Repacking on demand for speed-sensitive channel assignment

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Repacking on demand for speed-sensitive channel assignment

Hsien-Ming Tsai

a

, Ai-Chun Pang

b

, Yung-Chun Lin

c

, Yi-Bing Lin

c,d,*

a

Quanta Research Institute, Quanta Computer Inc., No. 4, Wen Ming 1 St., Kuei Shan Hsiang, Tao Yuan Shien, Taiwan

b

Graduate Institute of Networking and Multimedia, Department of Computer Science and Information Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei, Taiwan

c

Department of Computer Science and Information Engineering, National Chiao Tung University, 1001 Ta Hsueh Rd., Hsinchu, Taiwan

d_{Providence University, Chung-chi Rd., Shalu Taichung, Taiwan}

Received 17 December 2003; received in revised form 27 July 2004; accepted 27 July 2004 Available online 14 October 2004

Responsible Editor: E. Chong

Abstract

In mobile telecommunications networks, radio channels are limited resources that should be carefully allocated. To increase network capacity, channel assignment can be exercised in a hierarchical cellular network (HCN). By consid-ering the speeds of MSs, we propose an HCN channel assignment approach called repacking on demand (RoD). A sim-ulation model is developed to investigate the performance of RoD and some previously proposed approaches for HCN. Our study quantitatively shows that RoD signiﬁcantly outperforms the previous proposed approaches.

Keywords: Channel assignment; Channel repacking; Hierarchical cellular network; Repacking on demand

1. Introduction

One of the most important issues in cellular net-work operation is capacity planning. Especially when subscriber population grows rapidly, a cellular service provider needs to increase its network capac-ity eﬀectively. One possible solution is to deploy

hierarchical cellular network (HCN) [6,11,13]. As

shown in Fig. 1, an HCN consists of two types of

base stations (BSs): micro BSs and macro BSs. A micro BS with low power transceivers provides small radio coverage (referred to as microcell), and a macro BS with high power transceivers provides large radio coverage (referred to as macrocell). The microcells cover mobile stations (MSs) in heavy tele-traﬃc areas. A macrocell is overlaid with several microcells to cover all MSs in these microcells.

In an HCN, radio channels must be carefully as-signed to reduce the numbers of new call blockings

*

Corresponding author. Address: Department of Computer Science and Information Engineering, National Chiao Tung University, 1001 Ta Hsueh Rd., Hsinchu, Taiwan. Tel.: +886 3 5731842; fax: +886 3 5724176.

E-mail address:liny@csie.nctu.edu.tw(Y.-B. Lin).

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as well as handoﬀ call force-terminations. Several channel planning and assignment approaches have

been proposed for HCN [1,5,18,14–17]. Some of

them assign channels according to the received

radio signal strength [5,15]. Other approaches

[1,14,16,17] reduce call blocking and force-termi-nation through repacking techniques. Repacking is the process of switching a connected call from a macrocell to a microcell and vice versa. Based on the repacking techniques, our previous work

proposed Repacking on Demand (RoD) [7,18] to

improve the network performance. RoD was orig-inally designed to eﬃciently allocate the radio channels for wireless local loop (WLL) systems

[7]. In [18], we studied the RoD performance for

mobile networks without considering the MS speeds. In this paper, we investigate repacking per-formance improvement by considering the moving speeds of MSs.

Speed-sensitive channel assignment and repack-ing (includrepack-ing Macro-to-micro and micro-to-Macro repacking) are exercised to satisfy the following criteria:

Criterion 1. Calls for slow MSs tend to be assigned with microcell channels so that the ‘‘global resources’’ of macrocells can be eﬀectively shared by calls in the blocked microcells (i.e., the microcells without any idle channel).

Criterion 2. Calls for fast MSs tend to be assigned with macrocell channels so that the number of handoﬀs can be reduced.

Following Criterion 1, Macro-to-micro (M-to-m) repacking may be exercised to switch a call for a slow MS from the macrocell to the microcell.

In this case, the ‘‘global resources’’ of macrocells can be eﬀectively shared by calls in the blocked microcells. Following Criterion 2, the micro-to-Macro (m-to-M) repacking may be exercised to switch a call for a fast MS from the microcell to the macrocell. In this case, the number of han-doﬀs can be reduced.

In this paper, we consider several speed-sensi-tive channel assignment and repacking approaches for HCN: Always Repacking (AR; or take-back in

[1,8,14]) always exercises repacking as soon as some speciﬁc events (e.g., a channel is released at the micro/macrocell) occur to reduce the num-ber of handoﬀs for calls of fast MSs. However, al-ways exercising repacking may degrade the system performance on call blocking probability and force-termination probability in some cases (e.g., the number of macrocell channels is small). In Partial Repacking on Demand (PRoD; or

preemp-tion in[19,20]), only calls with slow speeds are

re-packed on demand. When the proportion of fast calls increases, the eﬀect of slow-call repacking on performance improvement becomes insigniﬁ-cant. Therefore, in order to further reduce call blocking and force-termination, this paper im-proves the Repakcing on Demand (RoD) approach by considering repacking for both slow and fast MSs.

