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Performance Analysis

for Dual Band PCS Networks

Yi-Bing Lin, Senior Member, IEEE, Wei-Ru Lai, and Rong-Jaye Chen, Member, IEEE

AbstractÐIn a dual band personal communications services (PCS) network, heavy traffic areas are covered by microcells which overlay macrocells. In such a network, microcells and macrocells utilize different frequency bands. We propose an analytic model to study the performance of a dual band PCS architecture. Our model assumes that a PCS subscriber has a general residence time distribution in a microcell and the macrocell residence time distribution is derived from the microcell residence time distribution. An iterative algorithm is used to compute the overflow traffic from a microcell to its overlaid macrocell. Then, the call incompletion probability is computed by a macrocell model based on the overflow traffic. Our study indicates that the variance of the microcell residence time distribution and the number of microcells covered by a macrocell have significant effects on the call incompletion probability.

Index TermsÐDual band, GSM, handoff, macrocell, microcell, personal communications services.

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NTRODUCTION

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PERSONAL communications services (PCS) network provides communication services to mobile users via their mobile stations (MS) or handsets. When the PCS network attempts to deliver a call to an MS or the MS attempts to originate a call, the call is connected between the MS and the base transceiver station (BTS) through the radio channel within the radio coverage of the BTS or the cell. If no idle radio channel is available, the call is blocked. When the MS moves from a cell to another during the conversation, a handoff occurs. To continue the call, the MS requests an idle radio channel in the new cell to set up a new connection and drops the connection to the old cell. If the handoff is rejected because no idle radio channel is available in the new cell, the call is forced to terminate. PCS networks such as GSM [16] exercise the nonprioritized channel allocation scheme in which the BTS handles a handoff request exactly the same as a new call. In a PCS network, an important performance measure is the incom-pletion probability or the probability that a call is blocked or forced to terminate.

PCS channel assignment has been intensively studied in the literature [3], [4], [6], [10], [17], [21]. In heavy traffic areas, the new call blocking probability and the forced termination probability grow rapidly as traffic increases. Microcell/macrocell architectures have been proposed [2], [18], [8], [7] to reduce the call incompletion probability. In Taiwan, a dual band GSM microcell/macrocell cellular network has been deployed by Far EasTone Telecommuni-cations Co., Ltd. to improve the capacity of GSM services

[16]. This microcell/macrocell configuration (see Fig. 1) consists of two types of BTSs: The DCS1800 BTSs operate at 1.8 GHz and the GSM900 BTSs operate at 900 MHz. In this configuration, the GSM900 BTSs serve as the base stations for macrocells while the DCS1800 BTSs serve as the microcell base stations. Every macrocell overlays with one or more microcells (see Fig. 2a). A GSM900 BTS and its overlaid DCS1800 BTSs are connected to the same Base Station Controller (BSC), as illustrated in Fig. 1. The BSCs connect to mobile switching centers (MSCs) and the MSCs connect to the switches in the public switched telephone network. The reader is referred to [16] for the details of BSC and MSC.

In this dual band configuration, a new call arriving at the system is first handled by a microcell. If no channel is available in the microcell, the call is handled by the macrocell. The call is blocked if no channel is available in the macrocell (see Fig. 2b). During the conversation, an MS may move from a microcell to another microcell and the call will be handed over among microcells. During a microcell handoff, if no channel is available in the new microcell, the handoff call is handled by the macrocell. In a dual band network, it is important to determine the number of microcells covered by a macrocell to achieve good perfor-mance (i.e., low call incompletion probability). This paper studies how the microcell/macrocell configuration and the MS movement behavior affect the dual band network performance. We first derive the MS residence time distribution (the interval between the arrival and the departure of an MS in a cell) for a macrocell based on the microcell residence time distribution. By using the overflow traffic as the input, the call incompletion probability is computed in the macrocell model (see Fig. 2b).

