Dynamic Location Management with Caching in Hierarchical

Dynamic Location Management with Caching in Hierarchical Databases for Mobile Networks Chang Woo Pyo1 , Jie Li2 , Hisao Kameda2 , and Xiaohua Jia3 1

System & Information Engineering, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan 2

[email protected]

Institute of Information Science & Electronics, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan 3

flijie, [email protected]

Department of Computer Science, City University of Hong Kong, 83 Tat Chee Ave., Kowloon, Hong Kong [email protected]

In location management which is one of the key issues in mobile networks, the mobile user's location updates and the incoming calls behavior signi cantly aect the network performance. This paper purposes dynamic location management with caching in hierarchical databases for personal communication networks. The performance analysis of dynamic location management has more complicated problems compare to that of current static location management, since the location update area is dynamically changed by the mobile users mobility patterns and the loads of Home Location Register (HLR) and the Visitor Location Register (VLR), which are the indispensable databases storing the mobile user's location, may be adaptively changed by the user's mobility. We propose an analytical model to solve the movement complexity problems in practical HLR/VLR architectures. By using the analytical model, we provide formulas for the performance analysis and a minimization of the overall cost of the proposed scheme. And also, the eect of the caching method on dynamic location management and the performance comparison of the proposed scheme, the existing static scheme, and the dynamic scheme without caching are studied. Abstract.

1 Introduction One of the key issues in mobile networks is to economically and quickly transfer any form of information between any desired locations at any time for mobile subscribers. That is the reason why the eÆcient location tracking mechanism, Location Management, becomes one of the most important problems for mobile communication. Personal Communication Service (PCS) networks [2,5,6,7,8,9], which is one of mobile network, consist of the xed databases and switches, and a geographically service area. The area is divided into cells and the range of each cells is limited by the base station's radio power. It means that the network access point changes as the Mobile Terminal (MT) moves around several cells.

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Chang Woo Pyo, Jie Li, Hisao Kameda, and Xiaohua Jia

For correctly delivering calls, the PCS network needs to maintain the current location of the MT. How to perform location management so that the MT's can move freely in the wireless environment is one of the important problems in PCS networks. To perform location tracking of the MT's, two level hierarchical databases [5] such that two types of the location database, the Home Location Register (HLR) and the Visitor Location Register (VLR), are commonly used in PCS networks. The HLR stores the permanent information about the MT's which the primary subscription within the area. For each MT, it contains a pointer to the VLRs, which are distributively resided in the network, to assist routing incoming calls. The VLR contains temporary records for all MT's currently active within the service area. The VLR is used to retrieve information for handling calls to or from a visiting MT. The HLR and VLR are attached to an Mobile Switching Center (MSC) that represents the control element for wireless communications. All interesting PCS networks, IS-41 [3] and GSM [4], are based on two level hierarchical HLR/VLR architectures. There are two basic operations in location management : location update and call delivery. The location update is the process through which system tracks the current location of MT's that are not in conversation. The MT reports its up-to-date location information dynamically. The location information of each MT is stored in the location databases such as the HLR and the VLR. The location entries of an MT are updated when the mobile user performs a location update. The call delivery is the process of determining the serving VLR of the called MT based on the information at the location databases, and paging the cell location of the MT when a call for the MT is initiated. To facilitate the tracking of a moving MT, a PCS network is partitioned into many Location Areas (LA's). Each LA includes tens or hundreds of cells. In existing PCS networks, the size of an LA is xed. This means that the location update is performed whenever an MT crosses the LA boundaries. Under static location management [2], using a priori determined location update area, an MT located near the boundary of an LA may excessively perform the location update, as it moves back and forth between two LA's. That is reason of the high loads of network and the ineÆcient usage of the location databases. The static scheme is diÆcult to adapt eÆciently to mobile environment of being a various MT mobility patterns. And also, since the signaling messages concentrate on the HLR, as the number of MT's keeps increasing, the volume of signaling and database access traÆc may increase beyond the capacity of the network. The dynamic location update schemes [1,9] and the caching scheme [6,7] deal with the problems of the current static location management scheme. To support dynamic location update, the HLR/VLR architecture allows the system to page an MT within a cells subset called a Paging Area (PA), which size is less than or equals to that of an LA. The dynamic location update schemes focus on dynamically changing the size of paging areas according to MT mobility patterns and the interval between incoming calls. There are three types of the dynamic schemes : distance-based, movement-based, and time-based. It was pointed out