This paper is organized as follows. In Section 2, we describe speed-sensitive HCN channel assign-ment approaches including AR, PRoD and RoD. In Section 3, input parameters and output meas-ures for these channel assignment approaches are described. Section 4 compares RoD with AR and PRoD. Our study quantitatively shows that RoD outperforms AR and PRoD.

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2. Speed-sensitive channel assignment for HCN Based on the moving speeds of MSs, several speed-sensitive channel assignment approaches have been proposed. In these approaches, a call for a fast MS is referred to as a fast call, and a call for a slow MS is referred to as a slow call.

No Repacking (NR) [16]: This approach does not

perform repacking. Based on Criteria 1 and 2, the slow and fast calls are handled diﬀerently.

• NR for Slow Calls (Fig. 2). When a slow call

attempt is newly generated at or handed off to the ith microcell, the HCN first tries to assign a channel in the ith microcell (following Criterion 1) to the call attempt (Steps 1 and 2 in Fig. 2). If no idle channel is available in the ith microcell, the call attempt overflows to the macrocell that is overlaid with the ith microcell. If the macrocell has an idle channel, the HCN accepts the call (Steps 3 and 4). Oth-erwise, the call attempt is rejected; i.e., the new call is blocked or the handoff call is forced to

terminate (Step 5). Steps 1–4 inFig. 2are called

the NR Slow MS Channel Assignment

Procedure.

• NR for Fast Calls (Fig. 3). When a new or

han-doﬀ fast call attempt occurs in the ith microcell, the HCN ﬁrst tries to assign the call attempt a channel in the macrocell (following Criterion 2) that is overlaid with the ith microcell (Steps 1

and 2 inFig. 3). If no idle channel is available

in the macrocell, the call attempt overﬂows to the ith microcell, and the HCN tries to allocate the call a channel in the ith microcell (Steps 3 and 4). If no idle channel is available in the microcell, the call attempt is rejected (Step 5). When a call is complete or the MS moves out of a cell, the radio channel is reclaimed to the idle channel pool of the corresponding micro (macro) cell.

Always Repacking (AR) [1,8,14]: The repacking

mechanism is triggered when an occupied channel is released. Two repacking procedures are exer-cised in AR to reduce the force-termination prob-ability of fast calls. In Macro-to-micro (M-to-m) repacking, a slow call occupying a macrocell chan-nel is switched to an idle chanchan-nel in the microcell where the MS resides (following Criterion 1). In micro-to-Macro (m-to-M) repacking, a fast call occupying a microcell channel is switched to an

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idle channel of the overlay macrocell (following Criterion 2). During repacking, one or more calls may be available for switching. These calls are referred to as the repacking candidates. The AR channel assignment is the same as that for NR (seeFigs. 2 and 3). The AR repacking procedures (Fig. 4(a) and (b)) are repeatedly executed to check if one of the following situations occurs.

M-to-m Repacking (Fig. 4(a)): When a channel is

released at a microcell (Step 1 inFig. 4(a)), the

HCN checks if there is any slow M-to-m repacking candidate (which is a slow call) in

the corresponding macrocell (Step 2). If so, M-to-m repacking is exercised to switch one of the slow repacking candidates from the macro-cell to the micromacro-cell (Steps 3 and 4).

When an MS of a slow macrocell call moves across the boundary of two microcells in that

macrocell (Step 5 inFig. 4(a)), the HCN checks

if the call can be a slow M-to-m repacking

can-didate (Step 2 inFig. 4(a)). If so, Steps 3 and 4

in Fig. 4(a) are executed to perform M-to-m repacking.

m-to-M Repacking (Fig. 4(b)): When a channel is

released in a macrocell (Step 1 inFig. 4(b)), the

HCN checks if there is any m-to-M repacking candidate (which is a fast call ; Step 2). If so, m-to-M repacking is exercised (Steps 3 and 4). The following approach is proposed in this paper.

Repacking on Demand (RoD): In RoD, the HCN may trigger the M-to-m repacking when a call assignment (for a new call or a handoﬀ call) occurs (following Criterion 1). Note that Criterion 2 is used in channel assignment for fast calls, but is not used in repacking.

• RoD I: RoD for Slow Calls (Fig. 5).

Step RoD I.1: When a slow call attempt Cs is

newly generated at or handed oﬀ to the ith microcell, the HCN ﬁrst exercises the NR slow MS channel assignment procedure. If the call

attempt Csis assigned a channel, the procedure

exits.

Step RoD I.2: If no idle channel is found in Step RoD I.1, then following Criterion 1, the HCN checks if there is any slow M-to-m repacking candidate. If so, Step RoD I.3 is exe-cuted. Otherwise, Step RoD I.5 is exeexe-cuted. Steps RoD I.3 and 4: The HCN selects one of these calls to exercise M-to-m repacking, and the reclaimed macrocell channel is assigned to

the slow call attempt Cs. The procedure exits.

Steps RoD I.5 and 6: If no slow M-to-m repacking candidate is found, the HCN checks if there is any fast M-to-m repacking candidate. If so, go to Step RoD I.3. Otherwise, the call

attempt Cs is rejected.

