Optimal aggregation factor and clustering under delay constraints in aggregate sequential group...

Optimal aggregation factor and clustering under delay constraintsin aggregate sequential group paging schemes

Hung Tuan Do • Yoshikuni Onozato •

Ushio Yamamoto

Published online: 16 October 2009

� Springer Science+Business Media, LLC 2009

Abstract This paper considers several optimization

problems of sequential paging with aggregation mecha-

nism which has been shown to reduce significantly the

paging cost of a wireless communication system. An

important problem is to find the optimal aggregation factor

subject to a constraint on the average paging delay.

Another problem is, given a cost function that depends on

both paging cost and paging delay, how to find the optimal

aggregation factor to minimize that cost function. We have

formulated and shown that these can be solved nicely due

to the monotonicity and convexity of the average paging

cost function and paging delay function. We demonstrate

that the optimization problems of the aggregate factor and

subnet clustering are not separable. This leads to joint

optimization problems of aggregation factor and clustering

that are investigated in this paper. The paper presents dif-

ferent algorithms to solve these joint optimization prob-

lems using the monotonicity in the aggregation factor and

the number of clusters of the average paging cost and delay

with the unconstrained optimal clustering and the struc-

tures of the constrained optimal clustering.

Keywords IP paging � Paging cost � Paging delay �Joint optimization � Convex optimization

1 Introduction

In wireless mobile networks, paging is a process to deter-

mine the exact location of a specific mobile terminal (MT)

in Personal Communication Systems (PCSs) or a mobile

node (MN) in Mobile IP [1] when this terminal or node is

in stand-by state. Paging service is popularly deployed in

wireless networks for two major benefits [2, 3]: to reduce

location update cost and to save power consumption of

mobile terminals or nodes.

In paging, a mobile node is not required to register its

location per each inter-cell or inter-subnet move, but it

needs to update its location if it moves out of a determined

set of cells or subnets called a paging area (PA). In this

paper, we use the term PA for location area (LA) in PCSs

and registration area in Mobile IP networks. In each paging

cycle, the network sends paging request to a predefined

subnet of cells/subnets. If the mobile node is located, it

updates its location with the network and the paging pro-

cess is terminated successfully. Otherwise, the paging

process will continue with another subnet of cells/subnets

and so on. The paging latency (or delay) is defined as the

number of paging cycles needed to locate the mobile node.

Usually, there is a time constraint imposed on the paging

latency.

It is well-known that paging cost and location update cost

represent a trade-off in node paging. If the mobile node

updates its location frequently, the network can know its

location more precisely, and thereby the paging cost can be

reduced. Nevertheless, the location update cost certainly

increases in this case. On the other hand, if the mobile node

performs location update not frequently, a larger PA must

be paged to locate it when it is wanted. Thus, the paging cost

incurred will be larger. Also, there exists a more obscure

trade-off between paging cost and paging latency. Paging

H. T. Do

Department of Operations Management, Purdue University,

West Lafayette, IN, USA

e-mail: [email protected]

Y. Onozato � U. Yamamoto (&)

Department of Computer Science, Graduate School

of Engineering, Gunma University, Kiryu 376-8515, Japan


Y. Onozato


123

Wireless Netw (2010) 16:1427–1446

DOI 10.1007/s11276-009-0212-z

cost is proportional to the number of paging cycles and the

number of subnets paged during each cycle. With the same

paging algorithm, if the number of subnets being paged in

each cycle is larger (i.e. the subset of subnets being paged is

bigger) then the mobile node is more likely to be found

faster, yet the paging cost may be higher. An important area

of research on paging systems is to address these two main

trade-offs. Considering the two basic trade-offs in paging,

the crucial problems of research in paging are to minimize

the total signaling cost of both paging and location update

and to strike a balance between paging cost and paging

latency, which is specified by various constraints.

It should be noted that in PCSs, paging is implemented at

the link layer, while in Mobile IP networks, paging is

employed at the network layer (hence, named IP paging).

Some points of difference between layer 2 paging and layer 3

paging in implementation is investigated in Ref. [4]. Several

IP paging protocols have been proposed in [2, 3, 5, 6]. Ref-

erences [7, 8] provide good surveys on mobility management

in IP-based wireless systems. In [9], the authors proposed an

IP architecture for systems with different infrastructure.

Some Individual IP Paging schemes in which PA con-

struction is customized to each mobile node were first

proposed in [10]. In PCS networks, construction of the

registration areas adaptive to user mobility and call patterns

is introduced in [11, 12]. The idea of Individual Paging is

completely ramifying from the traditional paging, in which

PA is fixed and common to all mobile hosts. Some merits

of Individual IP paging are investigated in [13].

Most of research works in the literature are concerned

with Static Aggregate Paging schemes [14] that are inde-

pendent of MNs, i.e. the PAs are fixed and common to all

MNs [2, 3, 15]. An important problem in this context is to

construct an optimal PA to solve the trade-off between

paging cost and location update cost. In [15, 16], this

problem is addressed in various scenarios. For a compar-

ative study between Individual Paging and Static Aggre-

gate Paging schemes, please refer to [14].

However, for a given and fixed PA, a prominent problem

is to solve the trade-off between paging cost and paging

delay with multi-step paging. Multi-step (sequential) paging

is proposed to reduce the paging cost at the expense of

higher paging latency. In multi-step paging, all the subnets

of a PA are partitioned into groups. Upon a paging request,

all the subnets of the first group are paged at the same time

in the first round then, if the user can not be located, all the

subnets of the second group are paged, and so on. Usually,

multi-step paging algorithms can work in both PCSs and

wireless IP networks because they deal with clustering

subnets and specifying the paging sequence.

The works most related to our work here are [17–19].

The authors in [17] showed how location probabilities of

cells can be used to cluster the cells of a LA in PCSs into

groups so that the average paging cost is minimized subject

to delay constraint. This is the first paper investigating

sequential group paging in cellular networks. The optimi-

zation of clustering subject to a constraint on paging delay is

formulated and solved using a continuous density function

that approximates the discrete location probability distri-

bution of cells. The problem is then formulated as a convex

optimization problem. However, the clustering obtained is

just an approximate solution because the problem of con-

cern is discrete.

In [18], the optimization problems as presented in [17]

are solved using the structures of the optimal solution and

then dynamic programming is employed to solve the

problem recursively in polynomial time.

We introduced paging mechanisms with aggregation in

[19]. With an aggregation of paging requests, the system

can aggregate k paging requests looking for k mobile users

into one request packet to find all k users together, where k

is a control variable. This mechanism can significantly

reduce the paging cost, but at the same time may increase

the paging delay as demonstrated in [19]. The aggregation

of paging requests is especially useful when the rate of

incoming messages is large and some level of service delay

is acceptable at the beginning of a communication session.

When the call arrival rate to dormant MNs in the PA (i.e.

the paging request rate to the PA) is large, there will be a

queue of paging requests built up at the Paging Agent, and

hence, a batch processing of aggregate paging would be

necessary.

Reference [19] also proposes effective paging schemes

based on the balanced partitions of subnets. We can see

that the sequential group paging scheme considered in [17,

18] is a special case of sequential paging with aggregation

when the aggregation factor is equal to one. However,

when aggregation mechanism is employed the average

paging cost and delay functions become more complicated

and the structure of the optimal clustering is changed as

shown later in this paper.

For aggregate sequential group paging, there are several

important questions left unanswered in the literature. What

is the optimal aggregation factor subject to a maximum

average paging delay? Given a utility function what is the

value of the aggregation factor to minimize it under con-

straint on the average paging delay? More generally, what

is the best paging scheme regarding the aggregation factor,

the number of clusters, and clustering itself?

This paper concerns with these problems, more precisely

optimization problems of sequential paging with aggrega-

tion mechanism. An important problem is to find the

optimal aggregation factor k subject to a threshold of

paging delay. Another problem is given a cost function that

depends on both paging cost and paging delay, how to find

the optimal k to minimize that cost function. The problem

1428 Wireless Netw (2010) 16:1427–1446

123

of joint optimization of both aggregation factor k and

clustering is also a topic of this paper. The main contri-

butions of this paper are as follows:

1. Derivation of the average paging cost and delay

functions as functions of aggregation factor k and a

clustering which is by ordering property represented

by a partition vector v. Decreasing monotonicity of the

average paging cost function and increasing monoto-

nicity of the average paging latency with respect to k

are proven. These properties are essential in finding the

optimal aggregation factor k given a clustering in an

effective way.

2. Proof of the convexity of the average paging cost

function and the average paging delay function with

respect to k. These findings will formulate a convex

optimization problem if the utility function preserves

the convexity. Moreover, since k is an integer, if the

objective function is convex, we can use a very simple

algorithm to find the optimal k.

3. Proof of the optimal stochastic ordering of subnets

according to their location probabilities for a general k.

With this feature, a clustering can be represented

simply by a vector so called a partition vector, and

thus, a sequential paging strategy is represented by this

vector as well.

4. The unconstrained and constrained optimization of

subnet clustering given an aggregation factor k. The

monotonicity in the aggregation factor and the number

of clusters of the average paging cost function and

delay function for the unconstrained optimal clustering

is demonstrated. This helps to solve unconstrained

joint optimization problem with much less complexity.

5. The approaches to the joint optimization problems of

aggregation factor and clustering. The two optimiza-

tion problems are found interdependent instead of the

first possible intuition that they seem independent. In

this paper, the recursive structures of both uncon-

strained optimal clustering and constrained optimal

clustering are presented. These structures can be seen

as generalizations of the structures found in [18]. The

constrained joint optimization problem then can be

addressed using dynamic programming with polyno-

mial time complexity. We can do better with the

unconstrained optimization thanks to the monotonicity

in the aggregation factor and the number of clusters

mentioned above.

