Eliminating overflow for large-scale mobility databases in cellular telephone networks

Eliminating Overflow for Large-Scale Mobility

Databases in Cellular Telephone Networks

Yi-Bing Lin, Senior Member, IEEE

AbstractÐIn a cellular phone system, mobility databases called visitor location registers (VLRs) are used to temporarily hold the subscription information of the roaming users who visit the service area of the VLR. When the users leave the VLR area, the corresponding records in the VLR are deleted. Due to user mobility, the capacity of the VLR may not be large enough to hold information for all visitors in the VLR area at some time periods. This issue is called VLR overflow. This paper describes a record replacement policy to allow mobile users to receive services in the VLR overflow situation. We utilize analytic modeling to investigate the performance of the replacement policy. The study indicates that our approach effectively eliminates the VLR overflow problem with insignificant extra overhead.

Index TermsÐCellular telephone network, database overflow, home location register, large-scale database, visitor location register.

æ

1 I

NTRODUCTION

A

cellular telephone network provides telecommunica-tion services (telephone connectelecommunica-tions, data transmis-sion, multimedia, and Internet services) to roaming users who move around the service areas covered by the network. Through roaming agreement, cellular telephone networks belonging to different operators can interwork to offer services to users who move around various cellular telephone networks. For example, a cellular service sub-scriber of FarEasTone in Taiwan can use his/her handset to make/receive phone calls in England through Vodafone/ AirTouch, Cellnet (British Telecom), or other cellular operators. Cellular telephone networks use a distributed database architecture to support roaming of users. In this architecture, there are two types of databases: Home Location Register (HLR) and Visitor Location Register (VLR). When a user subscribes to the services of a cellular operator (called the home system of the user), a record is created in the operator's HLR. The record stores services (such as call waiting, call forwarding, voice mailbox, and so on) subscribed to by the user. Furthermore, the location information of the user is also kept in the record (to be elaborated). The typical size of an HLR in Taiwan is around a million records.

When the mobile user visits a cellular network other than the home system, a temporary record for the mobile user is created in the VLR of the visited system. The VLR temporarily stores subscription information (replicated from the HLR) for the visiting subscribers so that the visited system can provide services. In other words, the VLR is the location register other than the HLR used to retrieve information for handling calls to or from a visiting mobile user. The capacity of a typical VLR in Taiwan is

around 250,000-500,000 records. To track the location of a mobile phone, the mobile phone automatically reports its location (to both the visited VLR and the HLR) when it moves to a new location. This procedure is called registration and will be elaborated on in Section 2. To deliver a call to a mobile phone, the network retrieves the location information stored in the HLR and the VLR and the network sets up the trunk based on this location information.

Many studies have focused on normal mobile registra-tion and call setup procedures [2], [6], [1], [10] and failure restoration [4], [5]. Unlike the previous work, this paper studies the VLR database overflow problem. A VLR database overflows if the number of visiting customers exceeds the capacity of the VLR database. In this case, the incoming visitors cannot register using the standard registration procedure described in Section 2 and, thus, cannot receive cellular phone services. Note that HLR does not have the database overflow problem. The number of subscriber records in the HLR is known for an operator, which is the number of customers subscribing to the services of that specific operator. Thus, the HLR database capacity can be scaled and database overflow never occurs. On the other hand, the number of records in a VLR changes dynamically. This size increases when registrations occur and decreases when deregistrations occur. It is possible that many users enter a VLR in a short period. If the number of users in a VLR area is larger than the capacity of that VLR database, then the VLR database overflows and the incoming users cannot successfully perform registration. In this case, these users will not be able to receive services and are referred to as the overflow users.

In [9], we proposed an approach to resolve the VLR overflow issue. In our approach, when VLR overflows, the visited system still can provide services to incoming users. Our approach takes advantage of the distributed database structure of cellular phone network where the subscription information of a user is duplicated in both HLR and VLR. For overflow users (i.e., the users who do not have records

. The author is with the Department of Computer Science and Information Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC. E-mail: [email protected].

Manuscript received 5 May 2000; revised 29 Sept. 2000; accepted 8 Dec. 2000. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number 112059.

in the VLR), call setup can be complete by using the information stored in the HLR. However, extra cost is required in the call setup procedure of an overflow user (to be described in the next section). Thus, it is important to reduce the possibility of overflow call setup. This paper proposes a mechanism to achieve this goal and investigate the performance of this mechanism by an analytic model. The notation used in this paper is listed in the Appendix.

2 A S

OLUTION FOR

VLR O

VERFLOW

In this section, we first describe the standard registration, call origination, and call delivery procedures and then show how the VLR overflow mechanism is integrated into these procedures. The registration procedure is illustrated in Fig. 1 and is described in the following steps:

Step 1.1. Suppose that the home system of a mobile user is in Taiwan. When a mobile user moves from one visited system (e.g., Hong Kong) to another (e.g., London), the user's mobile phone automatically registers in the VLR in London. Note that the radio base stations connected to the mobile switch are partitioned into several location areas. To simplify our discussion, we assume that there is one location area per mobile switch. In registration, the addresses of the mobile switch and location area where the mobile phone resides are sent to the VLR.

