Performance Evaluation of a Hierarchical Cellular...

Performance Evaluation of a HierarchicalCellular System with Mobile Velocity-Based

Bidirectional Call-Overflow SchemeWenhao Shan, Pingzhi Fan, Senior Member, IEEE, and Yi Pan, Senior Member, IEEE

Abstract—With the increase of teletraffic demands in mobile cellular system, hierarchical cellular systems (HCSs) have been adopted

extensively for more efficient channel utilization and better GoS (Grade of Services). A practical issue related to HCS is to design a

scheme for controlling and allocating call traffic to different layers. There are several strategies to deal with this problem, such as no

call-overflow scheme, unidirectional call-overflow scheme and bidirectional call-overflow scheme. The objective of this paper is to

investigate a bidirectional call-overflow scheme, based on the velocity of the mobile making the calls. To ensure that hand off calls are

given higher priorities, it is assumed that guard channels are assigned in both macrocells and microcells. In order to evaluate the

performance of the new scheme and compare the performance of several related schemes, two new models based on a one-

dimensional Markov process are developed and analytical results are derived. Theoretical analysis and numerical evaluation show that

the proposed scheme outperforms others in terms of average new call blocking and hand off failure probability of the system. In

addition, when the teletraffic to the HCS reaches a certain grade, the GoS is insensitive to the maximum velocity and the velocity

threshold which is used to assign calls to different layers in our scheme.

Index Terms—Hierarchical cellular system, overflow, Markov process, guard channel.

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1 INTRODUCTION

RECENTLY, the demand for wireless communications hasgrown tremendously and a lot of fundamental chal-

lenges and issues on wireless networks and mobilecomputing have been identified, such as hierarchicalcellular network, location management, handover and calladmission, wireless error control and security, data man-agement, mobile agents, broadcast scheduling, routing inwireless ad hoc networks, fixed and dynamic channelassignment, multiple access schemes, MAC protocols,power saving issues, satellite, etc. [1]

In a wireless cellular network, a fixed number ofchannels (frequencies for FDMA, time slots for TDMA, orcode channels for CDMA) are assigned to a given cell. If achannel is used by a call, no other call can use the channelagain in the same cell at the same time. With the decrease inthe size of the cells, the system capacity increases because ofthe more efficient reuse of the frequencies in a given area.However, there is also an increase in the number of cellboundaries that a mobile unit crosses. These boundarycrossings stimulate hand off calls and location trackingoperations, which are very expensive in terms of time delayand communication bandwidth, hence limiting the callhandling capacity of a cellular system. One way ofcontrolling the increase of signaling traffic, while preser-ving the frequency reuse advantage of smaller cells, is to

adopt a hierarchical architecture. In a hierarchical cellularsystem (HCS),1 by using different antenna heights (often onthe same building or tower) and different power levels, it ispossible to provide “large” (macro) and “small” (micro)cells which are colocated at a single location, so a large cell(macrocell) is subdivided into smaller microcells [2] (seeFig. 1). Radio channels are allocated according to differentstrategies to macrocells and to microcells for efficientchannel reuse. The hierarchical cellular system can provideoverlaid microcells for high-teletraffic areas and overlayingmacrocells for low-teletraffic regions.

A call originates from a mobile and the mobile movesaround with different speeds. Based on different mobilevelocities and a predetermined velocity threshold, all thecalls are divided into two groups: fast calls and slow calls(including new calls and hand off calls). In general, the fastcalls are served by the macrocells, while the slow calls areserved by the microcells [3], [4].

In early schemes introduced in [5], [6], for eachindividual call, the serving layer does not change at all,e.g., fast (slow) calls are always served by the macrolayer(microlayer). The approaches to sharing the spectrumbetween two tiers were evaluated by I, et al. [5]. Murataand Nakano describe methods of selecting a cell (tier)during the call setup phase [6]. Both schemes in the abovepapers allow no overflow2 from one tier to another. It isobvious that the procedure of allocating calls in differentlayers is very important. Imperfect calls assignment willresult in some disadvantages, i.e., if the macrocell has no

72 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 14, NO. 1, JANUARY 2003

. W. Shan and P. Fan are with the Institute of Mobile Communications,Southwest Jiaotong University Chengdu, 610031, People’s Republic ofChina. E-mail: [email protected]; [email protected].

. Y. Pan is with the Department of Computer Science, Georgia StateUniversity, University Plaza, Atlanta, GA 30303-3083.E-mail: [email protected].

Manuscript received 31 May 2001; accepted 12 Feb. 2002.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number 116398.

1. To simplify the analysis, the HCS only including two layers(macrolayer and microlayer) is considered.

2. Overflow is the procedure that new or hand off calls blocked inmicrocells (macrocells) transfer to the corresponding macrocells(microcells).

1045-9219/03/$17.00 ß 2003 IEEE Published by the IEEE Computer Society

free channel, even though its overlaid microcell has manyfree channels, the fast calls will terminate.

Huang and Bhargava also consider and compare twostrategies for traffic management [7]. In strategy 1, themobile stations are divided into two groups based on theirspeed and then served by different layers to minimize thehand off rate. In strategy 2, the mobile stations may entereither layer, regardless of their speed. They also discusshow to determine the threshold speed to keep the trafficbalanced between the two layers. However, the hand offrate is not the only criterion to judge the systemperformance, although more hand offs can cause moreoverhead.