The rest of the paper is organized as follows. Section 2

describes the system model, assumptions and the formula-

tion of the average paging cost and paging delay. Section 3

presents the main properties of sequential paging with

aggregation that are essential in solving the optimization

problems. The optimization problems and approaches to

them are discussed in Sect. 4. We illustrate our results with

some numerical examples in Sect. 5. In Sect. 6, we discuss

relaxation of assumptions, extension of our analysis, and

some other related problems. Finally, Sect. 7 concludes the

paper.

2 System model and problem formulation

2.1 System model and assumptions

2.1.1 Network architecture

Our proposed model of sequential paging scheme with

aggregation is presented in Fig. 1. There is a paging agent

that initiates paging processes for each PA, preferably

located at a higher level in the network hierarchy. This

paging agent can be co-located with a router or even a

Home Agent (HA). When there comes an IP session to an

idle MN residing within the PA, the paging agent initiates a

paging request message that contains the IP address of the

wanted MN and sends it to the FAs of the subnets within

the first group g1. Within each subnet, upon a paging

request arrival, the Foreign Agent (FA) formulates and

broadcasts a paging message including the IP addresses

from the paging request message over the air to locate the

MN within their subnets. If the MN is not found, the paging

agent sends the request to g2, and so on until the MN is

located.

In aggregate sequential paging schemes, the paging

agent can aggregate k paging requests looking for k mobile

users into one aggregate paging request message to find all

k users together. In our proposed schemes, this can be

performed as follows: the paging agent first reads the

address fields of the incoming packets and then generate a

single message called aggregate paging request message

including all the addresses of the MNs. As proposed in

[6, 20], this implementation is readily accommodated since

for efficiency, it is allowed to have multiple IP addresses in

s1 si si+1 sN

g1 g2 gm

PA

Paging Agent

Paging requests

IP session arrivals

(1) (2) (m)

s1 si si+1 sN

g1 g2 gm

( )

Fig. 1 Proposed model of sequential paging scheme with aggregation

Wireless Netw (2010) 16:1427–1446 1429

123

the address field. Then the aggregate paging request mes-

sage is transmitted down to the subnets in the PA, cluster

by cluster, until all k users are located or the entire PA is

searched. An MN when getting the request destined for it

will change to active mode and perform a location update

to the system.

For example, there are two paging requests, which come

close to each other in time, to 2 MNs: X residing in group

g2 and Y dwelling in group g3 of the PA. When k = 1, the

system will first page for X by paging g1 then g2 and find X

in some subnet of g2. Next, the system pages all over again

for Y, starting from g1 to g2 and finally it finds Y in g3.

When k = 2 and suppose the paging agent aggregates these

two paging request, then the system will page for both X

and Y in one round. That is it pages g1, then g2 and finds X

there, then it finally pages g3 and finds Y there. Intuitively,

the paging cost in the first case will be significantly larger

than that in the second case.

The parameters used in our analysis are shown in Table 1.

The location probability of a subnet si is the probability that

an arbitrary MN residing in the PA will be found in si. This

location probability can be obtained from collective data,

and in this paper, we assume they are calculated and avail-

able. The location probability of paging group gj is the

probability that an arbitrary MN residing in the PA will be

found in gj, which can be calculated as the sum of the

location probabilities of all its subnets. Note that k also

represents the paging request rate. If a paging cycle time is

normalized to 1 and k[ 1, there will be a queue of paging

requests at the Paging Agent and therefore, a paging scheme

without aggregation could cause unacceptable paging delay.

We propose an improved version of aggregate sequen-

tial paging as follows. As a paging process goes, the IP

addresses of the MNs that are already located will be

removed by the paging agent from the list of k wanted

MNs’ IP addresses. Thus the sizes of paging requests sent

from the paging agent to FAs and paging messages

broadcasted by FAs are maximal for paging the first clus-

ter, but then reduced as the paging process goes. This

enhanced version definitely reduces the paging overhead

cost mentioned later in this section and in Sect. 6.2, while

offering the same average paging delay. Although an

analysis of this enhanced version would be out of scope of

this paper, the assumption of negligible overhead cost

would be more reasonable if the paging agent uses it.

Note that in this paper we use the abbreviation PA for

paging area, while the term paging agent is not abbreviated.

We also use the terms cluster and group, partition and

clustering pair-wise interchangeably.

2.1.2 Assumptions

Note that inter-PA mobility of MNs is handled by a handoff

procedure, which is usually required to be fast. Moreover,

inter-PA mobility, which is similar to inter-domain mobil-

ity, is relatively rare in practice. To focus on the optimi-

zation of PA clustering and paging order, we assume that

the handoff procedure is fast enough to assume zero missing

probability as in most of related works in the literature. In

other words, we assume that all the MNs are residing in the

PA during a paging process, i.e. any MN of interest will be

surely found in the PA. This also means no inter-PA

mobility occurs during a paging process. The assumption

can be formally stated in the following condition:

Xm

j¼1

PGðgjÞ ¼XN

i¼1

PSðsiÞ ¼ q ¼ 1 ð1Þ

In fact, this assumption is not required throughout our

formulation and analysis, and thus, can be easily relaxed

without changing our analysis. When there is a positive

paging probability, some MNs move out of the current PA

while the paging agent is paging for them, these MNs will be

required to make their registration with their new PAs and

subsequently, they belong to these new PAs. The current

paging agent will be updated with these address changes.

From now on their location management is not associated

with the current PA, and therefore, not relevant to our

formulation. When this assumption is relaxed (i.e. 0 B

q\ 1), the term ‘‘location probability distribution’’ should

become ‘‘truncated location probability distribution’’.

Arrival process of paging request packets is uniform or

Poisson with the mean rate k.

When aggregation of paging packets is employed, the

paging agent needs to process the incoming paging packets

by reading their MNs address fields and then creates an

aggregate packet including all the addresses of the aggre-

gated paging packets. This processing time can be assumed

to be proportional to the number of the paging packets, i.e.

the aggregation factor k. However, the processing time is

assumed negligible in this paper.

By including multiple addresses of MNs in an aggre-

gated paging message, we can increase paging packet size,

Table 1 Parameters in our analysis

Symbol Meaning

N Total number of subnets/cells in the PA

si Subnet i in the PA (1 B i B N)

m The number of paging groups in the PA (1 B m B N)

gj jth group in the paging sequence (1 B j B m)

PS(si) Location probability of subnet si

PG(gj) Location probability of paging group gj

CG(gj) Paging cost incurred by paging group gj

k Average call arrival rate to dormant MNs in the PA

1430 Wireless Netw (2010) 16:1427–1446

123

thus incurring some overhead paging cost. However, as in

the paging message structure proposed in [6, 20], the field

‘‘Mobile Nodes’ IP addresses’’ is readily designed to

accommodate multiple addresses. Furthermore, usually this

field may account for a small fraction of a paging message.

Also, as we will show when k C 2, the paging cost could

be reduced multiple times. Therefore, for simplicity, it is

reasonable to assume that the overhead incurred by sending

the aggregated packet down to the paging groups of sub-

nets is negligible. This assumption would be even more

reasonable if the enhanced version of aggregate paging

mentioned in Section Network Architecture is employed.

We discuss a relaxation of this assumption in Sect. 6.2.

2.2 Average paging cost and paging delay functions

Let C(k) and L(k) be the expected paging cost function and

expected paging delay function with respect to aggregation

factor k, respectively. Let (g1, g2, …, gm) be an m-partition

of the PA. As derived in [19], the expected paging cost per

each MN being paged can be calculated as follows:

CðkÞ ¼ 1

k

Xm

i¼2

Xi

t¼1

PGðgtÞ !k

�Xi�1

t¼1

PGðgtÞ !k

0

@

1

A

0

@

0

@

�Xi

t¼1

CGðgtÞ!þ PGðg1Þð ÞkCGðg1Þ

!ð2Þ

Note that once an aggregate paging request packet has

been generated and transmitted, it acts exactly as a non-

aggregate paging request, from a mobile user perspective.

Basically, in (2), C(k) is calculated from the probability of

finding k users in the first i clusters and the corresponding

paging cost of paging these i clusters. Note that i is the

smallest number of first clusters that k wanted users can be

found.

The average paging delay L(k) for an arbitrary user can

be seen as the summation of a half of aggregation delay and

the average paging delay without aggregation as follows:

LðkÞ ¼ LaggrðkÞ2

þ Ls ð3Þ

where Ls denotes the average paging delay due to sequen-

tial paging without aggregation, and Laggr(k) represents the

average aggregation delay. Ls is calculated as follows

[17, 18]:

Ls ¼Xm

i¼1

PGðgiÞi ð4Þ

For uniform or Poisson arrival process with rate k, the

average delay due to the aggregation Laggr(k) is (k - 1/k.

Note that we count the aggregation time since the time of

the first arrival. So the average delay L(k) is calculated as

follows:

LðkÞ ¼ k � 1

2kþXm

i¼1

PGðgiÞi ð5Þ

For Poisson arrival process with rate k, the aggrega-

tion time with factor k can be computed as the time to the

(k - 1)th arrival of paging packets Xk, and this is given by

a Gamma distribution with moment generating function as

follows:

fXk�1ðtÞ ¼ ke�kt ðktÞk�2

ðk � 2Þ! WXk�1ðsÞ ¼ k

k� s

� �k�1

So we have: LaggrðkÞ ¼ E Xk½ � ¼dWXk

ðsÞds s¼0 ¼ k�1

k

�� . As a

result, formula (5) is used for both uniform and Poisson

arrival processes.

