromEq. (21)wehave the following theorem.

Optimal Theorem. The minimum CTotal can be achieved at

a unique movement threshold dmin where dmin satisfies

following equation:

›CTotal

›d

��d¼dmin

¼RVLR

hLA

lgðhcell;hLA;dÞ

þRVLRgðhcell;l;dÞþRpollð6d23Þ¼0

Since Eq. (23) is difficult to solve, we propose Binary

Searching Algorithm I based on the following fact. From

Eq. (21), we know that ð›CTotal=›dÞld¼dminis a strict

increasing function of the movement threshold d: Therefore,

we can easily prove

›CTotal

›d¼

.0ifd.dmin

¼0 ifd¼dmin

,0ifd,dmin

8>><>>: ð24Þ

With above considerations, we provide Binary Searching

Algorithm I based on the fact that the first derivative at the

left endpoint has a different sign from the first derivative at

the right endpoint., where d is the precision and dp1: Step 4

is to find the integer value of dmin:

Binary Searching Algorithm I.

Step 1. Initially, let L ¼ 1 and R ¼ rmax; and they stand

for the left endpoint and right endpoint of the search

interval ½L;R�: rmax is the maximum number of rings in


an LA. If

›CTotal

›d

��d¼L

›CTotal

›d

��d¼R

. 0;

dmin will be either L or R: Then,

dmin ¼L if CTotalld¼L , CTotalld¼R

R Otherwise

(;

and go to step 4. Otherwise, let M ¼ ðL þ RÞ=2 and go to

step 2.

Step 2. If

›CTotal

›d

��d¼M

, d;

let dmin ¼ M; and go to step 4. Otherwise, go to step 3.

Step 3. Let M ¼ ðL þ RÞ=2: If

›CTotal

›d

��d¼M

¼ 0;

let dmin ¼ M; and go to step 4. Otherwise, if we have

›CTotal

›d

��d¼M

›CTotal

›d

��d¼R

. 0;

let R ¼ M and go to step 2. Otherwise let L ¼ M; and go

to step 2.

Step 4. Let L ¼ ceilingðdminÞ and R ¼ floorðdminÞ: We

have

dmin ¼L if CTotalld¼L , CTotalld¼R

R Otherwise

(:

The complexity of the above algorithm is log2 n; where n

is one over the precision, e.g. if the precision is 0.01, then

n ¼ 100:

3.3. The hybrid scheme

Similar to Ref. [5], we adopt the CMR to evaluate the

mobility of an MT. The CMR is defined as the ratio of the

residence time in an LA and the interarrival time for phone

calls on average, i.e. ð1=lLAÞ=ð1=lÞ ¼ l=hLA: From Lemma

4, we know that CMR is the same as one over the number of

LA boundary crossings between to phone calls. The smaller

the CMR is, the higher the mobility of an MT. Let IðeÞ

denote the indicator function, which equals 1 if e ¼¼ true;

otherwise it equals 0.

Let dmin denote the optimal movement threshold for the

dynamic movement-based threshold found by using the

Binary Searching Algorithm proposed in Section 3.2. Let

TCMR denote the hybrid threshold. The cost function of

the hybrid scheme, C00Total; is expressed as follows:

C00Total ¼ I

l

hLA

$ TCMR

� �CTotalld¼dmin

þ Il

hLA

, TCMR

� �C0

Total ð25Þ

Next, we will propose an algorithm to find the proper hybrid

threshold TCMR: Before pursuing such an effort, we need the

following theorem. Please note that both CTotalld¼dminand

C0Total are function of l=hLA:

Theorem II. The cost functions for the dynamic movement-

based scheme and the static location update are strict

decreasing functions of the CMR l=hLA: Furthermore, if

there is a value of l=hLA that satisfies the following

equation, the value is unique

CTotalld¼dmin¼ C0

Total ð26Þ

Proof. From Eqs. (19) and (20), both CTotalld¼dminand C0

Total

are strict decreasing functions of l=hLA: Therefore, the

second part of theorem can be also easily obtained. A

With Theorem II in mind, we can easily solve the

solution of Eq. (26) by the standard Newton approximation

method. The idea of the Newton approximation method is

stated as follows. Let f ðrÞ ¼ CTotalld¼dmin2 C0

Total; where

r ¼ l=hLA: Let r0 be the initial value of a first approximation

of the root r: Now point (r0; f ðr0Þ) is on the curve of the

function. The slope of the tangent line in (r0; f ðr0Þ) is f 0ðr0Þ:

This tangent line intersects the x-axis in a point with an x-

value r1: The value r1 is a better approximation of the root

than r0: From r1 you can find a better approximation r2; and

so on so forth. Keep doing so until some precision is met.

4. Numerical results

In this section, we evaluate the optimal dynamic

movement-based location update scheme, the static location

update scheme, and the hybrid scheme. For the demon-

stration purpose, we adopt following parameters for our

numerical results. Particularly, we have 1=hLA ¼ 1800:0 s,

1=hcell ¼ 120:0 s, RHLR ¼ 20; RVLR ¼ 20; and dpoll ¼ 1: We

have d ¼ 0:0001; which is the precision. We let 1=l change

in a range so that it is equivalent to say that the CMR, l=hLA;

changes in a range.

Fig. 2 shows the optimal threshold values for the

dynamic scheme, obtained by our Binary Searching

Algorithm, over 1=l; the mean of call interarrival times. In

this figure, 1=l range is from 10 to 1000 s. As illustrated, as

1=l increases, the optimal threshold increases. In the other

words, as the mobility increases, the optimal threshold

increases. This fact confirms our intuition.


