Integrated power and handoff control for next generation wirelessnetworks

Mehmet Akar

Published online: 12 September 2007

� Springer Science+Business Media, LLC 2007

Abstract In this paper, joint downlink power control and

handoff design is formulated as optimization problems that

are amenable to dynamic programming (DP). Based on the

DP solutions which are impractical, two new algorithms

suitable for next generation wireless networks are pro-

posed. The first one is an integrated hard handoff/power

control scheme that endeavors a tradeoff between three

performance criteria: transmitted power, number of hand-

offs, and call quality. The second is a soft handoff/power

control algorithm that also takes into account the additional

cost of utilizing soft handoff. The proposed algorithms

present a paradigm shift in integrated handoff/power con-

trol by capturing the tradeoff between user satisfaction and

network overhead, therefore enjoy the advantages of joint

resource allocation, and provide significant improvement

over existing methods. The achievable gains and the

tradeoffs in both algorithms are verified through

simulations.

Keywords Power control � Handoff � Handover �Resource allocation � Cellular communication systems �Hybrid systems � Wireless

1 Introduction

Due to its capacity and coverage advantages, code division

multiple access (CDMA) has become the leading technol-

ogy to be implemented in next generation wireless systems

[1–5]. The spectral efficiency and transmission quality in a

CDMA system (e.g., IS95, CDMA2000 and WCDMA) are

best achieved by proper implementation of power control

that regulates the signal to interference ratio at the receiver

end, and handoff that updates the assigned base station

based on channel conditions. In hard handoff, an existing

connection is broken before a new one is set up. Soft

handoff, on the other hand, allows the mobile to commu-

nicate with more than one base station at a particular

instant, and thus allows a smoother transition. Some

advantages of soft handoff include the elimination of the

ping–pong effect, macroscopic selection diversity, and

decrease in time constraints in the network, whereas the

disadvantages are increased implementation complexity, a

possible increase in downlink interference, and decreased

network resources [6].

The focus of this paper is joint power control and hard/

soft handoff design. There is a vast amount of literature on

allocation of transmit powers and base station assignment.

In the sequel, we discuss only those which are relevant to

the problems considered herein. A power control and cell-

site selection algorithm that minimized the total transmit-

ted power was first studied in [7, 8]. Later extensions to this

algorithm included channel allocation [9], and beamform-

ing [10]. The implementation of these algorithms lead to

unnecessary handoffs, which can be reduced by using the

hysteresis test [11, 12], the hysteresis/threshold algorithm

[13], or other hard handoff algorithms [14–16]. However,

these hard handoff algorithms do not incorporate transmit

powers in the decision process, and hence possible

advantages of joint resource allocation are not realized. In

an attempt to fulfill this objective in this paper, we propose

to determine the transmitted powers and make hard handoff

decisions from the solutions of optimization problems that

achieve a tradeoff between the following performance

criteria of interest: The expected number of signal

M. Akar (&)

Department of Electrical and Electronic Engineering, Bogazici

University, Bebek, Istanbul 34342, Turkey

e-mail: mehmet.akar@boun.edu.tr

123

Wireless Netw (2009) 15:691–708

DOI 10.1007/s11276-007-0069-y

degradations, number of handoffs, and total amount of

transmitted power. A signal degradation is said to occur if

the averaged signal strength for the mobile user at some

instant falls below some threshold. This may not neces-

sarily imply dropping the call; however, it is an indicator in

assessing the quality of service for the mobile user, e.g.,

low signal degradation would imply better call quality.

Clearly, the latter can be improved by increasing the

number of handoffs, and/or increasing the transmitted

power intended for the user in question. On the other hand,

a higher number of handoffs causes excessive switching in

the network, while higher transmitted power increases

interference for the other users, thereby reducing the net-

work capacity. As will be discussed in Sect. 3, the

proposed hard handoff/power control algorithm strives to

achieve a tradeoff between the above performance criteria

of interest.

Since CDMA gained popularity in early 1990s, there has

been numerous studies conducted on modeling and analysis

of soft handoff [17–24], that either concentrated on eval-

uating the performance of the algorithms in the standards

[21, 24], or analyzing the effects of various parameters

such as the speed of the vehicle, call duration, and the size

of handoff region on system performance [17–20, 22, 23].

An alternative approach to soft handoff design was carried

out in [14, 25–27] which motivates the problems consid-

ered in this paper, and therefore these works are discussed

in more detail below.

Following the framework of [12, 13] that consisted of a

single mobile moving from one base station towards

another, Zhang and Holtzman derived approximate

expressions for the expected number of base stations in the

active set, the expected number of active set updates, and

outage probability, and verified the accuracy of their results

through simulations in [25]. However, no optimization was

done. Considering a similar setting, Asawa and Stark for-

mulated both hard and soft handoff design as discounted

infinite horizon reward/cost stochastic optimization prob-

lems, where the reward was associated with good signal,

and the cost with soft handoff overhead [14]. Using

dynamic programming (DP) arguments in a limited look-

ahead context, both hard and soft handoff algorithms were

derived, and were shown to outperform the algorithm in the

IS-95 standard. Later, finite horizon versions of similar

handoff problems were studied in [15, 16, 26, 27] by

explicitly characterizing the cost criteria of interest to both

the mobile user and the network operator, and numerous

hard/soft handoff algorithms were proposed.

The present paper provides advances on [7–27] by

incorporating power control into a joint decision mecha-

nism that also involves handoff design, thereby taking

advantage of joint resource allocation. This is especially

critical for the implementation of soft handoff during

which several base stations transmit resulting in increased

downlink interference (unless soft handoff and power

control are used intelligently). There have been numerous

studies on downlink power control to decrease interference

during soft handoff [28–33] which concentrate on

improving the performance of the algorithms implemented

in 3G systems [34]. In the standards, outer and inner loops

are used to control the transmit power. Outer loop power

control sets the target for fast power control (inner loop) so

that the required quality is achieved. In IS-95, fast power

control is done every 1.25 ms only in the uplink, whereas

WCDMA supports fast power control at 1.5 kHz in both

directions. When implementing downlink power control,

each base station in the active set may interpret the power

control command that the mobile sends differently leading

to some base stations increasing their transmit powers

whereas others decrease theirs. This problem is referred to

as power drifting, and may degrade downlink performance.

In [28, 29], mechanisms to balance the transmitted powers

from different base stations were discussed. Although

balancing transmit powers maximizes diversity gain, it also

increases downlink interference. In [30], SSDT (Site

Selection Diversity Transmission) power control, which is

also adopted by 3GPP [34], is studied. In SSDT, only the

best base station transmits data, while other base stations in

the active set transmit only on the control channel. Hence

this scheme mitigates downlink interference only at the

expense of giving up the macro-diversity advantage of soft

handoff. A combination of these methods were also

investigated [31–33]. However, the research in [28–33] all

assume that the mobile is already in the soft handoff mode;

therefore joint design of handoff and power control was not

considered. In this paper, we realize this objective by

engineering handoffs and control transmit power so as to

capture the tradeoff between the diversity advantage of soft

handoff and network overhead. In this context, call quality

is an important criterion for the mobile user (and also for

the network operator not to cause dissatisfaction), and it

can be improved for a given user via three methods: (i) by

increasing the transmitted power, (ii) by frequently

updating the active set, or (iii) by employing the soft

handoff mode. It is important that these decisions be made

properly in order to improve the utilization of network

resources. With this in mind, we will show in Sect. 4 that it

might be advantageous to design a joint handoff/power

control algorithm from the solution of an optimization

problem that takes the above criteria of interest into

consideration.

The rest of the paper is organized as follows. In Sect. 2,

combined handoff and power control is modeled as a

hybrid system which involves finite state and continuous

state variables [26]. This model is used in Sects. 3 and 4 to

address power control jointly with hard and soft handoff,

692 Wireless Netw (2009) 15:691–708

123

respectively. Section 5 is devoted to possible extensions.

The proposed algorithms are tested numerically in Sect. 6,

and some concluding remarks are given at the end of the

paper.