For simplicity of presentation, we introduce the fol-

lowing notations. For a given set of subnets and its partition

(g1, g2, …, gl), define:

A0 ¼ 0;Ai ¼Xi

t¼1PGðgtÞ

Bi ¼Xi

t¼1CGðgtÞ

ð6Þ

Without loss of generality, assume the paging cost per

subnet is 1, then we have: Bi ¼Pi

t¼1 CGðgtÞ ¼Pi

t¼1 gtj j;where |gt| denotes the number of subnets in group gt.

In the new notation we express C(k) and L(k) for the set

of all subnets of the PA with a partition (g1,g2,…,gm) as

follows:

CðkÞ ¼ 1

k

Xm

i¼1

Aki �Ak

i�1

� �Bi

LðkÞ ¼Xm

i¼1

PGðgiÞiþLaggrðkÞ

2¼Xm

i¼1

Ai�Ai�1ð ÞiþLaggrðkÞ2

ð7Þ

where 0 ¼ A0 \ A1 \ � � � \ Am ¼Pm

j¼1 PGðgjÞ ¼PN

i¼1

PSðsiÞ ¼ q ¼ 1 and B1 \ B2 \ … \ Bm = N, Bj is a

positive integer.

3 Analysis

We can see that the average paging cost and paging delay

are functions of the location probability distribution of the

subnets, the clustering of subnets and the aggregation fac-

tor. In this section, we consider the most essential properties

and forms of optimality of sequential group paging with

aggregation. Note that when there is no aggregation, i.e.

k = 1, we have the sequential paging schemes as presented

in [17, 18]. From these findings, we will state and address

important optimization problems in Sect. 4.

Wireless Netw (2010) 16:1427–1446 1431

123

3.1 Monotonicity of the paging cost and paging delay

functions

Proposition 1 Given a fixed clustering, C(k) is a strictly

decreasing function of k, while L(k) is a strictly increasing

function of k.

Proof We will show that it holds for any positive real

number k. Then it also holds for any positive integer k.

dCðkÞdk¼ 1

k2

kXm

i¼1

Aki log Ai � Ak

i�1 log Ai�1

� �Bi

�Xm

i¼1

Aki � Ak

i�1

� �Bi

!

¼ 1

k2

Xm

i¼1

1� k log Ai�1ð ÞAki�1 � 1� k log Aið ÞAk

i

� �Bi

Let’s investigate the function f(x) = (1 – k log x)xk, x [(0, 1]. Take the first order derivative of this function:

df ðxÞdx¼ kxk�1 � kxk�1ðk log xþ 1Þ ¼ kxk�1ð�k log xÞ[ 0

Thus f(x) is a strictly increasing function. Since Ai-1 \ Ai,

we can see that the derivative of C(k) is negative, and

therefore, C(k) is strictly decreasing.

Recall that LðkÞ ¼ LSðkÞ þ LaggrðkÞ=2. For a fixed

clustering of subnets, LS(k) is a constant, while LaggrðkÞ ¼ðk�1Þ

k is linearly increasing in k. Hence, L(k) is strictly

increasing in k. h

3.2 Optimal paging sequence

When there is no aggregation employed, it is shown that

the optimal paging order, which minimizes both average

paging cost and average paging delay, is the descent order

of the subnet location probabilities [17, 18]. The proof is

fairly easy for k = 1. In this part, we will investigate this

optimal paging order for the paging with a general aggre-

gation factor k. To prove for a general k, we can argue that

it suffices to show for the case of two adjacent groups

because of bubble-sorting algorithm.

Proposition 2 For sequential paging with a general

aggregation factor k, paging in descent order of the loca-

tion probabilities of subnets minimizes both the average

paging cost and paging delay.

Proof Appendix.

From now on, we can restrict our discussion to ordered

sequence of subnets. A partition (g1,…,gm) is therefore

completely specified by a so-called partition vector

v = (v1,…,vm) = (|g1|,…,|gm|). Note that, given the location

probabilities of subnets, a clustering is also uniquely defined

by a group location probability vector a = (A1,…,Am) or a

group paging cost vector b = (B1,…,Bm), where Ai and Bj

are defined in (6). Hence, any one of these three vectors may

be used to represent a clustering.

Paging cost function and paging delay function now can

be expressed as the functions of the aggregation factor k

and the partition vector v: C(k, v) and L(k, v), respectively.

3.3 Convexity of paging cost and paging delay

The convexity of the average paging cost function and

affinity of the average paging delay function are essential in

formulating various optimizations as convex problems. The

Slater condition is easily satisfied and thus we have no

duality gap. As a result, the optimization problem can be

solved by the duality approach. Furthermore, for k is an

integer, a more effective iterative algorithm might be

employed to solve the optimization problems of concern in

Sect. 4.

Proposition 3 The average paging cost C(k) and the

average paging delay L(k) are convex functions in k.

Proof Appendix.

3.4 Monotonicity of the average paging cost and delay

functions for the unconstrained optimal clustering

As assumed, the sequence of subnets investigated in our

analysis is ordered in non-increasing location probabilities

of the subnets. A partition then is completely specified by a

partition vector v.

Let C*(k) = minvC(k,v), v*(k) = argminvC(k,v) where

the number of clusters m is fixed. We claim that this

optimal paging cost function with respect to v is monoto-

nous in k in the following proposition.

Proposition 4 For a fixed number of clusters, C*(k) is a

decreasing function of k.

Proof Let v*(k) be defined by a = (Ai), and v*(k ? 1) be

defined by ~a ¼ ð ~AiÞ where 1 B i B m. Let b = (Bi) and~b ¼ ð ~BiÞ be associated group paging cost vectors.

By Proposition 1, we have:

C�ðk þ 1Þ ¼ Cðk þ 1; v�ðk þ 1ÞÞ

¼ 1

k þ 1

Xm

i¼1

~Akþ1i � ~Akþ1

i�1

� �~Bi

� 1

k þ 1

Xm

i¼1

Akþ1i � Akþ1

i�1

� �Bi

� 1

k

Xm

i¼1

Aki � Ak

i�1

� �Bi ¼ Cðk; v�ðkÞÞ ¼ C�ðkÞ

1432 Wireless Netw (2010) 16:1427–1446

123

Before going to the joint optimization problem, we

present one more property of the sequential group paging

with aggregation. The paging cost and paging delay with

the optimal clustering are also monotonous with respect to

the number of clusters m, which is presented in the

following proposition.

Proposition 5 Let v*(m) = argminvC(k,v(m)), where k is

fixed. The optimal average paging cost is decreasing in m:

C(k,v*(m)) C C(k,v*(m ? 1)) while the optimal average

paging delay is increasing in m: L(k,v*(m)) B L(k,v*

(m ? 1)).

Proof Let v0(m ? 1) be the partition obtained from v*(m)

by moving the last subnet sN from the last cluster of v*(m)

to the new (m ? 1)th cluster. Suppose v*(m) is defined by

vectors (Ai) or (Bj).

Cðk; vðmþ 1ÞÞ ¼Xm�1

i¼1

Aki � Ak

i�1

� �Bi þ Am � PðsNÞð Þk�Ak

m�1

� �

� Bm � 1ð Þ þ 1� Am � PðsNÞð Þk� �

Bm

¼Xm�1

i¼1

Aki � Ak

i�1

� �Bi � Am � PðsNÞð Þk

� Akm�1 Bm � 1ð Þ þ Bm

¼Xm�1

i¼1

Aki � Ak

i�1

� �Bi þ Bm 1� Ak

m�1

� �

þ Akm�1 � Am � PðsNÞð Þk

C k; vðmÞð Þ ¼Xm�1

i¼1

Aki � Ak

i�1

� �Bi þ Ak

m � Akm�1

� �Bm

¼Xm�1

i¼1

Aki � Ak

i�1

� �Bi þ 1� Ak

m�1

� �Bm

So

C k; vðmÞð Þ � Ck; v0ðmþ 1ÞÞ ¼ Am � PðsNÞð Þk�Akm�1� 0

because Am - P(SN) C Am-1 C 0. Finally, we have the

following as required:

C k; vðmÞð Þ�C k; v0ðmþ 1Þð Þ�C k; vðmþ 1Þð Þ

h

3.5 Recursive structure of the optimal clustering

In the next part, we investigate the structure of the optimal

clustering with and without constraint on the average

paging delay. These structures are essential in solving the

unconstrained and constrained optimization problem of

clustering for a given k using recursive algorithms. Due to

the fact that the average paging cost and paging delay for a

(m ? 1)-partition can be separated into the paging cost and

delay of m-partition and an extra term, the optimization

problem can be solved recursively for the constrained case.

The recursive structures presented in this section might be

considered generalizations of those discussed in [18] where

no aggregation of paging requests is applied.

The following proposition states the structure of the

optimal clustering for the unconstrained optimization case.

Proposition 6 For a given aggregation factor k, if

v*(m) = (v1*,…,vm*) is the optimal m-group clustering of

subnets without a constraint on the average paging delay,

then v�ðm0Þ ¼ v�1; . . .; v�m0� �

is the optimal m0-group clus-

tering of the firstPm0

i¼1 v�i subnets, where 1 B m0 B m.

Proof By induction, if it holds for m0 = m - 1, then it

will hold for any 1 B m0 B m. Thus, it suffices to prove for

m0 = m - 1.

Suppose that there exists a partition v(m - 1) of the firstPm0

i¼1 v�i subnets such that C(k,v(m - 1)) \ C(k,v*(m -

1)). Let vðmÞ ¼ vðm� 1Þ; v�mðmÞ� �

; that is a m-group

partition vector with the first m - 1 elements are the

elements of v(m - 1) and the last group is the last element

of v*(m). A contradiction will result with the assumption

that v*(m) is the optimal partition.

C k; v�ðmÞð Þ ¼ 1

k

Xm�1

i¼1

Aki � Ak

i�1

� �Bi þ Ak

m � Akm�1

� �Bm

!