Title Suppressed Due to Excessive Length

3

that the movement-based method may be the most practical one, since it is eective and can be easily implemented under the framework of current PCS networks. In [9], the authors proposed an algorithm to directly determine the optimal size of location update areas in the movement-based location update scheme. However, the existing performance analysis of the dynamic scheme is diÆcult to use in PCS networks with practical HLR/VLR architectures, because it does not consider the movement of an MT among both PA's (Paging Areas) and LA's (Location Areas) at all. The fact that the size of PA's may change while the size of LA's is xed increases the complexity of the problem. The caching scheme may improve the system performance when some MT's are called several times from the same MSC in a short time interval. Especially when a particular MSC receives a large number of calls to a particular MT that belongs to a dierent MSC, the signaling and database lookup cost involved in delivering the call can be reduced by caching the location information (i.e. a mapping of MT's ID and the identity of the serving VLR) at the calling MSC. The caching scheme may be useful in highly decentralized wireless networks. However, the caching scheme has been proposed for static location update, where the size of LA's is xed. This paper purposes dynamic location management with caching in two level hierarchical database architectures. Another important contribution of the paper is establishing an analytical model for studying the performance of dynamic location management with caching under the practical HLR/VLR architectures. To evaluate the performance of the proposed location management scheme, it is necessary to carefully consider the movement between both PA's and LA's under HLR/VLR architectures since the size of PA's is changed while the size of LA's is xed in dynamic location management. This makes the analysis much more diÆcult. We establish a novel analytical model despite the complexity of modeling. And also, we study the eect of the caching scheme on the size of PA's and on the performance of the dynamic location management scheme. The caching method for call delivery using the cached information of a particular MT may aect the size of PA's and the performance of dynamic location management in contrast with static location management. The proposed analytical model enables us to derive the location update cost and the call delivery cost with dependent in the system parameters, including the mobility and the incoming call arrival pattern of each MT in detail. Our model provides the formulas to solve the minimizing problem of the overall cost that is a trade-o between the location update cost and the call delivery cost. By using the model, we present the performance evaluation and comparison of the proposed scheme and the existing location management schemes using various parameters. This studies show clearly how much the performance improvement can be achieved by the dynamic location management scheme with caching over the existing location management schemes. The rest of this paper is organized as follows. In Section 2, we describes the system for our proposed dynamic movement-based location management with caching. The cost evaluation for location update and call delivery is shown

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Chang Woo Pyo, Jie Li, Hisao Kameda, and Xiaohua Jia

in Section 3. We determine the optimal size of paging areas, and study the performance of our location management in Section 4 and 5. Section 6 concludes our paper.

2 System Description The existing PCS networks, IS-41 and GSM, use two types of network location database (HLR and VLR) organized in a two level data hierarchy. The location databases manage the current location of mobile terminals (MT's) in a service area. The location management functions are achieved by the exchange of signaling messages through a signaling network. Signal System 7 (SS7) [11] is the protocol used for signaling exchange. The SS7 signaling network connects the HLR, the several VLR's and MSC's. It represents an interface between the MT's (via base stations) and the backbone network, making the mobile services widely accessible to the public. As the MT's move around the wireless network area, the stored location data in databases may no longer be accurate. To ensure that calls can be delivered successfully, the databases should be dynamically updated. Location update is initiated by an MT when it reports its current location to the network. PCS networks currently require that the MT performs a location update whenever it enters a new Location Area (LA). Recall that each LA consists of a number of cells where each cell is served by an associated base station. In general, all base stations belonging to the same LA are connected to the same MSC. The process of dynamic location updates and call delivery give an account in detail. 2.1