2 System description

For simplicity of presentation, a system consisting of a

single mobile and two base stations is considered as in [12–

16, 25–27]; however, the ideas discussed in the sequel can

be generalized to multiple users and cell sites. As shown in

Fig. 1, it is assumed that the mobile is moving on a straight

line with a constant velocity v between two base stations B1

and B2 that are D meters apart. At each discrete-time

instant k, the signal received by the mobile from base

station Bi can be written as

siðkÞ ¼ �g log diðkÞ þ uiðk � 1Þ þ ziðkÞ; i ¼ 1; 2; ð1Þ

where di(k) is the distance between the mobile user and

base station Bi; ui(k–1) is the transmitted power from Bi (in

decibels); zi(k) is the shadowing term; and g is the path-loss

exponent.1 The shadow fading term zi(k) is modeled (in

decibels) as a log-normal Gaussian random process with

zero mean, and exponential autocorrelation function,

E ziðkÞziðk þ mÞ½ � ¼ r2ajmj; where r2 is the variance;

a ¼ e�ds=d0 is the correlation coefficient; ds is the

sampling distance; and d0 is the correlation distance [35].

The received signals are averaged to obtain the filtered

signals

�siðkÞ ¼ b�siðk � 1Þ þ ð1� bÞsiðkÞ; i ¼ 1; 2; ð2Þ

where b ¼ e�ds=dav ; and the parameter dav determines the

decay rate of the exponential averaging window.

Let XiðkÞ 2 f0; 1g; i = 1, 2 denote the state variable

describing whether the ith base station is in the active set or

not: If Xi(k) = 1 (Xi(k) = 0), then Bi is said to be (not) in the

active set at instant k. In addition, a switching input

Ui(k)[{0,1} is assumed to control the value of the state

Xi(k). If Ui(k) = 1 (Ui(k) = 0), the value of the state Xi(k) is

changed (unchanged).

Based on the above discussion, the state evolution can

be described by

xiðk þ 1Þ ¼ fiðxiðkÞ; uiðkÞ; ziðkÞÞXiðk þ 1Þ ¼ FiðXiðkÞ;UiðkÞÞ

�i ¼ 1; 2; ð3Þ

where xiðkÞ ¼ ½diðkÞ; siðkÞ; �siðkÞ�T is associated with Bi, and

the transition functions, fi(xi(k), ui(k), zi(k)), and Fi(Xi(k),

Ui(k)) are given by

fiðxiðkÞ;uiðkÞ;ziðkÞÞ¼diðkÞþð3�2iÞds

�glogdiðkþ1ÞþuiðkÞþziðkþ1Þb�siðkÞþð1�bÞsiðkþ1Þ

24

35;ð4Þ

FiðXiðkÞ;UiðkÞÞ ¼XiðkÞ; if UiðkÞ ¼ 0;

0; if UiðkÞ ¼ 1 and XiðkÞ ¼ 1;1; if UiðkÞ ¼ 1 and XiðkÞ ¼ 0:

8<:

ð5Þ

For convenience, the state and the input will be denoted by

hk ¼ ½xT1 ðkÞ; xT

2 ðkÞ;X1ðkÞ;X2ðkÞ�T ; and pk = [u1(k), u2(k),

U1(k), U2(k)]T, respectively. Given (3), the objective of the

present paper is to jointly determine the inputs ui(k)

(transmitted powers) and Ui(k) (handoff decisions) at each

instant k, in some optimal sense which will be defined later.

Note that the discrete state, X(k) = [X1(k),X2(k)], can take

values from the finite set f½0; 0�; ½1; 0�; ½0; 1�; ½1; 1�g (by

definition). However, throughout the paper it will be

assumed that the mobile user is connected to at least one

base station at all times, eliminating the discrete mode

[0,0].2 Two cases will be considered: (i) XðkÞ 2 Xh,

f½1; 0�; ½0; 1�g; and (ii) XðkÞ 2 Xs,f½1; 0�; ½0; 1�; ½1; 1�g:When XðkÞ 2 Xh; the user is associated with exactly one

base station at any given time, and only hard handoffs are

possible. When both stations are allowed to serve the user

at a given time, i.e., XðkÞ 2 Xs; then soft as well as hard

handoffs can occur. These two cases will be treated in

Sects.3 and 4, respectively.

Fig. 1 Simulation scenario

1 This discrete-time representation is obtained by sampling contin-

uous-time signal strength measurements [16], and it ignores the effect

of multi-path fading. Such an assumption is justified for two reasons:

(i) It is impractical to design handoff algorithms to compensate for

multi-path fading, and (ii) multi-path fading can also be compensated

by diversity combining and interleaving [15].

2 This assumption can be relaxed to accommodate call admission and

call removal.

Wireless Netw (2009) 15:691–708 693

123

3 Integrated power control and hard handoff

In this section, it is assumed that exactly one base station is

serving the mobile at all times, and our objective is to

determine the transmitted powers and jointly make hard

handoff decisions in some optimal sense to satisfy both the

mobile users and the network operator. Note that the

algorithm in [7, 8] can be used for combined power control

and cell-site selection. However, this algorithm is not

robust to fluctuations in the received power level, and

therefore may result in undesired switching in the network

as explained below. Consider the mobile when it is moving

near the boundary of two neighboring cells. In this case, the

combined power control and cell-site selection algorithm in

[7, 8] assigns the base station which needs to transmit the

lowest amount of power to the mobile. However, it is clear

that such a scheme will generate too many unnecessary

handoffs when a small increase in the controlling base

station power is still satisfactory to keep the existing link

intact. The unnecessary handoffs can be reduced by using

the hysteresis test [11, 12], the hysteresis/threshold algo-

rithm [13], or other hard handoff algorithms [14–16].

However, since none of these hard handoff algorithms

incorporate transmit powers in the decision process, the

advantage of joint resource allocation is not realized.

In this section, the objective is to jointly determine the

transmitted powers and make handoff decisions in order to

achieve a tradeoff between the following performance

indicators: Total number of signal degradations (NSD), total

number of handoffs (NH), and total amount of transmitted

power (PT). A signal degradation is said to occur at time k,

if the averaged signal strength for the mobile at that instant

falls below a threshold, D , and the total number of signal

degradations (NSD) is obtained by counting these events on

the way from B1 to B2. Occurrence of a signal degradation

does not necessarily mean that the call is lost, as we assume

that the threshold level, D, can be chosen to be higher than

the actual threshold below which the call is dropped.

Hence, the metric NSD is used to assess the call quality for

the mobile user, and low (high) NSD implies good (poor)

call quality. Note that alternative characterizations of call

quality may be possible. For instance, a service failure can

be assumed if the signal strength drops below a threshold

for more than a certain amount of time. Alternatively, the

duration that the signal is below the threshold can be

considered as a means to characterize call quality. There

are two main reasons why we do not consider such char-

acterizations in this paper: (i) They tend to result in

intractable problem formulations in our framework, and (ii)

more importantly, we believe that the metric NSD adapted

herein provides sufficient means to measure call quality.

The other key criteria of interest are the number of hand-

offs, and the amount of transmitted power. We consider the

number of handoffs (NH) as a performance measure, since

it is important to prevent excessive switching in the net-

work. Last but not least, minimizing transmitted power not

only decreases interference, but also increases network

capacity.

The three performance criteria of interest are mathe-

matically described as

NSD ¼XK

k¼1

IfdðhkÞ\Dg; ð6Þ

NH ¼XK�1

k¼1

IfUðkÞ6¼½0;0�g; ð7Þ

PT ¼XK�1

k¼1

10u1ðkÞ=10 þ 10u2ðkÞ=10h i

; ð8Þ

where If�g is the indicator function; d(hk) is the filtered

signal strength for the mobile in discrete mode X(k)3; and K

is the length of the finite horizon under consideration.

Given (6)–(8), the objective in this section is to derive an

integrated power control and handoff algorithm that

endeavors to minimize a weighted sum of the expected

values of NSD, NH, and PT.

Problem 1 Determine u ¼ ½uð1Þ; uð2Þ; . . .; uðK � 1Þ� and

U ¼ ½Uð1Þ;Uð2Þ; . . .;UðK � 1Þ� which minimize the cost

E½NSD þ cNH þ rPT � ð9Þ

subject to (3) and the power constraints ui (k) £ umax,

i = 1,2 where c is the nonnegative cost of switching, and r

is a nonnegative constant that determines the relative

weight of the total transmitted power.