¼ C k; v�ðm� 1Þð Þ þ 1

kAk

m � Akm�1

� �Bm

[ C k; vðm� 1Þð Þ þ 1

kAk

m � Akm�1

� �Bm

¼ C k; vðmÞð Þ

h

Next, we consider the structure of the optimal clustering

for the constrained optimization problem. Denote n as the

number of the first subnets out of our total N subnets

ordered in descending sequence of subnet location proba-

bilities. Let C_

ðk;m; n; hÞ be the optimal paging cost of

paging n first subnets with the number of clusters m, the

constraint on the average paging delay h, i.e.

C_

ðk;m; n; hÞ¼ min

v2Kðm;nÞCðk;m; n; v; hÞ subject to Lðk;m; n; vÞ� h

where K(m, n) is a set function that denotes the set of

all partitions of the first n subnets with m clusters. The

form of optimal clustering is stated in the following

proposition:

Wireless Netw (2010) 16:1427–1446 1433

123

Proposition 7 The following recursive equation holds:

C_

ðk;mþ 1; n; hÞ

¼ minj¼m;::;n

�C_

ðk;m; j; h� mþ 1ð Þ Amþ1 � Amð ÞÞ

þ1

kAk

mþ1 � Akm

� �n

�ð8Þ

Proof

Cðk;mþ 1; n; hÞ

¼ 1

k

Xm

i¼1

Aki � Ak

i�1

� �Bi þ Ak

mþ1 � Akm

� �Bmþ1

!

¼ 1

k

Xm

i¼1

Aki � Ak

i�1

� �Bi þ

1

kAk

mþ1 � Akm

� �n

Lðk;mþ 1; n; hÞ ¼Xm

i¼1

Ai � Ai�1ð Þi

þ Amþ1 � Amð Þ mþ 1ð Þ þ k � 1

2k� h

Xm

i¼1

Ai � Ai�1ð Þiþ k � 1

2k� h

� Amþ1 � Amð Þ mþ 1ð Þ

In order for C(k, m ? 1, n, h) to be minimized, it is nec-

essary that the first term in brackets be minimized, i.e. the

partition of all subnets in the first m clusters must be

optimal subject to the varied constraint h - (Am?1 - Am)

(m ? 1). h

4 Optimization problems and solutions

In the following parts, we consider several optimization

problems and propose different approaches using the

optimality properties and structures discussed in Sect. 3:

• Given a clustering and location probabilities of subnets,

find the optimal aggregation factor k under a constraint

on the average paging delay.

• Optimization of a joint objective function of both

paging cost and paging delay.

• Are aggregation factor optimization and clustering

optimization independent?

• Finding the optimal clustering given a specific value of

the aggregation factor k.

• Constrained joint optimization of aggregation factor

and clustering.

4.1 Optimal aggregation factor under a constraint

on the average paging delay with a fixed partition

Optimization problem in this section is to minimize C(k)

subject to L(k) B h. We can solve this problem for the

continuous variable x and then take k ¼ xb c. In this

case, the problem is to minimize C(x) subject to

L(x) B h.

As shown above, this is a convex optimization problem

with Slater condition [21] easily satisfied for h[ min

L(k) = 1. Note that L(k) is minimized when there is no

aggregation, i.e. k = 1 and the blanket polling is per-

formed, i.e. the number of groups m = 1.

So, we can solve it using standard procedure of duality

without duality gap. However, this optimization problem

can be solved more effectively for C(k) is monotonically

decreasing and L(k) is monotonically increasing.

As a result, the optimization problem k* = argminC(k)

subject to L(k) B h becomes the following:

k� ¼ max kjLðkÞ� hf g ð9Þ

Here L(k) is an affine function of k, so the solution is

readily obtained.

4.2 Joint objective function of paging cost and paging

delay with a fixed partition

For a total cost function that takes into account both the

average paging cost and delay, if the total cost function

preserves the convexity, then the optimization problem

under a constraint on the average paging delay can be

formulated as a convex optimization problem. For exam-

ple, that is to minimize F(k) = aC(k) ? bL(k) subject to

L(k) B h, where a, b C 0.

In the literature, a utility function is commonly assumed

a concave and increasing function. In our case this is

equivalent to a convex, decreasing total cost function or

disutility function. So, in fact, the assumption of convexity

of the total cost function is very reasonable and widely

used in practice.

In this case, F(k) is convex. For unconstrained problem,

we can solve by gradient descent method since F(x) is

differentiable. Finding the suitable step for the gradient

descent algorithm might be tough. Also, if the utility

function is differentiable, solving the equationdFðkÞ

dk ¼ 0

might not be straightforward.

For the constrained problem, the optimal solution can be

found by taking duality approach. Here the Slater condition

for zero duality gap easily holds.

Owing to the convexity of F(k) and k is an integer, we

can use the following simple iterative algorithm to find the

optimal k as follows:

We define a cost difference function between aggrega-

tion factor k and (k - 1) as follows:

dðkÞ ¼ CðkÞ � Cðk � 1Þ

Then the optimal kopt is defined as the following:

1434 Wireless Netw (2010) 16:1427–1446

123

kopt ¼1; if dð2Þ[ 0

min k�;max k : dðkÞ� 0f gf g; otherwize

ð10Þ

where k* is defined in (9).

4.3 Optimal clustering under limited aggregation factor

k, unconstrained expected paging delay

The constrained aggregation factor is due to the limited

capacity of the paging agent, for instance, the limited

buffer to store incoming paging requests.

When there is no constraint on the number of groups m,

the monotonicity in m of the optimal policies results in a

trivial scheme where m = N. So we are more interested in

the optimization problem where both k and m are con-

strained. Here is our optimization problem:

mink;m;v

Cðk;m;N; vÞ subject to v 2 Kðm;NÞ;

1� k� kmax; 1�m�mmax

By Proposition 5, due to the monotonicity of the optimal

average paging cost function in m, this optimal problem

simply becomes:

minv2Kðmmax;nÞ

Cðkmax;mmax;N; vÞ

With the structure of the optimal clustering as described

in Proposition 6, the solution to this optimization problem

can be solved via a quadratic-time algorithm of dynamic

programming. Let ~Cðk;m; nÞ, minv2Kðm;nÞ

Cðk;m; n; vÞ. Due to

the form of optimality in Proposition 6, we immediately

have the following recursive equation Vm, 1 B m B n - 1:

~Cðk;mþ 1; nÞ ¼ minj¼m;::;n

~Cðk;m; jÞ þ 1

kAk

mþ1 � Akm

� �Bmþ1

�

ð11Þ

where Amþ1 ¼Pn

t¼1 PSðstÞ, Am ¼P j

t¼1 PSðstÞ, and Bmþ1 ¼Pmþ1t¼1 CGðgtÞ ¼ n.

When there is only one cluster of all n subnets, we have

the initial condition for our recursive equation:

~Cðk; 1; nÞ ¼ 1

knXn

t¼1

PSðstÞ" #k

¼ n

kAk

mþ1 ð12Þ

Again this algorithm can be seen as a generalization of the

algorithm in [18].

The optimization problem here is in fact a joint opti-

mization problem without a constraint on the expected

paging delay. As we point out in the next section, opti-

mization of the aggregation factor and optimization of

clustering are not separable. Fortunately, the monotonicity

of the optimal policy dictates that the constraints on k and

m must be binding, and therefore we need to solve only for

the optimal clustering given k and m.

4.4 Joint optimal policy under constrained expected

paging delay

The first question might be of our concern is if the opti-

mization of the aggregation factor and optimization of

clustering are coupled or independent? Contrary to possible

intuition, they are in fact not separable. We can see this by

simply taking an example for the case of uniform distri-

bution of subnets. An example to illustrate this is described

in Sect. 5.

From the user viewpoint, in many scenarios, an impor-

tant metric of QoS is the maximal expected paging delay.

Thus, solving the optimization problem under this con-

straint is of our main concern in this section. However, this

optimization requires much more computation complex-

ity since the monotonicity of the optimal policy in k, m

is lost. Moreover, even for a fixed m, optimization of v

is not independent of k. Even worse, the convexity of

C(k,v*(k, m), m), where v*(k, m) is the optimal clustering

given k under a constrained expected paging delay, might

not be maintained.

The constrained joint optimization of aggregation factor

k and partition vector v can be formulated as:

mink;v;m

Cðk; vðk;mÞ;mÞ subject to Lðk; vðk;mÞ;mÞ� h

Here we assume for practical purpose that h 2 D, where D

is a finite set of discrete values of time.

To find the constrained optimal solution, it is necessary

to locate a feasible range of aggregation factor k. Appar-

ently, given a number of clusters m, the average paging

delay is maximized with the partition vmax(m) = (1,…,1,

N – m ? 1). From the constraint on the average paging

delay and the monotonicity of L(k), the maximal value of

k is given as follows:

kmaxðmÞ ¼ max kjLðk; vmaxðmÞÞ� hf g ð13Þ

Note that given a number of clusters m, L(k, vmax(m)) is

monotonically increasing with respect to k, thus finding

kmax(m) is straightforward. An algorithm then only needs to

work with the range of aggregation factor: 1 B k B kmax.

By Proposition 7, for a given (k, m), solution for v*(k, m)

can be derived with polynomial complexity by the fol-

lowing recursive equations with initial condition:

C_

ðk; 1; j; dÞ ¼jk

P ji¼1 PSðsiÞ

� �if Lðk; 1; jÞ ¼

P ji¼1 PSðsiÞ þ k�1

2k � d

1 otherwise

(

C_

ðk;mþ 1; n; dÞ ¼ minj¼m;::;n

C_

ðk;m; j; d � mþ 1ð Þ Amþ1 � Amð Þ þ nk Ak

mþ1 � Akm

� ��

ð14Þ

where Amþ1 ¼Pn

t¼1 PSðstÞ, Am ¼P j

t¼1 PSðstÞ, d 2 D.

