WiMAX Location Update for Vehicle Applications

DOI 10.1007/s11036-009-0171-8

WiMAX Location Update for Vehicle Applications

Yi-Bing Lin· Yung-Chun Lin

Published online: 26 May 2009

© Springer Science + Business Media, LLC 2009

Abstract In IEEE 802.16e mobile Worldwide

Inter-operability for Microwave Access (WiMAX), paging groups (PGs; groups of base stations) are used to identify the locations of mobile stations (MSs). An anchor paging controller (APC) is assigned to an MS to handle location tracking for the MS. During location update, the WiMAX network may or may not relocate the APC. This paper considers a linear WiMAX base station layout for vehicle applications, where a base station serves as a roadside unit, and a WiMAX MS installed in a vehicle serves as an onboard unit. In these vehicle applications, APC relocation may significantly affect the network traffic. This paper proposes an ana-lytic model to study the performance of the location up-date with/without APC relocation. Our study provides guidelines to utilize the APC relocation for vehicles with various moving behaviors.

Keywords intelligent transportation systems·

location update· mobility management ·

paging group

Y.-B. Lin (

_B

Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan, Republic of China e-mail: liny@csie.nctu.edu.tw Y.-C. Lin

Information and Communications Research Laboratories, Industrial Technology Research Institute,

Hsinchu, Taiwan, Republic of China e-mail: yjlin@csie.nctu.edu.tw

1 Introduction

IEEE 802.16e mobile Worldwide Interoperability for

Microwave Access (WiMAX) provides broadband

wireless services with wide service coverage, high data

throughput, and high mobility [9–11]. In Taiwan, an

important WiMAX usage is to provide broadband

ac-cess for vehicles. Figure 1illustrates Taiwan WiMAX

experience bus where people in the bus enjoy appli-cations such as GPS/Navigation, portable digital TV, multimedia player, and so on.

Figure 2 shows a simplified WiMAX network

ar-chitecture, which consists of the connectivity service

networks (CSNs; see Fig. 2a) and the access service

networks (ASNs; see Fig.2b). An ASN provides radio access (such as radio resource management, paging and location management) to the WiMAX mobile

sta-tion (MS; Fig. 2i). The ASN comprises ASN gateways

(ASN-GWs; see Fig. 2c) and WiMAX base stations

(BSs; see Fig.2d). Every ASN-GW connects to several

BSs. The ASN-GWs are also connected to each other to coordinate MS mobility. A CSN consists of network

nodes such as the mobile IP (MIP) [4] home agent

(HA; see Fig.2f), the authentication, authorization, and

accounting (AAA) server (see Fig. 2g) and the

dy-namic host configuration protocol (DHCP) server (see

Fig. 2h). The CSN provides IP connectivity (such as

Internet access and IP address allocation) to a WiMAX MS and interworks with the ASNs to support capabil-ities such as AAA and mobility management. Before an MS is allowed to access WiMAX services, it must be authenticated by the ASN-GW (which serves as the

authenticator) and the AAA server in the CSN.

In Taiwan, a linear layout with 27 WiMAX BSs has been deployed from Taipei to Taoyuan International

Fig. 1 (Left) WiMAX

experience bus; (right) broadband wireless access in the bus

Airport to cover highway and local road traffics (the distance is 30 km). The network is anticipated to extend to 679 BSs in north Taiwan for more general broadband wireless applications [5]. One of the applications aims for intelligent transportation systems (ITS) where the WiMAX BSs can be viewed as roadside units and the MSs are onboard units [2,3,6,13]. In ITS applications,

high mobility of vehicles may significantly affect net-work signalling overhead for location tracking, which is investigated in this paper.

We first introduce the WiMAX location tracking mechanism. In WiMAX, two subscriber modes charac-terize the activities of an MS attached to the network. In the normal mode, the MS sends or receives packets

Fig. 2 A simplified WiMAX

network architecture

CSN

PG 1

PG 2

PG 0

ASN-GW 0

PG 3

HA

a

f

g

h

e

c

d

b

IP Network

ASN

PC 0

ASN-GW 1

FA 0

PC 1

FA 1

CN

j

i

to/from a BS. When there is no data transmission for a period, the MS switches from the normal mode to the

idle mode to conserve resources.