Remark The formulation of Problem 1 is such as to find

the best integrated hard handoff/power control algorithm

that will minimize the weighted cost in (9). Using (9), the

resource allocation gain (RAG) of the proposed method

can be computed to be the difference between the cost

values achieved by the proposed algorithm and by any

other available technique for all of their parameter choices.

More formally, given two algorithms A1 and A2, the

resource allocation gain of A1 over A2 for the hard handoff

case can be defined as

RAGHðA1;A2Þ ¼Best value of (9) for A2

� Best value of (9) for A1:ð10Þ

In a similar manner, percentage gains can be defined, if

desired. In this paper, we will say that algorithm A1 is

more efficient than algorithm A2 if and only if

3 The effect of co-channel interference will be discussed in Sect. 5.

694 Wireless Netw (2009) 15:691–708

123

RAGH(A1,A2) [ 0. Detailed comparisons of several algo-

rithms will be made in Sect. 6.

The cost in (9) is the expected sum of the three per-

formance criteria of interest in this section, and its

minimization maximizes the call quality for the user while

minimizing the burden on the network, thereby achieving a

tradeoff between the expectations of the mobile users and

the network operator. Alternatively, this tradeoff can be

captured by formulating the problem as

minu;U

E½NH þ aPT � subject to E½NSD� � c; ð11Þ

where the constraint E[NSD] £ c ensures that the QoS

requirement is satisfied. Since the solution for the problem

in (11) can be obtained from the solution of Problem 1 for

appropriately chosen parameters c and r, we will focus on

Problem 1 in the sequel.

We first discuss a couple of limiting cases to gain some

insight into the solution of Problem 1. Consider r = 0. In

this case, since there is no cost associated with total power,

the base stations can transmit at the maximum allowable

level umax, and Problem 1 reduces to hard handoff design for

which several solutions were proposed in [14–16]. Another

limiting case is one when handoffs are free. Setting c = 0,

the optimization problem reduces to minimizing E[NSD +

rPT]. Below we show that the optimal handoff and power

control decisions can be made at each instant k, independent

of the past and future decisions. To this end, define

COST iðuhi; kÞ ¼ minuhi � umax

Q�Dþ miðuhi; kÞ

�r

� �

þ r10uhi=10; i ¼ 1; 2; ð12Þ

where mi (uhi,k), �r; and the function Q(�) are described by

miðuhi; kÞ ¼ b�siðkÞ þ ð1� bÞ½uhi � auiðk � 1Þ þ asiðkÞ� g log½diðk þ 1Þ=da

i ðkÞ��; ð13Þ

�r2 ¼ ð1� a2Þð1� bÞ2r2; ð14Þ

QðxÞ ¼ 1ffiffiffiffiffiffi2pp

Z 1x

e�t2=2dt: ð15Þ

Furthermore, let uH

hi denote the critical value of uhi for

which COST iðuhi; kÞ is attained.

Lemma 1 When hard handoffs are free (c = 0), then it is

optimal to connect to the base station BiH where iH is

obtained from iH ¼ arg mini¼1;2COST iðuhi; kÞ; and the

optimal power uiHðkÞ is given by uH

hiH:

Proof Since handoffs are free (c = 0), minimizing the

total cost in (9) is equivalent to minimizing

P½dð�sðk þ 1Þ;Xðk þ 1ÞÞ\D� þ r 10u1ðkÞ=10 þ 10u2ðkÞ=10� �

ð16Þ

at each step k. Given the current measurements, the above

minimization reduces to (12) for which the critical values

of uhi are umax and the two roots of the second order

equation, ln 1010

uhi þ ð�Dþmiðuhi;kÞÞ22�r2 � lnðkcÞ ¼ 0 where

kc ¼ 10ð1�bÞffiffiffiffi2pp

�r lnð10Þr : (

When handoffs are free, the optimal policy to Problem 1

is described by Lemma 1. However, if c = 0, there is a

conflict between future switching costs, and current call

quality and transmitted power; therefore the complete tra-

jectory of the mobile is necessary to derive the optimal

solution that is discussed next.

3.1 DP solution to Problem 1

The formulation of Problem 1 is so that its solution can be

obtained using DP techniques [36]. To this end, let

gKðhKÞ ¼ IfdðhK Þ\Dg ð17Þ

gjðhj; pjÞ ¼ IfdðhjÞ\Dg þ cIfUðjÞ6¼½0;0�gþ r 10u1ðjÞ=10 þ 10u2ðjÞ=10h i

; j ¼ 1; . . .;K � 1;

ð18Þ

and define the expected cost-to-go at time k as follows:

JkðhðkÞÞ ¼ minpk

EzðkÞjhkgKðhKÞ þ

XK�1

j¼k

gjðhj; pjÞ" #

: ð19Þ

The DP solution is obtained recursively by solving the

cost-to-go at each instant k:

JK�1ðhK�1Þ ¼ minpK�1

EzðK�1ÞjhK�1gKðhKÞ þ gK�1ðhK�1; pK�1Þ½ �

¼ IfdðhK�1Þ\Dg

þ minUðK�1Þ

OCiðK � 1Þ;OCjðK � 1Þ þ c� �

;

ð20Þ

where

OCiðK�1Þ¼ minuiðK�1Þ

P �siðkþ1Þ\D jhK�1ð Þþ r10uiðK�1Þ=10h i

;

i¼1;2:

ð21Þ

and for k¼K�2; . . .;1:

Wireless Netw (2009) 15:691–708 695

123

JkðhkÞ ¼ minpk

EzðkÞjhkJkþ1ðhkþ1Þ þ gkðhk; pkÞ½ �

¼ IfdðhkÞ\Dg þminUðkÞ

OCiðkÞ;OCjðkÞ þ c� �

; ð22Þ

where the indices i and j refer to the controlling and the

complementary base stations at time k, respectively, and

OClðkÞ is the expected optimal cost-to-go if the lth base

station were to serve the mobile at the (k + 1)th instant.

Theorem 1 Suppose the user is associated with base

station Bi at time k. Then a hard handoff from Bi to Bj is

optimal at time k + 1 if and only if

OCjðkÞ þ c\OCiðkÞ; ð23Þ

and Bj should transmit at uH

hj: Otherwise, it is optimal for Bi

to transmit at uH

hi:

The DP solution as described above can be obtained

numerically to determine the optimal handoff and power

control policy pH. This solution relies on prior knowledge

of the future trajectory of the mobile, which makes the DP

algorithm impractical. However, it still can be used as a

benchmark in comparison to other algorithms such as the

limited look-ahead policy discussed next.

3.2 A locally optimal hard handoff/power control

algorithm

One way to obtain a practical algorithm from the DP

solution is to limit the time horizon, and make a decision

by considering only a few number of stages [36]. The

simplest such scenario is to use a one-step lookahead

policy, where the decision process is restricted to times k

and k + 1 only. In this case, the problem reduces to

deciding whether to make a hard handoff at instant k or not,

and to choosing the transmitter power levels u1(k) and u2(k)

which can be computed as follows:

UHðkÞ ¼

½1; 1�; if XðkÞ ¼ ½1; 0� and COST 2

þc\COST 1;or XðkÞ ¼ ½0; 1� and COST 1

þc\COST 2;½0; 0�; otherwise,

8>>>><>>>>:

ð24Þ

uHðkÞ ¼

½uH

h1;�1�T ; if XðkÞ ¼ ½1; 0� and

UHðkÞ ¼ ½0; 0�;or XðkÞ ¼ ½0; 1� and

UHðkÞ ¼ ½1; 1�;½�1; uH

h2�T; otherwise:

8>>>><>>>>:

ð25Þ

In this paper, (24) and (25) together with COST i as defined

in (12) will be referred to as our joint hard handoff/power

control algorithm. In particular, (24) is the hard handoff

decision whereas (25) corresponds to power control. From

(24), we note that a hard handoff is made (i.e.,

U(k) = [1,1]), when the sum of expected cost of being

served by the complementary base station and the cost of

switching (c) is lower than the expected cost of being

served by the current controlling base station. As can be

noted from (12), these expected costs depend not only on

the expected signal quality, but also the expected trans-

mitter power levels. Once the decision whether to handoff

or not is made, the suboptimal downlink transmitted power

levels are given by (25). Recall that the power levels are

given in decibels, therefore the power value �1 corre-

sponds to no transmission, e.g., ½uH

h1;�1�T

denotes B1

transmitting with the power level uH

h1 while B2 does not

transmit. The beauty of this algorithm is that it incorporates

the effect of shadow fading, mobility and the cost of

switching which further reduces undesired switching

between the base stations (i.e., the ping-pong effect is

mitigated). While doing so, a slight increase in transmitter

power is used to keep the call quality at a satisfactory level.