Step 1.2. The new VLR then informs the mobile user's HLR of its current location, i.e., the address of the new VLR.

The HLR sends an acknowledgment, which includes the user's profile, to the new VLR.

Step 1.3. The new VLR then creates a record for the visiting user to store the profile received from the HLR. Then, the VLR informs the mobile phone of the successful registration.

Step 1.4. After Step 1.2, the HLR also sends a deregistration message to cancel the obsolete record of the mobile phone in the old VLR at Hong Kong. The old VLR acknowledges the deregistration.

To originate a call, the following steps are executed: Step 2.1. The mobile phone first contacts the mobile switch

in the visited cellular network.

Step 2.2. The call request is forwarded to the VLR for approval. For example, the user profile may indicate that the user is not allowed to make international calls. Thus, any attempt to make an international call will be rejected by the VLR.

Step 2.3. If the call is accepted, the mobile switch sets up the call to the called party following the standard PSTN (public switched telephone network) call setup procedure.

The call delivery (or call termination) procedure to a mobile phone is illustrated in Fig. 3 and it is discussed in the following steps:

Fig. 1. The mobile user registration process.

Step 3.1. If someone attempts to call a mobile subscriber, the call is forwarded to a gateway mobile switch.

Step 3.2. The gateway mobile switch queries the HLR to find the current VLR of the mobile phone. Then, the HLR queries the VLR of the mobile phone to obtain a routable address.

Step 3.3. The VLR searches the record for the called mobile subscriber. Based on the location information, the VLR creates the routable address and returns it to the gateway mobile switch through the HLR.

Step 3.4. Based on the routable address, a trunk (voice circuit) is then set up from the originating switch to the serving mobile switch. The serving mobile switch queries the VLR to find the location area of the mobile phone. The radio base stations in the location area then page the mobile phone and the call path to the mobile phone is eventually established.

Details of mobility management and call setup procedures can be found in [3], [11], [12].

As mentioned in the previous section, VLR overflow may occur in the existing mobile networks. To resolve this problem, one may overdimension the VLR capacity by, say, doubling the storage size. However, this solution may not be an appropriate option because the cost of memory is not the only concern in VLR database planning. The dominated costs also include replication for fault tolerance, database management, and others. These costs significantly increase as the size of VLR increases. In [9], we have proposed a solution to resolve the VLR overflow problem without increasing the VLR size. The solution modifies mobile registration, call origination, and call delivery procedures as follows:

Overflow Registration. Suppose that the mobile phone of user u initiates the registration procedure. At Step 1.3, if the VLR is full, then a record is selected for replacement. That is, an existing record in the VLR is deleted and the reclaimed storage is used to hold the record of u. In this case, the user of the replaced record becomes an overflow user. Alternatively, user u may be considered as the overflow user and no record replacement occurs. In this case, Steps 1.2-1.4 are executed as before except that, in Step 1.3, no record for u is created. In Step 1.4, if u is an overflow user at the old VLR, then no record cancellation occurs at that VLR.

Overflow Mobile Call Origination. When an overflow user u attempts to make an outgoing call, the VLR notices that

no record exists for u at Step 2.2. The VLR will request u to perform an overflow registration operation to create a record for u. (In this registration, u cannot be selected as the overflow user.) Then, normal call origination procedure is executed to set up the call.

Overflow Mobile Call Delivery. For an incoming call to an overflow user u, the VLR cannot find the record for u and, thus, cannot generate a routable address at Step 3.3. In this case, HLR will generate a routable address based on its knowledge of u's location [3]. Through a replace-ment at Step 3.3, the VLR creates a record for u to store subscription data as well as location information. For an overflow user, the costs of executing Steps 2.2 and 3.3 are higher than a normal mobile user (that is, an extra registration operation is required). Therefore, it is desirable to reduce these extra overheads. One possibility is to select an ªinactiveº record for replacement so that the correspond-ing overflow user does not have any call activity before the user leaves the VLR area. In [9], we consider the random replacement policy, where every record in the VLR is selected for replacement with the same probability. The perfor-mance of the random replacement policy is acceptable in a homogeneous environment where the mobile users have relatively low call activities. In reality, call activities of visiting users in a VLR area may vary significantly. If a record with low call activity is selected for replacement at Step 1.3, then it is more likely that the overflow user leaves the VLR without creating any call activity. To achieve this goal, we propose the inactive replacement policy that attempts to select records with low call activities for replacement. A period called inactive threshold is utilized to determine if a user is not active. If a user does not have any call activity during the inactive threshold, then he/she is considered inactive and the VLR record can be selected for replace-ment. The replacement policy is described as follows: The Inactive Replacement Algorithm.

Overflow Registration. At Step 1.3, let R be the set of records that do not have call activities for periods longer than the inactive threshold. If R ˆ ; (all records are active), then the user who initiates the registration procedure is considered the overflow user. If R 6ˆ ; (some records are inactive), then randomly select a record from R for replacement.

