Analysis of soft handoff algorithm for multi-cellular systems: A finite integral approach

INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMSInt. J. Commun. Syst. 2009; 22:863–884Published online 8 April 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/dac.1003

Analysis of soft handoff algorithm for multi-cellular systems:A finite integral approach

Aniruddha Chandra∗,† and Sanjay Dhar Roy

Department of Electronics and Communication Engineering, National Institute of Technology, Durgapur,M.G. Avenue, Burdwan—713209, West Bengal, India

SUMMARY

In this paper the performance of soft handoff algorithm based on pilot signal strength measurementshas been studied. In connection with the soft handoff it has been observed that the existing analyticexpressions often involve integration in infinite limits. The expressions containing such kind of integralshave been meticulously transformed into finite range equations. Probability of outage, number of basestations (BS) in the active set, and number of active set updates are some of the performance metricscommonly considered in soft handoff. Accurate closed forms for all these performance indicators havebeen computed which, in turn, simplifies the calculation of several other metrics entailing them. Moreover,the underlying system model takes into account of more than two BS (specifically three) extending thetraditional two BS model toward more complex but realistic characterization of soft handoff performanceanalysis framework. Copyright q 2009 John Wiley & Sons, Ltd.

Received 29 May 2008; Revised 4 December 2008; Accepted 16 January 2009

KEY WORDS: soft handoff; multiple base station; finite integral; Q function

1. INTRODUCTION

Future 4G wireless communication promises seamless connectivity, be it intra system or, intersystem, in an attempt to remain always best connected. This causes frequent change in radiochannels, popularly known as handoff. Intra system or horizontal handoffs are initiated due tocoverage loss in the present communication mode whereas inter system or vertical handoffsoriginate when a relatively preferred mode is detected [1]. In situations where mobility is theprime issue, we find that vertical handoffs are rare due to the absence of parallel wide-coveragestandards. The scenario is a dominatingly cellular mobile environment and horizontal handoffstake place as mobile station (MS) moves in or out of cell boundaries. To accommodate the ever

∗Correspondence to: Aniruddha Chandra, Department of Electronics and Communication Engineering, NationalInstitute of Technology, Durgapur, M.G. Avenue, Burdwan—713209, West Bengal, India.

†E-mail: [email protected], aniruddha [email protected]

Copyright q 2009 John Wiley & Sons, Ltd.

864 A. CHANDRA AND S. D. ROY

increasing subscribers, the geographical area covered by cells is diminishing as we move frommacrocell to microcell and further to picocellular architecture. In contrast to conventional cells, thenumber of cell boundary crossings thus increases making handoff the most frequently encounterednetwork function. To make the matter worse, handoffs, providing continuation of calls, have directimpact on the perceived quality of service (QoS) and should be nearly invisible to the mobilesubscriber [2, 3].

There are mainly two types of horizontal handoffs, viz. hard handoff and soft handoff. Hardhandoff relies on break-before-make mechanism during the switching between base stations (BSs),i.e. at a given instant MS is connected to a single BS only. This sometimes leads to the so-called ping-pong effect, the repeated handoff between two BSs caused by rapid fluctuations in thereceived signal strengths from both [3]. A definite hysteresis margin and signal strength averagingare required to overcome such problems. Soft handoff, on the contrary, follows make-before-break algorithm by connecting the MS with several BS. It eliminates the ping-pong effect andoffers smoother user communications without the clicks typical of hard handoff when speechtransmissions are stopped momentarily during handoffs. Although the system overhead is morein the case of soft handoff and it is inherently more complex, it can provide macro-diversity inshadowed regions. More importantly, soft handoff is necessary for systems requiring strict powercontrol. For power control to work properly, the user must attempt to be linked at all times to theBS from which it receives the strongest signal. If this does not happen, a positive power controlfeedback loop could inadvertently occur, causing system problems [4]. Code division multipleaccess (CDMA) is a worldwide accepted standard adopted for increasing system capacity as wellas offering high data rate and high QoS support to end users. However, CDMA suffers from highinterference and near-far effect, which calls for rigid power control. Thus, soft handoff becomes anatural choice for CDMA-based systems. With soft handoff there is a longer mean queuing timeto get a new channel from the target BS, hence this helps to reduce blocking probability. KeepingBS separations and transmitter powers fixed, soft handoff also allows more users for the samerequired EC/I0 (ratio of received energy per chip to total received spectral density) as overalluplink interference is reduced [3].

For hard handoff, the MS is handed over from the serving BS to the target BS normally duringcell boundary crossing. On the other hand, in soft handoff the current BS is changed to any ofthe candidate BS chosen from an active set. The active set is the set of BSs with which a user iscommunicating simultaneously at any given time. It may consist of one BS initially, but other BSsare added when the signal strengths received from them exceed a predefined add threshold Tadd.Conversely, a BS is removed from the active set when signal power received from it drops belowthe drop threshold Tdrop and remains lower than for at least M successive comparisons, whereM is a preset constant drop timer. Other parameters include NBS, which is a system resourceutilization metric denoting the expected number of BSs in the active set. It has a value NBS=1for hard handoff and NBS�1 as handoff gets softer and softer. NUP is the expected number ofchanges in an active set, a measure of network loading [4–7]. The tradeoff is between the numberin the set and the number of active set updates. It was shown that the concept of drop timer greatlydecreases the number of updates to the active set while only slightly increasing the average sizeof the set [3].