Later, in [8], Rappaport and Hu consider a system to beoperated in such a way that a call served at a givenhierarchical level will not request hand off to a cell that islower in the hierarchy. In their system, the microcellsreceive input streams of slow new and hand off calls,whereas overlaying macrocells receive input streams of fastnew and hand off calls as well as overflow trafficcomponents from subordinate microcells. At the same time,hand off calls are given priority for accessing to channels ateach level. Another similar scheme is also studied in [9],which studies the performance of three-layer HCS. Com-munication satellites at the highest level comprise a spacesegment. The satellite beams overlay clusters of terrestrialmacrocells and microcells. Guerin presents another ap-proach to study this kind of unidirectional overflow system[10].

For the nonprioritized scheme, Calin and Zeghlachedescribe an analytical model (without guard channels3)without macrocell to microcell hand offs in HCS [11]. Amore general study taking into account macrocell tomicrocell hand offs is conducted by simulation, not bytheoretic analysis, in their paper.

Chang et al. analyze an HCS with finite queues for newand hand off calls [12]. Different from [8], [9], [10], in theirpaper, all of the channels in overlaying macrocell are usedby the overflowing calls from microcells. Both the effect ofthe reneging of waiting new calls because of the callers’impatience and the effect of dropping of queued hand offcalls as the callers move out of the hand off area areconsidered. Besides, they investigate how the designparameters of buffer sizes and guard channel numbers inmacrocells and microcells affect the performance of theHCS. It is regrettable that the scheme will induce highertraffic load in the macrocell and thus deteriorate itsperformance.

In addition, because fuzzy logic control has beensuccessful in various applications, fuzzy algorithms have

also been employed to improve the cellular systemperformance [2], [13]. The approach used by Shum andSung aims mainly at layer selection in HCS and the fuzzyrules are constructed in order to reduce the hand off rateand the blocking probability [13]. However, anotherimportant element, hand off failure probability, is ignored.On the other hand, the scheme proposed by Lo et al.emphasizes a fuzzy channel allocation controller for HCS[2]. The general performance of this scheme is good exceptfor the forced termination probability of calls. Although, theabove two schemes allow the flexible assignment of callsbetween different layers, the mobile velocity, an importantfactor for layer selection, is not adequately considered.

Recently, Wie et al. presented an improved scheme

including both overflow and underflow4 [14]. Performance

characteristics for users with different motilities are

evaluated. The disadvantage of this system is that the

blocking probability for low mobility users is increased due

to the underflow scheme. Maheshwari and Kumar [15] put

forward a similar scheme a with repacking procedure.5 In

their paper, slow calls may be allowed to overflow to the

macrolayer, but may be repacked to vacated microcell

channels. The main contribution of their paper is to develop

an approximate analysis for calculating the probabilities of

calls blocking in a model of a microcellular network.

However, the fast call overflowing into the microcell is

not taken into account in these two schemes. It is obvious

that the environment is unfair to fast calls. In addition,

Valois and Veque introduce another policy relying on

speed-sensitive selection and a taking back capability of

overflowed calls in macrocell [16]. Moreover, they use an

enhanced mechanism based on preemption which allows a

call issued in a saturated microcell to preempt resource of

another call in the macrocell. The preempted a call is not

dropped but handed off in a nonsaturated microcell.

However, the scheme is costly in terms of signaling traffic

as it increases the hand off rate for those preempted calls.From the above discussions, it is clear that the existing

schemes have the following limitations: First, the fast callsand slow calls are not treated equally i.e., slow calls can sharemore channel resources, while fast calls cannot. Second, theexisting schemes are not very flexible. If many fast calls burstout, even though many free channels can be used inmicrocells, these calls have to be terminated. It is undesirableto see the extremely unbalanced traffic load. Third, thevelocity threshold, a factor by which we assign different callsin different layers, should be determined carefully andexactly, i.e., they are almost velocity-sensitive systems.Otherwise, unbalanced occupancy rate of channels arises.Because traffic load in different layers is a variable, thethreshold should be adjusted continually to gain bettersystem performance. To solve these problems, in this paper,a new general scheme for HCS, which allows calls tooverflow6 between the macrocells and microcells based ondifferent mobile velocity, is considered. In this scheme, the

SHAN ET AL.: PERFORMANCE EVALUATION OF A HIERARCHICAL CELLULAR SYSTEM WITH MOBILE VELOCITY-BASED BIDIRECTIONAL... 73

Fig. 1. Hierarchical cellular system.

3. A fraction of the total available channels in a cell is reservedexclusively for hand off requests from ongoing calls which may be handedoff in the cell.

4. Underflow is the procedure in which slow calls transferred to themacrocells return to microcells.

5. Similar to the concept of underflow.6. In this paper, for our purpose, we do not differentiate the overflow

and underflow, instead, unidirectional overflow or bidirectional overflow,indicating the calls to be transferred between microcell and macrocell in onedirection only or in both directions, will be used.

occurrence of a call being forced to terminate is considered tobe less desirable than that of blocking, so guard channelsreserved for special purpose are assigned in both microcellsand macrocells to ensure hand off calls’ priority [9], [10], [12].