Dynamic movement-based location update

In the dynamic movement-based location update scheme, the sub-areas in an LA called Paging Areas (PA's) allow to update the cell location of each MT's [9] from the reason that a large LA will result in a decrease in the cost of location update but an increase in the cost of call delivery. A location update occurs by an MT when the number of cell boundary crossing since the last location update equals the pre-de ned system threshold. We de ne the center cell to be the cell where the last location update occurred. We consider PCS networks with a hexagonal cell con guration for the purpose of demonstration. Other cell con guration (e.g., mesh con guration, etc.) can be considered in the same fashion. For the hexagonal cell con guration, cells are hexagonal shaped and each cell has six neighbors. There are many rings of cells in the con guration. The innermost ring (i.e., ring 0) consists of only the center cell. Ring 0 is surrounded by ring 1, which in turn is surrounded by ring 2, and so on. The number of cells in ring i, denoted by g (i), is given by

g(0) = 1; g(i) = 6i; i = 1; 2; 3; : : : : Note that the covering area within a distance d 1 from the center cell may cover more than one LA's. For the hexagonal cell con guratin, the number of

Title Suppressed Due to Excessive Length

5

HLR (H-5)

(H-3)

(H-4)

(H-2)

old VLR/MSC

new VLR/MSC (H-1) (V-1)

A

(V-2)

(V-3)

Move In

LA1 B CC

Fig. 1.

Move Out D

LA1

Dynamic location update in the two-level hierarchical databases

P

cells in a paging area, denoted by P A(d), is upper bounded by dj =01 g (j ). We P note that if the number of cells in an LA is much larger than dj =01 g (j ), the upper bound is a good approximation to P A(d). We use the upper bound as an approximation to P A(d),

P A(d) =

X1 d

i=0

g(i) = 1 + 3d(d 1):

(1)

In general, the size of a PA is less than or equal to its LA. In addition, the size of a PA is dynamically determined according to the relation between the MT mobility and call arrival patterns. As an MT has a high mobility, the size of a PA may be bigger. In contrast of dynamic location update, a PA and an LA have identical size for a system using static location update. In HLR/VLR architectures, the dynamic movement-based scheme updates location as a function of the number of PA's or/and LA's boundary crosses. To simplify the description and analysis of dynamic location update, we de ne HLR location update and VLR location update. Consider the movement of an MT, the MT freely travels in the wireless network, shown in Fig.1. The MT moves in an LA at A cell and freely travels and passes on B cell and C cell in the LA. And then, the MT moves out the LA at D cell. For example, assume that d is pre-de ned as 4 cells. If an MT moves in/moves out of an LA at A cell and D cell, the HLR updates the current serving VLR location of the MT and the current serving VLR updates the current residing cell location of the MT to correctly route the calls for the MT. And also, the previous serving VLR cancels the location information of the MT that is not useful at all. That we call the HLR location update. The precedure of signaling for the HLR location update is following the steps ((H 1) ! (H 2) ! (H 3) ! (H 4) ! (H 5)) shown

6

Chang Woo Pyo, Jie Li, Hisao Kameda, and Xiaohua Jia HLR (C-6) (C-5) called VLR/MSC

(C-7) (C-8)

called VLR/MSC

(C-4) (C-3)

calling VLR/MSC

(C-2) (C-1)

Paging Area LA2

Fig. 2.

LA1

Call delivery with caching in the two-level hierarchical databases

in Fig.1. To ensure that initiated calls from local networks can be delivered successfully, the HLR are dynamically updated the MT's location between LA's. In the other case, when an MT crosses 4 cells boundary in the same LA at B and C cell or crosses the LA boundary at A and D cell, then not the HLR but the VLR only updates the new cell location of the MT with the procedures (V 1) at A cell, (V 2) at B cell, and (V 3) at C cell. That we call the VLR location update. Note when an MT moves in or moves out of an LA, both HLR and VLR for the MT's current location are updated. VLR location update is an extra procedure compared with static location update. However, VLR location update is not complicated and does not signi cantly eect of the system performance since the processing time for VLR location update is short compared with its HLR location update. 2.2