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Fig. 4. Channel repacking procedures for AR (a) M-to-m repacking procedure for AR (b) m-to-M repacking procedure for AR.

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• RoD II: RoD for Fast Calls (Fig. 6).

Steps RoD II.1 and 2: When a fast call attempt

Cfis newly generated at or handed oﬀ to the ith

microcell, the HCN ﬁrst tries to assign a chan-nel in the macrocell that is overlaid with the ith microcell (following Criterion 2) to the call

attempt. If the call attempt Cf is assigned a

channel, the RoD procedure exists.

Step RoD II.3: If no idle macrocell channel is found in Step RoD II.1, then following Crite-rion 1, the HCN checks if there is any slow M-to-m repacking candidate. If so, the procedure proceeds to Step RoD II.4. Otherwise, Step II.6 is executed.

Steps RoD II.4 and 5: The HCN performs M-to-m repacking to generate a free macrocell

channel for Cfand the procedure exits.

Steps RoD II.6 and 7: If no slow M-to-m repack-ing candidate is found, the HCN tries to allocate a channel in the ith microcell to the call.

Steps RoD II.8 and 9: If no idle channel is available in the microcell, the HCN checks if there is any fast M-to-m repacking candidate. If so, go to Step RoD II.4. Otherwise, the call

attempt Cfis rejected.

In RoD, there are two alternatives to select the M-to-m repacking candidate at Steps RoD I.3 and RoD II.4. Random RoD (RoD-R) randomly selects a repacking candidate with the same proba-bility. Load Balancing RoD (RoD-L) selects the repacking candidate whose microcell has the least traﬃc load. Both RoD-R and RoD-L can be adopted by an HCN that utilizes radio systems

such as GSM/PCS1900[13]or WCDMA[6], where

the handoﬀ decision is made by the network.

Partial RoD (PRoD; or preemption in[19,20]) is a

special case of RoD, where Steps RoD I.5 (for slow calls) and RoD II.8 (for fast calls) are not executed. That is, PRoD only repacks slow calls.

Note that for both PRoD and RoD, Criterion 2 is used in channel assignment, but not in

Fig. 5. RoD I: RoD for slow MSs.

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repacking. Only AR uses Criterion 2 in repacking. The primary purpose is to reduce the number of force-terminations for fast calls. Our study in Sec-tion 4 indicates that AR does not achieve its goal to reduce the force-termination probability as compared with RoD.

3. System model for HCN channel assignment This section describes the input parameters and output measures for the HCN channel assignment model. For the demonstration purpose, we con-sider a wrapped mesh cell conﬁguration as shown in Fig. 7. This conﬁguration consists of four

macrocells. Each macrocell covers 4· 4 microcells.

The wrapped topology simulates unbounded HCN so that the boundary cell eﬀects can be ignored

[12]. Without loss of generality, the MS moves to

one of the four neighbor microcells with the same

probability (i.e., 0.25). Three types of input param-eters are considered.

• System parameters: Each macrocell has C radio channels, and each microcell has c radio channels.

• Traﬃc parameters: The call arrivals to a micro-cell (for both incoming and outgoing calls) form a Poisson stream with rate k. For fast MSs and slow MSs, the call arrivals rates are fk and

(1 f)k, respectively, where 0 6 f 6 1. The call

holding times have a Gamma distribution with

mean 1/l and variance Vc (the typical value

for 1/l is 1 min).

• Mobility parameters: The microcell residence times of slow (fast) MSs have a Gamma

distri-bution with mean 1/gs (1/gf) and variance Vm, s

(Vm, f).

The Gamma distribution is often used in mobile telecommunications network modeling (call block-ing analysis for PCS networks under general cell

residence time[2], teletraﬃc analysis and mobility

modeling for PCS network [3], analytical results

for optimal choice of location update interval for mobility database failure restoration in PCS

net-works[4]). It has been shown that the distribution

of any positive random variable can be approxi-mated by a mixture of Gamma distributions (see Lemma 3.9 in reversibility and stochastic networks

[9]). Several output measures are deﬁned in this

study:

Pb: the probability that a new call is blocked.

Pf: the probability that a successfully

con-nected call is forced to terminate because of handoﬀ failure.

Pﬀ: the probability that a successfully

con-nected fast call is forced to terminate because of handoﬀ failure.

Pnc: the probability that a new call is blocked

or a connected call is forced to terminate

H: the expected number of handoﬀs

(includ-ing repack(includ-ings) occurred dur(includ-ing a call.

Fig. 8 shows five types of handoffs. Handoff measures for these handoff types are defined as follows.

Macrocell Microcell

Fig. 7. Hierarchical cellular network with wrapped mesh conﬁguration.

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Hmm: the expected number of handoﬀs from a

microcell to another microcell during a

call (Fig. 8(a)).

HmM: the expected number of handoﬀs from a

microcell to a macrocell during a call (Fig. 8(b)).

HMm: the expected number of handoﬀs from a

macrocell to a microcell during a call (Fig. 8(c)).

HMM: the expected number of handoﬀs from a

macrocell to another macrocell during a

call (Fig. 8(d)).