Overflow Call Setup. In call origination or call delivery, if R 6ˆ ;, then a record in R is randomly selected for replacement. On the other hand, if R ˆ ;, then a record in VLR is randomly selected for replacement.

3 A

NALYTIC

M

ODEL FOR

R

ECORD

R

EPLACEMENT

This section investigates the performance of the replace-ment policies. Since a VLR typically consists of 250,000-500,000 records, it is very difficult, if not impossible, to model a VLR using the simulation approach. Thus, we propose analytic approaches to model the inactive replace-ment policy and the random replacereplace-ment policy. We first make the following assumption: Suppose that there are K classes of mobile users where the portion of class i users

in the VLR is i(1‡ 2‡ . . . ‡ K ˆ 1) and the call arrival

rate to a class i user is c;i.

3.1 The Inactive Replacement Policy

In the inactive replacement (IR) policy, a VLR record is considered inactive if no call for the corresponding mobile user arrives in a period tT (the inactive threshold). Let pIR

(p

IR) be the probability that, after an overflow registration

(call setup), the overflow user does not have any call activities before he/she leaves the VLR. Probabilities pIR

and p

IR are derived as follows:

Consider the timing diagram in Fig. 4. Suppose that a mobile user uiof class i enters the VLR at time t0and leaves

it at time t5. Then, the VLR residence time is m;iˆ t5ÿ t0,

which has a general distribution with density function fm;i…m;i†, expected value 1=m;i, and Laplace Transform

f m;i…s† ˆ

Z ₁

m;iˆ0

eÿstm;i_f

m;i…m;i†dm;i:

Suppose that a record replacement occurrs at time t3,

where t0< t3< t5. Since the replacement request stream is a

Poisson process, the request occurring at t3 is a random

observer of the period m;i. Then, tm;iˆ t3ÿ t0 has the

density function rm;i…tm;i† and Laplace Transform rm;i…s†.

From the excess life theorem [13], rm;i…tm;i† ˆ m;i

Z ₁ tˆtm;i fm;i…t†dt and r m;i…s† ˆ m;i_s   1 ÿ f m;i…s†  : …1† Suppose that the last call to ui before t3 occurs at time t1

(where t1< t0 or t0< t1< t3) and the first call to ui after t3

occurs at time t4 (where t4> t5 or t3< t4< t5). Since call

arrivals to ui are a Poisson process, c;iˆ t4ÿ t1 is

exponentially distributed with rate c;i. Again, since the

replacement request occurring at t3is a random observer of

the period c;i, the excess life tc;iˆ t3ÿ t1 has the density

function (from (1))

rc;i…tc;i† ˆ c;ieÿc;itc;i:

When the replacement request arrives at time t3, the VLR

record for ui is considered ªinactiveº if

tT  t3ÿ max…t0; t1†

(in Fig. 4, max…t0; t1† ˆ t1, tT ˆ t2ÿ t1, and t2< t3). Let p1;i

be the probability that a class i record is inactive. Then, tm;i> tT, tc;i> tT, and the corresponding probability is

expressed as

p1;iˆ Pr‰tm;i> tT and tc;i> tTŠ: …2†

We consider two cases:

Case 1. tTˆ 1=T is a fixed period and

Case 2. tT is exponentially distributed with mean 1=T.

For constant tT, p1;iˆ Z ₁ tm;iˆ1=T Z ₁ tc;iˆ1=T

rc;i…tc;i†rm;i…tm;i†dtc;idtm;i

ˆ eÿc;i=T‰1 ÿ R m;i…1=T†Š; …3† where Rm;i…1=T† ˆ Z _1=T tm;iˆ0

rm;i…tm;i†dtm;i:

For exponentially distributed tT,

p1;iˆ Z ₁ tm;iˆ0 Z _tm;i tTˆ0 Z ₁ tc;iˆtT

TeÿTtTc;ieÿc;itc;irm;i…tm;i†

dtc;idtTdtm;i ˆ Z ₁ tm;iˆ0 Z _tm;i tTˆ0

Teÿ…T‡c;i†tTrm;i…tm;i†dtTdtm;i

ˆ _ T T‡ c;i   Z 1 tm;iˆ0 1 ÿ eÿ…T‡c;i†tm;i h i

rm;i…tm;i†dtm;i

ˆ T T‡ c;i   1 ÿ r m;i…T‡ c;i†  : …4†

From (4) and (1), p1;iˆ T

…T‡ c;i†2

" #

T‡ c;iÿ m;i‰1 ÿ fm;i …T‡ c;i†Š

n o

: …5† Consider arbitrary N records in the VLR. From (3) and (5), the probability that these N records are ªactiveº can be expressed as 1…N† ˆ XN k1ˆ0 X Nÿk1 k2ˆ0