Several procedures defined in WiMAX are exercised when the MS is in the idle mode. For example, the

lo-cation update (LU) procedure is exercised for lolo-cation

tracking of an MS [1]. When there are incoming packets

for the idle MS, the paging procedure is exercised to alert the MS. Then the MS performs the idle mode exit procedure to return to the normal mode and starts data transmission. In the control plane, the paging controller

(PC) in an ASN-GW (see Fig.2c) handles the location

tracking and the paging operations for the idle MSs. All BSs connected to a PC are partitioned into several

paging groups (PGs; see Fig.2e). The PGs are used for MS tracking. For each MS in the idle mode, the network assigns a PC called the anchor PC (or APC) to the MS. Every APC is associated with a database called location

register (LR) that contains the MS tracking and paging

information of the idle mode MSs. The information includes the current PG where the MS resides, the paging parameters, the QoS profiles, etc. Suppose that an MS in the normal mode enters the idle mode through

PG 0 in Fig. 2, and PC 0 serves as the APC of the

MS. When this idle MS moves from PG 1 to PG 2, it performs LU procedure to inform its APC (i.e., PC 0) of the new location through PC 1. After location up-date, PC 1 serves as the relay PC for the MS and all PC-related control messages are delivered between PC

0(the APC) and the MS indirectly through PC 1.

When the MS enters the normal mode, data trans-mission with handover is supported by the MIP. In

Fig.2, assume that an MS first enters the WiMAX

net-work (in the normal mode) and obtains an IP address through ASN-GW 0. The MS then registers to the HA (see Fig.2f) in the CSN to indicate that the foreign agent

(FA) is FA 0 associated with ASN-GW 0 (see Fig.2c).

At this point, the MS can start data transmission with

a corresponding node (CN; see Fig.2j). When the MS

moves from BS 4 to BS 5, the local ASN-GW is changed from ASN-GW 0 to ASN-GW 1. The data between FA

0and the MS is then delivered through ASN-GW 0,

ASN-GW 1 and BS 5.

In WiMAX, the APC is only defined in the idle mode and does not play any role in the normal mode. When an MS switches from the idle mode to the normal mode, the APC is no longer associated with the MS; i.e., the MS information is removed from the LR of the APC (through the idle mode exit procedure). When the MS enters the idle mode again, a new APC is assigned to the MS by exercising the idle mode entry procedure. In the idle mode, WiMAX provides two alternatives to reassign the APC during the MS movement: static

or dynamic. When the MS moves from an old PG to a new PG, the APC can be dynamically reassigned during the LU procedure. If the APC reassignment occurs frequently, these APC/LR relocations result in extra network signalling overhead. On the other hand, if the APC reassignment occurs infrequently, the APC may be far away from the MS after several movements. In this case, long delays for message exchange will be expected in the LU procedures. This paper analyzes the cost for the APC/LR reassignment under Taiwan’s linear WiMAX BS layout for ITS.

This paper is organized as follows. In Section2, we

describe the LU procedure. Section 3 illustrates two

scenarios to study the cost for the APC/LR relocation

due to MS (vehicle) mobility. Section 4 numerically

compares these two scenarios.

2 The location update procedure

This section describes the location update (LU) pro-cedure exercised in the idle mode [10,11]. Figure 3a illustrates how the LU procedure is performed when an idle MS moves from a BS in PG 1 to another BS in PG 2. Without loss of generality, we assume that PC 0 is the MS’s APC, and both the APC (PC 0) and the FA (FA 0) are collocated in ASN-GW 0. We also assume that a specific ASN-GW serves as the idle MS’s authenticator. When the idle MS moves to ASN-GW

1, the relay PC (e.g., PC 1 in Fig. 3a) may become

the APC. For the description purpose, let SR be the

LU scenario with APC relocation and SW be the LU

scenario without APC relocation. The LU procedure

with APC relocation scheme SRis described as follows

(see SRin Fig.3a):

Step 1 The MS moves from PC 0 to PC 1. Through the new BS, the MS sends an

LU Request message to PC 1.

Step 2 PC 1 decides to relocate the APC and selects itself as the APC. The LU

Re-quest message is forwarded to PC 0

with a relocation indicator.