Fig. 3 shows the total cost of the dynamic scheme

over the call interarrival time for different thresholds. In

this figure, 1=l range is from 10 to 1000 s. As illustrated,

the optimal dynamic scheme has the lowest cost. The

figure also shows that as the call interarrival time

increases, the total cost of location update and paging

also increases.

Fig. 4 compares the total costs of the optimal dynamic

scheme, the static scheme and the hybrid scheme over the

call interarrival time. In this figure, 1=l range is from 10 to

1000 s. As illustrated, the optimal dynamic scheme and the

hybrid scheme has the same total cost since 1=l is small, and

they are both much better than the static scheme under such

a mobility range. The figure also shows that as the call

interarrival time increases, the total cost also increases.

Fig. 5 shows the optimal threshold values for the

dynamic scheme, obtained by our binary searching

algorithm, over 1=l; the mean of call interarrival times.

Fig. 5 is different from Fig. 2 with a different range of

1=l; i.e. from 1000 to 60,000 s. As illustrated, as 1=l

increases, the optimal threshold increases. In the other

words, when the mobility increases (CMR decreases), the

optimal threshold increases. This fact confirms our

intuition.

Fig. 6 compares total costs of the optimal dynamic

scheme, the static scheme and the hybrid scheme over the

call interarrival time. In this figure, 1=l range is from

1000 to 60,000 s. As illustrated, the optimal dynamic

scheme is better than the static scheme when 1=l is small

(the mobility is low), but worse than the static scheme

when 1=l is large (the mobility is high). The hybrid

scheme is the best scheme among the three schemes.

The hybrid scheme has the same total cost as that of the

optimal dynamic scheme when 1=l is small, and it has

Fig. 2. Optimal threshold vs. 1=l (s).

Fig. 3. Total cost vs. 1=l (s).

Fig. 4. Comparing total costs when the mobility is high.

Fig. 5. Optimal threshold vs. 1=l (s).


the same total cost as that of the static scheme when 1=l

is high. The figure also confirms Theorem II that the total

costs for the optimal dynamic scheme and the static

scheme are both strict decreasing functions of CMR

(increasing functions of 1=l), and they intersect at a

unique point. The figure also shows that the adopted

Newton approximation method can find the hybrid

threshold.

5. Conclusions

In this paper, we proposed a hybrid scheme for location

management for PCS networks. We analytically modeled

total cost functions of the dynamic movement-based

scheme, the static scheme, and the hybrid scheme. We

prove analytically that there is a unique optimal movement

threshold for the dynamic scheme that minimizes the total

cost of HLR location updates, VLR location updates, and

paging, per call arrival. An effective algorithm is proposed to

find the optimal movement threshold, and the Newton

approximation method was adopted to find the hybrid

threshold. Furthermore, the following results were observed:

† The proposed algorithm can effectively find the optimal

movement threshold.

† The optimal dynamic scheme is better than the static

scheme when the mobility is low, but is worse than the

static scheme when the mobility is high.

† The hybrid scheme is the best scheme among the three

kinds of schemes. The adopted Newton approximation

method can find the hybrid threshold.

Acknowledgements

The author would like to thank the anonymous

reviewers’ valuable comments that have significantly

improved the paper quality.

References

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[3] A. Bar-Noy, I. Kessler, M. Sidi, Mobile users: to update or not to

update?, ACM/Baltzer, Wireless Networks 1 (2) (1995) 175–186.

[4] J. Li, Y. Pan, X. Jia, Analysis of dynamic location management for PCS

networks, IEEE Transactions on Vehicular Technology September

(2002) 1109–1119.

[5] Y.B. Lin, Reducing location update cost in a PCS network, IEEE/ACM

Transactions on Networking 5 (1997) 25–33.

[6] S.M. Ross, Introduction to Probability Methods, Academic Press, New

York, 1997.

[7] I.F. Akyildiz, J. Ho, Y.B. Lin, Movement-based location update and

selective paging for PCS networks, IEEE/ACM Transactions on

Networking 4 (4) (1996) 629–638.

[8] Y. Fang, I. Chlamtac, Y. Lin, Portable movement modeling for PCS

networks, IEEE Transactions on Vehicular Technology 49 (4) (2000)

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Yang Xiao received his Ph.D. degree in computer science and

engineering from Wright State University. Dayton, Ohio. From 1991

to 1996, he had been a software engineer, a senior software engineer,

and a technical lead working in the computer industry. From 1996 to

2001, he had been awarded the DAGSI Ph.D. Fellowship, which

supported his five-year Ph.D. studies. From Aug. 2001 to Aug. 2002, he

worked at Micpro Linear-Salt Lake Design Center as a MAC architect

involving the IEEE 802.11 (Wireless LAN) standard enhancement

work. Since Aug. 2002, he has been an assistant professor of computer

science at The University of Memphis. He serves as the symposium co-

chair of Symposium on Data Base Management in Wireless Network

Environments in IEEE VTC 2003. He is/was a TPC member for

conferences: IEEE ICCCN 2003, Hearthcom 2003, SCI 2003, IEEE

ICCCN 2002, SCI 2002, and IEEE IC3N 2001. He is a voting member

of the IEEE 802.11 Working group, and a memeber of IEEE and ACM.

His current research interests include wireless LANs, wireless PANs,

and mobile cellular networks.

Fig. 6. Comparing total costs when the mobility is low.


Optimal location management for two-tier PCS networks

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