Call delivery with caching under dynamic location update

Two major steps involved in a call delivery are to determine the serving VLR of the called MT, call setup, and to locate the current cell of the called MT, paging. When an MT initiates a call in call delivery without caching for existing PCS networks, IS-41, the system performs the call setup between a calling MSC and a called MSC shown in Fig.2. When an MT initiates a call, the base station forwards the call to the MSC serving the MT and a HLR with (C 1) and (C 4). The HLR sends a location request message to the MSC serving the called MT with (C 5), and then the called MSC sends a Temporary Location Directory Number (TLDN) to the HLR with (C 6). The HLR forwards the TLDN to the calling MSC with (C 7). The calling MSC sets up the call connection to the called MSC using this TLDN with (C 8). The called MSC sends a polling message to all base stations in an LA. In call delivery with caching [6], each time when a call is attempted, the cached information is checked rst at a calling MSC. The MSC maintains the

Title Suppressed Due to Excessive Length

7

cached entries (i.e., MT's ID, updated VLR's ID, etc.,) for the MT's residing a local network. The calling MSC can see the VLR location for the called MT by the cached entries when a call initiates to the MT from that MSC. If the location information for the called MT is already cached and is still served in a same LA since the MT's last location is updated (i.e., cache hit), then the call is directly connected to the called MSC without looking up the called MT's location entry at the HLR with (C 2). In this case, the call delivery time may be signi cantly reduced by a cached information. If the called MT moves to another LA before a call at the calling MSC attempts to the MT, the cached data is not valid at all (i.e., cache miss), then the extra signal procedures like (C 2) and (C 3) are needed to the call setup. The call setup time may be longer than that of the call delivery without caching. A caching scheme has been proposed for the performance improvement of static location update [6]. We consider that the caching scheme also can be used for dynamic location update. In existing PCS networks based on static location management, system determines the cell location of the called MT that polling signals are broadcast to all cells within the residing LA of the called MT. Based on dynamic location management, otherwise, the MSC pages only the coverage area within a pre-de ned distance d, a Paging Area (PA). This dynamic paging may reduce the delay of polling time to locate the MT in contrast to static based paging where the size of PA's is equal to its LA's.

3 Analytical Model The factors attributing to the location update cost and the call delivery cost greatly depend on network topologies, the radio power of base stations and mobile terminals, the capacity of databases, and so on. For the purpose of cost analysis, we consider the costs of location update and call delivery between two calls interval, respectively. 3.1

Dynamic Location Update Cost

To establish the analytical model for dynamic movement-based location update with HLR/VLR database architecture, it is necessary to consider the movement among both PA's and LA's carefully. The size of PA's is changed while the size of LA's is xed in dynamic scheme, which makes the analysis diÆcult to be conducted. In HLR/VLR architecture, dynamic location update is involved in two kinds of location updates shown in previous section 2: HLR location update that updates the location data in the HLR, and VLR location update that updates the location data in the VLR. Let HU be the expected cost for performing HLR location update, and V U be the expected cost for performing VLR location update which account for the wireless and wireline bandwidth utilization and the computational requirements in order to process each location update. Thus, the expected overall location updates cost between two calls,

Chang Woo Pyo, Jie Li, Hisao Kameda, and Xiaohua Jia Next call

tc

Previous call

8

tc,1 tc,2 tc,K

t tm tM0 enter LA0

t MK

tM1 enter LA1 enter LA2 Fig. 3.

enter LA K

enter LA K+1

Timing diagram for (K )

denoted by CDU (d), is expressed by

CDU (d) = nHLR  HU + nV LR

 V U;

(2)

where d is the pre-de ned movement threshold and nHLR and nV LR are the average number of HLR/VLR location updates between two call arrivals, respectively. We derive nHLR and nV LR as follows. The average number of HLR location updates between two call arrivals should be calculated carefully. Consider the timing diagram shown in Fig.3 that an MT moves around K LA's boundary between consecutive two calls. Then, nHLR is expressed by The average number of HLR location updates

nHLR =

1 X

K =1

K(K );