Nÿk1ÿkX2ÿ


ÿkKÿ2 kKÿ1ˆ0 N k1k2


kKÿ1    Kÿ1Y iˆ1 i…1 ÿ p1;i†  ki ( ) ‰K…1 ÿ p1;K†ŠNÿk1ÿk2ÿ


ÿkKÿ1 …6† ˆ XK kˆ1 k…1 ÿ p1;k† " #N : …7†

The product term in (6) represents the probability that there are kiactive records for class i users andPKiˆ1kiˆ N. Note

that 1…1† is the probability that a record in the VLR is

active. Suppose that the size of the VLR is M (i.e., the VLR can hold at most M records). Let p2;0be the probability that

when a replacement request arrives, no inactive record is found in the VLR. Then, from (7),

p2;0ˆ 1…M† ˆ XK kˆ1 k…1 ÿ p1;k† " #_M ˆ  1…1† _M : …8† For a large-scale VLR (e.g., M ˆ 250; 000), even if an arbitrary record in the VLR is very likely to be active (e.g., 1…1† ˆ 0:999), the inactive replacement policy can almost

always find an inactive record (i.e., p2;0ˆ …0:999†250;000' 0).

Again, consider N arbitrary records in the VLR, which exclude class i records. Let 2…N; i† be the probability that

all these records are inactive, then 2…N; i† ˆ XN k1ˆ0 X Nÿk1 k2ˆ0


Nÿk1ÿkX2ÿ


ÿkiÿ2 kiÿ1ˆ0 X Nÿk1ÿk2ÿ


ÿkiÿ1 ki‡1ˆ0


X Nÿk1ÿk2ÿ


ÿkiÿ1ÿki‡1ÿ


ÿkKÿ2 kKÿ1ˆ0  N k1k2


kiÿ1ki‡1


kKÿ1    Y 1jKÿ1;j6ˆi jp1;j ÿ
kj " # …Kp1;K†Nÿk1ÿk2ÿ


ÿkiÿ1ÿki‡1ÿ


ÿkKÿ1 ˆ X 1kK;k6ˆi kp1;k !_N : …9† Let p2;i be the probability that, when a replacement

request arrives, a class i record is selected for replacement. Then, from (7) and (9),

p2;iˆ XM mˆ1 M m   Xm jˆ1 m j   j m   …ip1;i†j2…m ÿ j; i† " # 1…M ÿ m†: …10† In (10), the term 1…M ÿ m† is the probability that there are

M ÿ m active records in the VLR. The term …ip1;i†j2…m ÿ

j; i† is the probability that there are m inactive records in the VLR and j of them are for class i users. The term j=m is the probability that a class i inactive record is selected for replacement. Equation (10) is simplified as follows:

p2;iˆ XM mˆ1 M m   …ip1;i† X mÿ1 jÿ1ˆ0 m ÿ 1 j ÿ 1   …ip1;i†jÿ1 " X 1kK;k6ˆi kp1;k !_{mÿ1ÿ…jÿ1†}3 51…M ÿ m† ˆ P_Kip1;i jˆ1jp1;j ! XM mˆ1 M m   XK kˆ1 kp1;k !m ( XK kˆ1 k…1 ÿ p1;k† " #_Mÿm9₌ ; ˆ P_Kip1;i jˆ1jp1;j ! XM mˆ0 M m   XK kˆ1 kp1;k !m ( XK kˆ1 k…1 ÿ p1;k† " #_Mÿm ÿ XK kˆ1 k…1 ÿ p1;k† " #_M9₌ ; ˆ P_Kip1;i jˆ1jp1;j ! …1 ÿ p2;0†: …11† One of three situations occurs when a record replacement request arrives.

Situation 1. During an overflow registration (Step 1.3) or call setup (Steps 2.2 and 3.3), a record for an inactive class i user ui is selected for replacement at t3 in Fig. 4

and uiwill not make or receive calls before he/she leaves

the VLR (i.e., t

c;i> tm;i in Fig. 4). From the excess life

theorem [13], the distribution for t

m;iis the same as that

for tm;iand the distribution for tc;i is the same as that for

tc;i. Let p3;i be the probability that, after the record of a

class i user is replaced, the user does not have any call activity before he/she leaves the VLR. Thus, p3;iˆ

Pr‰t

m;i< tc;iŠ is expressed as

p3;iˆ Pr‰tm;i< tc;iŠ

ˆ Z ₁ t m;iˆ0 Z ₁ t c;iˆtm;i

rc;i…tc;i†rm;i…tm;i†dtc;idtm;i

ˆ m;i c;i   1 ÿ f m;i…c;i†  : …12†

Let pIRjrbe the probability that Situation 1 is true and the

overflow user (i.e., the user of the replaced record) does not have any call activity before the person leaves the VLR. From (11) and (12), pIRjris expressed as:

pIRjrˆ XK iˆ1 p3;ip2;i ˆXK iˆ1 im;ip1;i …PK jˆ1jp1;j†c;i " _# 1 ÿ f m;i…c;i†  …1 ÿ p2;0†:

Situation 2. During an overflow registration initiated by a user uiof class i, no inactive record is found in Step 1.3.