HR: the expected number of repackings

(including m-to-M and M-to-m

repac-kings) during a call (Fig. 8(e)).

From the above description, H can be expressed as

H ¼ Hmmþ HmMþ HMmþ HMMþ HR: ð1Þ

Based on above discussions, a discrete event simulation model for RoD-R is described in

Appendix A. Other strategies (such as NR, AR, PRoD and RoD-L) can be studied by similar

sim-ulation models, and the details are omitted. Table

1lists the notation described in this section, which

will be used in the remainder of this paper. Note that the default values for the input parameters

are also presented inTable 1.

Fig. 8. Handoﬀ types.

Table 1 Notation

Notation Description Default value for input parameter

C The number of radio channels in a macrocell 8

c The number of radio channels in a microcell 10

Traﬃc parameters

k The call arrival rate to a microcell 7l

f The proportion of fasts calls 9%

1/l The expected call holding time 1 min

Vc The variance for call holding times 1/l

2

Mobility parameters

1/gs The expected microcell residence time of slow MSs 10/l

1/gf The expected microcell residence time of fast MSs 1/5gs

Vm,s The variance for microcell residence times of slow MSs 100/l2

Vm,f The variance for microcell residence times of fast MSs Vm,s/25

Output measures

Pb The call blocking probability

Pf The force-termination probability

Pﬀ The force-termination probability of fast calls

Pnc The call incomplete probability

Hmm The expected number of microcell-to-microcell handoﬀs per call

HmM The expected number of microcell-to-macrocell handoﬀs per call

HMm The expected number of macrocell-to-microcell handoﬀs per call

HMM The expected number of macrocell-to-macrocell handoﬀs per call

HR The expected number of repackings

(including m-to-M and M-to-m repackings) during a call H The expected number of handoﬀs and repackings per call

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4. Results and discussions

We compare NR, AR, PRoD, RoD-R and RoD-L in terms of the output measures listed in

Table 1. In our numerical examples, the radio channel number is c = 10 for every microcell and the expected microcell residence time for slow MSs is 5 times the value for fast MSs (i.e., 1/

gs= 5/gf). The eﬀects of the input parameters are

described as follows.

Eﬀect of the macrocell channel number C. Fig. 9

(a)–(d) plot Pﬀ, Pb, Pfand Pnc as functions of C,

where the values for the input parameters except

C follow the default values listed inTable 1. These

ﬁgures show an intuitive result that for all

ap-proaches, Pﬀ, Pb, Pf and Pnc decrease as C

in-creases. We also observe that the Pﬀ, Pb, Pf and

Pncare more sensitive to the change of C for small

C values than for large C values. Since the macro-cell channels are the bottleneck resources when C

is small, increasing C signiﬁcantly reduces Pﬀ, Pb,

Pf and Pnc. Fig. 10(a)–(d) plot Hmm, HmM, HR

and H as functions of C, where the values for the input parameters except C follow the default

values listed inTable 1. Fig. 10(a) shows that for

Fig. 9. Eﬀects of macrocell channel number C on (a) force-termination probability Pﬀof fast calls, (b) new call blocking probability Pb,

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all approaches, Hmmis a decreasing function of C.

This phenomenon is due to the fact that when C increases, more calls (especially fast calls) will occupy macrocell channels, and the number of

microcell to microcell handoﬀs will decrease.Fig.

10(b) shows that for AR, HmM is an increasing

function of C. Increasing C results in more idle macrocell channels, and more calls (especially slow calls) will be handed oﬀ from microcells to

macro-cells. For NR, PRoD and RoD, HmM increases

and then decreases as C increases. When C is small, macrocell channels are the bottleneck re-sources and increasing C results in more calls

handed oﬀ from microcells to macrocells (espe-cially fast calls that overﬂow to microcells). When

C is large (C > 5 inFig. 10(b)), macrocell channels

are no longer the bottleneck resources and less fast calls overﬂow to microcells. In this case, increasing

C results in decreasing of HmM. The performance

ﬁgures for HMm and HMM are similar to that for

HmM, and the details are omitted.Fig. 10(c) shows

that for AR, HRincreases as C increases.

Increas-ing C results in more slow calls that would over-ﬂow to macrocells, and thus more calls are repacked. On the other hand, for PRoD and

RoD, HR increases and then decreases as C

Fig. 10. Eﬀects of the macrocell channel number C on (a) expected number Hmmof microcell to microcell handoﬀs, (b) expected

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increases. This non-trivial phenomenon is ex-plained as follows. When C is small (C < 5 for

RoD and C < 12 for PRoD inFig. 10(c)),

increas-ing C results in more M-to-m repackincreas-ing candi-dates, and more on-demand repackings are exercised. When C is large, macrocell channels are no longer the bottleneck resources. Increasing C results in less blockings as well as force-termina-tions, and less on-demand repackings are needed.

Therefore HRdecreases as C increases in this case.

Fig. 10(d) shows the net effects of repackings and all types of handoffs. In this figure, as C increases, H decreases for NR and increases for AR. For PRoD and RoD, H increases and then decreases as C increases.