(3)

where (K ) be the probability that there are K LA's boundary crossing between two call arrivals. For the analysis of (K ), we assume that the residence time tMi of an MT p in an LAi (0  i  K ) is an exponentially distributied variable with mean 1m and the call arrival to the MT p is a Poisson process with rate c . Let tc be the call inter-arrival time between the previous call and the next call to the MT p. Without loss of generality, we suppose that the MT p resides in an LA LA0 , when the previous call arrived. After the call, p visits another K LA's, and resides in the ith LA for a period tMi (0  i  K ). Let tm be the interval between the arrival of the previous call and the time when p moves out of LA0 . Let tc;i be the interval between the time that p enters LAi and the time that the next call arrives. Let fc (t) and gm (t) be the density function of tc and tMi , respectively. Note that E [tc ] = 1c and E [tMi ] = 1m . And then,

fc(t) = c e c t ; gm (t) = m e m t : From the memoryless property of the exponential distribution, tc;i and tm have the same exponential distribution as tc and tMi , respectively.

Title Suppressed Due to Excessive Length enter LA1

Next call

tc

Previous call

enter LA0

tc,1 tc,2

9

enter LA K

tc,k t

ts ts0

tsk

ts1

enter cell 0

enter cell 2

enter cell1

Fig. 4.

enter cell k+1

enter cell k

Timing diagram for 1 (k)

The probability (K ) is derived for K  1 as follows. Note that K = 0 means that there is no LA boundary crossing, and K  1 means that there are K LA's boundary crossing. For K  1 :

(K ) = P r[tm + tM1 +


+ tMk = P r[tc > tm ]  =

m

m + c

Y

K

1

i=1

i=1

< tc  tm + tM1 +


+ tMK ]

P r[tc;i > tMi ]

!

Y1 m

K

1

!

 P r[tc;K  tMK ]

c c m K = ( ) : (4) m + c m + c m + c m + c

Substituting (4) into (3), nHLR is calculated as follows.

nHLR =

1 X K =1

K(K ) =

1 X K =1

K(

m K m c )( ) = : m + c m + c c

(5)

Deriving the average number of VLR location updates between two call arrivals is the most diÆcult in the analysis. The calculation of the average number of VLR location updates needs to be concerned with the movement among both PA's and LA's into account. Note that the size of LA is xed while PA is changed following the value of threshold d. Suppose that the MT p resides in the LA LA0 , when the previous call arrived. The average number of VLR location updates per call arrival, denoted by nV LR , can be expressed as follows, The average number of VLR location updates

nV LR =

1 X K =0

nV LRK
(K );

(6)

where nV LRK is the average number of VLRs when the MT p receives the next call in arbitrary LA LAK (K = 0; 1; 2; :::).

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Chang Woo Pyo, Jie Li, Hisao Kameda, and Xiaohua Jia

Let d be the pre-de ned movement threshold and  be the probability of the MT's movement among both PA's and LA's. Then, nV LRK is given by,

nV LRK =

1 X l=1

l

X

(l+1)d

1

j =ld

(j ); K  0:

(7)

We consider the MT p does not move out a PA or crosses some PA's and LA's between two calls. In order to calculate the value of nV LRK , it is necessary to obtain the following four probabilities which can describe the movement of an MT among both PA's and LA's. The four probabilities are given by 1. 1 (k ): The probability that there are k cell boundary crossings within the LA0 , where the previous phone call arrived when the MT receives the next phone call in the same LA. In this case, K = 0, i.e., there is no LA boundary crossing. 2. 2 (k ): The probability that there are k cell boundary crossings within LA0 when the MT enters LA1 . In the case, K > 0, i.e., there are LA boundary crossings. 3. 3 (k ): The probability that there are k cell boundary crossings within LAi during period tMi (1  i  K 1; K > 0). 4. 4 (k ): The probability that there are k cell boundary crossings after entering the last LAK (K > 0) until the next call arrival. We proceed to the calculation of these four probabilities. We assume that the cell residence time follows the Gamma distribution. The Gamma distribution with appropriate parameters can be transformed important distributions such as exponential and Erland distributions. Assume that the probability density function of the cell residence time has Laplace-Stieltjes transform fm (s) with mean 1s and variance  . Let tsi be an independent identically distributed random variable with a general distribution Rm (tsi ), the density function rm (tsi ) and the Laplace-Stieltjes Transform

fm (s) =

Z1

t=0

e

st r

m (t)dt =



s s + s



; =

1

2s

:

(8)

From using time diagram shown in Fig.4, we can obtain the probabilities

1 (k), 2 (k), 3 (k), and 4 (k), respectively.