Traditionally, the research focus was on the uplink performance of the CDMA in the presenceof soft handoff. The statement can be supported by noting a flurry of open technical contributions[8–11] compared with the fewer literatures available on the downlink performance [12]. Further,research on the tradeoffs and parameter settings has been mostly in the form of simulation studies

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Commun. Syst. 2009; 22:863–884DOI: 10.1002/dac

ANALYSIS OF SOFT HANDOFF ALGORITHM 865

[13, 14] except few [15]. There exist other unaddressed issues as well. For example, Zhang andHoltzman [16, 17] analyzed CDMA soft handoff algorithm based on pilot signal strength measure-ments considering relative drop threshold (Tdrop) and absolute add threshold (Tadd) for the twoBS model. Their model considers just two BS and a user moving between them in a straight line,without any interference. There is a parametric tradeoff curve between NUP and NBS, where Mis the parameter. By increasing Tdrop, NUP decreases significantly, while NBS increases slowly.Earlier Gudmundson [18] proposed a correlation model for log-normal shadow fading based onexperimental results. Using the shadow fading model, Vijayan and Holtzman [19] laid the frame-work for analysis of handoff algorithm with two BS. In general, there are two metrics that are usedto determine the quality of a channel in order to do a handoff-received signal strength indicator(RSSI) and quality indicator (QI). RSSI gives a measure of received signal strength, whereas QIestimates the signal to interference and noise ratio (SIR) [1]. IS 95 and later systems use SIRmeasurements rather than pilot signal strength. Albeit, research in this field is still open. Analysis ofsoft handoff algorithm has been performed by Stuber [20] for the two BS model considering abso-lute threshold for both Tadd and Tdrop. The work is perhaps more justifiable considering that suchabsolute thresholds will not change the performance considerably, though the analysis becomesfar simple.

We have extended the model proposed by Stuber [20] for three BSs since it gives a more gener-alized approach for the performance analysis of soft handoff algorithm [21, 22]. Both Zhang andHoltzman [16] and Stuber [20] considered integral solutions for different performance parametersin infinite domain that involves certain approximation. We propose integral solution for perfor-mance parameters in the finite domain making analysis very accurate and simple. Our contributionsin this paper are two-fold. First, we extended the two BS model with soft handoff to the morerealistic three BS model and analyzed the performance of the algorithm for the extended model.Second, we proposed closed-form solutions for soft handoff performance measure. To the best ofour knowledge, the proposition is new and a stride toward accurate analytic modeling.

The remainder of the paper is organized as follows. In Section 2 we present the basics of thesystem that is modeled. Section 3 gives analytical details along with the calculation of probability ofoutage (Pout), average number of BS in active set (NBS), and average number of active set updates(NUP) considering handover based on absolute pilot signal strength measurements. Transformationof the related integrals to the finite range is given in Section 4. In Section 5, selected numericalresults are illustrated to substantiate the efficacy of the analysis. The paper concludes with asummary of the present work and future extensions in Section 6.

2. SYSTEM DESCRIPTION

In our assumed model, one BS is equivalent to one cell covered by omni-directional antenna. EachBS is added to or dropped from an active set independent of other BSs since we consider absolutethresholds for Tadd and Tdrop. We consider Pout, NBS and NUP as performance metrics for ouranalysis. If NBS is increased, more diversity gain is achieved at the cost of more system overhead.If NBS is kept small then there would be more NUP. One has to choose proper Tadd and Tdropthresholds to keep them optimum. This paper focuses on the selection of proper design parametersfor soft handoff.

The traditional two BS model [20] along with the proposed multi-cellular model is described inFigures 1(a) and (c). As found from Figure 1(c), three BSs, BS A, BS B and BS C are separated



D m

eter

(a) (b)

(c) (d)

Figure 1. Traditional two BS model: (a) layout diagram with two BS labeled as A and B and (b) signalstrengths at MS with correlated shadow fading. Proposed hexagonal multi-cellular model with three BS:(c) layout diagram with three BS labeled as A, B, and C and (d) signal strengths at MS with correlatedshadow fading. The MS moves from BS A to BS B at a constant velocity v m/s. BSs are separated fromone another by D meter. k=1,2, . . . ,D/ds −1(ds =1m) are the measurement points for the MS wherethe pilot signal strength measurements are performed for finding the suitability of addition or deletion of

a particular BS from the active set [23].

by D meters from each other. The MS is moving from BS A to BS B at constant speed (vm/s).The variations in the pilot signal strengths received by the MS are shown for the two models inFigures 1(b) and (d). Assuming log-normal shadowing, the pilot signal strength received from thethree BSs (in dB) at a distance d from BS A can be expressed as follows:

SA(d) = K1−K2 log10(d)+x1(d) (1a)

SB(d) = K1−K2 log10(D−d)+x2(d) (1b)

SC (d) = K1−K2 log10

⎛⎝√(

D

2−d

)2

+ 3

4D2

⎞⎠+x3(d) (1c)



where SA, SB and SC are received signal from BS A, BS B and BS C , respectively, and K1,K2=10� are path loss coefficients with � being the path loss exponent. In addition, x1(d), x2(d)

and x3(d) are independent zero mean stationary Gaussian processes with identical standard devi-ation � accounting for the shadowing. The functions x1(d), x2(d) and x3(d) are assumed tohave exponential correlation as proposed by Gudmundson [18] based on experimental results, i.e.E{x1(d1)x1(d2)}=E{x2(d1)x2(d2)}=E{x3(d1)x3(d2)}=�2 exp(−ds/d0), where d0 is the correla-tion distance, which determines the decaying factor for correlation. The effect of any small-scalemultipath fading effects is averaged out since it has shorter correlation distance compared withshadow fading and hence ignored.