Comparison of NR, AR, PRoD, R and RoD-L. Fig. 9(a)–(d) compare NR, AR, PRoD,

RoD-R and RoD-RoD-L on Pﬀ, Pb, Pfand Pnc, respectively.

These ﬁgures show that RoD-R and RoD-L have

smaller Pﬀ, Pb, Pf and Pnc values than NR, AR

and PRoD. Furthermore, AR has higher Pncthan

NR when C is small, and the opposite result is

ob-served when C is large (C > 12.5 inFig. 9(d)). This

non-trivial phenomenon is explained as follows. Consider the case when C is much less than the number of fast calls. In AR, m-to-M repacking is always exercised, and therefore macrocells have less idle channels in AR than in NR. In this case,

the Pnc value is higher for AR than for NR. On

the other hand, when C is large, the eﬀect of

M-to-m repacking for slow calls becomes more signif-icant. Thus macrocells have more idle channels in AR than in NR. Speciﬁcally, if the HCN is

engi-neered at Pnc= 2% (see the horizontal dashed line

inFig. 9(d)), C = 5.5 for RoD, C = 13 for PRoD, C = 14.5 for AR, and C = 15 for NR. Thus RoD can save at least 7 macrocell channels over other approaches.

Fig. 10(c) shows that HR,RoDR> HR,RoDL>

HR,AR> HR,PRoD> HR,NR= 0 when C is small

(C < 12). When C is large (C > 12), HR,AR>

HR,RoDR > HR,RoDL> HR,PRoD> HR,NR= 0.

Similar phenomena for H are observed in Fig.

10(d). Although NR, AR and PRoD have smaller

HR and H values than RoD, the increase of HR

and H does not results in the increase of Pﬀ, Pb,

Pfand Pncin RoD (seeFig. 9(a)–(d)).

Eﬀect of the proportion f of fast MSs.Fig. 11plots

Pncas a function of f, where the values for the

in-put parameters except f follow the default values

listed inTable 1. In the ﬁgure, the call arrival rates

for fast and slow MSs are fk and (1 f)k,

respec-tively. This ﬁgure shows that for all approaches,

Pncincreases as f increases. Increasing of fast calls

results in the increase of handoﬀs and hence force-terminations. This ﬁgure also shows that RoD approaches (i.e., RoD-R and RoD-L) are less sensitive to f than other approaches. Further-more, when f is large (f > 8% for AR and f > 18% for PRoD), AR and PRoD have higher

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Pnc values than NR. The reason is that for AR,

when f is large, the eﬀect of m-to-M repackings is more signiﬁcant than that of M-to-m repac-kings. Thus, less idle channels are available in macrocells for AR than for NR, and more calls are blocked or forced to terminate in AR. For PRoD, when f is large, more fast calls repack slow calls, which occupy more macrocell channels. Therefore, more slow calls are blocked or forced to terminate.

Eﬀect of the MS mobility (i.e., mean microcell

res-idence times).Fig. 12plots Pncas a function of the

microcell mobility rates (i.e., gs for slow MSs and

gf= 5gs for fast MSs), where the values for the

input parameters except 1/gs and 1/gf follow the

default values listed inTable 1. This ﬁgure shows

thatPnc increases as gs increases. This ﬁgure also

shows that to keep the same Pnc performance,

(e.g., Pnc= 7%), RoD can support much faster

MSs (at least 23 times the gs value) than other

approaches.

Eﬀect of the arrival rate k. Fig. 13 plots Pnc as a

function of k, where the values for the input parameters except k follow the default values listed inTable 1. This ﬁgure shows that Pncincreases as k

increases. It also shows that to keep the same Pnc

performance (e.g., Pnc= 2%), RoD can support

more call arrivals (at least 18%) than other approaches.

Eﬀect of the variance Vcfor the call holding times.

Fig. 14 plots Pncand H as functions of Vc, where

the values for the input parameters except Vc

fol-low the default values listed in Table 1. Fig. 14

shows that Pnc and H decrease as Vc increases.

Note that, for the call holding time distributions with the same mean value 1/l, the standard

devia-tion r¼ ffiffiffiffiffiffiVc

p

. By the ChebyshevÕs Inequality, the probability that the call holding times are out of

range ½1=l ð5 ffiffiffiffiffiffiVc

p

Þ=3; 1=l þ ð5 ffiffiffiffiffiffiVc

p

Þ=3 is

smal-ler than 36% for all Vc values. For example, if

Vc= 100/l2, then ð5 ffiffiffiffiffiffiVc

p

Þ=3 ¼ 50=3l and the probability that the call holding time exceeds 53/

3l is smaller than 36%. As Vcincreases, more long

and short call holding times are observed. More short call holding times implies that more calls are completed before next new call attempts arrive or next handoﬀ attempts are exercised. Thus the numbers of blocked calls, force-terminated calls and handoﬀs decrease.

Eﬀect of the variances for the microcell residence

times.Fig. 15plots Pncand H as functions of

var-iances (i.e., Vm,sfor slow MSs and Vm,f= Vm,s/25

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for fast MSs) for the microcell residence times, where the values for the input parameters except

Vm,s and Vm,f follow the default values listed in

Table 1. Fig. 15 shows that Pnc and H decrease

as Vm,s increases. From the residual life theorem

[10], the mean value of the ﬁrst microcell residence

time increases as Vm,sincreases, which implies that

more calls will complete in the ﬁrst microcell be-fore they are handed oﬀ to the next cells.