8  (k) = s [1 f ( )]2 [f ( )]k 1 ; k = 1; 2; 3::: > > < 12 (k) = [1c fm (mm )][c fm (mm )]ck ; k = 1; 2; 3::: k > > : 3 (k) = [1s fm (m )][f2m (m )] ; kk =1 1; 2; 3::: 4 (k) = c [1 fm (c )] [fm (c )]

; k = 1; 2; 3:::

From the probabilities of the MT's movement, the average number of VLR location updates per call arrival is given by,

nV LR =

1 X

K =0

V LRK


(K )

Title Suppressed Due to Excessive Length

9  08  1 < K [fm (m )]d + = X c m K 1 [fm (m )]d @ = : cs [1 fm (c )] 1[fm[f(mc()]dc )]1d ;  m + c ( m + c ) K =0 =

m [fm (m )]d  [f ( )]d 1 + s [1 fm (c )] m c : d c 1 [fm(m )] c 1 [fm (c )]d

11

(9)

To sum up, we have the dynamic location update cost between two calls interval as follows:

0

1

d

m [fm (m )] +  CDU (d) = HU  m + V U  @ sc 1 [fm (m )]d [fm (c )]d 1 A c c [1 fm (c )] 1 [f ( )]d

m c

3.2

(10)

Call delivery Cost with Caching

Call delivery involves in the call setup between the calling MSC and the called MSC and paging the cell location of an MT in an LA. Under dynamic location management, it is necessary to consider the size of paging areas carefully because the size of paging areas is dynamically changed while the size of LA's is xed. The performance of call delivery with dynamic location management with caching may be depended on the size of paging areas. Let Pk denote the probability that the cached location information for the called MT is correct. Thus, Pk is de ned the cache hit ratio. Alternatively, at steady state, Pk denotes the probability that the MT has not moved out an LA since the last call. Consider the previous section in Fig.2 to derive Pk . From our assumption of section 3, the LAi residence times tMi (0  i  K ) and tm for an MT are an exponentially distributied with mean 1m and the call arrival interval tc to the MT is a Poisson process with rate c . Thus, we have

Pk = P r[tc  tm ] =

Z1 Z

tc =tm

tm =0 tc =0

c e c tc m e m tm dtc dtm =

c :(11) c + m

If the cache information is correct, the cost of call setup is smaller than that of without caching. However, if the cache information becomes invalid, the cost of call setup with caching is greater than that of without caching. Let DC be the cost for performing a call setup when the cached location entry is correct. Let SC be the cost for performing a call setup when the cached location entry is invalid. Under dynamic location update, the call delivery cost for the cache hit, denoted by CH (d), and the call delivery cost for the cache invalid, denoted by CB (d), are given by

CH (d) = DC + Cp (d); CB (d) = SC + Cp (d);

(12) (13)

where Cp (d) is the expected paging cost per call arrival. From the equation (1), the expected paging cost per call is given by

Cp (d) = P  P A(d) = P  (1 + 3d(d 1));

(14)

12

Chang Woo Pyo, Jie Li, Hisao Kameda, and Xiaohua Jia

where P is the cost for performing the polling a cell. The location information of an MT has to be cached at an MSC only if the MT changes its LA less frequently than it receives calls from that MSC. In order to determine which MT's location information to cache, we use the system parameters called Local-Call-to-Mobility Ratio (LCMR) in caching the location information. LCMR is the ratio between the number of calls originating from an MSC to the number of times the MT changes its service area as seen by that MSC. That is the small value of LCMR (= mc ) means the high mobility that the MT has, and vice versa. The location information of an MT is cached at an MSC, if the LCMR maintained for the MT at the MSC is larger than a threshold derived from the link and database access cost of the network. A call delivery cost saving with the caching scheme is achieved by caching location information if

Pk CH (d) + (1 Pk )(CH (d) + CB (d)) < CB (d);

(15)

Simpli ng the above equation (15),

Pk >

CH (d) : CB (d)

(16)

If the cache hit ratio, Pk , is larger than a cost saving, CCHB ((dd)) , the system will cache the MT's location information. Consequently, the mean call delivery cost denoted by CDC (d) can be expressed.