Step 3 PC 0 retrieves the Authorization Key (AK) context from the authentica-tor through the Context Request/

Response message exchange.

Steps 4 and 5 The LU result (with the AK context and the MS’s record data) is delivered to PC 1 by the LU Response message. PC 1 stores the received information (including the MS’s new location) in its LR. The LU Response message

Fig. 3 Flows of Inter-PC LUs

with signalling costs (a–c)

PG 2

PG 2

PG 3

Ctx Req/Resp

S

S

ASN-GW 1 PC 1 (old APC) ASN-GW 0 FA 0 1 1 5 5 8 2 4 9 6 7 1 2 3 4 6 9 7 8 5 LU Req Ctx Req/Resp LU Resp

CMAC Upd Req CMAC Upd Resp

LU Cnf PC Relocate Ind/Ack PC Relocate Ind/Ack LU Cnf new BS new BS MS ASN-GW 0 1 1 3 3 6 2 4 5 1 2 3 4 5 6 LU Req LU Resp CMAC Upd Req CMAC Upd Resp LU Cnf

(a) Inter-PC LU “Out” from PC 0 to PC 1

ASN-GW 1 ASN-GW 2 2 4 1 1 5 5 8 ASN-GW 0 3 FA 0 9 67 Relay Function 6 7 PC 2 (new APC) PC 1 (old APC) new BS (c) Inter-PC LU from PC 1 to PC 2 ASN-GW 0 ASN-GW 2 PC 0 (APC) 2 3 4 8 1 1 5 5 8 6 7 Relay Function 7 6 1 2 3 4 6 7 8 5 LU Req Ctx Req/Resp LU Resp CMAC Upd Req CMAC Upd Resp LU Cnf new BS PC 2 (relay PC) ASN-GW 1 1 2 3 4 6 9 7 8 5 LU Req Ctx Req/Resp LU Resp CMAC Upd Req CMAC Upd Resp LU Cnf PC Relocate Ind/Ack PC Relocate Ind/Ack LU Cnf CI=14 C*=16 CI=9 C*=12 PG 4 PG 4 PG 3 PG 1 PG 1

(b) Inter-PC LU from PC 1 “Into” PC 0 Relay Function 6 7 Relay Function 5 4 PC 0 (APC) A u th en ti ca to r PC 0 (new APC) 10 11 11 10 3 MS 11 10 11 10 ASN-GW 1 PC 1 old BS old BS old BS old BS MS MS A u th en ti ca to r A u then ti cato r Authe n ti ca to r PG 1 PG 1 LU Req ASN-GW 0 ASN-GW 1 PC 0 (old APC) PC 1 (new APC) 2 3 4 FA 0 9 MS new BS 1 1 5 5 8 6 7 6 7 Relay Function 1 2 3 4 6 9 7 8 5 Ctx Req/Resp LU Resp CMAC Upd Req CMAC Upd Resp LU Cnf PC Relocate Ind/Ack PC Relocate Ind/Ack LU Cnf ASN-GW 0 ASN-GW 1 PC 0 (APC) 2 3 ₄ 8 MS 1 1 5 5 8 6 7 Relay Function 7 6 1 2 3 4 6 7 8 5 LU Req Ctx Req/Resp LU Resp CMAC Upd Req CMAC Upd Resp LU Cnf new BS PC 1 (relay PC) CO=16 CO=12 PG 2 PG 2 11 10 11 10 old BS old BS A u th enti cat or A u th enti ca tor

then carries the AK context (to be retrieved by the new BS) and the LU result (for the MS).

Steps 6 and 7 To prevent the replay attack, the new BS updates Cipher-based Message

Au-thentication Code (CMAC) Key Count

(the latest counter value for the MS’s request message) with the authentica-tor through the CMAC Update

Re-quest/Response message exchange.

Step 8 The new BS notifies PC 1 of the com-pletion of the LU procedure by an LU

Confirm message.

Steps 9 and 10 PC 1 informs both the MS’s FA (i.e., FA 0) and authenticator about the APC relocation through the PC

Relo-cation IndiRelo-cation/Acknowledge

mes-sage exchange.

Step 11 PC 1 informs PC 0 that the LU proce-dure is completed by the LU Confirm

message. After receiving the message, PC 0 removes the MS’s record from its LR.