There-fore, both Pncand H drop as Vm,sincreases.

Fig. 13. Eﬀects of the arrival rate k on incomplete probability Pnc.

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5. Conclusions

By considering the moving speeds of MSs, this paper proposed the repacking on demand (RoD) ap-proach for channel assignment in the hierarchical cellular network (HCN). We developed simulation models to investigate the RoD performance on the

blocking probability Pb, the force-termination

prob-ability Pf, the incomplete probability Pncand the

ex-pected number of handoﬀs H during a call (for both slow and fast calls). We compared RoD with other HCN channel assignment approaches including No Repacking (NR), Always Repacking (AR) and Partial RoD (PRoD). Our study indicated that • If the requests for the macrocell channels can

not be satisﬁed (e.g., when the number of the macrocell channel is small, the call arrival rate and MS mobility are high, and so on), macro-cell channels are the bottleneck resources. In

this case, RoD signiﬁcantly reduces Pb, Pf,

and Pncas compared with other approaches.

• The Pb, Pf, and Pncperformance for RoD is not

sensitive to the proportion of fast call arrivals as compared with other approaches. That is, the increase of fast calls does not aﬀect RoD as much as other approaches.

• With the same Pncperformance, RoD can

sup-port much faster MSs and/or more call arrivals than other approaches.

• In RoD, Random RoD (RoD-R) and Load

Balancing RoD (RoD-L) have the same Pb,

Pf, and Pnc performance. Note that in

repack-ing, macrocell is a resource pool used to adjust traﬃc load of each microcell. That is, repacking already conducts the load balancing function to eﬀectively balance the system workload, and the

load-balancing improvement by RoD-L

becomes insigniﬁcant. Therefore, the perform-ance resulted from random selection for repack-ing candidates (i.e., RoD-R) is similar to that for RoD-L. This result is very important for network operators because RoD-R is much eas-ier to implement than RoD-L.

Acknowledgments

We would like to thank the anonymous reviewers. Their comments have signiﬁcantly improved the quality of this paper. This work was sponsored in part by Intel, Microsoft, National Science Council under contracts 2213-E-002-083, NSC93-2213-E-002-025 and NSC93-2213-E-009-100.

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Appendix A. Simulation model for RoD-R

This appendix describes the discrete event simu-lation model for RoD-R. In this model, three types of events are deﬁned to represent call arrival, call completion, and MS movement. The following at-tributes are deﬁned for an event e:

• The type attribute indicates the event type. An Arrival event represents a new call arrival. A Move event represents an MS movement from one cell to another. A Complete event represents a call completion.

• The ts attribute indicates the time when the event occurs.

• The tc attribute indicates the time when the call corresponding to e will complete. Note that tc P ts.

• The mc attribute indicates the microcell where the MS (corresponding to this event) resides. • The ma attribute indicates the macrocell where

the MS (corresponding to this event) resides. • The speed attribute indicates the moving speed

(i.e., Slow or Fast) of the MS (corresponding to this event).

• The is_macro attribute indicates whether the call corresponding to this event occupies a macrocell channel. If a macrocell channel is occupied, is_macro = 1. Otherwise, is_macro = 0.

In the simulation model, an array mc_ch[i] is used to represent the number of the idle channels of microcell i. Another array ma_ch[j] is used to represent the number of the idle channels of macrocell j. Two variables new_mc and new_ma are used to indicate the target micro and macro cells where the call (corresponding to current event) is newly generated from or handed oﬀ to. The output measures of the simulation are the number N of total call arrivals during the

simula-tion, the number Nb of blocked calls, the number

Nf of force-terminated calls, the number NR of

repackings, the number Nhof handoﬀs, the

num-ber Nsf of successfully connected fast calls, and

the number Nﬀ of force-terminated fast calls.

From the above output measures, we can

compute Pb¼ Nb N ; Pf¼ Nf N Nb ; Pnc¼ Nbþ Nf N ; Pff ¼ Nff Nsf ; HR¼ NR N Nb and H¼ Nh N Nb : ð2Þ A simulation clock is maintained to indicate the simulation progress, which is the timestamp of the event being processed. All events are inserted into the event list, and are deleted/processed from the event list in the non-decreasing timestamp order.

Figs. 16 and 17 show the simulation flow chart for RoD-R. In this flow chart, Step 1.1 initializes the input parameters. Step 1.2 generates the first Arrival events for each microcell and inserts these events into the event list. In Steps 1.3 and 1.4, the next event e in the event list is processed based on its type. There are three cases:

Case I. e.type = Arrival: At Step 1.5, if

N Nb> N*(in our simulation, N* = 6· 105·

64), then Step 1.6 computes the output measures

using (2), and the simulation terminates.