For inclusion or deletion of a particular BS from the active set, the received pilot signal strengthis sampled every ts seconds or equivalently after a sampling distance of ds =vts considering theconstant MS velocity vm/s. We assume pedestrian movement speed of v=2m/s (7.2km/h) andts is 480ms (nearly equal to 0.5 s) so that ds is normalized to 1m. Thus, the received pilot signalstrengths from BSs sampled at kds (=k) distance corresponding to ti can be found by simplysubstituting d with k in (1a)–(1c) as given below:

SA(k) = K1−K2 log10(k)+x1(k) (2a)

SB(k) = K1−K2 log10(D−k)+x2(k) (2b)

SC (k) = K1−K2 log10

⎛⎝√(

D

2−k

)2

+ 3

4D2

⎞⎠+x3(k) (2c)

All these sampled signal values are Gaussian distributed random variables S�(k)∼N (�S�(k),�)

with mean �S�(k) and variance �2. Although the standard deviation remains constant, mean values�S�(k) for the pilot signal strengths from three BSs vary differently and are given as

�SA(k) = K1−K2 log10(k) (3a)

�SB (k) = K1−K2 log10(D−k) (3b)

�SC (k) = K1−K2 log10

⎛⎝√(

D

2−k

)2

+ 3

4D2

⎞⎠ (3c)

The active set maintained by MS should have sufficient number of BSs for macro-diversity, at thesame time if active set contains more BSs as it would increase system overhead. Interestingly, ifthe active set is too small then that will increase the number of the active set updates. Hence softhandover algorithm is designed to maintain an optimum active set. The standard algorithm for softhandoff based on these notations [20] is stated in Figure 2.

It is worth mentioning that the soft handoff strategy depicted above is applicable not only incellular personal mobile communications but also the same basic principle is followed in a wirelesslocal area networks (WLAN). When any mobile data transceiver, typically a laptop or a personaldigital assistant is moving away from a serving network access point (NAP) to the vicinity ofsome other NAP, handoff might occur. A handover algorithm takes collected measurement data asinput and outputs whether or not a handover should take place. This measurement data typicallyinclude the currently received signal strength, the current load on the serving NAP, the signal to



Figure 2. Soft handoff strategy based on absolute pilot signal strength measurements.

interference ratio, the bit-error rate, the carrier interference ratio, and so on [24]. In a typical IEEE802.11 network, these handoffs are realized in hard mode. In future 4G WLAN standards although,soft handoff is supported which enables the mobile device to be associated with several NAPs atthe same time and thus realizing smooth, uninterrupted and secure transition. With soft handoffit is also possible to speed up by changing the order of things, for example one could performauthentication before connecting to the new access point (pre-authentication) or take care aboutthe re-authentication etc.

3. ANALYTICAL MODEL

In order to model the soft handoff algorithm analytically we would first assess the probability ofinclusion of any BS to the active set in a quantitative manner. The inclusion essentially occurs atone of the discrete time instants or measurement points. Suppose BS �, �∈{A, B,C} was not inthe active set at (k−1) time instant. This indicates that the pilot signal strength at (k−1)th instantwas less than the add threshold Tadd or S�(k−1)<Tadd. Now BS � will be in the active set atinstant k if S�(k)�Tadd, i.e. the received signal strength at MS exceeds add threshold. When boththese events occur consecutively, we detect an inclusion of the particular BS to the active set. Thecorresponding probability may be written as [16, 20]

PBS�,add=Prob〈S�(k)�Tadd|S�(k−1)<Tadd〉= Prob〈S�(k)�Tadd, S�(k−1)<Tadd〉Prob〈S�(k−1)<Tadd〉 (4a)

The numerator and denominator may be evaluated as

Prob〈S�(k)�Tadd, S�(k−1)<Tadd〉=∫ Tadd

−∞Q

(Tadd−�S�(k)−�t+��S�(k−1)

�

)fS�(k−1)(t)dt (4b)



and

Prob〈S�(k−1)<Tadd〉=1−Q

(Tadd−�S�(k−1)

�

)(4c)

where Q(·) is the Gaussian Q function [25], � is the correlation coefficient and k is the currentsample being investigated. In addition, fS�(i)(t) denotes the probability density function (pdf) ofa Gaussian distributed random variable t∼N (�S�(i),�) with mean �S�(i) and variance �2, i.e.

fs�(i)(t)=1√2��2

exp

(− (t−�s�(i))

2

2�2

)(5)

Thus, from (4b) and (4c), we obtain

PBS�,add= 1

1−Q


�

) ∫ Tadd

−∞Q

(Tadd−�S�(k)−�t+��S�(k−1)

�

)fS�(k−1)(t)dt (6)