Note that the handoff decision in (24) is a comparison

type algorithm, and reduces to the hysteresis test in [11, 12]

by letting COST i ¼ ��si; and to the hard handoff algo-

rithms in [14–16] by letting COST i ¼ Q �Dþmiðumax;kÞ�r

� �;

i ¼ 1; 2: A simple comparison of these expressions with

COST i as in (12) reveals that the proposed algorithm

enjoys the advantages of joint resource allocation, while

other hard handoff algorithms [11, 12, 14–16] integrated

with power control (25) do not.

Another conceptually noteworthy point is that the

decisions that are made by (24) and (25) are not indepen-

dent of each other, but require joint optimization using

different techniques due to the nature of the variables

involved. In particular, power control is continuous, and we

have used techniques of differentiation given a fixed

assignment of base stations. However, among a finite

number of such possible assignments, one needs to decide

on whether it would be advantageous to make a handoff,

taking into account also the cost of switching as well. Thus,

the algorithm is hybrid in the sense that the decision is

based on both the continuous (e.g., power) and discrete

(base station assignment) variables.

4 Integrated power control and soft handoff

Due to the nature of CDMA signaling, further performance

improvements over joint power control and hard handoff

algorithms are possible by considering soft handoff and

power control. Although soft handoff provides macro-

scopic diversity, it is also more complex to implement, and

is likely to increase downlink interference. The increased

complexity in implementation stems from the extra

696 Wireless Netw (2009) 15:691–708

123

parameters that are incorporated into the standards. For

instance, the three parameters in IS-95A are [37]: Add

threshold (Ta), drop threshold (Td), and drop timer thresh-

old (Td,t). In IS-95A, a base station is added to the active set

when the pilot strength exceeds Ta, and it is dropped from

the active set only when the pilot stays below Td for at least

Td,t seconds. The IS-95A algorithm was later modified in

IS-95B by allowing dynamic thresholds. The soft handover

algorithm in WCDMA is more complicated, and uses the

following five parameters [5, 34]: Add hysteresis (ha), drop

hysteresis (hd), replacement hysteresis (hr), threshold for

soft handover (Ts), and time to trigger ðDTÞ. The WCDMA

handover algorithm operates as follows:

(Addition) A base station is added to the active set if the

active set is not full, and its pilot exceeds best–pilot–

Ts + ha for a period of DT, where best–pilot is the best

pilot already in the active set.

(Removal) A base station is dropped from the active set

if its pilot stays below best–pilot–Ts + hd for a period of

DT .

(Replacement) If the active set is full and best candidate

pilot is better than worst–pilot–Ts + hr for a period of

DT , where worst–pilot is the worst pilot already in the

active set.

When implementing soft handoff, outer and inner loops

are used to control the transmit power. Outer loop power

control sets the target for fast power control (inner loop) so

that the required quality of service is achieved. In IS-95,

fast power control is done every 1.25 ms only in the uplink,

whereas WCDMA supports fast power control with

1.5 kHz in both directions. As discussed in the introduc-

tion, a vast amount of research has been reported on power

control for soft handoff [28–33]; however all assume that

the mobile is already in the soft handoff mode, and

therefore a truly joint design of handoff and power control

was not considered.

The objective in this section is to engineer handoffs and

control transmit power so as to capture the tradeoff

between the diversity advantage of soft handoff and net-

work overhead. Once again, call quality is the important

criterion for the mobile user, which can be improved by

utilizing soft handoff. However, when X(k) = [1,1], there is

increased complexity in implementation and decreased

network resources, simply because there are more base

stations that are transmitting signals. Obviously, this is

undesirable unless the call for the mobile of interest is

about to be lost. In this section, we characterize the per-

formance criteria of interest to the mobile user and the

network operator, and incorporate them in an optimization

problem to design a joint handoff/power control algorithm.

The call quality will be accessed through the signal

degradation criterion in (6) with a slight modification on

the characterization of dð�sðkÞ;XðkÞÞ: In particular, selec-

tion diversity will be assumed in the soft handoff mode,

and hence dð�sðkÞ;XðkÞÞ is given by dð�sðkÞ;XðkÞÞ¼ max �s1ðkÞ; �s2ðkÞ½ � when X(k) = [1,1]. Other combining

techniques such as equal gain and any other weighted

combining of the signals are also possible [26]. The call

quality for the mobile may be improved via three methods:

(i) by increasing the transmitted power, (ii) by frequently

updating the active set, or (iii) by employing the soft

handoff mode. Associated with these three methods will be

three performance criteria: Total power used (PT), total

number of active set updates (NU), and total number of base

stations in the active set (NB). The total power used (PT) for

the downlink is still characterized by (8). Similarly, NU and

NB are expressed as

NU ¼XK�1

k¼1

IfUðkÞ6¼½0;0�g; ð26Þ

NB ¼ K þXK

k¼1

IfXðkÞ¼½1;1�g: ð27Þ

Note that NU replaces the total number of handoffs, which

is considered in the hard handoff algorithm design in the

previous section. Even though NU and NH have the same

description, NU can conceptually be viewed as an extension

of NH since the discrete state space is larger for the joint

soft handoff and power control design. On the other hand,

the total number of base stations in the active set (NB) is the

additional criterion that will be taken into account in this

section. Since we assume that the mobile is connected to at

least one base station at each step, the constant K (total

number of steps from B1 to B2) is added to the expression

for NB. Considering this additional performance measure is

important from the network operator’s point of view in

order to penalize unnecessary use of the soft handoff mode.

The objective in this section is to derive an algorithm that

endeavors to jointly minimize a weighted sum of the

expected values of NSD, NU, NB and PT.

Problem 2 Determine u ¼ ½uð1Þ; uð2Þ; . . .; uðK � 1Þ� and

U ¼ ½Uð1Þ;Uð2Þ; . . .;UðK � 1Þ� which minimize the cost

E gKðhKÞ þXK�1

j¼1

gjðhj; pjÞ" #

ð28Þ

with gKðhKÞ ¼ IfdðhK Þ\Dg þ cbIfXðKÞ¼½1;1�g; and gj(hj,pj)

for j ¼ 1; . . .;K � 1 as

gjðhj; pjÞ ¼IfdðhjÞ\Dg þ cuðjÞIfUðjÞ6¼½0;0�gþ r 10u1ðjÞ=10 þ 10u2ðjÞ=10h i

þ cbIfXðjÞ¼½1;1�g;

ð29Þ

Wireless Netw (2009) 15:691–708 697

123

subject to (3) and the power constraints ui(k) £ umax where

cuðjÞ ¼ch; if UðjÞ ¼ ½1; 1�;cs; if UðjÞ2f½0; 1�; ½1; 0�g;

and the constants r,

ch, cs, and cb are all nonnegative.

The cost in (28) is the expected sum of the four per-

formance criteria of interest in this section, and its

minimization maximizes the call quality for the user

while minimizing the network overhead. In this formu-

lation, the weight factor r determines the relative

importance of the transmitted power, whereas cb is the

extra cost of utilizing soft handoff.4 Similarly, ch and cs

are the switching costs that are used to penalize the active

set updates; a cost of ch is incurred if a hard handoff is

performed, and a switching cost of cs (cs £ ch) is assumed

when a base station is added to, or dropped from the

active set.

Remark Analogous to the hard handoff case, given two

soft handoff/power control algorithms A1 and A2, the

resource allocation gain of A1 over A2 can be defined as

RAGSðA1;A2Þ ¼Best value of (28) for A2

� Best value of (28) for A1: ð30Þ

We will say that algorithm A1 is more efficient than

algorithm A2 if and only if RAGS(A1,A2) [ 0.