CDC (d) = Pk CH (d) + (1 Pk )(CH (d) + CB (d)):

(17)

To sum up, the expected total cost from (9) and (17) denoted by T CDC (d) is given by

T CDC (d) = CDU (d) + CDC (d)

1 0 m [fm (m )]d +  d = HU  m + V U  @ c 1 [fm (m )] [fm (c )]d 1 A c

s

fm (c )] 1 [fm (c )]d + Pk CH (d) + (1 Pk )(CH (d) + CB (d)); c [1

(18)

where d ( 1).

4 Optimal Paging Area In this paper, we obtain the optimal paging area through minimizing the total cost. The total cost has a dierent value through the size of a paging area determined by the threshold d. Note d is a positive integer. We have Minimize[T CDC (d)]:

(19)

Title Suppressed Due to Excessive Length 10

10

Non-caching DC= 1 DC=20 DC=50

VU= 5 VU=10 VU=15

8

Optimal threshold

(d)

(d)

8

Optimal threshold

13

6

4

4

2

2

0 0.1

6

1

10

Local-Call-to-Mobility-Ratio (LCMR)

Fig. 5.

0 0.1

1

10

Local-Call-to-Mobility-Ratio (LCMR)

Optimal threshold d with P = 1

If we take d as a real number, functions CDU (d) and CDC (d) are twice differentiable. To derive the rst derivative and the second derivatives of CDU (d), and arrange them. We have

0 (d) < 0; C 0 (d) > 0; C 00 (d) > 0; C 00 (d) > 0: CDU DC DU DC That is, CDU is a decreasing and convex function and CDC is an increasing and convex function. Since the function T CDC (d) is convex, the value of d is the unique solution to minimize total cost, T CDC (d), if it satis es the following dierential condition,

0 (d) = C 0 (d) + C 0 (d) = 0: T CDC DU DC

(20)

To determine the optimal threshold d, we use the binary search algorithm since it has simple and clear structure. For the purpose of analysis, we use the system parameter called Local-Call-to-Mobility-Ratio (LCMR), which is the ratio between the number of calls originating from an MSC and the number of location updates of a called MT as seen by that MSC. That is the small value of LCMR (= mc ) means the high mobility that the MT has, and vice versa. Note that the mean LA residence time (= 1m ) shall be larger than the mean cell residence time (= 1s ). It is known that the size of paging areas is greatly aected by VU and P. The eect of VLR location update cost VU and polling cost P on the threshold d is shown in Fig.5.(A). Three VU values, 5, 10, and 15 are considered with P = 1 and other parameters are set to HU = 200, DC = 20 and SC = 200. As the LCMR increases from 0.1 to 30, the value of d decreases like a stairway. Intuitively, the high LCMR will cause a large threshold d, that means increasing

14

Chang Woo Pyo, Jie Li, Hisao Kameda, and Xiaohua Jia

the size of a paging area (PA) according to decrease the value of LCMR. It is also shown that an increase in VU may cause an increase in the size of a paging area. Fig.5.(B) shows the eect of caching in the dynamic movement-based caching scheme on the size of a paging area. Three DC values, 1, 20, and 50 are considered with SC = 200 and other parameters are set to P = 1, V U = 10, and HU = 200. The optimal paging area with caching is smaller than that of without caching. A decrease in the value of DC causes an smaller the value of d between LCMR = 0:3 and LCMR = 5. It means that the caching scheme has a most eect on the dynamic movement-based location updates between LCMR = 0:3 and LCMR = 5. Otherwise, the higher LCMR (LCMR  0:3) or the lower LCMR (LCMR  5), the caching scheme has a limited eect on the size of a paging area.