Next, we would study the situation when a BS is dropped from the active set and calculatethe respective probabilities. A BS, BS � is dropped from the active set if its pilot signal strengthdrops below Tdrop for consecutive M samples. Hence, the probability that BS � is dropped fromthe active set at k given that it was in active set at (k−1) [16, 20] as

PBS�,drop = Prob〈S�(k−M)<Tdrop|S�(k−M−1)�Tdrop〉k∏

n=k−M+1

×Prob〈S�(n)<Tdrop|S�(n−1)<Tdrop〉 (7a)

or equivalently,

PBS�,drop= P1k∏

n=k−M+1P2 (7b)

where

P1 = 1

Q

(Tdrop−�S�(k−M−1)

�

) ∫ ∞

Tdrop

[1−Q

(Tdrop−�S�(k−M)−�t+��S�(k−M−1)

�√1−�2

)]

× fS�(k−M−1)(t)dt (7c)

and

P2= 1

1−Q

(Tdrop−�S�(n−1)

�

) ∫ Tdrop

−∞

[1−Q

(Tdrop−�S�(n)−�t+��S�(n−1)

�√1−�2

)]fS�(n−1)(t)dt (7d)

The active set keeps the record of the number of BSs that are in active communication with theconcerned MS. BS � will be in the active set of MS at kth instant either if it was in the active set



at (k−1)th instant and it is not dropped from the active set at kth instant or if it was not in theactive set at (k−1)th instant but is being added at kth instant. Thus, the probability that the activeset contains BS � is

PBS�(k)= PBS�(k−1)(1−PBS�,drop(k))+PBS�,add(k)(1−PBS�(k−1)) (8)

where PBS�(k) refers to the probability that the active set contains BS � at the kth instant. It isobvious that PBSA(0)=1 and PBSB (D/ds)=1.

3.1. Probability of outage

Here, outage refers to the fact that MS is not connected to any of the BSs. MS’s active connectionwill be dropped if the active set does not contain any BS [20]. For the three BS case it is

Pout(k)=(1−PBSA(k))(1−PBSB (k))(1−PBSC (k)) (9)

3.2. Active set size

Active set size is the number of BSs with whom MS is connected at a given time. At any instantk, it is simply the addition of PBS�(k)s; �∈{A, B,C}. Accordingly the expected number of BSsin the active set considering all sampling instants is [20]

NBS= 1

D/ds

D/ds∑k=1

(PBSA(k)+PBSB (k)+PBSC (k)) (10)

3.3. Number of active set updates

The number of active set updates for BS � will be obtained from PBS�,add and PBS�,drop. Theactive set is updated if any new BS is added to it or any BS presently in it, is dropped. Thus, themean number of updates for BS�, considering all sampling instants, is

NBS�UP= 1

D/ds

D/ds∑k=1

(PBS�,add(k)+PBS�,drop(k)) (11)

The total number of active set updates will be the addition of active set updates for all threeBSs. Hence, the number of updates may be found as

NUP=NBSAUP+NBSBUP+NBSCUP (12)

4. FINITE INTEGRAL SOLUTIONS

In the classical definition of Gaussian probability integral function, popularly known as Q function,the argument appears in the integration limit and not in its integrand. This makes computation ofthese functions quite difficult and when one needs to work out any integration involving them, themethod of exchanging sequence of integration cannot possibly be used in the expressions involvingmultiple integrals. Alternate representations of these functions have the integration limits inde-pendent of function arguments. This feature greatly simplifies the evaluation process. In addition,integrations are defined over a finite range, which make the numerical calculation much easier.



For the three BS case, the probability of adding and dropping a specific BS �∈{A, B,C} fromthe active set is given in (5) and (7a)–(7d). From (5) and (7c)

P1 = 1√2��2Q


�

) ∫ ∞

Tdrop

[1−Q

(Tdrop−�S�(k−M)−�t+��S�(k−M−1)

�√1−�2

)]

×exp

⎛⎜⎝−

(t−�S�(k−M−1)

)22�2

⎞⎟⎠ dt (13)

Using the relation Q(−x)=1−Q(x), (13) can be simplified as

P1= 1√2��2Q


�

) ∫ ∞

TdropQ

(�t−M1

�√1−�2

)exp

(− (t−�S�(k−M−1))

2

2�2

)dt (14)

where M1=Tdrop−�S�(k−M)+��S�(k−M−1). Now substituting x=(�t−M1)/(�√1−�2), the

probability P1 becomes

P1=√1−�2

2�

[�Q


�

)]−1 ∫ ∞

M3

Q(x)exp

(− (x−M2)

2

2M24

)dx (15)

where,

M2= �

�√1−�2

{�S�(k−M−1)−

M1

�

}, M3= �Tdrop−M1

�√1−�2

and M4= �

�√1−�2

Let M ′3=|M3|, hence the integration in (15) can be written as a sum of two individual integrations as

IP1 =∫ M ′

3

M3

Q(x)exp

(− (x−M2)

2

2M24

)dx+

∫ ∞

M ′3

Q(x)exp

(− (x−M2)

2

2M24

)dx (16)

The first integral has finite limits, covers the negative range of values for x and vanishesfor M3�0, while the second integral is defined for x�0 but has infinite limits. However, thisdecomposition enables us to write the Q function in the second integral in terms finite integralusing (A4). With the help of this alternate representation and interchanging the order of integration,the second integral can be further simplified as