Hence, the procedure we can do is to check all vectors

(k,m) in the range: 1 B m B N, 1 B k B kmax(m) and solve

for v*(k, m) with each pair and calculate the corresponding

Wireless Netw (2010) 16:1427–1446 1435

123

paging cost. Finally, find the policy with the minimal

paging cost among all these. Note that our algorithm is still

in polynomial time.

The method by using an approximate continuous prob-

ability density function as presented in [17] does not work

for a general value of k because the Lagrange function is

not always convex. This fact will be discussed in more

detailed in Sect. 6. Moreover, the optimal structure [18] is

difficult to be obtained for a general k.

5 Numerical examples

In this section, we implement our algorithms that use the

forms of optimality described in Sects. 3 and 4, in particular,

the monotonic and recursive forms of optimal clustering. We

show some examples which use the combinations of two

types of location probability distribution (uniform and

binomial) and two types of optimization problems (uncon-

strained and constrained). In these examples, we basically

consider a PA that consists of N = 10 subnets. The incom-

ing calls destined for idle mobile nodes in the PA arrive

according to a uniform or a Poisson point process with the

average rate k.

5.1 Examples of unconstrained optimization

First, we show the numerical examples of unconstrained

optimization for two types of location probability distri-

butions, uniform and binomial. The paging request rate is

k = 2.0.

Optimal clusterings for different numbers of clusters

with the uniform location probability distribution are

shown in Table 2. For k = 1, the clusterings are consistent

with the results in [19] where it is stated that for the case of

uniform distribution of location probabilities and no paging

request aggregation, the optimal clustering is balanced.

Figures 2 and 3 depict average paging costs and paging

delays against the number of groups for optimal cluster-

ings, respectively.

Optimal clusterings for different number of clusters with

the binomial location probability distribution are shown in

Table 3.

Figures 4 and 5 depict average paging costs and paging

delay against the number of groups for optimal clusterings,

respectively.

These examples obviously show advantages of paging

with aggregation. While average paging cost can reduce

several times, the corresponding average paging delay may

vary much less for a high value of k = 2.0. In particular,

the average paging delay is not necessarily increasing in k

since the paging delay element Ls is in fact reducing in k.

Figures 2 and 4 confirm Proposition 4 stating that the

optimal paging cost function is decreasing in k. Proposition

5 is confirmed via Figs. 2, 3, 4, and 5 since the optimal

paging cost function is strictly decreasing in the number of

clusters m, while the corresponding paging delay is strictly

increasing in m. Examination of the optimal clusterings for

Table 2 Optimal clusterings with uniform location probability distribution

Value of k Patterns of optimal clustering

1 (10), (5,5), (4,3,3), (3,3,2,2), (2,2,2,2,2), (2,2,2,2,1,1), (2,2,2,1,1,1,1), (2,2,1,1,1,1,1,1), (2,1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

2 (10), (7,3), (5,3,2), (4,2,2,2), (4,2,2,1,1), (4,2,1,1,1,1), (3,2,1,1,1,1,1), (3,1,1,1,1,1,1,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

3 (10), (7,3), (6,2,2), (6,2,1,1), (5,2,1,1,1), (4,2,1,1,1,1), (4,1,1,1,1,1,1), (3,1,1,1,1,1,1,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

4 (10), (8,2), (7,2,1), (6,2,1,1), (6,1,1,1,1), (5,1,1,1,1,1), (4,1,1,1,1,1,1), (3,1,1,1,1,1,1,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

Fig. 2 Average paging cost (uniform)

Fig. 3 Average paging delay (uniform)

1436 Wireless Netw (2010) 16:1427–1446

123

different values of k shows that group location probabilities

tend to concentrate more on first groups as k increases.

Given a location probability distribution of subnets, we

conjecture that for optimal clusterings, the group location

probability distribution with k1 is stochastically larger than

the group location probability distribution with k2 if

k1 \ k2. A formal definition and good treatment of sto-

chastic order in usual sense can be found in [22]. Intui-

tively, this property, if it holds, implies that group location

probabilities concentrate more on first groups for a larger

value of k.

5.2 Examples of constrained optimization

Next, we show the numerical examples of constrained opti-

mization for two types of location probability distributions,

uniform and binomial, like as in the examples of uncon-

strained optimization. In these examples, we assume that due

to hardware constraint, the paging agent can aggregate at

most 5 paging requests, i.e. aggregation factor k B 5. The

incoming paging rate k = 2.0 and the step of discrete levels

of time is 0.25.

The optimal clusterings with uniform location proba-

bility distribution are presented in Table 4. Figures 6 and 7

show the optimal average paging cost and optimal aggre-

gation factor vs. upper bounds of the average paging delay,

respectively.

The optimal clusterings with binomial location proba-

bility distribution are presented in Table 5. The minimal

average paging costs and the optimal aggregation factors

for different upper bounds of average paging delay are

plotted in Figs. 8 and 9, respectively. When the upper

bound is 1, average paging cost is minimized with k = 1

and the optimal clustering is 1-group clustering for both

examples. Obviously, the optimal average paging costs are

decreasing in upper bounds of the average paging delay.

We can see that with aggregation we can control both

the aggregation factor k and the number of clusters m to

minimize average paging cost while satisfying the con-

strained average paging delay. The aggregate paging

mechanism offers a very low and stable paging cost in both

Figs. 6 and 8.

Table 3 Optimal clustering with binomial location probability distribution

Value of k Patterns of optimal clustering

1 (10), (4,6), (2,3,5), (1,2,2,5), (1,2,2,2,3), (1,1,1,2,2,3), (1,1,1,1,1,2,3), (1,1,1,1,1,1,1,3), (1,1,1,1,1,1,1,2,1), (1,1,1,1,1,1,1,1,1,1)

2 (10), (5,5), (3,2,5), (3,2,2,3), (2,1,2,2,3), (2,1,1,1,2,3), (2,1,1,1,1,1,3), (1,1,1,1,1,1,1,3), (1,1,1,1,1,1,1,2,1), (1,1,1,1,1,1,1,1,1,1)

3 (10), (5,5), (5,2,3), (3,2,2,3), (3,1,1,2,3), (3,1,1,1,1,3), (2,1,1,1,1,1,3), (2,1,1,1,1,1,2,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

4 (10), (5,5), (5,2,3), (3,2,2,3), (3,1,1,2,3), (3,1,1,1,1,3), (3,1,1,1,1,2,1), (2,1,1,1,1,1,2,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

Fig. 4 Average paging cost (binomial)

Fig. 5 Average paging delay (binomial)

Table 4 Optimal clustering with uniform location probability

distribution

Delay upperbound (step 0.25) Optimal clustering

0–0.75 None

1.00–2.00 (10)

2.25 (9,1)

2.5 (8,2)

2.75 (7,2,1)

3–3.25 (7,1,1,1)

3.5–3.75 (6,1,1,1,1)

4–4.50 (5,1,1,1,1,1)

4.75–5.5 (4,1,1,1,1,1,1)

5.75–6.25 (3,1,1,1,1,1,1,1)

6.5–7.25 (2,1,1,1,1,1,1,1,1)

7.5 and above (1,1,1,1,1,1,1,1,1,1)

Wireless Netw (2010) 16:1427–1446 1437

123

Figure 9 shows that for constrained optimization, the

optimal aggregation factor is not necessarily increased in

the upper bound of the average paging delay, given a

discrete step of time. This is due to the dependency of the

average paging cost and delay on k, the clustering of the

PA, and the tradeoff between the average paging cost and

delay. For a given discrete step of time, when the upper

bound of delay is increased step by step, sometimes the

average paging cost can be reduced more by changing the

clustering than by increasing k. We have also observed that

a stochastic order of group location probability distribution

as k increases may not exist.

Now, we demonstrate the impact of aggregation with

another example where the paging agent has a larger

capacity that allows it to aggregate up to 10 paging requests,

i.e. aggregation factor k B 10, and the average incoming

paging rate is lower k = 0.5. The location probability dis-

tribution is binomial.

In this case, when k increases, the aggregation delay

greatly increases, consequently, the average paging delay is

increased significantly. We plot the optimal average paging

cost and aggregation factors in Figs. 10 and 11, respectively.

From Figs. 8 and 10, we can see that the benefit of

aggregation is less for a lower paging request rate. Yet,

Fig. 6 Average paging costs (uniform)

Fig. 7 Optimal aggregation factors (uniform)

Table 5 Optimal clustering with binomial location probability

distribution

Delay upper bound

(step 0.25)

Optimal clustering Optimal

kOptimal

paging cost

0–0.75 None

1.00 (10) 1 10.0000

1.25 (10) 2 4.9902

1.50 (10) 3 3.3236

1.75 (10) 4 2.4902

2.00 (10) 5 1.9903

2.25 (5,5) 4 1.7038

2.50 (6,4) 5 1.4199

2.75 (5,2,3) 5 1.2278

3.00 (5,1,1,3) 5 1.1725

3.25 (4,1,1,1,3) 5 1.1420

3.50 (3,1,1,1,1,3) 5 1.1176

3.75 (3,1,1,1,1,2,1) 5 1.0990

4.00–4.50 (3,1,1,1,1,1,1,1) 5 1.0899

4.75–5.50 (2,1,1,1,1,1,1,1,1) 5 1.0861

5.75 and above (1,1,1,1,1,1,1,1,1,1) 5 1.0859

Fig. 8 Average paging costs (binomial)

Fig. 9 Optimal aggregation factors (binomial)

1438 Wireless Netw (2010) 16:1427–1446

123

aggregation paging is still very effective because it can

bring about great paging cost reduction for any upper

bounds of average paging delay that is larger than or equal

to 1.5 where 1 is the minimal paging delay. The aggrega-

tion factor k also grows slowly as the upper bound of

average paging delay increases.