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The microcell model to be presented in this section generalizes our previous work [14]. Three assumptions are used in this paper:

. Y.-B. Lin and R.-J. Chen are with the Department of Computer Science and Information Engineering, National Chaio Tung University, Hsinchu, Taiwan, ROC. E-mail: {liny, rjchen}@csie.nctu.edu.tw.

. W.-R. Lai is with the Department of Information Management, Chin-Min College, Tou-Fen, Miao-Li, Taiwan, ROC.

E-mail: wrlai@mis.chinmin.edu.tw. Manuscript accepted 24 Nov. 1999.

For information on obtaining reprints of this article, please send e-mail to: tc@computer.org, and reference IEEECS Log Number 108280.

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Assumption 1. The phone call arrivals to an MS are a Poisson process.

Assumption 2. The call holding time (the interval between the time the conversation starts and the time it completes) is exponentially distributed.

Assumption 3. The microcell residence times of an MS have a general distribution.

Assumption 1 is required to derive the interval between the time of a call arrival and when the MS leaves the microcell (to be elaborated in Section 2.1). This assumption also guarantees that the net call arrival process to a microcell is

Poisson. Note that, as long as the number of MS in a microcell is sufficiently large, the net call arrivals to a microcell are Poisson even if the call arrivals to individual MS are not. Assumption 2 does not hold for wireline calls. Many telephone companies offer flat rate charging for local phone calls and, so, very long call holding times are observed (e.g., computer dial-up connection for several days). Thus, long tails may be expected in the call holding time distributions. In this case, Lognormal or Gamma distributions are more appropriate to approximate the call holding time distribution. Cellular phone calls, on the other hand, are charged based on the air-time usage, and the

Fig. 1. Dual band GSM network.

Fig. 2. Homogeneous microcell/macrocell layout and channel assignment procedure. (a) Microcell/macrocell configuration. (b) The channel assignment procedure.

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users do not hold unnecessary long calls. Data measured from real GSM network operation indicated that exponen-tial call holding time distribution is reasonable and the mean call holding time is typically less than one third of that for wireline calls.

2.1 The MS Residence Time Distributions

Consider the timing diagram in Fig. 3. From Assumption 1, the call arrivals to an MS are a Poisson process with arrival rate . That is, tp has a density function eÿtp. From

Assumption 3, the MS residence time tmat microcell Cihas

a general density function fm;i…tm† and the distribution

Fm;i…tm† ˆRˆ0tm fm;i…†d. Let m be the interval between

when a call arrives and when the MS moves out of Ci. Let

rm;i…m† be the density function of m. If fm;i…t† ˆ fm;j…t† for

all i; j, then rm;i…t† ˆ rm;j…t† can be derived by using the

excess life property [5]. Unfortunately, the excess life property does not apply when fm;i…t† 6ˆ fm;j…t† and the

residence time distribution should be derived as follows: From the definition of density function, we have

rm;i…m† ˆ limd

m!0

Pr‰mdmwhere mˆ tmÿ txjtm> txŠ

dm : …1†

From Assumption 1 and the memoryless property of the exponential distribution, tx has the same exponential

distribution as tp(where txand tpare illustrated in Fig. 3).

Thus, (1) is rewritten as rm;i…m† ˆ limd m!0 Pr‰mdmwhere mˆ tmÿ txand tm> txŠ dm ˆ Z 1 txˆ0 eÿtxf m;i…tx‡ m†dtx …2† ˆ em f m;i…† ÿ Z m t0ˆ0fm;i…t 0†eÿt0 dt0   : …3† Equation (3) is derived from (2) by using the Laplace transform rule P5.2.1 in [19], where

f m;i…s† ˆ

Z 1

mˆ0

fm;i…m†eÿsmdm

is the Laplace transform for the MS residence time distribution Fm;i…m†. The probability Pr‰tm> txŠ is

Pr‰tm> txŠ ˆ Z 1 tmˆ0 Z tm txˆ0 fm;i…tm†eÿtxdtxdtm ˆ 1 ÿ f m;i…†: …4† From (3) and (4) and since