Although, Lo et al. [17] propose a similar scheme namedcombined channel assignment (CCA), its performanceanalysis is mainly based on simulation, with or withoutguard channels. Furthermore, they do not consider thenumber of hand offs and the mobile velocity which haveeffects on the system GoS. Cimone et al. also introduceanother similar scheme in multimedia traffic [18] withsimulation analysis, where the microcells handle all types ofmultimedia traffic (voice, video, data) and macrocells handlereduced quality video. Their scheme allows bidirectionaloverflow without layer selection of the originated call cell (theinitial layer selection), i.e., all the new calls are directedprimarily to the microlayer. So does the scheme proposed byLi et al. [19]. As reported in our previous work [20], theperformance (including the number of hand offs per call andthe unsuccessful call probability) of the scheme with initiallayer selection is better than that of the scheme without initiallayer selection. So, initial layer selection based on mobilevelocity and predetermined velocity threshold is adopted inour scheme. Our scheme provides a very flexible way to dealwith unbalanced and variational teletraffic.

In order to investigate and compare the related schemes,two one-dimensional Markov process models are derivedand investigated. Theoretical analysis and numericalevaluation show that the proposed scheme outperformsothers in terms of the average new call blocking probability,hand off failure probability and the probability of unsuc-cessful hand off of the system.

The major contributions of our paper have three folds.First, a simple but accurate analytical model is developed toevaluate the performance of the proposed scheme. Becausemany existing channel assignment schemes are only specialcases of the new scheme, the analytical model developed isapplicable to them as well. Therefore, fair comparisons canbe carried out easily among the different schemes. Second,more factors are considered to analyze the system perfor-mance, including the mobile velocity and the number ofhand offs. Third, the performance comparison of the threeschemes is carried out. Our analytical results indicate thatthe new scheme proposed in this paper is better than theexisting ones in terms of blocking probability of new callsand failure probability of hand off calls, etc. In addition, ouranalytical model can easily be extended to predicting theperformance of multilayer hierarchical cellular systems.

The remainder of this paper is organized as follows:Section 2 describes the notation and basic assumptions usedin our analysis, Section 3 describes three different call-overflow scheme, Section 4 analyzes the proposed schemeby using one-dimensional Markov process models, Section 5evaluates and compares numerically the new one with thetwo related call handling schemes, and Section 6 concludesthis paper.

2 BASIC NOTATION AND ASSUMPTIONS

In this section, basic notation and some assumptions used inour model and analysis in the following sections aredescribed. We define new calls as calls newly originatedfrom a cell and hand off calls as existing calls in the system

which seek a new channel due to cell boundary crossings.

The scenario with which we are concerned is that there is a

macrocellular network, with a given frequency allocation to

each cell. Each macrocell is then microcellized and the

original frequencies assigned to each cell are partitioned

between the microcells and the original macrocells. A call

that is handled by a channel in a macrocell is said to be in

the macrolayer, while a call that is handled by a channel in a

microcell is said to be in the microlayer. In order to make

our analysis tractable, we also simplify our model and make

some assumptions about properties of the incoming calls

and hand off calls. Some random processes and the

corresponding notation used in the following sections are

defined and listed below:

1. The velocity vof a mobile is a random variable which isuniformly distributed on the interval [0, Vmax], whereVmax is the maximum speed of the mobiles;

2. The lifetime for both fast and slow calls t� is assumedto be a random variable, having the same negativeexponential distribution with the mean 1=�;

3. The new call arrival process offered to a given cell ateach system level is considered to be a Poissonprocess. The mean new slow call arrival rate to eachindividual microcell is denoted by �sn, and the meannew fast call arrival rate to each individual macrocellis denoted by �fh,

4. The hand off call arrival process occurring in a givencell at each system level is assumed to be a Poissonprocess. The mean hand off slow call arrival rate toeach individual microcell is denoted by �sh and themean hand off fast call arrival rate to an individualmacrocell is expressed as �fn;

5. The dwell time is the mobile residence time in agiven cell. For a slow mobile, its dwell times inmicrocell and macrocell, tsm and tsM , are differentrandom variables and have the negative exponentialdistribution with the mean 1=�sm and 1=�sM ,respectively. For a fast mobile, its dwell times inmicrocell and macrocell, tfm and tfM are differentrandom variables and have the negative exponentialdistribution with the mean 1=�fm and 1=�fM ,respectively;

6. It is assumed that each microcell in the HCS has Cm,channels including Cg guard channels, and eachmacrocell has CM channels including CG guardchannels. The guard channels, are reserved for handoff calls due to their higher priority than the newcalls. The purpose is to reduce the termination(failure) probability of hand off calls.