Before we dwell into the general solution of Problem 2,

we discuss a couple of limiting cases. First consider r = 0.

In this case, since there is no cost associated with total

power, the base stations can transmit at the maximum

allowable level umax, and the problem reduces to the soft

handoff design considered in [14, 26, 27]. Another limiting

case is one when active set updates are free. Setting ch =

cs = 0, the optimization problem reduces to minimizing

E[NSD + rPT + cb NB]. To state the solution in this case, let

COST i be as in (12), and define COST 12 by

COST 12 ¼ minus1;us2 � umax

Y2

i¼1

Q�Dþ miðusi; kÞ

�r

� �

þ rX2

i¼1

10uH

si =10 þ cb;

ð31Þ

where mi (usi,k) is given by (13), and uH

si is the value of usi,

i = 1,2 for which COST 12 is achieved.

Lemma 2 When active set updates are free, i.e., ch =

cs = 0, then

(i) it is optimal that only Bi should transmit at uH

hi if and

only if COST i�minfCOST j; COST 12g; and,

(ii) it is optimal to be in the soft handoff mode if and only

if COST 12�minfCOST 1; COST 2g; and the base

stations B1 and B2 should transmit at uH

s1 and

uH

s2;respectively.

Proof Suppose ch = cs = 0. Then minimizing (28) over K

steps is equivalent to minimizing

P½dð�sðk þ 1Þ;Xðk þ 1ÞÞ\D� þ r 10u1ðkÞ=10 þ 10u2ðkÞ=10� �

þ cbIfXðkþ1Þ¼½1;1�g ð32Þ

at each step k. Given the current state, the above

minimization can be done by taking the minimum of

the expected cost (31) in the soft handoff mode and the

costs in (12). In minimizing (31), the critical points can

be obtained from the roots of two coupled nonlinear

equations

ln 10

10us1 þ

ð�Dþ m1ðus1; kÞÞ2

2�r2

� ln kcQ�Dþ m2ðus2; kÞ

�r

� � �¼ 0; ð33Þ

ln 10

10us2 þ

ð�Dþ m2ðus2; kÞÞ2

2�r2

� ln kcQ�Dþ m1ðus1; kÞ

�r

� � �¼ 0: ð34Þ

(

Lemma 2 describes the optimal policy to Problem 2

when the active set updates are free. If this is not the

case, then there is a conflict between future switching

costs, and current call quality, transmitted power and cost

of being in the soft handoff mode. Hence information

about the future is needed to compute the optimal DP

solution.

4.1 DP solution to Problem 2

For notational convenience, let

OC12ðK�1Þ¼ minu1ðK�1Þ;u2ðK�1Þ

"cbþP maxf�s1;�s2g\D jhK�1ð Þ

þrX2

i¼1

10uiðK�1Þ=10

#:

ð35Þ

Then the expected cost-to-go at time k as defined in (19)

can be computed recursively as follows:

4 Since the possible adverse effect of downlink interference is already

captured in the total transmitted power, cb is used to model additional

signaling and computational cost associated with utilizing soft

handoff.

698 Wireless Netw (2009) 15:691–708

123

where OCiðkÞ and OC12ðkÞ are the expected optimal costs if

only Bi or both base stations were to serve the mobile at the

next instant k + 1, respectively.

Theorem 2 Suppose the user is associated with base

station Bi at time k. Then

(i) a hard handoff from Bi to Bj is optimal at time k + 1 if

and only if

OCjðkÞ þ ch\minfOCiðkÞ;OC12ðkÞ þ csg ð38Þ

and the controlling base station Bj transmits at uH

hj:

(ii) a soft handoff is optimal (i.e. both base stations serve

the mobile) at time k + 1 if and only if

OC12ðkÞ þ cs\minfOCiðkÞ;OCjðkÞ þ chg ð39Þ

and the base stations B1 and B2 transmit at uH

s1 and uH

s2;

respectively.

Theorem 2 part (i) provides the necessary and sufficient

conditions for performing a hard handoff at time k. The

second part of the theorem, part (ii), delineates the condi-

tions for adding a base station to the active set, and hence

for soft handoff to be optimal. The following theorem

states similar conditions for dropping a base station from

the active set.

Theorem 3 Suppose that both base stations are in the

active set at time k. Then base station j is dropped from the

active set if and only if

OCiðkÞ þ cs\minfOC12ðkÞ;OCjðkÞ þ csg; ð40Þ

and it is optimal that base station Bi transmits at uH

hi:

Otherwise, staying in the soft handoff mode is optimal.

4.2 A locally optimal handoff/power control algorithm

As in the previous section, the DP solution requires the

complete trajectory of the mobile in advance which makes

it impractical. By restricting the decision process to times k

and k + 1 only, the following algorithm is obtained:

JK�1ðhK�1Þ ¼minpK�1

EzðK�1ÞjhK�1gKðhKÞ þ gK�1ðhK�1; pK�1Þ½ �

¼ IfdðhK�1Þ\Dg þ cbIfXðK�1Þ¼½1;1�g

þ

minUðK�1Þ OCiðK � 1Þ;OCjðK � 1Þ þ ch;OC12ðK � 1Þ þ cs

� �;

if only Bi is serving the mobile at time K � 1;

minUðK�1Þ OCiðK � 1Þ þ cs;OCjðK � 1Þ þ cs;OC12ðK � 1Þ� �

;

if both base stations are serving the mobile at time K � 1;

8>>><>>>:

ð36Þ

JkðhkÞ ¼minpk

EzðkÞjhkJkþ1ðhkþ1Þ þ gkðhk; pkÞ½ �

¼ IfdðhkÞ\Dg þ cbIfXðkÞ¼½1;1�g

þ

minUðkÞ OCiðkÞ;OCjðkÞ þ ch;OC12ðkÞ þ cs

� �;

if only Bi is serving the mobile at time k;

minUðkÞ OCiðkÞ þ cs;OCjðkÞ þ cs;OC12ðkÞ� �

;

if both base stations are serving the mobile at time k;

8>>><>>>:

ð37Þ

Wireless Netw (2009) 15:691–708 699

123

u�ðkÞ ¼½uH

h1;�1�T ; if XHðk þ 1Þ ¼ ½1; 0�;

½ �1; uH

h2�T ; if XHðk þ 1Þ ¼ ½0; 1�;

½uH

s1; uH

s2�T ; otherwise;

8<: ð42Þ

where XH

i ðk þ 1Þ ¼ FiðXiðkÞ;UH

i ðkÞÞ; i = 1,2. Note that

the algorithm in (41)–(42) is a generalization of the inte-

grated hard handoff/power control algorithm given in (24)

and (25) (This can be seen by choosing cb ¼ 1). In par-

ticular, (41) corresponds to the handoff decision. In (41),

the input selection U(k) = [1,1] corresponds to hard hand-

off, and the discrete inputs, U(k) = [1,0] and U(k) = [0,1],

correspond to adding/dropping base stations, B1 and B2, to/

from the active set, respectively (soft handoff). From (41),

we note that a switching from one discrete mode to another

occurs only when the expected cost of being served in the

candidate discrete mode plus the cost of switching to that

mode is less than both the expected cost of being served in

the current mode and the expected cost of being served in

the third mode plus the cost of switching to the third mode.

On the other hand, depending on which base stations are

going to be in the active set, the suboptimal downlink

transmitted power levels are given by (42), e.g., in the soft

handoff mode, the transmit powers are uH

s1 and uH

s2 for B1

and B2, respectively.

The handoff decision in (41) has a structure similar to

the ones studied in [14, 26]. In particular, the simplified

algorithm in [14] is obtained by letting COST i ¼ �Ri;

i ¼ 1; 2; and COST 12 ¼ �R12 � cb (Ri and R12 are the

rewards of being served by Bi only and both base

stations, respectively). On the other hand, if

COST i ¼ Q �Dþmiðumax;kÞ�r

� �; i ¼ 1; 2; COST 12 ¼ COST 1�

COST 2 þ cb; then (41) reduces to the soft handoff algo-

rithm in [26]. Once again, a simple comparison of these

expressions with COST i in (12) and COST 12 in (31)

reveals that the proposed algorithm enjoys the advantages

of joint resource allocation, while other soft handoff

algorithms [14, 26, 27] do not.