Other-wise, Step 1.7 generates the next Arrival event

e1 for the same microcell (i.e., e1.mc = e.mc,

e1.ma = e.ma, e1.speed = e.speed and e1

.is_ma-cro = 0), and sets the target cell where the incom-ing call arrives (new_mc = e.mc and new_ma =

e.ma). The timestamp of e1 equals e.ts plus the

inter call arrival time generated by a random

num-ber generator. Then e1 is inserted into the event

list, and the simulation proceeds to execute

Algo-rithm A inFig. 17.

In Algorithm A (seeFig. 17), the HCN tries to

allocate a channel for the call. The steps are de-scribed as follows. If e.speed = Slow at Step 2.1, Step 2.2 checks if microcell new_mc has idle chan-nels (i.e., mc_ch[new_mc] > 0). If so, a channel is as-signed to the incoming call. Then mc_ch[new_mc] is decremented by 1 at Step 2.3 and this algorithm terminates at exit B(i.e., the call is assigned a chan-nel). At Step 2.2, if microcell new_mc has no idle channel (i.e., mc_ch[new_mc] = 0), the call attempt overﬂows to macrocell new_ma. Then Step 2.4 checks if macrocell new_ma has idle channels (i.e., ma_ch[new_ma] > 0). If so, a macrocell channel is assigned to the call at Step 2.5. In this case, ma_ch-[new_ma] is decremented by 1, and

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e.is_macro is set to 1. Then this algorithm termi-nates at exit B. If macrocell new_ma has no idle channel (i.e., ma_ch[new_ma] = 0) at Step 2.4, then Step 2.6 checks if there is any slow M-to-m repack-ing candidate in macrocell new_ma. If so, Step 2.7 randomly selects a slow M-to-m repacking candi-date (i.e., a call corresponding to a Complete or

Move event e3) to be handed oﬀ from macrocell

new_ma to microcell e3.mc (i.e., e3.is_macro is set

to 0), and the reclaimed macrocell channel is as-signed to the incoming call (i.e., e.is_macro is set

to 1). Step 2.7 also decrements mc_ch[e3.mc] by

1, and increments both NRand Nhby 1. Then this

algorithm terminates at exit B. If no slow M-to-m repacking candidate is found at Step 2.6, Step 2.8 checks if there is any fast M-to-m repacking can-didate in macrocell new_ma. If so, Step 2.7 is exe-cuted to randomly select a fast M-to-m repacking candidate, and this algorithm terminates at exit B. Otherwise, if no fast M-to-m repacking candidate is found at Step 2.8, the incoming call is not as-signed any idle channel. In this case, this algo-rithm terminates at exit C (i.e., no channel is assigned to the call).

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For the case of e.speed = Fast at Step 2.1, Step 2.9 checks if macrocell new_ma has idle channels (i.e., ma_ch[new_ma] > 0). If so, Step 2.5 is exe-cuted, and the algorithm terminates at exit B. Otherwise, Step 2.10 checks if there is any slow M-to-m repacking candidate in macrocell new_ma. If so, Step 2.7 is executed, and this algorithm ter-minates at exit B. If no slow M-to-m repacking candidates are found in macrocell new_ma, then Step 2.11 checks if microcell new_mc has idle chan-nels (i.e., mc_ch[new_mc] > 0). If so, Step 2.3 is executed, and this algorithm terminates at exit B. If microcell new_mc has no idle channel at Step 2.11, Step 2.12 checks if there is any fast M-to-m repacking candidate in macrocell new_ma. If so, Step 2.7 is executed, and this algorithm terminates at exit B. Otherwise, the incoming call is not as-signed any idle channel, and the this algorithm ter-minates at exit C.

If any channel is assigned to the incoming call in Fig. 17 (i.e., Algorithm A exits from B), Step

1.8 updates the Nsfvalue (i.e., if this is a fast call,

increment Nsf by 1). Step 1.8 then computes the

call completion time e.tc as e.ts plus the call hold-ing time. Step 1.8 also determines the MS move

time Thwhen the MS moves out of the microcell.

Th equals e.ts plus the cell residence time. Then

Step 1.9 determines the next event (i.e., a Move event or Complete event) for the call

correspond-ing to e. If the MS will move to another cell after

call completion (i.e., e.tc 6 Th), then Step 1.10 is

executed to generate a Complete event e2 where

e2.ts = e.tc, e2.mc = e.mc, e2.ma = e.ma, e2

.spee-d = e.spee.spee-d an.spee-d e2.is_macro = e.is_macro. Event

e2 is inserted into the event list. Otherwise, if the

MS moves to another cell before the call

comple-tion (i.e., e.tc > Th) at Step 1.9, then Step 1.11 is

executed to generate the next Move event e2 with

the timestamp Th for this call (i.e., e2.ts = Th,

e2.tc = e.tc, e2.mc = e.mc, e2.ma = e.ma, e2

.spee-d = e.spee.spee-d an.spee-d e2.is_macro = e.is_macro). Event

e2is inserted into the event list. On the other hand,

if HCN has no idle channel for the incoming call (i.e., Algorithm A exits from C), the incoming call

is blocked and Nbis incremented by 1 at Step 1.12.