I2,P1 = 1

�

∫ �/2

0

∫ ∞

M ′3

exp

(− x2

2sin2 �

)exp

(− (x−M2)

2

2M24

)dx

︸︷︷︸I2in,P1

d� (17)

The inner integral may be expressed in the following form:

I2in,P1 =exp

(− M2

2

2(M24 +sin2 �)

)∫ ∞

M ′3

exp

(− (x−M5)

2

2M26

)dx (18)



which when evaluated gives

I2in,P1 =√2�M6 exp

(− M2

2

2(M24 +sin2 �)

)Q

(M ′

3−M5

M6

)(19)

where

M5= M2 sin2 �

(M24 +sin2 �)

and

M6= M4 sin�√M2

4 +sin2 �

Inserting (16), (17) and (19) in (15) gives

P1 =√1−�2

2�

[�Q


�

)]−1{∫ M ′

3

M3

Q(x)exp

(− (x−M2)

2

2M24

)dx

+√2

�

∫ �/2

0M6 exp

(− M2

2

2(M24 +sin2 �)

)Q

(M ′

3−M5

M6

)d�

}(20)

Using the definitions of different parameters Mi =1,2, . . . ,6, and after considerable algebraicmanipulation (see Appendix B) we finally get Equation (21) given at the bottom of the page.

From (5) and (7d), we observe that the integral∫ Tdrop−∞ fs�(n−1)(t)dt , with a change of vari-

able y=−t , can be restated as (2��2)−1/2∫∞−Tdrop

exp(−(y−{−�s�(n−1)})2/2�2)dy, which whensolved gives Q((−Tdrop+�S�(n−1))/�). Further, as Q(−x)=1−Q(x), the integral is equal to1−Q((Tdrop−�S�(n−1))/�). Thus, P2 can be simplified as

P1 =√1−�2

2�

[�Q


�

)]−1⎧⎨⎩∫ |L|/(�

√1−�2)

L/(�√

1−�2)Q(x)

× exp

(− (x�

√1−�2−�S�(k−M)+Tdrop)2

2�2�2

)dx+

√2

�

∫ �/2

0

�sin�√sin2 �+�2 cos2 �

×exp

⎛⎜⎝−

(�S�(k−M)−Tdrop

)22�2(sin2 �+�2 cos2 �)

⎞⎟⎠Q

×⎛⎝ |L|(sin2 �+�2 cos2 �)−(1−�2)(�S�(k−M)−Tdrop)sin2 �

�√1−�2�sin�

√sin2 �+�2 cos2 �

⎞⎠ d�

⎫⎬⎭ (21)



where L=Tdrop(�−1)+�S�(k−M)−��S�(k−M−1)

P2 = 1− 1√2��2

[1−Q


�

)] ∫ Tdrop

−∞Q

(Tdrop−�S�(n)−�t+��S�(n−1)

�√1−�2

)

×exp

(− (t−�S�(n−1))

2

2�2

)dt (22)

Also from (5) and (6)

PBS�,add = 1√2��2

[1−Q


�

)] ∫ Tadd

−∞Q

(Tadd−�S�(k)−�t+��S�(k−1)

�√1−�2

)

×exp

⎛⎜⎝−

(t−�S�(k−1)

)22�2

⎞⎟⎠ dt (23)

Now from (22) and (23), we observe that both the integrations involve a common form

I =∫ T

−∞Q

(T −�S�( j)−�t+��S�( j−1)

�√1−�2

)exp

(− (t−�S�( j−1))

2

2�2

)dt (24)

Substituting z=T − t we can write the integral as

I =∫ ∞

0Q

(C1+�z

�√1−�2

)exp

(− (z−C2)

2

2�2

)dz (25)

where C1=T (1−�)−�S�( j)+��S�( j−1) and C2=T −�S�( j−1). Further substituting x=(C1+�z)/

(�√1−�2) the integration takes the form

I = �√1−�2

�

∫ ∞

C3

Q(x)exp

(− (x−C4)

2

2C25

)dx (26)

where

C3= C1

�√1−�2

, C4= C1+�C2

�√1−�2

and C5= �√1−�2

By letting C ′3=|C3| and mimicking the approach for calculation of P1, the integration in (26) can

also be written as a sum of two individual integrations as

I = �√1−�2

�

{∫ C ′3

C3

Q(x)exp

(− (x−C4)

2

2C25

)dx+

∫ ∞

C ′3

Q(x)exp

(− (x−C4)

2

2C25

)dx

}(27)



The first integral I1 can be readily evaluated by numerical integration and for the second part,an alternate finite integral representation can be derived using (A4) as

I2= 1

�

∫ �/2

0

∫ ∞

C ′3

exp

(− x2

2sin2 �

)exp

(− (x−C4)

2

2C25

)dx

︸︷︷︸I2,in

d� (28)

The inner integral may be expressed in the following form:

I2,in=exp

(− C2

4

2(C25 +sin2 �)

)∫ ∞

C ′3

exp

(− (x−C6)

2

2C27

)dx (29)

which, when evaluated, the overall integration takes the form

I2=√2

�

∫ �/2

0C7 exp

(− C2

4

2(C25 +sin2 �)