Note that, without aggregation, the paging cost for any

individual MN is bounded below by 1. However, with

aggregation, the average paging cost per MN is calculated

by dividing the paging cost incurred when paging an

aggregate paging request by the aggregation factor k. This

point is demonstrated in (2). As a result, the average paging

cost (per MN) could be smaller than 1, and this clearly

shows a benefit of aggregation.

The processing capability of the paging agent is an

important factor to aggregate paging system, which can be

measured by the step of discrete levels of time it uses for

calculation and optimization, and the capacity of storing

and aggregating multiple paging requests. To investigate

the impact of the processing capacity of the paging agent,

we further calculate the results for the discrete steps 0.1

and 0.01, and the maximal values of the aggregation factor

5 and 10. As described in our algorithms, the computa-

tional complexity is therefore increasing in the maximal

value of k and decreasing in the value of time step. Our

results for these scenarios are presented in Figs. 12, 13, 14,

15, 16, 17, 18, 19.

Examination of these graphs and our corresponding

program running times reveals an important point: although

with a smaller step of time the results are smoother, the

average paging cost experiences only very minor changes

while the computational complexity increases significantly.

In fact, a paging system requires real time processing, thus

Fig. 10 Average paging costs (binomial, k B 10, k = 0.5)

Fig. 11 Optimal aggregation factors (binomial, k B 10, k = 0.5)

Fig. 12 Average paging costs (binomial, k B 5, k = 2.0, step 0.1)

Fig. 13 Optimal aggregation factors (binomial, k B 5, k = 2.0, step

0.1)


Wireless Netw (2010) 16:1427–1446 1439

123

high computational complexity is not justifiable. Our

aggregate paging system works pretty well with a coarse

step of 0.25 or higher (and for a much larger number of

subnets in the PA). In these cases, the optimal clusterings

can be computed instantly.

It is worth noting that the graphs of the optimal k

are stepped curves in Figs. 11, 13, 15, 17 and 19, while

they are almost monotonous in Figs. 7 and 9. In our

optimization problem described in Sect. 4.4, there is a basic

tradeoff: the average paging cost C(k) will be reduced by

increasing k, but at the same time the average paging delay

will be increased. In (5) of the average paging delay, the

second term is independent of k, but dependent of the

clustering of the PA. For a fixed clustering, when k

increases, the first term increases at the rate that depends on

the value of k For a large (e.g. k = 2), L(k) increases

slowly in k (e.g. with the marginal rate of 0.25 for k = 2,

which is equal to the discrete time step of Figs. 7 and 9).

For a small k (e.g. k = 0.5), L(k) increases sharply in k

(e.g. with the marginal rate of 1.0 for k = 0.5).

Now, the relationship between the marginal rate of L(k)

and the discrete time step used by the paging agent will

make graphs step-wise or rapid rise. If the marginal step is

smaller or equal to the discrete time step as in Figs. 7 and

9, then when the upper bound of average paging delay is

increased by one step, the average cost can be brought

down by increasing k (at least by 1) if keeping the clus-

tering unaltered. If the marginal step is larger than the steps

of discrete time as in Figs. 11, 13, 15, 17 and 19, then when

the upper bound of average paging delay is increased by

one step of time, increasing k to reduce C(k) will violate the

constraint, hence resulting in stepped curves.


0.01)



0.1)



0.01)

1440 Wireless Netw (2010) 16:1427–1446

123

6 Discussion

6.1 Optimization of clustering and aggregation factor

We verify the correlation between the optimal clustering

and the aggregation factor by a simple example. Let’s

investigate an example with N = 10, m = 2, k = 2. The

optimal clustering is v*(k) = (n1*, n2*).

For k = 1, we have

Cð1; vÞ ¼ 1

N

X2

i¼1

ni

Xi

j¼1

nj ¼1

Nn1n1 þ n2ðn1 þ n2Þð Þ

¼ 1

Nðn1 þ n2Þ2 � n1n2

� �¼ 1

NðN2 � n1n2Þ

Cði; v�Þ�Cði; vÞ, n�1n�2� n1n2

It is easy to verify that v�ð1Þ ¼ arg min Cð1; vÞ ¼ fvjðn1;

n2Þ ¼ arg max n1n2g ¼ ð5; 5Þ:This confirms the fact that the optimal clustering is bal-

anced for the uniform distribution of location probability.

For k = 2,

Cð2; v�Þ�Cð2; vÞ

,n�21 n�1 þ ðn�1 þ n�2Þ2 � n�21

� �n�1 þ n�2� �

¼ n�31 þ N3 � Nn�21 � n31 þ Nn2

1

,n�21 ðn�1 � NÞ� n21ðn1 � NÞ

,n�21 n�2� n21n2

From this, the optimal partition vector v can be readily

derived: v* = (7,3). Apparently, the optimal clustering has

been changed when k varies from 1 to 2. Note that these

optimal partitions are consistent with our numerical results

in Sect. 5.

6.2 Overhead paging cost

Without generality, let’s normalize the cost of paging 1

subnet with k = 1 to be 1. Assume that the cost of paging a

subnet is proportional to the length of the paging message.

The cost of paging a subnet with aggregation factor k could

be calculated as: 1 ? a(k - 1) where

a ¼ size of an IP address

size of a paging message with k ¼ 1:

According to the structures of paging request message

and paging message in [20], a is pretty small fraction. Note

that there is no change in the formulas of L(.) when we

relax the assumption of negligible overhead paging cost,

and hence, all the properties of L(.) we have stated so far

are still valid.

Let Bi(k) be the cost of paging i groups from 1 to i

with aggregation factor k and Bi,Bð1Þi , we have Bi

(k) =

[1 ? a(k - 1)]Bi. Let CH(.) be the function of average

paging cost, taking into account the overhead paging cost.

Equation (6) now becomes:

CHðkÞ ¼ 1

k

Xm

i¼1

Aki � Ak

i�1

� �BðkÞi

¼ 1

k

Xm

i¼1

Aki � Ak

i�1

� �Bð1Þi 1þ aðk � 1Þ½ �

¼ 1þ aðk � 1Þ½ �CðkÞ

¼ 1� ak

Xm

i¼1

Aki � Ak

i�1

� �Bi þ a

Xm

i¼1

Aki � Ak

i�1

� �Bi

ð15Þ

Unfortunately, the monotonicity of CH(k) in k, as stated in

Propositions 1 and 4, and the convexity of CH(k) with respect

to k, as stated in Proposition 3, do not hold in general.

However, the recursive forms of optimality for unconstrained

optimization problem and constrained optimization problem,

as stated in Propositions 6 and 7, and the monotonicity of the

optimal average paging cost function in the number of

groups, as stated in Proposition 5, continue to hold. The

proofs are pretty simple with a little modification from the

case of negligible overhead paging cost. Due to space

limitation, we omit these proofs in this paper.

The algorithms for the optimization problems in Sects.

4.1 and 4.2 now requires a check over all possible values of

k: 1,…, k*, where k* can be specified similarly as in (13).

Of course the computational complexities are still poly-

nomial. The optimization problem in Sect. 4.3 also requires

a check over all possible values of k: 1,…, kmax. For the

problem in Sect. 4.4, since there is no optimal structure

concerning aggregation factor k in this problem, the algo-

rithm there is still applicable, i.e. we still can solve the

optimization problems in polynomial time.

Given possible huge benefit of aggregation while the

overhead cost is just a small fraction of average paging

cost, we argue that aggregation is recommendable for a

high incoming paging rate and a large PA.

6.3 Inter-PA mobility and node missing problem

It should be an interesting and practical research problem

to consider the case where node missing and inter-PA

mobility are included. In this scenario, the node missing

probability depends on paging delay and therefore it is a

function of the aggregation factor k. To investigate this

problem, some mobility model, PA shape and size need to

be assumed. Also we need to investigate neighbor PAs of

the current PA. The objective is to evaluate boundary

moving in and moving out rates. However, our model of

location probability can no longer be used. We need to

develop a new model to investigate clustering and paging

sequence for dynamic sets of subnets.

Wireless Netw (2010) 16:1427–1446 1441

123

6.4 Convexity of paging cost function

in approximation method

Now we verify the convexity in the approximation method

[17] for a general k. Given a location probability distri-

bution sorted in descending order, we can approximate it

with a decreasing probability density function h(x) which is

comparable to the given discrete distribution, and h(x) is

differentiable. The detail of this approximation is discussed

in [17]. Now the average paging cost and paging delay can

be calculated as:

CðxÞ ¼Xm

i¼1

xi

Zxi

0

hðtÞdt

0

@

1

Ak

�Zxi�1

0

hðtÞdt

0

@

1

Ak0

@

1

A

LðxÞ ¼ d þXm

i¼1

i

Z xi

xi�1

hðtÞdt

where d = (k - 1)/2k. LetdHðxÞdx¼ hðxÞ, we can present

paging cost and delay functions as:

CðxÞ ¼Xm

i¼1

xi HkðxiÞ � Hkðxi�1Þ� �

LðxÞ ¼Xm

i¼1

i HðxiÞ � Hðxi�1Þð Þ þ d

Now we form the Lagrange function and seek to minimize

it over x: Q = C(x) ? a(L(x) - h), a C 0.

Differentiate Q with respect to xn and set to zero, we

have:

oQ

oxi¼ HkðxiÞ � Hkðxi�1Þ þ xikHk�1ðxiÞhðxiÞ

� xiþ1kHk�1ðxiÞhðxÞ � ahðxiÞ¼ HkðxiÞ � Hkðxi�1Þ þ hðxiÞ

�kHk�1ðxiÞxi

� kHk�1ðxiÞxiþ1 � a�¼ 0

However, the most important point is to check if Q is

convex.