Pr‰mˆ tmÿ txjtm> txŠ ˆPr‰mˆ tPr‰tmÿ txand tm> txŠ m> txŠ ; we have rm;i…m† ˆ em f m;i…† ÿ Rm t0ˆ0fm;i…t0†eÿt 0 dt0 h i 1 ÿ f m;i…†

and its Laplace transform is r m;i…s† ˆ Z 1 mˆ0 rm;i…m†eÿsmdm ˆ R1 mˆ0fm;i…†em eÿsm dmÿ R1 mˆ0e…ÿs†m Rm t0ˆ0fm;i…m†eÿt 0 dt0dm h i 1ÿfm;i…† …5† ˆ  …s ÿ †‰1 ÿ f m;i…†Š ( ) f m;i…† ÿ fm;i …s†  : …6† The first term of (6) is derived from the first term of (5) by using the Laplace transform rule P.1.3 in [19]. The second term of (6) is derived from the second term of (5) by using the Laplace transform rules P1.1.3 and P1.1.6 in [19].

In a practical PCS environment, the number of MSs in a cell is expected to be much larger than the number of channels, and the traffic (the net call arrival rate) to the microcell is engineered at one or two percent blocking probability, which implies that the call arrival rate per MS  ! 0 for a fixed net call arrival stream [20]. If fm;i…t† is a

Gamma density function [9] with the shape parameter i

and mean 1=i (i.e., with the scale parameter ii), then

f

m;i…s† ˆ s‡ iiii

  i

. By using the l'Hospital's rule, we showed [14] that, for Gamma residence time distribution,

lim !0r  m;i…s† ˆsi  1 ÿ f m;i…s†  ; …7†

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which is the same as the r

m…s† function for the

homo-geneous PCS network (i.e., when fm;i…t† ˆ fm;j…t† ˆ fm…t†

for i 6ˆ j) [15]. In other words, (7) indicates that if the call activity per user is not heavy (but the net call traffic to a cell can be either light or heavy), then the mdistribution for the

heterogeneous network can be approximated by the homogeneous network.

We are particularly interested in the Gamma distribu-tion. It has been shown that the distribution of any positive random variable can be approximated by a mixture of Gamma distributions (see Lemma 3.9 in [11]). One may also measure the MS residence times in a real PCS network and the measured data can be approximated by a Gamma distribution as the input to our handoff model. In a heterogeneous PCS network, it suffices to use the Gamma distributions with different shape and scale parameters to represent different MS residence time distributions. 2.2 The Handoff Traffic

Consider a group of M microcells, where every macrocell covers N microcells (i.e., there are M=N macrocells, where M >> N). Let pi;jbe the probability that an MS moves from

microcell Ci to microcell Cj where 1  i; j  M. It is clear

that pi;iˆ 0. The handoff call arrival rate h…i† to Ci is

derived as follows:

Assume that the new call blocking probability po…i† and

the forced termination probability pf…i† for a microcell Ci

are known. We will show how to compute po…i† and pf…i† in

Section 2.5.

Let pvo…j† (pvh…j†) be the probability that a new call

(handoff call) is not complete before the MS moves out of the the microcell Cj. From Assumption 2, the call holding

time is exponentially distributed with the density function fc…tc† ˆ eÿtc. Then, pvo…j† ˆ Z 1 tˆ0 Z 1 tcˆt fc…tc†rm;j…t†dtcdt ˆ r m;j…†: …8† For a handoff call, the time t0

cbetween the arrival time of

the handoff call at Cjand its completion time has the same

density function as fc (the memoryless property of the

exponential distribution). Similar to the derivation for (8), we have pvh…j† ˆ Z 1 tˆ0 Z 1 t0 cˆt fc…t0c†fm;j…t†dt0cdt ˆ fm;j …†: …9†