To further simplify our analysis, we also make the

following assumptions. Many of the assumptions have been

used in much of the literature and are reasonable in the

sense that the analytical results obtained remain to be

accurate as shown in [2], [14], [15]. Here are additional

assumptions used in our analysis:

1. When a new call originates, its layer selection isbased on the comparison of the initial velocity withthe predetermined velocity threshold vth. Furtherdiscussion on identifying the mobility class of a callis given in [21], [22];


2. The mobile velocity does not change greatly duringits call lifetime such that the velocity of slow calls isalways less than the threshold vth and the velocity offast calls is always higher than the threshold. Inreality, mobiles do not move with constant speeds. Aspeed change occurs when a mobile moves from amore crowded area to a less crowded area or if amobile encounters a traffic signal. However, usuallya speed change is small considering the fact that acall normally lasts a short period of time;

3. A macrocell is overlaid completely by n microcellsand there is no macrocell-only or microcell-onlyarea;

4. The channel allocation of the HCS is fixed;5. The HCS considered is a system with homogeneous

traffic load in statistical equilibrium state.

3 VELOCITY-BASED BIDIRECTIONAL

CALL-OVERFLOW SCHEME

Before presenting the proposed bidirectional call-overflow

scheme, it is useful, for comparison purposes, to describe

the two typical existing schemes: no call-overflow scheme,

denoted as Scheme I, and the unidirectional call-overflow

scheme, denoted as Scheme II.

3.1 No Call-Overflow Scheme (Scheme I)

Calls cannot overflow between the macrocells and micro-

cells [4], [5], [6].When a new call originates, the corresponding layer is

selected by a predetermined threshold. If the velocity of the

mobile originating the call is bigger than the threshold, it is a

fast call and will be served by macrocells. Otherwise, it is a

slow call and will be served by microcells during its lifetime.

After the cell selection, the call should only be handled by the

cells of the same layer, i.e, when a new fast (slow) call

originates from a cell and the number of engaged channels in

the current macrocell (microcell) is equal to or greater than

CM ÿ CG ðCm ÿ CgÞ, this new call will be blocked. Otherwise,

the call is allocated an idle channel according to the fixed

channel allocation method. On the other hand, if all the

channels in the macrocell (microcell) are used, the existing

fast (slow) hand off calls from the adjacent macrocells

(microcells) will be forced to interrupt, resulting a hand off

failure. The no call-overflow scheme is shown in Fig. 2.

3.2 Unidirectional Call-Overflow Scheme(Scheme II)

Slow calls can overflow from microcells into macrocells inthe HCS [8], [9], or vice versa, but only in one direction.

In this scheme, when a new slow call originates and thenumber of engaged channels in the current microcell is equalto or greater than Cm ÿ Cg, this call will overflow into thecorresponding macrocell. If this macrocell has free channelsexcept guard channels, an idle channel will be allocated to thecall; otherwise, the call will be blocked. If all the channels ofthe target macrocell and microcell which the mobile willmove into are used, the slow hand off calls will be terminated.The unidirectional call-overflow scheme is shown in Fig. 3.Similarly, one can analyzes a unidirectional call-overflowscheme defined from macrocells into microcells.

3.3 Bidirectional Call-Overflow Scheme (Scheme III)

Fast or slow calls can overflow between macrocells andmicrocells.

The difference between Scheme I and this scheme is that,when a new fast (slow) call originates and the number ofengaged channels in the current macrocell (microcell) isequal to or greater than CM ÿ CG ðCm ÿ CgÞ, this call willoverflow into the corresponding microcell (macrocell). Ifthis microcell (macrocell) has free channels, excludingguard channels, a suitable channel will be allocated to thiscall. Otherwise, the call is blocked. On the other hand, if allthe channels including guard channels, in the targetmicrocell and macrocell into which the mobile will moveare engaged, the fast (slow) hand off calls will be forced toterminate. The bidirectional call-overflow scheme is shownin Fig. 4. Hence, in our new scheme, as long as there is a freechannel either in microcell or macrocell, a new or hand offcall will not be blocked, thus increasing the channelutilization and reducing the blocking probability of newcalls and failure probability of hand off calls. This intuitiveidea will be verified through our rigorous analysis in thenext section.

4 THEORETICAL PERFORMANCE ANALYSIS

In this section, theoretical analysis for Scheme III will becarried out in detail. Because Schemes I and II can beregarded as special cases of Scheme III, it is easy to derivetheir results from the analytical results of Scheme III. Thus,we will concentrate on the analysis of Scheme III in thefollowing discussion. Clearly, different types of call trafficor arrival processes should be considered separately.Analysis will be started from the microcell and then weproceed upward to the macrocell.

4.1 Microcell Level

If a call is allocated to a microcell channel, this channelwould later be released upon


Fig. 2. No call-overflow scheme.

Fig. 3. Unidirectional call-overflow scheme.

1. the completion of the slow (fast) call in the microcell,2. the departure of the slow (fast) call from the current

microcell, no matter, whether the hand off is success-ful or not.