)Q

(C ′3−C6

2C7

)d� (30)

where

C6= C4 sin2 �

C25 +sin2 �

and C7= C5 sin�√C25 +sin2 �

From (27) and (30), using the definitions of different parameters Ci , i=1,2, . . . ,7, and aftersome mathematics (see Appendix C) we finally get the following:

I = �√1−�2

�

∫ |H |/(�√

1−�2)

H/(�√

1−�2)Q(x)exp

(− (x�

√1−�2−T +�S�( j))

2

2�2�2

)dx

+√2(1−�2)

�

∫ �/2

0


exp

(− (T −�S�( j))

2

2�2(sin2 �+�2 cos2 �)

)

×Q

⎛⎝ |H |(sin2 �+�2 cos2 �)−(1−�2)(T −�S�( j))sin

2 �

�√1−�2�sin�

√sin2 �+�2 cos2 �

⎞⎠ d� (31)

where H =T (1−�)−�S�( j)+��S�( j−1).Thus from (22), (24) and (31),

P1 = 1−[�

{1−Q


�

)}]−1⎧⎨⎩√1−�2

2�

∫ |HP2|/(�√

1−�2)

HP2/(�√

1−�2)Q(x)

× exp

⎛⎜⎝−

(x�√1−�2−Tdrop+�S�(n)

)22�2�2

⎞⎟⎠ dx



+√1−�2

�

∫ �/2

0

sin�√sin2 �+�2 cos2 �

exp

(− (Tdrop−�S�(n))

2

2�2(sin2 �+�2 cos2 �)

)

×Q

⎛⎝ |HP2|(sin2 �+�2 cos2 �)−(1−�2)(Tdrop−�S�(n))sin

2 �

�√1−�2�sin�

√sin2 �+�2 cos2 �

⎞⎠ d�

⎫⎬⎭ (32)

where HP2=Tdrop(1−�)−�S�(n)+��S�(n−1).Similarly from (23), (24) and (31),

PBS�,add =[�

{1−Q


�

)}]−1⎧⎨⎩√1−�2

2�

∫ |Hadd|/(�√

1−�2)

Hadd/(�√

1−�2) Q(x)

×exp

(− (x�

√1−�2−Tadd+�S�(k))

2

2�2�2

)dx

+√1−�2

�

∫ �/2

0

sin�√sin2 �+�2 cos2 �

exp

(− (Tadd−�S�(k))

2

2�2(sin2 �+�2 cos2 �)

)

× Q

⎛⎝ |Hadd|(sin2 �+�2 cos2 �)−(1−�2)(Tadd−�S�(k))sin

2 �

�√1−�2�sin�

√sin2 �+�2 cos2 �

⎞⎠d�

⎫⎬⎭ (33)

where Hadd=Tadd(1−�)−�S�(k)+��S�(k−1).

5. RESULTS AND DISCUSSIONS

To validate the analytic expressions obtained in Sections 3 and 4, selected numerical results areplotted in this section for the following parameter values (unless otherwise stated explicitly),standard deviation of shadow fading, �=8dB, correlation coefficient, �=0.99, distance betweenBSs, D=2000m, correlation distance, d0=20m, velocity of mobile station, v=1m/s, samplingtime, ts =1s, sampling distance, ds =vt s =1m, Tadd=−120dB and Tdrop=−136dB, drop timer,M=5s. We also assume that K1=0 and K2=40 with path loss exponent �=4.Figure 3 shows the probability of adding a BS for a specific add threshold (Tadd=−120dB).

As is evident from the plot, the maximum add probability occurs at the boundary of the cells. The



Figure 3. Probability of adding a BS to active set as a function of distance from BS A for Tadd=−120dB.

add probability is zero very near to the serving cell which is obvious, as near to the serving BS,signal strength will be large and rarely falls below Tadd.

The probability of dropping a BS from the active set is depicted in Figure 4 for Tdrop=−136dBand M=5. We can find that the maximum drop probability occurs closer to the boundary of thecells for the chosen parameter values. The drop probability is zero very near to the target celli.e. far away from the serving cell. This can be explained as at those instants MS may need theconnectivity with the target cell only while the serving cell’s signal strength is decreased to a largeextent.

In both the figures, for BS C , the drop and add probability values are always nonzero. The dropprobabilities for all BSs are very small (<0.09) for the chosen drop threshold. It will be more ifTdrop is increased. For example, we shall find large drop probability for Tdrop=−129dB, whichmay lead to call drop.

Figure 5 shows assignment probabilities for all three BSs. The assignment probability for BSC is minimum compared with BS A or B as it is far from the considered linear MS path. TheBS A assignment probability decreases whereas the BS B assignment probability increases as MSmoves away from A toward B. When add and drop thresholds are properly chosen, assignmentprobabilities never become zero. But larger Tadd and Tdrop values (e.g. Tadd=−105dB and Tdrop=−129dB) results in a situation similar to hard handoff, which forces the assignment probabilitiesto zero when the MS is far from the serving BS.