The LU procedure without APC relocation (see SW

in Fig.3a) is similar to that with APC relocation, except that Steps 9 and 10 are eliminated. For SWin Fig.3a, PC

1serves as the relay PC, and no relocation indicator is

added in the LU Request message in Step 2. Therefore, the network needs not to transfer the MS data to the new APC/LR in Step 4.

3 Analytic analysis

This section studies the expected location update cost during the idle period. When the MS enters the idle mode, we assume that PC 0 is assigned as its APC and is collocated with its FA (FA 0). A location update can be an inter-PC LU (e.g., from BS 4 to BS 5 in

Fig.2) or an intra-PC LU (e.g., from BS 2 to BS 3 in

Fig.2), and the associated costs are elaborated below.

In our study, the network signalling cost is measured through the messages exchanged between the network nodes.

• CO: the cost of an inter-PC LU “Out” from PC 0 (to

any neighboring PC; see Fig.3a). For SR, CO= 16.

For SW, CO= 12.

• CI: the cost of an inter-PC LU from the neighboring

PCs “Into” PC 0 (see Fig.3b). For SR, CI= 14. For

SW, CI= 9.

• C∗_{: the cost of an inter-PC LU without involving PC}

0(see Fig.3c). For SR, C∗= 16. For SW, C∗= 12.

• c: the cost of an intra-PC LU within PC 0 (see Fig.4a). For both SRand SW, c= 9.

• c∗_{: the cost of an intra-PC LU in any PC other than}

PC 0 (see Fig.4b). For SR, c∗= 9. For SW, c∗ = 12.

• C: the expected cost of an LU in the idle mode. We propose an analytic model to study the expected

LU cost C for both SR and SW. Figure 5 shows the

MS movement state-transition diagram with the linear

configuration of PGs and PCs. Figure5a illustrates the

state-transition diagram for the MS movement among PGs. We use a one-dimensional linear layout of PGs to represent an extension of Taiwan’s WiMAX-based

ITS deployment described in Section 1, where state i

represents that the MS visits PG i (−∞ < i < ∞). The

MS moves from state i to states i+ 1 and i − 1 with

probabilities p and 1− p, respectively. Let Pk

j be the

probability that from PG 0, the MS visits PG j after k

PG movements, where k≥ 0. From the state-transition

diagram in Fig.5a, Pk

j is derived as follows. For j≥ 0

Pk_j=  k k− j 2  pjp(1−p) k− j 2 , k≥ j≥0 and k−j is even 0, otherwise . (1) Note that Eq.1is partially validated for j= 0 and p =

1− p = 0.5 [7]. Due to the symmetric PG layout, from

Eq.1, Pk_j for j< 0 can be expressed as

Pk_j =  k k+ j 2  (1 − p)− j_{p(1 − p)}k+ j2 , k ≥ − j > 0 and k + j is even 0, otherwise . (2) Let Pk

i, j be the probability that starting from PG i, the

MS visits PG j after k PG movements. From Eqs.1and

2, the probability Pk_{i, j}can be generalized as

Pk i, j= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  k k−( j−i) 2  pj−ip(1 − p) k−( j−i) 2 , _j≥ i, k ≥ | j − i| ≥ 0 and k − | j − i| is even  k k+( j−i) 2 

(1 − p)−( j−i)_{p(1 − p)}k+( j−i)2 , j < i, k ≥ | j − i| > 0 and k − | j − i| is even

0, otherwise

. (3)

Based on the above model, we define five parameters to compute C as follows:

• O(k): the expected number of the inter-PC

move-ments “Out” from PC 0 (to PC−1 or PC 1) during

k PG movements of the MS.

• I(k): the expected number of the inter-PC

move-ments from PC−1 or PC 1 “Into” PC 0 during k PG

movements of the MS.

• ∗_{(k): the expected number of the inter-PC}

ments without visiting PC 0 during k PG move-ments of the MS.