The rate of the handoff calls moving out of Cjconsists of the

rate of new calls and the accepted move-in handoff calls that are not complete before the MSs move out of cell Cj. Let

o…j† be the rate of the new calls in cell Cj. Following the

derivation in [13], this rate is

o…j†‰1 ÿ po…j†Špvo…j† ‡ h…j†‰1 ÿ pf…j†Špvh…j†: …10†

From (8), (9), and (10), the rate of the handoff calls moving into Ci is h…i† ˆ X 1jM pj;i  o…j†‰1 ÿ po…j†Šrm;j…† ‡ h…j†‰1 ÿ pf…j†Šfm;j …†  : …11†

For a homogeneous PCS network, we have f

m;i…s† ˆ fm;j …s† ˆ fm…s†; rm;i…s† ˆ rm;j…s† ˆ rm…s†;

and o…i† ˆ o…j† ˆ o. Then,

po…j† ˆ po…i† ˆ po; pf…i† ˆ pf…j† ˆ pf;

h…i† ˆ h…j† ˆ h, and (11) is rewritten as

hˆ o…1 ÿ po†rm…† ‡ h…1 ÿ pf†fm…†

ˆ …1 ÿ po†rm…†o

1 ÿ …1 ÿ pf†fm…†;

…12† which is the the same as the hequation derived in [13].

2.3 The Expected Channel Occupation Times When a channel of Ciis assigned to a new call, the channel

is released if the call completes or the MS moves out of the microcell. Let ton…i† be the channel occupation time of a new

call. Then,

ton…i† ˆ min…tc; m†:

For the example in Fig. 3, ton…i† ˆ m. The expected channel

occupation time of a new call is derived as E‰ton…i†Š ˆ Z 1 tcˆ0 Z tc mˆ0 mrm;i…m†eÿtcdmdtc ‡ Z 1 tcˆ0 Z 1 mˆtc tcrm;i…m†eÿtcdmdtc ˆ1   1 ÿ r m;i…†  : …13†

Let toh…i† be the channel occupation time of a handoff

call. Similar to (13), the expected channel occupation time of a handoff call is E‰toh…i†Š ˆ1  1 ÿ f m;i…†  : …14† Note that E‰toh…i†Š 6ˆ E‰ton…i†Š except for the case when

fm;i…t† is an exponential density function.

2.4 The Call Incompletion Probability

A call is not complete if it is a blocked new call or a forced-terminated handoff call. Note that a call may make several successful handoffs before it is forced to terminate. Let pnc…i† be the incompletion probability of a (new) call

initiated at Ci, and ~pnc…i† be the incompletion probability

of a handoff call at Ci. Then, we have

pnc…i† ˆ po…i† ‡ ‰1 ÿ po…i†Špvo…i†

X

1jM

pi;j~pnc…j†

! …15†

~pnc…i† ˆ pf…i† ‡ ‰1 ÿ pf…i†Špvh…i†

X

1jM

pi;j~pnc…j†

! : …16†

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Equation (15) implies that a new call at Ciis incomplete

if it is blocked at Ci because no idle channel is available

(with probability po…i†) or the call is connected (with

probability 1 ÿ po…i†), but the MS leaves Ci before the call

is complete (with probability pvo…i†), and the call remains

incomplete after the MS moves into a new microcell (with probability P1jMpi;j~pnc…j†). Similarly, (16) implies that a

handoff call at Ci is not complete if no idle channel is

available when the MS arrives at Ci(with probability pf…i†)

or if the handoff call is accommodated at Ci (with

probability 1 ÿ pf…i†), but the MS leaves Ci before the call

is complete (with probability pvh…i†), and the call is still not

complete after the MS moves into a new microcell (with probability P1jMpi;j~pnc…j†). In a homogeneous PCS

net-work, pnc…i† ˆ pnc…j† ˆ pnc; ~pnc…i† ˆ ~pnc…j† ˆ ~pnc,

~pncˆ pf‡ …1 ÿ pf†pvh~pncˆ1 ÿ …1 ÿ ppf f†pvh

and

pncˆ po‡ …1 ÿ po†pvo~pncˆ po‡h

opf: …17†

2.5 The Iterative Algorithm

This section shows how to iteratively compute pnc…i† using

the equations derived in the previous section. The iterative algorithm has the same steps as the homogeneous algo-rithm in [13] except that the equations used in every step are different. Consider a network of M=N macrocells where every macrocell covers N microcells (where M >> N). Algorithm 1 (Microcell).