The probability of not having sufficient signals from all three BSs or the probability of outagefor different drop timer values are plotted in Figure 6. The set of curves have a common peak at theboundary of cells served by BS A and B, i.e. at a distance of 1000m from BS A. This is because atthe boundary signal strengths from both BSs are low causing maximum outage probability. FurtherPout becomes more when M is lowered. Pout can be reduced if we provide more macro-diversityby selecting small Tadd and small Tdrop, so that any BS may be included to the active set veryquickly and that BS remains in the active set for a long time. In the presence of shadowing, for



Figure 4. Probability of dropping a BS from active set as a function of distancefrom BS A for Tdrop=−136dB and M=5.

Figure 5. Probability of assignment as a function of distance from BSA for Tadd=−120dB and Tdrop=−136dB and M=5.

lower add thresholds (Tadd=−120dB) outage may occur even near to BS A because the respectiveBS (BS A) is added to the active set of MS very early but may not remain in active set as its pilotsignal strength falls below Tadd with MS moving away from BS A toward the cell boundary.



Figure 6. Probability of outage as a function of distance from BS A for Tadd=−105dB andTdrop=−129dB and M=2,3,4 and 5.

Figure 7. Average number of BSs as a function of average number of active set updates for varying valuesof M and different Tadd and Tdropvalues.

Figure 7 shows the tradeoff between NBS and NUP for different add and drop threshold values.As seen from the figure, keeping Tdrop and Tadd fixed, NBS decreases whereas NUP increases fordecreasing M values and vice versa. The general trend shows that when NBS is small NUP comesdown very fast but at the higher range it almost saturates to a constant value.



Table I. NBS and NUP for different parameter values of M , TADD, and TDROP.

M Tadd Tdrop NBS NUP

5 −115 −130 2.5978 0.60605 −112 −138 1.8656 0.56525 −116 −132 2.5991 0.65455 −120 −138 2.5600 0.88695 −120 −136 2.4837 0.75113 −105 −129 1.0592 0.45383 −110 −136 1.4697 0.59692 −120 −136 2.2183 1.0655

Figure 8. Probability of particular BS in active set as a function of probability of active set updates forTadd=−120dB and Tdrop=−136dB and increasing value of M .

Best threshold values are chosen for both NBS and NUP minimum. A compromise has to bemade between NBS and NUP for optimum performance. Both of them increase if Tadd is lowered(see Table I). If more diversity is needed we should decrease Tadd, Tdrop threshold and/or M . Inpractice, if the active set size is two, among all BSs the best two BSs will be included in theactive set. This reduces overhead in the system. If Tadd is increased (e.g. −105dB), NBS and NUPdecrease and that leads to less macro-diversity. On the other side, both NBS and NUP are increasedif Tdrop is decreased. If Tdrop is increased then the probability of outage will increase considerably.Thus, selection of proper add and drop thresholds along with drop timer is vital. The selectionbecomes easier with our finite integral approach.

The tradeoffs for different BSs are depicted separately in Figure 8. Both BS A and BS Bshow similar tradeoff curves. The probability of a BS being in the active set increases when theprobability of dropping the particular BS is decreased.



Table I shows the few results obtained using the analytical model considered here. We find thatthe situation is similar to the hard handoff scenario with M=3 and Tadd=−105dB, Tdrop=−129dBwhen NBS=1.0592 i.e. MS is connected to a single BS only.

6. CONCLUSIONS AND FUTURE WORK

This paper presents an accurate method to choose handover design parameter for soft handoffanalysis. It uses an analytical method for finding the probability of outage, the number of activeset updates and the number of BSs in the active set for a soft handoff algorithm. It has beenobserved that keeping Tdrop and Tadd fixed, NBS decreases whereas NUP increases for decreasing Mvalues and vice versa. Proper threshold values may be obtained from the tradeoff curves and alsoconsidering the macro-diversity requirement. Absolute pilot signal strength measurement-basedsoft handoff algorithm gives an idea about Tadd, Tdrop and M values and such an algorithm maybe analyzed using finite integral solutions as shown in the current literature. The soft handoffalgorithms that are based on relative and absolute measurements based (or signal to interferencebased), although, result in too complex finite integral formulations for more than two BS. In suchcases simulation analysis may be an easier solution.

On the other hand, a more general solution may be obtained when the motion of the MS is notconstrained to the straight line joining BS A and BS B. In fact the present work may be extendedto encompass many different cases, three cells and the MS moving toward the interception of thecells, four cells and the MS moving between BS A and BS B, or the MS is moving while itmaintains a fixed angle with the line joining BS A and BS B.

APPENDIX A: ALTERNATE REPRESENTATION OF Q FUNCTION

From Equation (7.4.11) [26] we have

∫ ∞

0

exp(−at2)

t2+x2dt= �

2xexp(−ax2)erfc(

√ax), a>0, x>0 (A1)

where erfc(·) denotes complementary error function. Let x=1, then the integral representation oferfc(·) becomes

erfc(√a)= 2

�

∫ ∞

0

exp(−a(1+ t2))

1+ t2dt (A2)

With a change of variable t=cot�,

erfc(z)= 2

�

∫ �/2

0exp

(− z2

sin2 �

)d�, z>0 (A3)



Noting that Q(z)=(1/2)erfc(z/√2) the same can be written in terms of Q function

Q(z)= 1

�

∫ �/2

0exp

(− z2

2sin2 �

)d�, z>0 (A4)

as first devised by Craig [27].