4.3 Computational issues

The implementation of the algorithm in (41)–(42) requires

the computations of uH

hi and uH

si ; i = 1,2. The power levels

uH

hi; i = 1,2 can be directly computed as described in Sect. 3.

On the other hand, the candidate solutions for uH

si ; i = 1,2

must be obtained from the roots of (33) and (34), which does

not seem to be analytically possible, since these equations

are coupled and highly nonlinear. Therefore, we will resort

to nonlinear programming techniques [38]. In particular, we

will use Newton’s method to compute uH

s1 and uH

s2: To this

end, let pn = [us1(n),us2(n)]T, where usi(n) is an estimate for

uH

si ; i = 1,2 at the nth iteration. The gradient descent type

algorithm that will be used in this paper is given by

pnþ1 ¼ pn � anDnrJðpnÞ ð43Þ

where an is the positive step size, Dn is a positive definite

matrix, and J(pn) is the argument of (31). The simple

choice of Dn = I leads to the steepest descent algorithm,

however for faster convergence, we use Dn ¼ r2JðpnÞð Þ�1

provided that r2JðpnÞð Þ�1is positive definite. If

r2JðpnÞð Þ�1is not positive definite, then Dn is taken to be

0.001I. There are also numerous ways of choosing the step

size an. In this paper, the step size is successively reduced

according to the Armijo rule [38].

Using the boundedness properties of the function J(pn)

(which can easily be derived from (31)), it can be proven

that the sequence pn converges [38]. However, one setback

is that the function (31) is not convex in us1 and us2,

therefore any iterative method could converge to a local

minimum which makes the initial guess for the iterative

algorithm crucial. A sample plot of the function in (31) for

the values of u1; u2 2 ð�1; umax ¼ 105� is shown in Fig. 2.

It is also clear from this plot that the cost criterion in (31) is

bounded both from above and below. Hence, it can be

concluded that the sequence pn converges [38]. Further-

more, it is not difficult to see that the function J(pn) is

convex in a region ½u01; umax� � ½u02; umax� for some u01� umax

and u02� umax: Therefore a good starting point for the

U�ðkÞ ¼

½1; 1�; if XðkÞ ¼ ½1; 0� and COST 2 þ ch\minfCOST 1; COST 12 þ csg;or XðkÞ ¼ ½0; 1� and COST 1 þ ch\minfCOST 2; COST 12 þ csg;

½1; 0�; if XðkÞ ¼ ½0; 1� and COST 12 þ cs\minfCOST 2; COST 1 þ chg;or XðkÞ ¼ ½1; 1� and COST 2 þ cs\minfCOST 12; COST 1 þ csg;

½0; 1�; ifXðkÞ ¼ ½1; 0� and COST 12 þ cs\minfCOST 1; COST 2 þ chg;or XðkÞ ¼ ½1; 1� and COST 1 þ cs\minfCOST 12; COST 2 þ csg;

½0; 0�; otherwise,

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

ð41Þ

700 Wireless Netw (2009) 15:691–708

123

algorithm is usi = umax, i.e., we initialize with the maxi-

mum power levels.5 Although we have not managed to

prove analytically that the algorithm converges to the

global minimum, the desired results are always obtained in

our simulations as will be demonstrated in Sect. 6.

5 Extensions

In this section, we elaborate on extensions to the algo-

rithms discussed in the previous two sections. It is

noteworthy to point out that parallel results can be pre-

sented for the uplink as well. In particular, a model similar

to (3) can be developed, and the cost criteria of interest can

be modified appropriately to handle the challenges in the

reverse link. However, these details will be omitted in this

paper. Instead, we will focus on the effects of multiple base

stations and interference.

5.1 Extension to multiple base stations

In case of N base stations (N [ 2), the proposed algorithms

can still be used by suitably modifying the cost expres-

sions, COST 1; COST 2 and COST 12: To this end, we start

with integrated power control and hard handoff design. We

first compute COST i; i ¼ 1; 2; . . .;N from (12), and then

evaluate

BcðkÞ ¼ arg mini¼1;...;N

COST i; ð44Þ

where Bc(k) is the index of candidate controlling base

station at the next instant. If COST BcðkÞ þ c\COST BðkÞ(where B(k) is the index of the controlling base station),

then a handoff is performed to the base station with the

index Bc(k). If a handoff is made, then the transmit power is

uH

BcðkÞ; otherwise it is uH

BðkÞ: As such, the computational

complexity of extending the algorithm to N base stations is

linear in the number of base stations, whereas it is more

intensive for soft handoff.

For integrated soft handoff/power control in the case of

N base stations, one needs to minimize the cost function,

Yi¼1;...;N

Qmiðusi; kÞ � D

�r

� �þX

i¼1;...;N

ri10usi=10 þ cbjUðkÞj;

ð45Þ

where |U(k)| denotes the norm of U(k). Since U(k) assumes

2N different values in the case of N base stations, we need

to solve 2N optimization problems in general. It is shown in

[39] that the largest diversity gain is obtained in going from

N = 1 to N = 2, and diminishing returns are obtained with

increasing N. This is not only true of the selective com-

bining method (i.e., taking the maximum of the received

signals) adapted in this paper, but also is typical for all

diversity techniques [39]. Therefore, it is reasonable to use

the algorithm in (42)–(42) with the two base stations that

have the strongest pilot signals.

5.2 Effect of interference

In the previous sections, integrated handoff and power

control design was carried out by neglecting the possible

effect of co-channel interference. This section is devoted to

relaxing that assumption. Assume that there are N cells,

and each cell serves Mi (i ¼ 1; . . .;N) mobile users. The

candidate mobile is still considered to be moving away

from base station B1 and getting closer to base station B2.

First consider cell 1, and the M1 users it is serving. Let

u1j be the power intended for the jth user from base station

B1. For convenience, the candidate mobile is referred to as

the number 1 user, and u11 corresponds to the power

transmitted by the base station intended for the candidate

mobile. Based on this setting, the total interference expe-

rienced by the candidate mobile stemming from signals

intended for other users in the cell is given byPM1

j¼2 q211;1j10ðu1j�glogd1þz1Þ=10; where q11,1j is the cross-cor-

relation between the pulse shapes or spreading waveforms

for the candidate mobile and the jth interferer; d1 is the

distance between the candidate mobile and B1; and z1 is the

shadowing effect on the communication link between the

candidate mobile and B1.6,7 Similarly, the candidate mobile

experiences the interferencePMl

j¼1 q211;lj 10ðulj�g log dlþzlÞ=10

−100−50

050

100150

−100−50

050

100150

0

0.2

0.4

0.6

0.8

1

1.2

1.4

u1

u2

Dow

nlin

k so

ft co

st

Fig. 2 Sample function for downlink soft cost criterion

5 Another plausible starting point for the algorithm is usi ¼ uH

hi :

6 In this expression, u1j, d1, and z1 are all functions of time; however

the time index has been dropped for simplicity of the presentation.7 This model is quite general and meant to capture many of the non-

idealities experienced in practical systems in a simple manner. For

example, even if orthogonal spreading codes are employed in a

spread-spectrum system, these codes cease to be orthogonal after

passing through a multipath channel.

Wireless Netw (2009) 15:691–708 701

123

from the lth cell, where dl is the distance between the

candidate mobile and Bl; and zl is the shadowing effect on

the communication link between the candidate mobile and

Bl. The assumption on zl is that it is normally distributed

with zero mean and variance r2zj: Note that it may not be

possible to model zl as a correlated Gaussian random var-

iable as we did in Sect. 2. Now, the carrier-to-interference-

plus noise ratio (CINR) for the mobile in the discrete mode

X(k) = [1, 0] is

where r2n is the noise variance at the receiver. A signal

degradation event in this mode is said to occur if this

quantity falls below the threshold D. Similarly, a signal

degradation occurs in the mode X(k) = [0,1] if

falls below D. In the soft handoff mode, X(k) = [1,1], a

signal degradation event occurs if both of the quantities in

(46) and (47) fall below D.