According to the notation defined in Section 2, the

occupancy time of microcell channel tm is the smaller one of

the call holding time and the mobile residence time (fast calls

or slow calls) in the microcell. That is tm ¼ minft�; trmg, where

trm is the alternative to tsm and tfm. And from this equation,

we have

Probftm � tg ¼ 1ÿ Probðt� > tÞ and ðtrm > tÞ; t � 0:

According to Probability Theory, the probability densityfunction of the channel occupancy time distribution in themicrocell is given by

f�mðtÞ ¼ ð�þ �rmÞeÿð�þ�rmÞt t � 0

where 1=�rm is the mean of trm.The call flow to an individual microcell can be divided intothe following parts:

1. slow new calls whose arrival rate is �sn;2. slow hand off calls whose arrival rate is �sh;3. fast new calls which are blocked in the overlaying

macrocell and transferred to the microcell whosearrival rate is PMb�fn=n,7 where PMb is the prob-ability of fast calls being blocked in the macrocell;

4. fast hand off calls which are denied by the overlayingmacrocell and transferred to the microcell, whosearrival rate is PMhf�fh=n, where PMhf is the prob-ability of fast calls being denied by the macrocell.

It is assumed that �mn is the total new calls arrival rate inthe m icroce l l , tha t i s , �mn ¼ �sn þ PMb�fn=n and�mh ¼ �sh þ PMhf�fh=n, which is the total hand off callarrival rate in the microcell. According to the abovediscussion, the call state process in microcell can bemodeled by a Markov process with sþ 1 states, wheresðiÞ represents the state that i channels have been used inthe microcell. When i 2 f0; 1; . . .Cm ÿ Cg ÿ 1g, the transi-tion rate from state sðiÞ to sðiþ 1Þ is given by �mn þ �mh.Otherwise, the transition rate will be �mh. The reason forthat is, when the number of occupied channels is equal to orbigger than Cm ÿ Cg, only hand off calls can be served bythe microcell. Besides, the transition rate from state sðiþ 1Þto sðiÞ is given by ðiþ 1Þ�m. Based on the state diagramshown in Fig. 5, it can be shown that the steady stateprobability P

ðmÞi of state sðiÞ is given by [23]

PðmÞi ¼

�mnþ�mhi�m

PðmÞiÿ1 0 � i < Cm ÿ Cg

�mhi�m

PðmÞiÿ1 Cm ÿ Cg � i � Cm:

(ð1Þ

or

PðmÞi ¼

ð�mnþ�mhÞii!�im

PðmÞ0 0 � i < Cm ÿ Cg

ð�mnþ�mhÞcmÿcg �iÿcmþcgmh

i!�imPðmÞ0 Cm ÿ Cg � i � Cm:

8<: ð2Þ

In addition, PðmÞi should satisfy

XCmi¼0

PðmÞi ¼ 1: ð3Þ

4.2 Macrocell Level

If a call is allocated to a macrocell channel, this channelwould be released upon

1. the completion of the slow (fast) call in themacrocell,

2. the departure of the slow (fast) call from the currentmacrocell no matter the hand off succeeds or not.

According to the notation defined in Section 2, themacrocell channel occupancy time tM should be the smallerone of the call holding time and the mobile residence time(fast calls or slow calls) in the macrocell. That istM ¼ minft�; trMg, where trM is the alternative to tsM andtfM . From this equation, we have

ProbftM � tg ¼ 1ÿ Probfðt� > tÞ and ðtrM > tÞ; ðt � 0Þ;

Then, the probability density function of the channeloccupancy time distribution is

f�M ðtÞ ¼ ð�þ �rMÞeÿð�þ�rM Þt t � 0;

where 1=�rM is the mean of trM .Similarly, the call flow to an individual macrocell can be

divided into the following parts:

1. fast new calls whose arrival rate is �fn,2. fast hand off calls whose arrival rate is �fh,3. slow new calls which are blocked in the overlaid

microcell and transferred to the macrocell, whosearrival rate is nPmb�sn,8 where Pmb is the probabilityof slow calls being blocked in the microcell,

4. fast hand off calls are denied by the overlaid

microcell and transferred to the macrocell, whosearrival rate is nPmhf�sh, where Pmhf is the probability

of slow calls being denied by the microcell.


Fig. 4. Bidirectional call-overflow scheme.

7. For a homogeneous HCS, overflow calls from one macrocell will bedistributed into n microcells on average.

8. Overflowing calls from n microcells will be handled by the singleoverlaying macrocell.

It is assumed that �Mn is the total new calls arrival rate in

the macrocell, that is, �Mn ¼ �fn þ nPmb�sn and �Mh, which

equals �Mh ¼ �fh þ nPmhf�sh is the total hand off calls

arrival rate in the macrocell. According to the above

description, the state of macrocell can be modeled by a

Markov process with S þ 1 states, where SðjÞ represents

that j channels have been used in the macrocell. When j 2f0; 1; . . .CM ÿ CG ÿ 1g; the transition rate from state SðjÞ to

Sðjþ 1Þ is given by �Mnþ �Mh

. Otherwise, the transition

rate is �Mn. The reason for that is when the number of

occupied channels is equal to or bigger than CM ÿ CG, only

hand off calls can be served by the macrocell. In addition,

the transition rate from state Sðjþ 1Þ to SðjÞ is given by

ðjþ 1Þ�m. Considering the state diagram of Fig. 6, the

steady state probability PðMÞj can be derived as follows:

PðMÞj ¼

�Mnþ�Mh

i�mPðMÞjÿ1 0 � j < CM ÿ CG

�Mh

j�MPðMÞjÿ1 CM ÿ CG � j � CM

(ð4Þ

or

PðMÞj ¼

ð�Mnþ�MmhÞj

j!�jM

PðMÞ0 0 � j < CM ÿ CG

ð�Mnþ�MhÞcMÿcG �iÿcMþcGMh

j!�jM

PðMÞ0 CM ÿ CG � j � CM:

8><>:ð5Þ

On the other hand, PjðMÞ should satisfy the following

constraint

XCMj¼0

PðMÞj ¼ 1: ð6Þ

4.3 Probability Calculation

The parameters Pmb, Pmhf , PMb, and PMhf can be deter-mined as follows:

Pmb ¼XCm

i¼CmÿCgPðmÞi ð7Þ

Pmhf ¼ PCm ð8Þ

PMb ¼XCM

j¼CMÿCGPðMÞi : ð9Þ

PMhf ¼ PCM ð10Þ

Based on (2), (3), (5), (6), (7), (8), (9), and (10), PðmÞ0 , P

ðMÞ0 ,

Pmb, Pmhf , PMb, and PMhf can be solved.

4.4 Performance Measures

The following performance measures will be derived in ouranalysis and used for comparisons. These are typicalperformance measures used in the literature.

1. The average new call blocking probability of theHCS, Pb, is the ratio of the number of blocking newcalls (including slow and fast calls) and that of allnew calls in this system, which is

Pb ¼n�snPsb þ �fnPfb

n�sn þ �fn: ð11Þ

2. The average hand off failure probability of the HCSsystem, Phf , is the ratio of the number of failed hand


Fig. 6. State diagram in macrocells for Scheme III.Fig. 5. State diagram in microcells for Scheme III.

TABLE 1Parameters and Their Values

off calls (including slow and fast calls) and that of allhand off calls in this system, which is

Phf ¼n�shPshf þ �fhPfhf

n�sh þ �fh; ð12Þ

where Psb and Pfb are the blocking probability of

slow and fast calls, respectively; Pshf and Pfhf are

the hand off failure probability of slow and fast calls,

respectively. In our scheme, fast and slow calls can

share all the channels in different layers, so

Psb ¼ Pfb ¼ PmbPMb; Pshf ¼ Pfhf ¼ PmhfPMhf :

3. The number of successful hand off per call, nh, isalso an important parameter because the excessivehand offs give rise to huge overhead.

For a slow new call, it does not hand off because

of the following situations:

. the call is blocked in both the microcell andmacrocell,

. the call life time is smaller than its dwell time,

. the call has a failure hand off to next cell.Therefore, the probability of slow call with-

out hand off, Ps0, is

Ps0 ¼ PmbPMb þ ð1ÿ PmbÞ½P ðt� < tsmÞþP ðt� � tsmÞPmhfPMhf � þ Pmbð1ÿ PMbÞ½P ðt� < tsMÞ þ P ðt� � tsMÞPmhfPMhf �:

ð13Þ

The probability of slow call carrying out one

successful hand off, Ps1, is

Ps1 ¼ ð1ÿ Ps0Þð1ÿ PmhfÞ½P ðt� < tsmÞþP ðt� � tsmÞPmhfPMhf � þ Pmhfð1ÿ PMhfÞ½P ðt� < tsMÞ þ P ðt� � tsMÞPmhfPMhf �:

ð14Þ

Similarly, the probability of slow (fast) call

having xth successful hand off can be derived.

Then, nh can be described by (15)


Fig. 7. Performance comparison of slow calls.

Fig. 8. Performance comparison of fast calls.

nh ¼n�sn

Pxi¼0

xPsx þ �fnPxi¼0

xPfx

n�sn þ �fn; ð15Þ

where Pfx is the probability of fast call which

has x successful hand off.

4. Besides, whether a call is blocked or has a failurehand off, it will be unsuccessful. Therefore aparameter Pu named the unsuccessful call probabil-ity is introduced below[20]

Pu ¼XNi¼0

Pi; ð16Þ

where Pi is

P0 ¼ Pb þ ð1ÿ PbÞPhfP1 ¼ ð1ÿ P0ÞPhf

..

.

Pi ¼ ð1ÿ Piÿ1ÞPhf

8>>><>>>: ð17Þ

It is obvious that more successful hand offs willresult in higher Pu.

5 NUMERICAL AND SIMULATION ANALYSIS

Numerical and simulation analyzes are given in this sectionto compare the performance of the three schemes. Allparameters and their values needed in the numericalevaluation are shown as in Table 1. The evaluation resultsare shown in Figs. 7, 8, 9, 10, 11, 12, 13, and 14.

Fig. 7 shows that the blocking probability and hand offfailure probability of slow calls increase with the augmentof the total new call arrival rate �t.

9 Owing to the fact thatslow new calls and hand off calls can overflow into theoverlaid macrocell and use its free channels, both theblocking and hand off failure probability of slow calls inScheme II and III are far smaller than that in Scheme I. Onthe other hand, the two factors Psb and Pshf in Scheme II area bit smaller than those in Scheme III, the reason being thatthe fast calls are introduced in the microcells and this makesthe blocking probability ðPmbÞ and the hand off failure


Fig. 9. Average performance comparison for different schemes.

Fig. 10. Average performance comparison for different schemes.