In this situation, ui is considered as the overflow user

and no replacement occurs. Let pIRjnr be the probability

that Situation 2 is true and ui does not have any call

activity before the person leaves the VLR (i.e., m;i< c;i

in Fig. 5). We have

Pr‰m;i< tc;iŠ ˆ fm;i …c;i† …14†

and pIRjnr is expressed as

pIRjnrˆ XK iˆ1 ip2;0Pr‰m;i< tc;iŠ ˆ XK iˆ1 ip2;0fm;i …c;i†: …15†

Situation 3. During an overflow call setup, no inactive record is found at Steps 2.2 or 3.3. In this situation, a record in the VLR is randomly selected for replacement. Let p

IRjnr be the probability that Situation 3 is true and

the user of the replaced record does not have any call

activity before the person leaves the VLR. In Fig. 4, Situation 3 is true if for a class i record selected for replacement (i.e., tc;i< tT) and tc;i> tm;i. From (12),

p IRjnrˆ XK iˆ1 ip2;0p3;iˆ XK iˆ1 ip2;0m;i c;i   1 ÿ f m;i…c;i†  : …16†

Fig. 5. Timing diagram (II).

Fig. 6. Timing diagram (III).

In the inactive replacement policy, the probabilities pIR

(p

IR) that, after an overflow registration (call setup), the

overflow user does not have any call activity before the person leaves the VLR are expressed as

pIRˆ pIRjr‡ pIRjnr and pIRˆ pIRjr‡ pIRjnr: …17†

3.2 Random Replacement Policy

In the random replacement (RR) policy, let pRR be the

probability that, after an overflow registration or call setup, the user of the replaced record does not have any call activity before he/she leaves the VLR. The probability pRR

is derived as follows: During an overflow registration or call

Fig. 8. Effects of T on pIRjr(c;1ˆ 0:01m; c;2ˆ 500m; 1ˆ 0:01).

setup, a record of a class i user ui is randomly selected for

replacement with probability i. Consider the timing

diagram in Fig. 6. Suppose that ui's record is replaced at

time t3. If tm;i< tc;i, then ui does not have any call activity

before he/she leaves the VLR. From the excess life theorem, t

m;i has the density function rm;i…tm;i† (see (1)) and tc;i has

the same density function as c;i. This situation is exactly the

same as that in Fig. 4 and Pr‰t

c;i> tm;iŠ ˆ p3;i as given in

(12). Thus, pRRˆ XK iˆ1 ip3;i ˆXK iˆ1 im;i c;i   1 ÿ f m;i…c;i†  : …18†

4 N

UMERICAL

E

XAMPLES

Based on the derivations in Section 3, we discuss how tT

(T), c;i, i, and fm;iaffect the performance of the overflow

record replacement policies. To simplify our discussion, we consider two classes of users. Class 1 users have low call activities and class 2 users have high call activities (i.e., c;2>> c;1). We also assume that VLR residence time

distributions for both classes are the same, which have a Gamma distribution with mean 1=m, variance vm, and the

Laplace transform f m…s† ˆ _{1 ‡ }1 mvms   1 2mvm : …19†

In this case, the VLR residence times for both class 1 and 2 users are represented by the random variable m and the

excess lives of the VLR residence times are represented by the random variable tm(with the Laplace Transform rm…s†).

The Gamma distribution is selected because it can be used to approximate many other distributions as well as

measured data [7], [8]. Note that, for a Gamma distribution, we observe the following:

Fact 1. f

m…s† increases as vmincreases.

Fact 2. r

m…s† increases as vmdecreases.

Fact 1 states that, for a Gamma distributed m, when vm

increases, more short mare observed (from (14) and (19))

and it is more likely that m< tc;i. Fact 2 is derived from (1)

and Fact 1, which implies that, as vmincreases, more long tm

are observed.

Effects of tT distribution. Based on (3) and (5), Fig. 7 shows

the effects of the tT distribution on the probability p1;i

that a class i record is selected for replacement. In this figure, T ˆ 0:001mand vmˆ 100=2m.

The figure indicates that p1;i is a decreasing function

of c;i. That is, as the call activities for a class i user

increase, this user is unlikely to be selected for replace-ment. For c;i m (i.e., class i users have high call

activities), both the exponential and the fixed inactive

thresholds yield p1;i' 0. For c;i< 0:01m (i.e., class i

users have low call activities), the p1;i value yielded by

the exponential threshold is larger than that yielded by the fixed threshold. Thus, compared with the fixed tT,

the exponential tT will select inactive users with a larger

probability. With fixed tT, it is more likely that no user

can be selected for replacement because the yielded p1;iis

small. An implication of this figure is that, for both fixed and exponential tT, the same trend of the p1;icurves (see

Fig. 7) is observed. Thus, we can consolidate our efforts on using the exponential tT for our study and the general

conclusions are also valid for fixed tT.