5 Performance study We compare the total cost between dynamic location management with caching and dynamic location management without caching to static location management without caching shown in Fig.6. The total cost of static location update without caching denoted by T CSNC (r) is given by

T CSNC (r) = HU 

m + SC + P  (1 + 3r(r 1)); c

(21)

where r is the radius of LA from a center cell to the LA's boundary. The total cost of dynamic movement-based location update without caching denoted by T CDNC (d) is given by

8 9 m [fm (m )]d + < =  T CDNC (d) = HU  m + V U  sc 1 [fm (m )]d [fm (c )]d 1 : c [1 fm(c )] 1 [fm (c )]d ; c +SC + P  (1 + 3d(d

1))

(22)

In HLR/VLR architectures, the HLR location update cost is larger than the VLR location update cost (i.e., HU  V U ), the call setup cost for the correct cached data is smaller than the call setup cost for the invalid cached data. (i.e., SC  DC ), and the polling cost P is the smallest. For the purpose of demonstration, we set to the parameters which are the value of P = 1, V U = 10, HU = 200. DC = 20 for caching scheme and SC = 200 for non-caching scheme are set. In the static location management, the radius of LA (r) is set to 15 cells. From Fig.6.(A), the total cost of the dynamic location update schemes is smaller than the static location update schemes. And the dynamic movementbased caching scheme has the best cost performance within the location management schemes. Specially, if the value of LCMR is bigger, this means a mobile user has a low mobility, the cached information has a more eect on the dynamic location management scheme, therefore the total cost can be greatly reduced by

Title Suppressed Due to Excessive Length 3000

2500

Total Cost

2000

1500

1000

500

0 0.1

1.4

(CDC(d) + CDU(d)) (CDNC(d) + CDU(d)) (CSNC(r) + CSU(r))

Dynamic_Caching/Static_NoCaching

caching-dynamic non-caching-dynamic non-caching-static

1

10

Local-Call-to-Mobility-Ratio (LCMR)

Fig. 6.

15

1.2 LA=5 1 LA=10 0.8 LA=15 0.6 LA=20 0.4 LA=25 0.2

0 0.1

1

10

Local-Call-to-Mobility-Ratio (LCMR)

Cost comparison of location management schemes

dynamic caching caching. Fig.6.(B) shows the ratio of total cost static noncaching changing the radius of LA from r = 5 to r = 25. As the served LA by a VLR is bigger and bigger and the MT has a low mobility, the proposed scheme is a better performance compared with the static location management without caching scheme. There is the reason that the paging area can be dynamically changed by the MT mobility and incoming calls patterns. This make a cost balance between location update cost and call delivery cost.

6 Conclusion In this paper, we propose dynamic location management scheme with caching in PCS networks with practical HLR/VLR architectures. To evaluate the performance of the scheme, we establish an analytical model. The model successfully handles the additional complexity problems created by dynamically changing the paging area (PA) while the location area (LA) is xed. The model is applied to obtain the formulas to evaluate the costs of location update and call delivery with simplicity, and the formulas are easily used in the performance evaluation. By using the analytical model and formulas, we studied the eects of changing the parameters of location update costs, call delivery costs, paging costs, LocalCall-to-Mobility Ratio (LCMR) on the system performance. We determine the optimal size of a paging area, which minimizes the overall cost that is sum of location update cost and call delivery cost of the interval between two calls. The size of paging areas can be dynamically changed by an MT mobility pattern and the incoming calls of the MT. That size decreases as the local-call-to mobility ratio increases, and vice versa. The dynamic location management scheme has the high performance of location management in mobile networks with a various MT

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Chang Woo Pyo, Jie Li, Hisao Kameda, and Xiaohua Jia

mobility pattern. The caching method signi cantly reduces the network overload by frequent incoming calls from the remote network, and has a better eect on dynamic location management compared with static location management if the MT's have the low mobility pattern. The dynamic location management with caching method has a more higher performance in case that a network has a large service area and the mobile terminals have a low mobility pattern than the static location management and the dynamic location management without caching method.

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