9. The total new call arriving rate consists of both the arriving rate to onemacrocell and to seven microcells.

probability ðPmhfÞ in the microcell higher and, finally,results in the higher Psb and Pshf in Scheme III.

A similar situation also appears in Fig. 8, which showsthe relation among blocking probability of fast calls ðPfbÞ,hand off failure probability of fast calls ðPfhfÞ; and �t inthree schemes. Pfb and Pfhf in Scheme III are the smallest.This phenomenon can be explained by the fact that fast callscan overflow into the microcells and share their channelresource. Meanwhile, fast calls cannot overflow into thecorresponding microcells in Scheme II and, compared withScheme I, the heavier load will be handled by the macrocell,i.e., slow calls can share the macrocells’ channels, but fastcalls cannot share the microcells’ channels. Therefore, theunidirectional overflows restriction which is unfair to fastcalls result in the increase of Pfb and Pfhf . Considering therelation among parameters in Scheme II, the betterperformance of slow calls is obtained at the cost of reducingthe fast calls performance. Among the three schemes, theperformance of fast calls in Scheme III is the best, whichresults from the balanced traffic load.

According to Figs. 7, 8, and (11), (12), the averageblocking and hand off failure probability of the system can

be deduced, see Fig. 9. It is obvious that, with the givenconditions (�sh ¼ 0:5�sn and �fh ¼ 0:5�fn), the wholeperformance of Scheme III is the best. The major reason isthe balanced call-overflow. Note that as the increase of totalnew call arriving rate, comparing to Scheme I, the super-iority of Scheme II reduces, progressively. If the proportionof new calls to hand off calls is changed, i.e., �sh ¼ 0:3�snand �fh ¼ 0:3�fn (see Fig. 10), Scheme III also has the lowestPb and Phf . Hence our scheme can ensure the best GoScomparing with Schemes I and II.

Here the simulation is used to validate the theoreticalanalysis, i.e., the successful hand off per call (see Fig. 11).The results show that our analytical model is more reliable.Obviously, in Scheme III, as increase �t, nh decreases. Thatis because heavier teletraffic will make the call hand offmore difficult. The fact that all the calls share the channelresources in both macrocells and microcells brings aboutthe highest number of successful hand offs per call, nh.Moreover, nh of Scheme II is lower than that of Scheme Iand the reason is that, when slow calls overflow to themacrocell, their dwell time will be prolonged and nh willdecrease. Obviously, more hand offs can result in increased


Fig. 11. Number of successful hand off per call in different schemes.

Fig. 12. Pu versus total arriving rate in trhee schemes.

overhead and resource consumption potentially, so there isa tradeoff between the GoS and the number of hand offs. Ifthe teletraffic is not very heavy, there is little differenceamong nh in the three schemes, i.e., when the total new callarriving rate is 70 calls/min, nh of Scheme III is only biggerthan that of Scheme II by nearly 0.2. But, if the teletraffic isheavy, the increment of GoS may not make up theincrement of the overhead, then another parameter, Pu(the unsuccessful call probability) which relates to nh, Pband Phf , is introduced to evaluate the system performance.According to the values of nh, Pb, and Phf , Pu can becalculated as shown in Section 4.

From Fig. 12, Pu of the Scheme III is the lowest amongthree schemes. But, if �t reaches 80 calls/min, the status thatPu of three schemes are all bigger than 10 percent means theGoS of HCS is not satisified. In the situation, theperformance of the whole system needs other methods toimprove, for example, cell splitting, sectoring, etc.

To obtain better system performance in Scheme III, thevelocity threshold may be adjusted by assigning thearriving calls properly to different layers. From Fig. 13,when �t is below 60 calls/min, increasing Vth can make theHCS hold more users, that is, assigning more calls into the

microcells can bring about higher capacity. On the contrary,when �t ¼ 60 calls=min, the effect of decreasing Vth is notnotable, and this means the user capacity of HCS isinsensitive to Vth. If �t ¼ 60 calls=min, Vth should bereduced to reduce the number of hand offs and to achievebetter GoS. On the other hand, given a tolerant Pu, themaximum new call arriving rate of Scheme III with differentVth can be obtained, i.e., when Pu equals 0.1 percent andVth ¼ 70 km=h, the maximum �t is nearly 56 calls/min.

Furthermore, given Pu ¼ 0:1%, if Vmax in the HCS growsfrom 100 km/h to 140 km/h, the maximum �t will decreasefrom 55 to 51 calls/min (Fig. 14). When Vmax ¼ 120 km=h, Puis insensitive to the change of Vmax, so the situation is�t ¼ 64 calls=min.

6 CONCLUSION

Aiming at overcoming the shortcomings of Scheme I (no

call-overflow scheme) and Scheme II (unidirectional call-

overflow scheme), a bidirectional call-overflow scheme

(Scheme III) based on mobile velocity is proposed and

analyzed. By using two one-dimensional Markov processes,


Fig. 13. Pu versus total arriving rate with different Vth (Scheme III).

Fig. 14. Pu versus total arriving rate with different Vmax (Scheme III).

a theoretical and numerical analysis is carried out. The

results show that Scheme III has a better characteristic in

balancing the teletraffic load between macrocells and

microcells. Naturally, all the slow and fast calls in our

scheme can share the common channel resources provided

by the two layers, thus giving the best performance

compared with Scheme I and Scheme II.