Effects of the mean inactive threshold 1=T. Figs. 8, 9, 10,

and 11 show the effects of the expected length of inactive threshold tT or 1=T. Fig. 8 shows the probability pIRjr

that, when a record is selected for replacement, the corresponding user does not have any call activity before the person leaves the VLR (in the inactive replacement policy). The figure indicates that pIRjrincreases and then

decreases as T increases. This phenomenon is explained

as follows: When tT is long (T< 10ÿ7m), it is difficult

to find any inactive record. As tT decreases

(10ÿ7_{m< T < m), more inactive records for users}

with low call activities can be found, but it is still unlikely to find inactive records with high call activities. Thus, for T < m, pIRjrincreases as T increases. When

tT is small (T > m), more records for users with high

call activities are considered inactive and these records are likely to be selected for replacement. This phenom-enon becomes more significant as tTdecreases (i.e., as T

increases). Since these replaced users (with high call activities) are likely to have call activities before they leave the VLR, pIRjr decreases as T increases. Fig. 8 also

indicates that pIRjr increases as vm increases. As vm

increases, it is more likely to find inactive users with long VLR residence times and a larger pIRjris expected. From

(4) and Fact 2, Fig. 9 indicates that, when vm increases,

p1;1increases. Consequently, we have (from (11)).

Fact 3. p2;1increases as vmincreases.

On the other hand, from (12) and Fact 1, we observe that

Fact 4. p3;i decreases as vmincreases.

Thus, from (13) and Facts 3 and 4, the net effect is that pIRjr decreases as vmdecreases.

Fig. 10a plots pIRjnr as a function of T and vm. The

figure indicates that pIRjnris a decreasing function of T.

As T increases, it is more likely that, in the overflow

situation, the registration of an incoming user can find an inactive record for replacement. In this case, p2;0

decreases and pIRjnrdecreases (see (15)). We also observe

that, for T < 10ÿ3m, pIRjnr is an increasing function of

vm. In this case, p1;i (and, thus, p2;i) is large and is not

sensitive to vm. As vmincreases, fm…c;i† increases (Fact 1)

and pIRjnr increases. When T is large, p1;i with smaller

vmdrops much faster than that with larger vm(see Fact 2

and (4)).

Fig. 10b shows the effects of T on pIR. From (17), the

shapes of the pIRcurves are determined by the shapes of

the pIRjr and pIRjnr curves. The resulting pIR curves

increase and then decrease as T increases.

Fig. 11a plots p

IRjnras a function of T and vm. In this

case, results similar to Fig. 10a are observed. One exception is that p

IRjnr increases as vm decreases due to

Fact 2 and (16).

From Fig. 10b and Fig. 11b, the Tvalues that yield the

best pIR and pIR performance are in the range

‰10ÿ3_{m; 10}ÿ1_{mŠ. In this T range, pIR' p}

IR. Thus, in

the remainder of this section, we consider T ˆ 10ÿ3m

and only illustrate the pIR performance.

Effects of c;1. From Fact 3, Fig. 12a indicates that p2;1

increases as vmincreases.

Fig. 12b indicates that p3;1decreases as c;1increases.

That is, for a replaced record, when the intercall arrival times to a user become shorter, it is more likely that the next call arrives before the user leaves the VLR. Thus, p3;1

decreases. The figure also indicates that p3;1 increases as

vmdecreases. This phenomenon is due to Fact 4.

Based on (17), Fig. 13 plots pIR and pRR curves. It is

clear that, when c;1increases, the inactive users become

more active. If these inactive users are selected for replacement, it is more likely that they will have call activities before they leave the VLR. Thus, both pIR and

pRRdecrease as c;1 increases.

We note that both p2;2ˆ 1 ÿ p2;0ÿ p2;1 and p3;2 are

small (see Fig. 12). Thus, pIR' p2;1p3;1. Since p2;1 is an

increasing function and p3;1 is a decreasing function

against vm, the resulting pIR curve increases and then

decreases as vmincreases. Similarly, pRR' 1p3;1, which

is a decreasing function of vm.

Effects of c;2. Consider Fig. 14a. When c;2 decreases, it

becomes likely that records for class 2 users are selected for replacement. Consequently, p2;1 (see (10)) is an

increasing function of c;2. From Fact 3, Fig. 14a shows

that p2;1is an increasing function of vm.

Fig. 14b plots p3;i curves. From (12) and since the

class 1 call activities are independent of the class 2 call activities, p3;1is not affected by c;2. From (12) and due to

the high call activities of class 2 users, p3;2' 0.

Fig. 15a shows that probability pIR decreases as c;2

decreases. When c;2decreases, the active users become

less active (but they are still more active than the inactive users). In this case, the inactive replacement policy has a larger opportunity to select the active users for replace-ment, which results in a smaller pIR.

Similar to the reasoning given in the discussion for the effect of c;1, the results observed in Fig. 14 imply that

pIR' p2;1p3;1. That is, in Fig. 15, the pIR curve is the

ªproductº of the p2;1 and p3;1 curves. Thus, Fig. 15a

shows that pIR increases and then decreases as vm

increases.