Ctx Req/Resp Ctx Req/Resp

S

S

(a) Intra-PC LU within PC 0

(b) Intra-PC LU outside PC 0 new BS ASN-GW 0 PC 0 (APC) 1 1 3 3 6 2 4 5 1 2 3 4 5 6 LU Req Ctx Req/Resp LU Resp CMAC Upd Req CMAC Upd Resp LU Cnf ASN-GW 1 PC 1 (APC) 2 1 1 3 3 6 4 5 Relay Function 5 4 1 2 3 4 5 6 LU Req Ctx Req/Resp LU Resp CMAC Upd Req CMAC Upd Resp LU Cnf new BS

c

=9

c

=12

c=9

PG 1

PG 3

PG 3

LU Cnf

c=9

PG 1

PG 0

PG 0

PG 2

PG 2

Fig. 4 Flows of intra-PC LUs with signalling costs (a, b)

0 1

...

...

PC 0

PC 1

...

PC -1

...

...

...

PC -n

...

...

...

PC n

-2

-1

0

1

₂

i

-i

(a) State-transition Diagram for the MS Movement between PGs

(b) The Linear PC Configuration where the Size of a PC is N

• π(k): the expected number of the intra-PC move-ments within PC 0 during k PG movemove-ments of the MS.

• π∗_{(k): the expected number of the intra-PC}

move-ments in any PC other than PC 0 during k PG movements of the MS.

Figure5b illustrates a PC configuration with linear

PG layout to derive the above five parameters. In this

figure, every PC consists of N PGs (N≥ 1). For

exam-ple, PC n covers the PGs from nN to(n + 1)N − 1, and

PC−n covers the PGs from −nN to −(n − 1)N − 1.

O(k) is derived as follows. Starting from PG i, the

probability that the MS moves out from PC 0 (i.e., from

PG 0 to PG−1 or from PG N − 1 to PG N) at the

m-th PG movement is(1 − p)P_im_,0−1+ pPm_i,N−1−1 . There-fore, during k PG movements, the expected number of inter-PC movements out from PC 0 can be expressed as

k m=1  (1 − p)Pm−1 i,0 + pPmi,N−1−1 

. For simplicity, assume that the MS initially resides in any of the N PGs in PC

0with the same probability 1/N. Then

O(k) =  1 N N_−1 i=0 k  m=1  (1− p)Pm−1 i,0 + pPmi,N−1−1  . (4)

Similar to the derivation of Eq.4, we have

I(k) =  1 N N_−1 i=0 k  m=1  pPm_i,−1−1+ (1 − p)Pm_i,N−1  .(5) ∗_{(k) is derived as follows. Starting from PG i}

(0≤ i < N), after k PG movements, PCs ±_Nk are

the farthest PCs that can be visited by the MS, and the probability to make an inter-PC movement

be-tween PCs −l, −l + 1 or between PCs l, l − 1 at the

m-th movement ispPm_{i,−(l−1)N−1}−1 + (1 − p)Pm_{i,−(l−1)N}−1 +



pPm_i,lN−1−1 + (1 − p)Pm_i,lN−1, where 2≤ l ≤_Nk. In this case, the probability that at the m-th movement, the MS makes an inter-PC LU without involving PC

0can be expressed as k N  l=2   pPm_{i,−(l−1)N−1}−1 + (1 − p)Pm_{i,−(l−1)N}−1  +pP_i,lN−1m−1 + (1 − p)Pm_i,lN−1 . (6) From the definition for∗(k) and Eq.6, we have

∗_{(k) =}1 N ⎧⎪_⎨ ⎪ ⎩ N−1  i=0 k  m=1 k N  l=2  pP_{i,−(l−1)N−1}m−1 + Pm_i,lN−1−1  + (1 − p)Pm_{i,−(l−1)N}−1 + Pm_i,lN−1  ⎫ ⎪ ⎬ ⎪ ⎭. (7) Note that for_Nk≤ 1, ∗(k) = 0.

To compute π(k), we classify the intra-PC

move-ments inside PC 0 into two types.

• First-type: the MS moves to the boundary PGs of

PC 0 (i.e., the MS moves from PG N− 2 to PG

N− 1 or from PG 1 to PG 0 in Fig.5b).

• Second-type: the MS moves to the non-boundary

PGs (e.g., the MS moves from PG N− 1 to PG

N− 2 or from PG 0 to PG 1 in Fig.5b with N≥ 3).