Although call overflow can produce higher overhead

and more hand offs, by simulation, there is little difference

among three schemes from the aspect of the number of

successful hand off per call when the teletraffic is not very

high. Furthermore, this scheme can be easily realized in

actual wireless HCS and bring higher system capacity and

better QoS.At the same time, the relation between the mobile

velocity (including the velocity threshold and the maximummobile speed) and the GoS is analyzed. When the total callarriving rate is in the given range, increasing Vth ordecreasing Vmax can make the HCS capacity higher andachieve better GoS. Otherwise, the GoS is insensitive to thechange of Vth and Vmax.

Our analytical models can also be extended to analyzingmultilayer HCS. As more and more wireless systems aredeployed, the limited number of channels becomes evenmore scarce, the multiplayer HCS consisting of three ormore layers is feasible when the stratosphere cellularmobile system and satellite mobile system are taken intoaccount. In the multiplayer HCS, our scheme is moreflexible to deal with bursting out teletraffic in differentlayers and getting better system performance.

ACKNOWLEDGMENTS

This work was supported in part by the National ScienceFoundation of China (NSFC) under grant no. 6985102/69931050 and in part the US National Science Foundationunder grants CCR-9211621, OSR-9355040, and ECS-0196569.

REFERENCES

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Wenhao Shan received the BS, MS, and PhDdegrees from Southwest Jiaotong University,People’s Republic of China, in 1996, 1999, and2002, respectively. He has worked on severalprojects, including a National Natural ScienceFoundatioin of China (NSFC) and a HuaweiTechnology research project. He has publishedmore than 10 research papers in variousjournals and international conferences. He iscurrently engaged in research on next genera-

tion mobile and personal communications systems, including channelresource management and advanced CDMA techniques.


Pingzhi Fan received the MS degree in com-puter engineering from Southwest JiaotongUniversity, People’s Republic of China, in1987, and the PhD degree in electrical engineer-ing from Hull University, United Kingdom, in1994. From 1987 to 1992, he was a lecturer andassociate professor of computer engineering atSouthwest Jiaotong University. From 1994 to1997, he was a postdoctoral researcher at HullUniversity and a research fellow at Leeds

University, United Kingdom. Since 1997, he has been a visitingprofessor at Leeds University. Since 2000, he has been a guestprofessor at Shanghai Jiaotong University, China. In 1997, he joined theSchool of Computer and Communications Engineering, SouthwestJiaotong University, where he is currently a professor of electricalengineering and the director of the Institute of Mobile Communmica-tions. He is the first author of the book, Sequence Design forCommunications Applications, published in English in 1996 by ResearchStudies Press and John Wiley & Sons Ltd. He has also published oneedited book in English, two textbooks, and more than 150 journal andreferred conference papers. He has served as general chairman,session chairman or technical committee member of more than eightinternational conferences. He was a recipeient of the ORS Award,United Kingdom, and the National Science Foundation Award forOutstanding Young Scientist, PRC. He is currently a senior member ofthe IEEE. His research interests include wireless mobile and personalcommunications, wireless multimedia systems, computer networks,information security, and signal design and processing.

Yi Pan received the BEng degree in computerengineering from Tsinghua University, China, in1982 and the PhD degree in computer sciencefrom the University of Pittsburgh, Pennsylvania,in 1991. Currently, he is an associate professorin the Department of Computer Science atGeorgia State University. Previously, he was afaculty member in the Department of ComputerScience at the University of Dayton, Ohio. Hisresearch interests include parallel algorithms

and architectures, optical communication and computing, wirelessnetworks, high-performance data mining, distributed computing, taskscheduling, and networking. He has published more than 120 researchpapers, including more than 50 papers in international journals. He hasreceived many awards including the Outstanding Scholarship Award ofthe College of Arts and Sciences at the University of Dayton (1999), aJapanese Society for the Promotion of Science Fellowship (1998), anAFOSR Summer Faculty Fellowship (1997), US National ScienceFoundation (NSF) Research Opportunity Awards (1994 and 1996),and the best paper award from International Conference on Paralel andDistributed Processing Techniques and Applications (PDPTA ’96). Hisresearch has been supported by the NSF, the AFOSR, the US Air Force,and the state of Ohio. Dr. Pan is currently an associate editor of theIEEE Transactions on Systems, Man, and Cybernetics, area editor-in-chief of the Journal of Information, editor of the journal of Parallel andDistributed Computing Practices, associate editor of the InternationalJournal of Parallel and Distributed Systems and Networks, and serveson the editorial board of The Journal of Supercomputing. He has servedas a guest editor of special issues for several journals and as generalchair, program chair, vice program chair, publicity chair, session chair,and as a member of the steering, advisory, and program committees fornumerous international conferences and workshops. He is an IEEEComputer Society Distinguished Visitor, a senior member of the IEEEand a member of the IEEE Computer Society. He is listed in Men ofAchievement, Marquis Who’s Who in America, and Marquis Who’s Whoin Midwest.

. For more information on this or any computing topic, please visitour Digital Library at http://computer.org/publications/dlib.


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