Let z(t) = tx ? (1 - t)y where t [ [0,1] and x, y be two

m-dimensional vectors. Q(x) is convex iff Q(z(t)) = Q(t) is

convex in t. We convert the m-dimensional convex prob-

lem into one dimensional problem. Let D = x - y, then

z = t(x - y) ? y = tD ? y.

oQ

oxioxj¼

kHk�1ðxiÞhðxiÞ þ h0ðxiÞ kHk�1ðxiÞxi � kHk�1ðxiÞxiþ1 � a� �

þhðxiÞ kðk � 1ÞHk�2ðxiÞhðxiÞðxi � xiþ1Þ þ kHk�1ðxiÞ� �

; j ¼ i�kHk�1ðxiÞhðxiÞ; j ¼ iþ 1

�kHk�1ðxi�1Þhðxi�1Þ; j ¼ i� 1

0; otherwise

8>>>><

>>>>:

o2QðtÞot2

¼Xm

i;j¼1

o2QðzÞoxioxj

DiDj ¼Xm

i¼1

o2QðzÞox2

i

D2i þ 2

Xm�1

i¼1

o2QðzÞoxioxiþ1

DiDiþ1

¼Xm

i¼1

kHk�1ðxiÞhðxiÞ þ h0ðxiÞ kHk�1ðxiÞxi � kHk�1ðxiÞxiþ1 � a� �

þhðxiÞ kðk � 1ÞHk�2ðxiÞhðxiÞðxi � xiþ1Þ þ kHk�1ðxiÞ� �

( )D2

i � 2Xm�1

i¼1

kHk�1ðxiÞhðxiÞDiDiþ1

¼Xm

i¼1

h0ðxiÞ kHk�1ðxiÞxi � kHk�1ðxiÞxiþ1 � a� �

D2i þ

Xm

i¼1

2kHk�1ðxiÞhðxiÞD2i

þXm

i¼1

k k � 1ð ÞHk�2ðxiÞh2ðxiÞðxi � xiþ1ÞD2i � 2

Xm�1

i¼1

kHk�1ðxiÞhðxiÞDiDiþ1

¼Xm

i¼1

h0ðxiÞ kHk�1ðxiÞxi � kHk�1ðxiÞxiþ1 � a� �

D2i þ kHk�1ðx1Þhðx1ÞD2

1 þ kHk�1ðxmÞhðxmÞD2m

þXm�1

i¼1

kHk�1ðxiÞhðxiÞðDi � Diþ1Þ2 þXm

i¼1

kðk � 1ÞHk�2ðxiÞh2ðxiÞðxi � xiþ1ÞD2i

1442 Wireless Netw (2010) 16:1427–1446

123

For k = 1, we have

o2QðtÞot2

¼Xm

i¼1

h0ðxiÞ xi � xiþ1 � að ÞD2i þ hðx1ÞD2

1

þ hðxmÞD2m þ

Xm�1

i¼1

hðxiÞ Di � Diþ1ð Þ2

h(x) is a decreasing function sodhðxÞdx� 0. Since xi B xi?1

and a C 0, the first term is non-negative and henceo2QðtÞ

ot2 � 0 and Q(t) is convex in t. This is consistent with the

result in [17].

However, we can show that this is not true in general.

o2QðtÞot2

¼Xm

i¼1

h0ðxiÞ kHk�1ðxiÞðxi � xiþ1Þ � a� �

D2i

þ kHk�1ðx1Þhðx1ÞD21

þ kHk�1ðxmÞhðxmÞD2m

þXm�1

i¼1

kHk�1ðxiÞhðxiÞ Di � Diþ1ð Þ2

þXm

i¼1

k k � 1ð ÞHk�2ðxiÞh2ðxiÞ xi � xiþ1ð ÞD2i

If we choose x and y arbitrarily close, then |Di| and |Di -

Di?1| become arbitrarily small. The last term is a quadratic

of k, while the other terms are linear of k. When k getting

large enough, the last term will dominate the expression,

and since this term is negative, the sum becomes negative

with a sufficiently large k.

7 Conclusions

Sequential group paging with aggregation mechanism has

been demonstrated to save the paging cost critically and be

very implementable. However, the performance of these

systems is very sensitive to the aggregate factor and subnet

clustering. This fact calls for a demand to find the optimal

aggregate factor and optimal clustering from different

perspectives requirements. It turns out that the optimization

problem of these systems, which involves the trade-off

between paging cost and paging delay and the interde-

pendence of optimal aggregate factor and optimal subnet

clustering, is quite complicated.

In this paper, we have formulated and solved different

optimization problems of sequential group paging with

aggregation. The average paging cost function C(k,v) is

proven to be a decreasing function of the aggregation factor

k, while the average paging delay L(k,v) is an increasing

function of k. These two functions are shown to be convex

in k. These features help us find the optimal k under a

maximal paging delay constraint easily. For a total cost

function that preserves convexity, we can readily formulate

and solve a convex optimization problem. An important

property of sequential paging with aggregation about the

optimal paging order of subnets is verified. Paging in the

descent order of location probabilities of subnets is shown

to minimize both the average paging cost and paging delay.

With this, we only need to work with partition vectors.

It has been shown in this paper that the optimization of

aggregation factor k and clustering are not separable.

Therefore, we need to address a joint optimization prob-

lem. Using the properties of optimal schemes, namely

monotonicity in the aggregation factor and the number of

groups and the recursive structure, the joint optimization

problem without a constraint on the expected paging delay

can be solved in polynomial time. The joint optimization

problem under constrained expected paging delay is much

harder to solve due to the loss of monotonicity and con-

vexity. However, by specifying the feasible ranges and

employing the recursive form stated in Proposition 7, the

problem is still solvable in polynomial time.

As shown in our numerical examples, by aggregation,

the paging cost can be reduced significantly (several times)

while the average paging delay does not increase much

for a high or moderate paging rate. This huge benefit

makes aggregation paging justifiable and the assumption of

negligible overhead paging cost less influential to our

results.

Appendices

Proof of proposition 2

The problem to be considered is as follows.

Suppose we have two partition vectors v = (|g1|,…,|gm|)

and v0 = (|g01|,…,|g0m|) in which gi : g0i for i = j, l such

that 1 B j\ l B m and PG(gj) B PG(gl). In partition v0, gj

and gl are swapped, all other elements of v and v0 are the

same. We want to see if C(k,v) C C(k,v0) and L(k,v) C

L(k,v0).

Average paging cost function

Cðk; vÞ�Cðk; v0Þ

,Xm

i¼1

Aki � Ak

i�1

� �Bi�

Xm

i¼1

A0ki � A0ki�1

� �Bi

,Xj�1

i¼1

Aki � Ak

i�1

� �Bi þ Ak

j � Akj�1

� �Bj

Wireless Netw (2010) 16:1427–1446 1443

123

þXl�1

i¼jþ1

Aki � Ak

i�1

� �Bi þ Ak

l � Akl�1

� �Bl

þXm

i¼lþ1

Aki � Ak

i�1

� �Bi�

Xj�1

i¼1

A0ki � A0ki�1

� �Bi

þ A0kj � A0kj�1

� �Bj þ

Xl�1

i¼mþ1

A0ki � A0ki�1

� �Bi

þ A0kl � Akl�1

� �Bl þ

Xm

i¼lþ1

A0ki � A

0ki�1

� �Bi

, Akj � Ak

j�1

� �Bj þ

Xl�1

i¼jþ1

Aki � Ak

i�1

� �Bi

þ Akl � Ak

l�1

� �Bl� A

0kj � A

0kj�1

� �Bj

þXl�1

i¼jþ1

A0ki � A

0ki�1

� �Bi þ A

0kl � A

0kl�1

� �Bl

, Akj � A

0kj

� �Bj þ A

0kl�1 � Ak

l�1

� �Bl

þXl�1

i¼jþ1

Aki � Ak

i�1

� �� A

0ki � A

0ki�1

� �� Bi� 0 ð16Þ

Note that in the above derivation, we use Bi = Bi0 for i and

Aj-1 = A0j-1, Al = A0l.For k = 1, we have Ai - Ai-1=PG(gi) = PG(g0i) =

A0i-1, so the above (16) becomes simply the following:

Aj � A0j� �

Bj þ A0l�1 � Al�1ð ÞBl� 0

, PGðgjÞ � PGðglÞ� �

Bj þ PGðglÞ � PGðgjÞ� �

Bl� 0

, PGðglÞ � PGðgjÞ� �

Bl � Bj

� �� 0

This inequality obviously holds.

In general, the inequality is not simplified as for k = 1,

and hence more difficult to verify.

Instead of considering the above inequality, we consider

the case where l = j ? 1, i.e. two groups are adjacent. All

unordered partition can be transformed to the descent order

partition using this basic swap operation as in the bubble-

sorting algorithm. Thus, it suffices to show the inequality

for adjacent groups.

Notice that we can write Ai = ai ? PG(gj), A0i = ai ?

PG(gl) for j B i B l - 1 where

ai ¼ Aj�1 þXi

t¼jþ1PGðgtÞ:

, aj þ PGðgjÞ� �k� aj þ PGðglÞ

� �k� �

Bj

þ al�1 þ PGðglÞð Þk� al�1 þ PGðgjÞ� �k

� �Bl

þXl�1

t¼jþ1

at þ PGðgjÞ� �k� at�1 þ PGðgjÞ

� �k� ��

� at þ PGðglÞð Þk� at�1 þ PGðglÞð Þk� ��

Bt� 0

For l = j ? 1, the above becomes:

aj þ PGðgjÞ� �k� aj þ PGðglÞ

� �k� �

Bj

þ aj þ PGðglÞ� �k� aj þ PGðgjÞ

� �k� �

Bl� 0

, aj þ PGðglÞ� �k� aj þ PGðgjÞ

� �k� �

Bl � Bj

� �� 0

This also obviously holds.