For N= 1, there are no intra-PC movements and

π(k) = 0. For N = 2, all PGs are boundary PGs, and

no second-type movements are made. For N≥ 2,

starting from PG i, the expected number of the first-type movements during k PG movements is

k

m=1



(1 − p)Pm−1

i,1 + pPmi,N−2−1 . For N≥ 3, the

ex-pected number of the second-type movements during

k PG movements is k_m=1N_j=1−2Pm i, j. Therefore, π(k) can be expressed as π(k) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, N= 1  1 N N_−1 i=0 k  m=1  (1 − p)Pm−1 i,1 + pPmi,N−2−1   , N= 2  1 N ⎧_⎨ ⎩ N−1  i=0 k  m=1 ⎡ ⎣(1 − p)Pm−1 i,1 + pPim,N−2−1 + N−2  j=1 Pm_{i, j} ⎤ ⎦ ⎫ ⎬ ⎭ , N≥ 3 . (8)

From Eqs.4,5,7and8, we have

π∗_{(k) = k − }

O(k) − I(k) − ∗(k) − π(k). (9)

Equations4,5,7and8are validated against the simu-lation experiments (see Table1) [12], where the differ-ences are within 2%. Note that due to the symmetric PG

Table 1 Comparison

between analytic and simulation results (k= 39) p 0.5 0.6 0.7 0.8 0.9 1.0 N= 1 O(k) (analytic) 5.0148 3.9794 2.4783 1.6666 1.25 1 O(k) (simulation) 4.9899 3.9488 2.473 1.6729 1.2513 1 Error(%) 0.50 0.77 0.21 0.38 0.54 0 I(k) (analytic) 4.0148 2.9794 1.4783 0.6666 0.25 0 I(k) (simulation) 3.9899 2.9488 1.473 0.6729 0.2513 0 Error(%) 0.62 1.03 0.36 0.54 0.54 0 ∗_{(k) (analytic) 29.9703 32.0412 35.0434 36.6668 37.50 38} ∗_{(k) (simulation) 30.0203 32.1024 35.054 36.6541 37.4973 38} Error(%) 0.16 0.19 0.03 0.03 0.01 0 N= 2 O(k) (analytic) 4.5148 3.6036 2.2599 1.5416 1.1944 1 O(k) (simulation) 4.4939 3.5678 2.2529 1.5452 1.1961 1 Error(%) 0.46 0.99 0.31 0.23 0.14 0 I(k) (analytic) 3.6402 2.6613 1.2645 0.5416 0.1944 0 I(k) (simulation) 3.6179 2.6246 1.258 0.5452 0.1961 0 Error(%) 0.61 1.38 0.51 0.66 0.87 0 ∗_{(k) (analytic) 11.345 13.2352 15.9756 17.4168 18.1111 18.50} ∗_{(k) (simulation) 11.3908 13.3037 15.988 17.4095 18.1086 18.50} Error(%) 0.40 0.52 0.08 0.04 0.01 0 π(k) (analytic) 4.5148 3.4794 1.9783 1.1666 0.75 0.50 π(k) (simulation) 4.4921 3.4449 1.9755 1.1714 0.7517 0.50 Error(%) 0.05 0.99 0.14 0.41 0.22 0 N= 9 O(k) (analytic) 2.312 2.1995 1.7973 1.3815 1.1406 1 O(k) (simulation) 2.3131 2.1983 1.7931 1.3810 1.1406 1 Error(%) 0.05 0.06 0.23 0.03 0.01 0 I(k) (analytic) 1.7991 1.4715 0.8366 0.3822 0.1406 0 I(k) (simulation) 1.7974 1.4678 0.8339 0.382 0.1406 0 Error(%) 0.09 0.25 0.32 0.04 0.04 0 ∗_{(k) (analytic) 0.2222 0.6623 1.6995 2.5697 3.0521 3.3333} ∗_{(k) (simulation) 0.2219 0.6675 1.7063 2.5698 3.052 3.3333} Error(%) 0.17 0.79 0.40 0 0 0 π(k) (analytic) 23.0427 18.8287 11.7555 7.4215 5.2461 4 π(k) (simulation) 22.9828 18.7273 11.7141 7.4246 5.2476 4 Error(%) 0.26 0.54 0.35 0.04 0.03 0

structure in each PC, the effect for p is identical to that for 1− p. Therefore, it suffices to consider 0.5 ≤ p ≤ 1 in our study. Our simulation program is written in C++, which works as follows:

Step 1. Decide the PC size N, the right move proba-bility p and the number k of PG movements. (e.g., k= 39 for Table1).