Case II. e.type = Move: Step 1.13 selects the next microcell new_mc and its macrocell new_ma for the MS corresponding to event e. At Step 1.14, if the call of the MS occupies a microcell channel (i.e., e.is_macro = 0), the occupied channel of microcell e.mc is released (mc_ch[e.mc] is incre-mented by 1 at Step 1.15). Then the HCN tries to allocate a channel for the call, and the

simula-tion proceeds to execute Algorithm A in Fig. 17

as described in Arrival event. If the handoﬀ call is assigned a channel (i.e., Algorithm A exits from

B), Nh is incremented by 1 at Step 1.16. Step

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1.17 updates the current cell for the call (i.e., e.mc = new_mc and e.ma = new_ma) and computes

the next MS move time Th for the call. Then the

simulation proceeds to execute Steps 1.9 and 1.10 (or 1.11). On the other hand, if HCN has no idle channel for the call (i.e., Algorithm A exits from C), the call is forced to terminate. Then Step 1.18

is executed to increment Nf by 1 and update the

Nﬀ value (i.e., if this is a fast call, increment Nﬀ

by 1).

If the call occupies a macrocell channel at Step 1.14, then Step 1.19 checks if the MS is moving out of its macrocell e.ma. If so (i.e., new_ma 5 e.ma), the call is handed oﬀ to the new cell. Step 1.20 increments ma_ch[e.ma] by 1; i.e., the occupied channel of e.ma is released. Then the simulation proceeds to execute Algorithm A. If the handoﬀ call is assigned a channel (i.e., Algorithm A exits from B), Steps 1.16, 1.17, 1.9 and 1.10 (or 1.11) are then executed. Otherwise, if HCN has no idle channel for the call (i.e., Algorithm A exits from C), Step 1.18 is then executed. If the MS does not move out of its macrocell (i.e., e.ma = new_ma) at Step 1.19, the simulation proceeds to execute Step 1.17 and then Steps 1.9 and 1.10 (or 1.11). Case III. e.type = Complete: At Step 1.21, if the call occupies a macrocell channel (i.e., e.is_ma-cro = 1), then Step 1.22 is executed to increment ma_ch[e.ma] by 1. Otherwise, Step 1.23 is executed to increment mc_ch[e.mc] by 1.

To accommodate RoD-L, we only need to

modify Step 2.7 of the ﬂowchart in Fig. 17. In

our simulation experiments, the conﬁdence inter-vals of the 99% conﬁdence levels are within 3% of the mean values in most cases. The simulation models are partially validated by the analytic

model in [7]for the no mobility case. The details

are omitted.

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Hsien-Ming Tsai was born in Tainan, Taiwan, R.O.C., in 1973. He received the double B.S. degrees in Computer Science and Information Engineering (CSIE) and Communication Engi-neering, the M.S. degree in CSIE, and the Ph.D. degree in CSIE from National Chiao-Tung University (NCTU), Taiwan, in 1996, 1997, and 2002, respectively. He is currently a senior research specialist in Quanta Research Institute, Quanta Computer Inc. His research interests are in the areas of cellular protocols (UMTS/GPRS/GSM/DECT), cellu-lar multimedia (MPEG-4 Audio/Speech), and embedded sys-tems. He is an IEEE member.

Ai-Chun Pang was born in Hsinchu, Taiwan, R.O.C., in 1973. She received the B.S., M.S. and Ph.D. degrees in Computer Science and Information Engineering from National Chiao Tung University in 1996, 1998 and 2002, respectively. She joined the Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan, as an Assistant Professor in 2002. From August 2004, She also serves as an Assistant Professor in Graduate Insti-tute of Networking and Multimedia, National Taiwan Uni-versity, Taipei, Taiwan. Her research interests include design and analysis of personal communications services network, mobile computing, voice over IP and performance modeling.

Yung-Chun Lin was born in Kao-hsiung, Taiwan, R.O.C., in 1978. He received the B.S. and M.S. degrees in Computer Science and Information Engineering (CSIE) from National Chiao-Tung University (NCTU), Tai-wan, in 2001, 2003, respectively. He is currently pursuing the Ph.D. degree in CSIE. His research interests include design and analysis of a personal communications services network, the cellular protocols (UMTS/GPRS/ GSM), and mobile computing.

Yi-Bing Lin received his BSEE degree from National Cheng Kung University in 1983, and his Ph.D. degree in Computer Science from the University of Washington in 1990. From 1990 to 1995, he was with the Applied Research Area at Bell Communica-tions Research (Bellcore), Morristown, NJ. In 1995, he was appointed as a professor of Department of Computer Science and Information Engineering (CSIE), National Chiao Tung Uni-versity (NCTU). In 1996, he was appointed as Deputy Director of Microelectronics and Infor-mation Systems Research Center, NCTU. During 1997–1999, he was elected as Chairman of CSIE, NCTU. His current research interests include design and analysis of personal communications services network, mobile computing, distrib-uted simulation, and performance modeling. He has published over 150 journal articles and more than 200 conference papers. He is an Adjunct Research Fellow of Academia Sinica, and is Chair Professor of NCTU and Providence University. He serves as consultant of many telecommunications companies including FarEasTone and Chung Hwa Telecom. He is an IEEE Fellow and an ACM Fellow.