Inclusion of the effect of interference in the definition of

the signal degradation event ultimately changes the first

terms in the cost criteria (12) and (31). Under the assump-

tion that z1; z2; . . .; and zN are independent, this modification

is facilitated by noting that z1; z2; . . ., and zN are jointly

Gaussian with individual means mzjand variances r2

zj: Then

the first term in (12) is seen to be replaced by (48)

u11� g logd1þ z1� 10log r2nþXM1

j¼2

q211;1j10ðu1j�g logd1þz1Þ=10

þXN

l¼2

XMl

j¼1

q211;lj10ðulj�g logdlþzlÞ=10

!; ð46Þ

u21 � g log d2 þ z2 � 10 log r2n þ

XM2

j¼2

q221;2j10ðu2j�g log d2þz2Þ=10 þ

XN

l¼1;l6¼2

XMl

j¼1

q221;lj10ðulj�g log dlþzlÞ=10

!; ð47Þ

R1�1 � � �

R1�1

QNj¼1;j6¼i

1ffiffiffiffi2pp

rzj

eðzj�mzj

Þ2=2r2zj � Q

mzi�10 log diþ10 log

10ðu11�DÞ=10�PMi

m¼2

q2i1;im

10uim=10

r2nþPN

l¼1;l 6¼i

PMi

m¼2

q2i1;im

10uim=10

0BB@

1CCA

rzj

0BBBBBBBBB@

1CCCCCCCCCA

dzi�1dziþ1. . .dzN ; ð48Þ

702 Wireless Netw (2009) 15:691–708

123

where N–1 inner integrals exist. Recall that the objective is

to find the value of ui1, i = 1,2 which minimize (12) with the

first term replaced. However, in this case the stationary

points cannot be easily obtained as opposed to the case in

Sect. 3. Instead, the gradient descent type algorithm in (43)

needs to be used to find the minimizing value of the power

control variable. Similarly, for the soft handoff mode the

first term in (31) is to be replaced by (49)

where s1c and s2c are obtained from

s1c

s2c

�¼10log D�1

D

r2nþPNl¼3

PMl

j¼1

q211;lj10ðulj�g logdlþzlÞ=10

r2nþPNl¼3

PMl

j¼1

q221;lj10ðulj�g logdlþzlÞ=10

26664

37775

0BBB@

1CCCA;

ð50Þ

and

is assumed to be invertible.

6 Numerical results

In this section, we present some numerical results for the

algorithms proposed in Sects. 3 and 4 to demonstrate the

possible gains achievable by joint resource allocation. The

setup in Fig. 1 is simulated as the mobile user moves away

from B1 towards B2 for the parameter set given in Table 1

[14–16]. The sampling distance ds = 10 m corresponds to

the speed of 72 km/h at a sampling period of 0.5 s. The

signal predictor xðk þ 1Þ ¼ xðkÞ is used in the implemen-

tation of the algorithms. Although performance may

slightly be improved by using more sophisticated predic-

tors, such gains have been shown to be small [15];

therefore, we opt to use the current measurements for

simplicity. As the shadow-fading variance is assumed to be

8 dB, averaged results from Monte Carlo simulations will

be presented for several sets of parameters.

Z 1�1� � �Z 1�1

YNj¼3

1ffiffiffiffiffiffi2pp

rzj

eðzj�mzj

Þ2=2r2zj Q

mz1� s1c

rz1

� �Q

mz2� s2c

rz2

� �dz3. . .dzN ; ð49Þ

DD ¼10ðu11�D�10 log d1Þ=10 �

PM1

j¼2

q211;1j10ðu1j�g log d1Þ=10 �

PM2

j¼2

q221;2j10ðu2j�g log d2Þ=10

�PM1

j¼2

q211;1j10ðu1j�g log d1Þ=10 10ðu21�D�10 log d2Þ=10 �

PM2

j¼2

q221;2j10ðu2j�g log d2Þ=10

26664

37775; ð51Þ

Table 1 Simulation parameters

D = 2000 m base station separation

umax = 105 dBm maximum allowable power

g = 30 dB path-loss exponent

r = 8 dB standard deviation of the fading process

d0 = 30 m correlation distance

ds = 10 m sampling distance

dav = 10 m averaging distance

D ¼ 0 dB threshold of the signal degradation

0 200 400 600 800 1000 1200 1400 1600 1800 200055

60

65

70

75

80

85

90

95

100

105

d1(k)

Dow

nlin

k tr

ansm

ittal

pow

ers

u*h1

u*h2

u*s1

u*s2

Fig. 3 Suboptimal downlink transmitted power levels

Wireless Netw (2009) 15:691–708 703

123

We first examine the transmitted powers. Figure 3

shows uH

hi and uH

si ; i = 1,2 as a function of the distance from

B1 under the assumption that distance information is known

and there is no shadowing. The interesting observation is

that uH

s1 is equal to uH

h1 (similarly for uh2H and uH

s2) till the

mobile gets close to the boundary of the cells, then there is

a neighborhood around the boundary in which the subop-

timal soft power levels, uH

si ; i = 1,2 are significantly lower

than the suboptimal hard power levels, uH

hi; i = 1,2. From

Fig. 3, we also note that 10uH

s1=10 þ 10uH

s2=10� 10min uH

hi=10;

which supports the intuition that the proposed scheme

strives to minimize interference caused to users that are

close to the boundary of cells.

6.1 Results for integrated hard handoff and power

control

We first evaluate the performance of the algorithms pro-

posed in Sect. 3, and compare them with the commonly

used hysteresis test [11, 12] combined with power control

in (25). In the hysteresis test, a handoff is made when the

pilot from the new base station exceeds the current signal

level by a hysteresis margin of h decibels [11, 12]. The

hysteresis algorithm with the power levels uH

hi; i = 1,2 is

tested for various values of h in the interval [0, 15] deci-

bels. Then for given values of c and r, the cost in (9) for the

hysteresis test is evaluated by determining the minimum

cost over all values of h that are considered. This operation

is carried out to ensure that the best performance of the

hysteresis test combined with power control in (25) is

achieved.

Figure 4 shows the results for the DP solution (from

Theorem 1), its suboptimal version (24)–(25), and the

hysteresis handoff algorithm with power control (25). As

predicted by Lemma 1 (when c = 0), the proposed

algorithm (24)–(25) performs the same as the DP solution.

Furthermore, as the cost of switching (c) increases, so does

the difference between the average cost of the DP solution

and its locally optimal version. This is simply due to the

fact that the solution to Problem 1 requires the future state

information which the DP algorithm utilizes, whereas its

suboptimal version does not. Figure 4 also demonstrates

that the proposed method outperforms the hysteresis

handoff algorithm with power control (25) in terms of

minimizing the cost in (9). In particular, we have

RAGH(Algorithm (24–25), Hysteresis with PC) [ 0

which indicates that the proposed algorithm may improve

utilization of network resources while maximizing user

satisfaction.

Figure 5 shows the tradeoff between the three per-

formance criteria of interest for the integrated hard

handoff/power algorithm in (24)–(25). This plot is

obtained by testing the simulation scenario for varying

cost of switching (c) and transmitted power weight factor

(r). Points A, C, D, and F on this graph characterize the

extremes for the algorithm, e.g., point A (high r and

high c) corresponds to low transmit power, low handoffs,

which therefore result in a greater number of signal

degradations and poorer call quality. On the other hand,

point C is obtained by choosing a high value for r and

c = 0. This choice of constants also leads to low power

(because of high r), but a higher number of handoffs

compared to point A, because handoffs are not penalized

as heavily. Similar explanations can be made about the

other sharp points on the curve in Fig. 5, and these

results are tabulated in Table 2, which is useful in

gaining insight into the choice of these parameters c and

r under different channel and network conditions in a

real system. For instance, if there is a maximum number

of signal degradations requirement, such as the one in

00.2

0.40.6

0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10−12

0

0.5

1

1.5

2

2.5

3

3.5

4

rc

Ave

rage

cos

t

DP algorithm

Suboptimal algorrithm

Hysteresis algorithm with PC

Fig. 4 Comparison of hard handoff/power control algorithms

00.5

11.5

22.5

0

5

10

15118

119

120

121

122

123

124

Average number of signal degradations

Average number of handoffs

Ave

rage

am

ount

of t

rans

mitt

ed p

ower

A

B C

D

E

F

G

Fig. 5 Tradeoffs involved in the proposed hard handoff/power

control algorithm (24)–(25)

704 Wireless Netw (2009) 15:691–708

123

(11), this constraint can be satisfied by appropriately

picking the weights c and r, i.e., by operating on a point

in Fig. 5 (e.g. B, or G) which meets the QoS require-

ment, while minimizing the network overhead.