Step 2. An MS is initially assigned to the PGs in PC 0 with the same probability.

Step 3. According to the right move probability p, we randomly generate a series of PG movements for the MS.

Step 4. Log the numbers of each type of the LUs during the k PG movements.

Step 5. Repeat Steps 2–4 for one million times.

Step 6. Use the numbers of LU types obtained from Step 4 to calculateI(k), O(k), ∗(k), π(k)

andπ∗(k).

Let Pr[K = k] be the probability that during the idle period t, the MS makes k PG movements. Let t have the Gamma distribution [8,12] with density function f(.),

mean 1/μ, variance V and Laplace transform

f∗(s) =  1 Vsμ + 1 1 Vμ2 . (10)

Suppose that the residence time of the MS at a PG

is exponentially distributed with the mean 1/λ. Then,

Pr[K = k] can be expressed as Pr[K = k] =  ∞ t=0Pr[k LUs in t] f(t) dt =  ∞ t=0 (λt)k k! ! e−λtf(t) dt = _λk k!  _∞ t=0 e−λttkf(t) dt = (−λ)k k! ! dk_f∗_(s) dsk !"" "" s=λ . (11)

From Eq.10, for k≥ 1, we have

dkf∗(s) dsk = (−1) k_Vk_μk  1 Vμs + 1 1 Vμ2+k × k # l=1  1 Vμ2 + l − 1 . (12)

Substituting Eq.12into Eq.11, Pr[K = k] is expressed as Pr[K = k] = ⎧ ⎪ ⎨ ⎪ ⎩ $ 1 Vμλ+1 % 1 Vμ2 , for k= 0 $ λk k! % Vkμk $ 1 Vμλ+1 % 1 Vμ2+k&k l=1 $ 1 Vμ2 + l − 1 % , for k ≥ 1 = ⎧ ⎪ ⎨ ⎪ ⎩ $ 1 Vμλ+1 % 1 Vμ2_, for k= 0  1 Vμ2+k−1 k  (Vμλ)k$ 1 Vμλ+1 % 1 Vμ2+k , for k ≥ 1 . (13)

From Eqs. 4, 5, 7–9and 13, the normalized expected

cost C of an LU can be expressed as

C= ∞  k=1  1 k ' COO(k) + CII(k) + C∗∗(k) + c π(k) + c∗_π∗_(k)](_{Pr[K = k]. (14)}

We note that Eq.14is affected byλ/μ (the update rate

or the speed to move out of a PG) and p (the moving

direction of a vehicle). An MS with largeλ/μ and p

represents a vehicle moving in highway. An MS with

smallλ/μ and p represents a vehicle moving in local

roads.

4 Performance evaluation

This section studies the performance for both SR and

SW. We investigate the effects of parameters N (the PC

size), p (the right move probability),λ/μ (the update

rate or the speed of the vehicle) and V (the variance of idle period) on the expected signalling cost of a location

update (LU) operation. Let CRand CWbe the LU costs

(the expected number of messages sent in an LU) for

SRand SW, respectively. We note that the costs of the

paging and idle mode exit procedures for both SRand

SWare the same and are not considered in this paper.

Figure6plots CR and CW as functions of N, p and

λ/μ, where the idle period is exponentially distributed

(V= 1/μ2_).

In this figure,λ/μ = 1.85, 24, 240, p = 0.5, 0.8, and

N ranges from 1 to 20. We first point out several facts.

From Fig.6, we have

Fact 1. As N increases, more intra-PC LUs (with lower costs) and less inter-PC LUs (with higher costs) are exercised.

Fact 2. Whenλ/μ is very small, few LUs occur in an idle period, and it is more likely that the MS only makes intra-PC LUs within PC 0. When

λ/μ is large, more LUs outside PC 0 occur.

Fact 3. If an MS makes more inter-PC LUs (with costs CI, COor C∗; see Fig.3), then CR> CW. Fact 4. If an MS makes more intra-PC LUs outside PC 0 (with cost c∗; see Fig.4), then CR< CW.