Fig. 15b shows that pRR increases as c;2 decreases.

Since the random replacement policy selects records randomly for replacement, when call activities for some users (either classes 1 or 2) are reduced, we expect a lower probability that the replaced users will have call

activities before they leave the VLR. Thus, pRRincreases

as c;2decreases.

Effects of 1(2). Fig. 16 plots pIRand pRR as functions of

1. It is obvious that pIR increases as 1 increases. This

figure indicates that pIRis not sensitive to 1when vmis

large. When vm is large, there is a better opportunity to

find users with larger VLR residence times. In this case, even if 1is small, the inactive replacement policy is still

very likely to find a class 1 user with tm;1> tT. Thus,

Fig. 16a shows that both small and large 1yield similar

pIR performance when vm is large. Fig. 16b shows the

trivial result that that pRR increases as 1increases.

Effects of the replacement policies. Figs. 13, 14, 15, and 16 show that the inactive replacement policy significantly outperforms the random replacement policy. These figures indicate that, even if the portion of the users with low call activities is small (i.e., 1 is small), the

inactive replacement policy still performs very well. On

the other hand, random replacement policy shows poor performance when 1 is small. We note that, in the

random replacement policy, the replaced users are selected from all users with the same probability. The inactive replacement policy tends to select the users with low call activities (see (2) and (11)). Thus, pIR> pRR is

expected.

5 C

ONCLUSIONS

This paper investigates the overflow issues for large-scale mobility databases in cellular phone networks. With record replacement, we can provide communication services to more users than can be accommodated in a mobility database (specifically, visitor location register or VLR). It is important to select ªappropriateº VLR records for replacement to reduce the possibility of overflow operations in the future. The inactive replacement policy was proposed to replace a record that does not have any call activity longer than a period called inactive threshold. We

compared the inactive replacement policy with the random replacement policy. Our study indicated that the inactive replacement policy significantly outperforms the random replacement policy (by reducing over 90 percent of the overflow call setups). We also observed that, under the input parameter values considered in this paper, an appropriate value for inactive threshold is about 1,000 times that of the mean VLR residence time.

As a final remark, we note that an interesting replace-ment policy not studied in this paper is the most-idle policy, where the most idle record is selected for replacement. In a real mobile network, the records of a huge VLR are physically stored in several separated databases. The most-idle policy will need to access all databases for a replacement (although techniques may exist to speed up the process), while the inactive policy only need to access some databases independently. The statistics from real mobile networks indicate that, in high traffic situations, several

call/registration requests will arrive simultaneously. In this case, the inactive policy allows us to:

1. quickly return several replaced records from a database (so that multiple overflow requests can be served simultaneously) and/or

2. process in parallel on several databases to speed up the execution.

In this case, the most-idle policy is not as efficient as the inactive replacement policy. However, it would be interest-ing to investigate the performance of the most-idle policy, which will be a future research direction.

A

PPENDIX

N

OTATIONS

. i: the portion of class i users in the VLR

. fm;i…m;i†: the density function for the m;idistribution

. f

m;i…s†: the Laplace Transform for the m;idistribution

. K: the number of classes of mobile users . c;i: the call arrival rate to a class i user

. 1=m;i: the expected VLR residence time for a class i

user

. p1;i: the probability that a class i record is inactive

. p2;0: the probability that, when a replacement request

arrives, no inactive record is found in the VLR . p2;i: the probability that, when a replacement request

arrives, an inactive class i record is selected for replacement

. p3;i: the probability that, after the record of a class i

user is replaced, the user does not have any call activity before he/she leaves the VLR

. pIRˆ pIRjr‡ pIRjnr: the probability that, after an

overflow registration, the overflow user does not have any call activities after he/she leaves the VLR (the inactive replacement policy)

Fig. 16. Effects of 1 on pIRand pRR(Tˆ 0:001m; c;1ˆ 0:01m; c;2ˆ 500m). (a) The inactive replacement policy. (b) Random replacement

. p

IRˆ pIRjr‡ pIRjnr: the probability that, after an

overflow call setup, the overflow user does not have any call activities after he/she leaves the VLR (the inactive replacement policy)

. pIRjnr: the probability that, in an overflow

registra-tion, the user ui who initiated the registration is

considered as the overflow user and uidoes not have

any call activity before the person leaves the VLR (the inactive replacement policy)

. p

IRjnr: the probability that, after an overflow

regis-tration, the overflow user does not have any call activities after he/she leaves the VLR (the inactive replacement policy)

. pIRjr: the probability that, at a replacement, no

inactive record is found in the VLR and a record is randomly selected for replacement and the corre-sponding user does not have any call activity before the person leaves the VLR (the inactive replacement policy)

. pRR: the probability that, in the random replacement

policy, after an overflow registration or call setup, the user of the replaced record does not have any call activity before he/she leaves the VLR

. rm;i…tm;i†: the density function for the tm;idistribution

. r

m;i…s†: the Laplace Transform for the tm;idistribution

. tm;i: the excess (residual) life of m;i

. c;i: the intercall arrival time to a class i user

. m;i: the VLR residence time for a class i user

. 1…N†: the probability that arbitrary N records in the

VLR are active

. 2…N; i†: the probability that N arbitrary records

(excluding class i records) in the VLR are inactive . tT: the inactive threshold

A

CKNOWLEDGMENTS

The author would like to thank the the three anonymous reviewers for their valuable comments. This work was supported in part by MOE Program of Excellence Research under contract 89-E-FA04-4, InterVideo, Ericsson, National Science Council under contract NSC 89-2213-E-009-203, and the Lee and MTI Center for Networking Research, NCTU.