6.2 Results for integrated soft handoff/power control

For integrated soft handoff and power control, we first

compare the performances of the DP solution (from The-

orems 2 and 3), and its suboptimal version given in Sect. 4.

The simulation scenario is tested for several values of the

parameters involved, and the achieved cost as described in

(28) is plotted in Figs. 6 and 7, from which it is noted that

the DP algorithm performs the same as its suboptimal

version when the active set updates are free (as predicted

by Lemma 2). The performances are also close when the

transmit power weight factor r is low, in which case the

transmit powers are not given as much importance. Hence,

it is advantageous for the mobile to go into the soft handoff

mode as soon possible, and therefore soft handoff is used

extensively. However, with increasing r and nonzero cs, the

gap between the performances of the DP solution and its

suboptimal version also increases. This is intuitive since

the optimal solution to Problem 2 (given by the DP algo-

rithm in Theorems 2 and 3) requires the complete future

state, which is not utilized by the suboptimal solution.

On the other hand, the suboptimal algorithm (41)–(42) is

the way to go in practical systems for computational rea-

sons. We now compare its performance with that of the

handoff policy proposed in the IS-95A standard [37].

Although downlink power control is not implemented in

IS-95, the IS-95A handoff algorithm is used here with the

power levels uH

hi and uH

si ; i ¼ 1; 2 to illustrate the benefits of

joint resource allocation.

The IS-95A handoff algorithm is implemented as in

[14], i.e., a base station is added to the active set if its pilot

exceeds a threshold Ta, and is dropped from the active set if

its pilot falls below another threshold Td. Note that power

control was not considered in [14]. As in [14], two cases

for the parameters Ta and Td were considered: (i) Ta ¼umax � g logðD=2Þ þ csoft � 21, Td ¼ umax � g logðD=2Þþcsoft � 24, and (ii) Ta ¼ umax �g logðD=2Þ þ csoft � 20,

Td ¼ umax � g logðD=2Þ þ csoft �25. The simulation results

for these two cases are shown in Figs. 6 and 7 for various

values of cs, ch = 2cs, cb and r. The cost in (28) for the IS-

95 algorithm is evaluated by testing numerous values of the

parameter csoft in the interval [0, 10], and then by taking

the lowest outcome. As in [14], since the values of Ta and

Td depend on the values of csoft, the plots in Figs. 6 and 7

also show the performance of the IS-95A algorithm com-

bined with power control for varying parameters Ta and Td.

In both Figs. 6 and 7, we note that the performance of the

IS-95A algorithm does not change significantly for the two

cases considered; both being outperformed by the algo-

rithms in Sect. 4. Note that RAGS(Algorithm (41–42), IS-

95 algorithm with PC) [ 0. Once again, this demonstrates

the advantage of joint resource allocation using the pro-

posed scheme.

Table 2 Critical points of Fig. 5

Point c r NSD NH P

A High High High Low Low

B Medium High Medium Medium Low

C Low High Low High Low

D Low Low Low High Medium

E Medium Low Low Medium High

F High Low High Low High

G Medium Medium Low Medium Medium

00.1

0.20.3

0.40.5

00.5

11.5

2x 10

−12

0

2

4

6

8

10

12

rcs

Ave

rage

cos

t

DP algorithm

Suboptimal algorithm

IS95 algorithm−parameter sets (i)−(ii)

Fig. 6 Comparison of soft handoff/power control algorithms: cb = 0,

ch = 2cs

00.10.20.30.40.50

0.51

1.52 x 10

−12

2

4

6

8

10

12

14

16

rcs

Ave

rage

cos

t

DP algorithm

Suboptimal algorithm

IS95 algorithm−parameter sets (i)−(ii)

Fig. 7 Comparison of soft handoff/power control algorithms: cb =

0.01, ch = 2cs

Wireless Netw (2009) 15:691–708 705

123

Finally, Figs. 8–11 illustrate the four performance cri-

teria of interest as the cost of switching, cs and the weight

factor for the transmit power, r, are varied. As cs is

increased, the cost of switching becomes more expensive

leading to a decreased number of active set updates

(Fig. 10) and therefore, degraded call quality (Fig. 8). In

order to minimize the degradation in call quality, soft

handoff is utilized more extensively for lower cs (Fig. 9),

and hard handoff combined with increased transmitted

power is used for higher cs (Figs. 9 and 11). On the other

hand, increasing r leads to lower transmit powers (Fig. 11),

which is a major cause of poor call quality (Fig. 8).

However, this adverse effect is mitigated by frequently

updating the active set (Fig. 10) or by entering into the soft

handoff mode (Fig. 9).

In concluding this section, we re-emphasize that we

have presented the simulation results for all meaningful

sets of control parameters r and cs for the problem in

hand, and demonstrated the tradeoffs involved. Clearly,

for the specific choice of the algorithm parameters in a

real network, field tests/measurements have to made

before one can decide which parameter set is the most

convenient for the network operator while meeting the

user demands.

7 Conclusions

In this paper, a framework for joint resource allocation

for next generation wireless networks has been devel-

oped. In particular, integrated power control and handoff

design has been formulated as finite horizon optimization

problems whose solutions were obtained using DP

techniques. Since the optimal solutions require future

state information, locally optimal versions were also

presented for ease of implementation. The proposed

algorithms achieve a tradeoff between user perceived call

quality and network overhead. Simulation results dem-

onstrate these tradeoffs and the achievable gains through

joint resource allocation. The algorithms have the

00.1

0.20.3

0.4

00.5

11.5

22.5

3

x 10−11

0

10

20

30

40

50

60

70

80

csr

Ave

rage

Num

ber

of S

igna

l Deg

rada

tions

Fig. 8 Average number of signal degradations as cs and r are varied

00.050.10.150.20.250.30.350.4

0

1

2

3x 10

−11

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

cs

r

Ave

rage

Num

ber

of B

ase

Sta

tions

in th

e A

ctiv

e S

et

Fig. 9 Average number of base stations in the active set as cs and rare varied

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

01

23

x 10−11

0

5

10

15

20

25

30

cs

r

Ave

rage

Num

ber

of A

ctiv

e S

et U

pdat

es

Fig. 10 Average number of active set updates as cs and r are varied

00.1

0.20.3

0.4

00.5 1

1.52

2.53x 10

−11

92

94

96

98

100

102

104

106

108

cs

r

Ave

rage

Am

ount

of T

rans

mitt

ed P

ower

Fig. 11 Average amount of transmitted power as cs and r are varied

706 Wireless Netw (2009) 15:691–708

123

flexibility to incorporate online estimates of the param-

eters involved, such as the speed of the mobile, and the

parameters of shadow fading. The performance of the

algorithms depend on the sampling interval, ds. Clearly,

too frequent sampling will result in a heavy computation

load, especially for the soft handoff case, where the

optimal power values are obtained through iterative

method. On the other hand, sampling too infrequently

will increase handoff delay [16], i.e., the decision to

switch to the correct base station will be delayed, which

in turn leads to inefficient use of resources and possible

degradation of call quality. One open research topic is to

show the convergence of the gradient descent algorithm

used in Sect. 4 to the global minimum of the function in

(31).

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Author Biography

Mehmet Akar received his B.S.

(1994) and M.S. (1996) degrees

from Bilkent University, Ankara,

Turkey, and his Ph.D. degree

(1999) from the Ohio State Uni-

versity, all in Electrical

Engineering. Subsequently, he

worked at Yale University, The

University of Southern Califor-

nia, and National University of

Ireland, Maynooth. Currently, he

is with the Department of Elec-

trical and Electronic Engineering,

Bogazici University, Istanbul, Turkey. His research interests include

wireless resource allocation, distributed decision making in communi-

cation networks, and stability and control of hybrid systems.

708 Wireless Netw (2009) 15:691–708

123