Effects of p In practice, a small p (e.g., p= 0.5 in

Figs.6a, c and e) represents the movement of a

pedes-trian or a vehicle in local roads, and a large p (e.g.,

p= 0.8 in Figs.6b, d and f) represents the movement

of a vehicle in highways. When p increases, the MS tends to move to one direction, and more LUs without involving PC 0 are exercised. That is, Fact 4 becomes more significant as p increases, and CWincreases faster

Fig. 6 CRand CWfor

V= 1/μ2(a–f)

Effects of N andλ/μ Figure6shows an intuitive result

that CRand CW decrease as N increases (Fact 1). This

figure also shows that CR and CW increase as λ/μ

increases. Whenλ/μ is very small (see Fig.6a and b),

the expected LU cost is small (Fact 2). The increase of

λ/μ results in more LUs with higher costs.

Figure6indicates that SW outperforms SR when N

or λ/μ are small. Conversely, SR is better than SW

when both N and λ/μ are large. When N is small,

CR> CW (Facts 1 and 3). When λ/μ is small, more

LUs around PC 0 are exercised (Fact 3), and CR> CW.

When bothλ/μ and N are large, more intra-PC LUs

outside PC 0 are exercised, and CR< CW (Facts 1, 2

and 4). Therefore, for highway traffic (i.e.,λ/μ and p

are large) and when the PC size is sufficiently large (e.g., N> 3 in our examples), SR is better than SW.

For local road traffic (i.e., λ/μ and p are small), SW

outperforms SR.

Effects of V Figure 7 shows that CR and CW are

decreasing functions of the variance V of idle periods. When V is small, less long and short idle periods are

observed, and CR/CW are mainly affected by the

Fig. 7 CRand CWfor

p= 0.8 and λ/μ = 24 (a, b)

V is large, many short idle periods with few LUs are

observed (i.e., Pr[K = 0] is large in Eq.14), and CR/CW

significantly decrease as V increases. In this case, CR≈

CW.

5 Conclusions

This paper studied the WiMAX-based ITS systems where the base stations serve as roadside units and the mobile stations installed in vehicles serve as the onboard units. We investigated the impact of vehicle mobility on location update of two APC relocation scenarios for WiMAX-based ITS: with APC relocation

(SR) and without APC relocation (SW). An analytic

model was proposed to model the expected location update cost C for the one-dimensional WiMAX paging group configuration. The analytic results were validated against the simulation experiments. Our study indicates the following results:

• For a vehicle or a pedestrian in local roads, SW

outperforms SR.

• For a vehicle in highway, SRoutperforms SWwhen

the PC size is sufficiently large.

These results provide guidelines to activate the APC mechanism for vehicles with various moving behaviors.

Acknowledgements This work was sponsored in part by NSC 97-2221-E-009-143-MY3, NSC 97-2219-E009-016, Far Eastone Telecom, Chung Hwa Telecom, ICL/ITRI, ITRI/NCTU Joint Research Center and MoE ATU.

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Yi-Bing Lin is Dean and Chair professor of College of Computer

Science, National Chiao Tung University (NCTU). He is a senior technical editor of IEEE Network. He serves on the editorial boards of IEEE Transactions on Wireless Communications and IEEE Transactions on Vehicular Technology. He is General or Program Chair for prestigious conferences including ACM MobiCom 2002. He is Guest Editor for several first-class jour-nals including IEEE Transactions on Computers. Dr. Lin is the authors of the books Wireless and Mobile Network

Architec-ture (Wiley, 2001), Wireless and Mobile All-IP Networks (John

Wiley, 2005), and Charging for Mobile All-IP

Telecommunica-tions (Wiley, 2008). He received numerous research awards

in-cluding 2005 NSC Distinguished Researcher and 2006 Academic Award of Ministry of Education. Dr. Lin is an ACM Fellow, an AAAS Fellow, and an IET Fellow.

Yung-Chun Lin received the B.S. and M.S. degrees in

Com-puter Science and Information Engineering from National Chiao Tung University (NCTU), Taiwan, in 2001, 2003, respectively. He is currently pursuing the Ph.D. degree in Computer Sci-ence. His research interests include design and analysis of a personal communications services network, the cellular protocols (UMTS/GPRS/GSM), and mobile computing.