R

EFERENCES

[1] I.F. Akyildiz and J.S.M. Ho, ªDynamic Mobile User Location Update for Wireless PCS Networks,º ACM-Baltzer J. Wireless Neworks, vol. 1, no. 1, pp. 187-196, 1995.

[2] I. Chlamtac, T. Liu, and J. Carruthers, ªLocation Management for Efficient Bandwidth Allocation and Call Admission Control,º Proc. IEEE Wireless Comm. and Networking Conf., Sept. 1999. [3] ETSI/TC, ªMobile Application Part (MAP) Specification,

Version 4.8.0,º Technical Report, Recommendation GSM 09.02, ETSI, 1994.

[4] Y. Fang, I. Chlamtac, and H. Fei, ªAnalytical Results for Optimal Choice of Location Update Interval for Mobility Database Failure Restoration in PCS Networks,º IEEE Trans. Parallel and Distributed Systems, vol. 11, 2000.

[5] Y. Fang, I. Chlamtac, and H. Fei, ªFailure Recovery of HLR Mobility Databases and Parameter Optimization for PCS Net-works,º J. Parallel and Distributed Computing, vol. 60, pp. 431-450, 2000.

[6] J.S.M. Ho and I.F. Akyildiz, ªLocal Anchor Scheme for Reducing Signaling Cost in Personal Communication Networks,º IEEE/ ACM Trans. Networking, Oct. 1996.

[7] N.L. Johnson, Continuous Univariate Distributions-1. John Wiley & Sons, 1970.

[8] N.L. Johnson, Continuous Univariate Distributions-2. John Wiley & Sons, 1970.

[9] Y.-B. Lin, ªOverflow Control for Cellular Mobility Database,º IEEE Trans. Vehicular Technology, vol. 49, no. 2, pp. 520-530, 2000. [10] Y.-B. Lin, W.R. Lai, and R.J. Chen, ªPerformance Analysis for Dual Band PCS Networks,º IEEE Trans. Computers, vol. 49, no. 2, pp. 148-159, 2000.

[11] Mehrotra, GSM System Engineering. Artech House, 1997. [12] M. Mouly and M.-B. Pautet, The GSM System for Mobile

Communications. Palaiseau, France: M. Mouly, 49 rue Louise Bruneau, 1992.

[13] S.M. Ross, Stochastic Processes. John Wiley & Sons, 1983. Yi-Bing Lin received his BSEE degree from National Cheng Kung University in 1983 and his PhD degree in computer science from the University of Washington in 1990. From 1990 to 1995, he was with the Applied Research Area at Bell Communications Research (Bellcore), Morristown, New Jersey. In 1995, he was appointed a professor in the Department of Computer Science and Information Engineering (CSIE), National Chiao Tung University (NCTU), Taiwan. In 1996, he was appointed deputy director of the Microelec-tronics and Information Systems Research Center, NCTU. During 1997-1999, he was elected chairman of CSIE, NCTU. His current research interests include design and analysis of personal communications services network, mobile computing, distributed simulation, and perfor-mance modeling.

Dr. Lin is an associate editor of IEEE Network, an editor of the IEEE Journal on Selected Areas in Communications: Wireless Series, an editor of IEEE Personal Communications Magazine, an editor of Computer Networks, an area editor of ACM Mobile Computing and Communication Review, a columnist of the ACM Simulation Digest, an editor of the International Journal of Communications Systems, an editor of ACM/Baltzer Wireless Networks, an editor of Computer Simulation Modeling and Analysis, an editor of the Journal of Information Science and Engineering, program chair for the Eighth Workshop on Distributed and Parallel Simulation, general chair for the Ninth Workshop on Distributed and Parallel Simulation, program chair for the Second International Mobile Computing Conference, guest editor for the ACM/Baltzer MONET special issue on personal communications, a guest editor for the IEEE Transactions on Computers special issue on mobile computing, and a guest editor for IEEE Communications Magazine's special issue on active, programmable, and mobile code networking. He is the author of the book Wireless and Mobile Network Architecture (coauthored with Imrich Chlamtac, published by John Wiley & Sons). He received 1998 and 2000 Outstanding Research Awards from the National Science Council, Republic of China, and 1998 Outstanding Youth Electrical Engineer Award from CIEE, Republic of China. He is an adjunct research fellow of Academia Sinica and a senior member of the IEEE.