APPENDIX B: CALCULATION OF TERMS CONTAINING Mi COEFFICIENTS

The different Mi (i=1,2, . . . ,6) values required for calculation of P1 are enlisted below for thereader’s convenience

M1 = Tdrop−�S�(k−M)+��S�(k−M−1) (B1)

M2 = �

�√1−�2

{�S�(k−M−1)−

M1

�

}= �S�(k−M−1)−Tdrop

�√1−�2

(B2)

M3 = Tdrop(�−1)+�S�(k−M)−��S�(k−M−1)

�√1−�2

= L

�√1−�2

(B3)

where L=Tdrop(�−1)+�S�(k−M)−��S�(k−M−1) and M ′3=|L|/�√1−�2. In addition,

M4 = �

�√1−�2

(B4)

M5 = M2 sin2 �

M24 +sin2 �

=√1−�2 sin2 �

�(sin2 �+�2 cos2 �){�S�(k−M)−Tdrop} (B5)

M6 = M4 sin�√M2

4 +sin2 �= �sin�√

sin2 �+�2 cos2 �(B6)

Next we show the calculation of various terms containing these coefficients, namely

− (x−M2)2

2M24

= − (x�√1−�2−�S�(k−M)+Tdrop)2

2�2�2(B7)

− M22

2(M24 +sin2 �)

= − (�S�(k−M)−Tdrop)2

2�2(sin2 �+�2 cos2 �)(B8)



M ′3−M5

M6= 1

�√1−�2�sin�

√sin2 �+�2 cos2 �

×{|L|(sin2 �+�2 cos2 �)−(1−�2)(�S�(k−M)−Tdrop)sin2 �} (B9)

APPENDIX C: CALCULATION OF TERMS CONTAINING Ci COEFFICIENTS

The different Ci (i=1,2, . . . ,7) values required for calculation of PBS�,add and P2 are enlistedbelow

C1 = T (1−�)−�S�( j)+��S�( j−1) (C1)

C2 = T −�S�( j−1) (C2)

C3 = T (1−�)−�S�( j)+��S�( j−1)

�√1−�2

= C1

�√1−�2

(C3)

C ′3 = |C1|

�√1−�2

(C4)

C4 = C1+�C2

�√1−�2

= T −�S�( j)

�√1−�2

(C5)

C5 = �√1−�2

(C6)

C6 = C4 sin2 �

C25 +sin2 �

=√1−�2

�

sin2 �

sin2 �+�2 cos2 �{T −�S�( j)} (C7)

C7 = C5 sin�√C25 +sin2 �

= �sin�√sin2 �+�2 cos2 �

(C8)

Now we show the calculation of various terms containing these coefficients,

− (x−C4)2

2C25

= −

(x− T −�S�( j)

�√1−�2

)2

2�2

1−�2

=− (x�√1−�2−T +�S�( j))

2

2�2�2(C9)

− C24

2(C25 +sin2 �)

= − (T −�S�( j))2

2�2(sin2 �+�2 cos2 �)(C10)



C ′3−C6

C7= 1


×{ |T (1−�)−�S�( j)+��S�( j−1)|

�√1−�2

−√1−�2

�

sin2 �(T−�S�( j))

(sin2 �+�2 cos2 �)

}

= |C1|(sin2 �+�2 cos2 �)−(1−�2)sin2 �(T −�S�( j))

�√1−�2�sin�

√sin2 �+�2 cos2 �

(C11)

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AUTHORS’ BIOGRAPHIES

Aniruddha Chandra received his BE (Hons.) degree in Electronics and CommunicationEngineering and ME degree in Communication Engineering from Jadavpur University,Kolkata, India, in 2003 and 2005, respectively, and is currently pursuing a PhD there.

He joined the Electronics and Communication Engineering department, NationalInstitute of Technology (NIT) Durgapur in 2005 as a lecturer. His research interestsinclude diversity combining, modulation techniques, location management and datasecurity.

Mr Chandra has co-authored a book titled Analog Electronic Circuits and published18 research papers in reputed journals and peer-reviewed conferences. He has alsodelivered several invited lectures including IEEE COMSOC lecture meetings at JU,Kolkata. He was ranked sixth in the state in the X standard Board Exam and receivedNTSE and EFIP scholarships from NCERT and MHRD, Govt. of India, respectively.He is a member of IEEE (Communication Society), IAENG and has also served as areviewer for IEEE Potentials and Elsevier Computer and Electrical Engineering. He islisted in the forthcoming 2009 issue of Marquis Who’s who in the World.

Sanjay Dhar Roy received his BE (Hons) degree in Electronics and CommunicationEngineering in 1997 from Jadavpur University, Kolkata, India, and his MTech degree inTele-Communication Engineering in 2008 from NIT Durgapur. He is currently pursuinghis PhD at NIT Durgapur.

He worked for Koshika Telecom Ltd. from 1997 to 2000. After that he joined Elec-tronics and Communication Engineering department, National Institute of TechnologyDurgapur as a lecturer in 2000 and is currently a senior lecturer there. His researchinterests include radio resource management, handoff, smart antenna techniques, beam-forming, multi-user detection and optical communication.

Mr Dhar Roy is a member of IEEE (Communication Society) and is a reviewer ofIET Communications and journal of Progress in Electromagnetic Research.


Analysis of soft handoff algorithm for multi-cellular systems: A finite integral approach

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