Average paging delay function

The average paging delay function consists of an affine

function of k, which is not dependent on partition v, and a

function dependent on v, but independent of k. The proof is

essentially the same as for k = 1.

Lðk; vÞ ¼Xm

i¼1

PGðgiÞiþLaggr

2¼Xm

i¼1

Ai � Ai�1ð ÞiþL

aggr

2

� Lðk; v0Þ ¼Xm

i¼1

PGðg0iÞiþL0aggr

2¼Xm

i¼1

A0i � A0i�1

� �i

þL0

aggr

2

, Aj � Aj�1

� �jþ Al � Al�1ð Þl� A0j � A0j�1

� �j

þ A0l � A0l�1

� �l

, Aj � A0j

� �jþ A0l�1 � Al�1

� �l� 0

, PGðglÞ � PGðgjÞ� �

l� jð Þ� 0

Proof of Proposition 3

Consider the function C(x), where x is a positive real

number, we will show that it is convex in x.

Let x1, x2 be positive real numbers and x = ax1 ? (1 -

a)x2, where a [ [0,1]. We need to show that C(x) B aC(x1) ? (1 - a) C(x2).

CðxÞ ¼ Cðax1 þ 1� að Þx2Þ

¼ 1

ax1 þ 1� að Þx2

Xm

i¼1

Aax1þð1�aÞx2

i � Aax1þð1�aÞx2

i�1

� �Bi

!

� ax1

Xm

i¼1

Ax1

i � Ax1

i�1

� �Bi þ

1� ax2

Xm

i¼1

Ax2

i � Ax2

i�1

� �Bi

, LHS� 1

x1x2

�Xm

i¼1

Ax1

i � Ax1

i�1

� �ax2Bi

þXm

i¼1

Ax2

i � Ax2

i�1

� �1� að Þx1Bi

�

¼ 1

x1x2

Xm

i¼1

�ax2Ax1

i þ 1� að Þx1Ax2

ið Þ

� ax2Ax1

i�1 þ ð1� aÞx1Ax2

i�1

� ��Bi

1444 Wireless Netw (2010) 16:1427–1446

123

, x1x2

Xm

i¼1

Aax1þ 1�að Þx2

i � Aax1þ 1�að Þx2

i�1

� �Bi

� ax1 þ 1� að Þx2ð ÞXm

i¼1

�ax2Ax1

i þ ð1� aÞx1Ax2

ið Þ

� ax2Ax1


i�1

� ��Bi

,Xm

i¼1

x1x2Aax1þð1�aÞx2

i Bi �Xm

i¼1

x1x2Aax1þð1�aÞx2

i�1 Bi

�Xm

i¼1

ax1 þ 1� að Þx2ð Þ ax2Ax1

i þ ð1� aÞx1Ax2

ið ÞBi

�Xm

i¼1

ðax1 þ ð1� aÞx2Þ ax2Ax1


i�1

� �Bi

,Xm

i¼1

�x1x2A

ax1þð1�aÞx2

i � ax1 þ ð1� aÞx2ð Þ

ax2Ax1

i þ 1� að Þx1Ax2

ið Þ�

Bi

�Xm

i¼1

�x1x2A

ax1þð1�aÞx2

i�1 � ax1 þ 1� að Þx2ð Þ

ax2Ax1


i�1

� ��Bi ð17Þ

Let h Aið Þ ¼ x1x2Aax1þð1�aÞx2

i � ax1ð1� aÞx2ð Þ ax2Ax1

i þ�

1� að Þx1Ax2

i

�; then it is easy to see that the following is a

sufficient condition for the above (17):

Xm

i¼1

hðAiÞ � hðAi�1Þð Þ� 0 ð18Þ

Again we investigate the first derivative of h(Ai):

dhðAiÞdAi

¼ x1x2 ax1 þ 1� að Þx2ð ÞAax1þ 1�að Þx2�1i

� ax1x2 ax1 þ 1� að Þx2ð ÞAx1�1i

� ax1 þ 1� að Þx2ð Þ 1� að Þx1x2Ax2�1i

¼ x1x2 ax1 þ 1� að Þx2ð ÞAi

� Ahx1þ 1�hð Þx2

i � aAx1

i þ 1� að ÞAx2

ið Þ� �

Note that the function ax is convex in x for a [ (0,1], so

by definition, we have

Aax1þð1�aÞx2

i � aAx1

i þ 1� að ÞAx2

i

This implies thatdhðAiÞdAi� 0. Consequently, h(Ai) is non-

increasing function. As Ai-1 \ Ai, we have (18). This implies

that (17) holds, and consequently, C(x) is convex in x.

References

1. Perkins, C. E. (2002). Mobile IP. IEEE Communications Magazine,

May 1997, updated version, IEEE Communications Magazine.

2. Zhang, X., Castellanos, J., Campbell, A. (2002). Design and

performance of mobile IP paging, ACM mobile networks and

applications (MONET). Special Issue on Modeling Analysis andSimulation of Wireless and Mobile Systems, 7(2), 127–141.

3. Ramjee, R., Li, L., Porta, T. L., & Kasera, S. (2002). IP paging

services for mobile hosts. Wireless Networks, 8, 427–441.

4. Zhang, T., Li, S.-W., Ohba, Y., & Nakajima, T. (2002). A flexible

and scalable IP paging protocol. Proceedings of IEEE GLOBE-COM, 1, 630–635.

5. Gustafsson, E., Jonrsson, A., Perkin, C. (2000). Mobile IP

regional registration. Internet Draft, draft-ietf-mobileip-reg-tun-

nel-02.txt, March.

6. Haverinen, H., Malinen, J. (2000). Mobile IP regional paging.

Internet Draft, draft-haverinen-mobileip-reg-paging-00.txt.

7. Akyildiz, I. F., Xie, J., Mohanty, S. (2004). A survey of mobility

management in next-generation all-IP-based wireless systems.

IEEE Wireless Communications, 11(4), 16–28.

8. Reinbold, P., & Bonaventure, O. (2003). IP micro-mobility pro-

tocols. IEEE Communications Survey and Tutorials, 5(1), 40–57.

9. Liebsch, M., & Lamparter, B. (2005). A generic IP paging

architecture and protocol. Computer Networks, 49, 427–448.

10. Castellucia, C. (2001). Extending mobile IP with adaptive indi-

vidual paging: A performance analysis. ACM Mobile Computingand Communication Review, 5(2), 14–26.

11. Xie, H., Tabbane, S., Goodman, D. J. (1993). Dynamic location

area management and performance analysis. In Proceedings of

43rd IEEE Vehicular Technology Conference, pp. 536–539.

12. Varsamopoulos, G., & Gupta, S. K. S. (2004). Dynamically

adapting registration areas to user mobility and call patterns for

efficient location management in PCS networks. IEEE/ACMTransaction on Networking, 12(5), 837–850.

13. Do, H. T., & Onozato, Y. (2005) IP paging scheme adaptive to

mobile host parameters. IEICE Transactions on Fundamentals,E88-A(4), 948–953.

14. Do, H. T., & Onozato, Y. (2007). A comparison of different

paging mechanisms for mobile IP. ACM Journal of WirelessNetworks, 13(3), 379–395.

15. Xie, J. & Akyildiz, I. F. (2002). Novel distributed dynamic location

management scheme for minimizing signaling costs in mobile IP.

IEEE Transactions on Mobile Computing, 1(3), 163–175.

16. Choi, T., Kim, L., Nah, J., & Song, J. (2004). Combinatorial

mobile IP: A new efficient mobility management using mini-

mized paging and local registration in mobile IP environments.

Wireless Networks, 10(3), 311–321.

17. Rose, C., & Yates, R. (1995). Minimizing the average cost of

paging under delay constraints. ACM Journal of Wireless Net-works (WINET), 1, 211–219.

18. Krishnamachari, B., Gau, R.-H., Wicker, S., & Haas, Z. (2004).

Optimal sequential paging in cellular wireless networks. ACMJournal of Wireless Networks (WINET), 10, 121–131.

19. Hung, T. D. & Onozato, Y. (2005). Flexible and effective multi-

step ip paging schemes. IEEE 62nd Vehicular Technology Con-

ference (VTC2005-Fall), Vol. 2, pp. 831–835, Sep. 2005, Dallas,

Texas, USA.

20. Haverinen, H., & Malinen, J. (2006). Method and apparatus for

mobile internet protocol regional paging. US Patent 7,142,520,

issued Nov. 28, 2006.

21. Boyd, S., & Vandenberghe, L. (2004). Convex optimization.

Cambridge: Cambridge University Press.

22. Shaked, M., & George Santhikumar, J. (2006). Stochastic orders(1st ed.). Berlin: Springer.

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Author Biographies

Hung Tuan Do obtained his Ph.

D in Electronics and Computing

at Gunma University, Japan in

2005. Currently, he is working

towards a Doctoral Degree in

Operations Management at

Purdue University, USA. His

research interests include Opti-

mizations, Performance Analy-

sis of Network Systems, Supply

Chain Management.

Yoshikuni Onozato is a Pro-

fessor with Gunma University.

His research interests are in

satellite systems, computer

communication networks and

distributed computing systems

and span the entire spectrum

from the design and perfor-

mance evaluation of these sys-

tems to their implementation.

He is a member of IEEE, ACM,

IPSJ, ORSJ and IEICE.

Ushio Yamamoto received a

Ph.D. degree in information

science from Tohoku University

in 1997. He is now an associate

professor in Dept. of Computer

Science, Gunma University. He

has been engaged in research on

wireless ad hoc networks, mul-

timedia communication net-

works, load balancing on

computer networks. He is a

member of IPSJ and JSAI.

1446 Wireless Netw (2010) 16:1427–1446

123

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