Input parameters: pi;j (the routing probability), o…i† (the

new call arrival rate),  (the call completion rate), ci (the

number of channels in BTS i), fm;i…t† (the MS residence

time density function), and rm;i…t†, where 1  i; j  M.

Output measures: h…i† (the handoff call arrival rate), po…i†

(the new call blocking probability), pf…i† (the forced

termination probability), and pnc…i† (the call incompletion

probability), where 1  i  M.

Step 1. Select initial values for h…i† where 1  i  M.

Step 2. For 1  i  M, compute po…i† and pf…i†. In the

nonprioritized scheme, the handoff calls and the new call attempts are not distinguishable, and po…i† ˆ pf…i†. In

such a handoff scheme, channel allocation can be modeled after an M=G=c=c queue [12] with the number of servers as ci and the job arrival rate as o…i† ‡ h…i†

[13]. The load onto Ci is computed using (13) and (14):

…i† ˆ o…i†E‰ton…i†Š ‡ h…i†E‰toh…i†Š

and

pf…i† ˆ po…i† ˆ B……i†; ci† ˆ …i† ci=c

i!

Pci

jˆ0……i†j=j!†

: Step 3. For 1  i  M, h;old…i† h…i†.

Step 4. For 1  i  M, compute h…i† by using (11).

h…i† ˆ X 1jM pj;i  o…j†‰1 ÿ po…j†Šrm;j…† ‡ h…j†‰1 ÿ pf…j†Šfm;j …†  :

Step 5. If there exists i such that jh…i† ÿ h;old…i†j > h…i†,

then go to Step 2. Otherwise, go to Step 6. Note that  is a predefined value.

Step 6. Compute pnc…i† by using (8), (9), (15), and (16)

pnc…i† ˆ po…i† ‡ ‰1 ÿ po…i†Šrm;i…†

X

1jM

pi;j~pnc…j†

!

~pnc…i† ˆ pf…i† ‡ ‰1 ÿ pf…i†Šfm;i …†

X

1jM

pi;j~pnc…j†

! : The above iterative procedure has been intensively used and validated by experiments [13].

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This section derives the expected macrocell residence times to be used in the macrocell model in the next section. To simplify the description, we assume that the microcells are homogeneous (i.e., (12) and (17) hold) with the notations oˆ o…i†, hˆ h…i†, poˆ po…i†, pncˆ pnc…i†,

 ˆ i, fm…t† ˆ fm;i…t†, and rm…t† ˆ rm;i…t†. To accommodate

heterogeneous microcells, the modifications on the model are straightforward but tedious.

In a homogeneous dual band GSM network, every radio coverage of a GSM900 BTS (a macrocell) overlays N ˆ 3n2ÿ 3n ‡ 1 DCS1800 BTSs (microcells) for some n (in

Fig. 2a, n ˆ 4). If a call cannot be served at a microcell, then it is switched to a macrocell. Since there are N identical microcells in a macrocell, the overflow new call arrival rate to a macrocell is

voˆ Npoo: …18†

Similarly, the overflow handoff call arrival rate to a macrocell is

vh ˆ Npoh: …19†

Note that pf ˆ po because the handoff calls and the new

calls are indistinguishable from each other in GSM. Suppose that an MS resides in a macrocell for a period tM. During tM, the MS visits k microcells (i.e., the MS moves

k steps), and the MS resides at microcell i for a period ti.

Then, tMˆ t1‡ t2‡ ::: ‡ tkhas the density function

fM…k†…tM† ˆ Z tM t1ˆ0 Z tMÿt1 t2ˆ0 . . . Z tMÿt1ÿ:::ÿtkÿ2 tkÿ1ˆ0 fm…t1†fm…t2†:::fm…tkÿ1†  fm…tMÿ t1ÿ t2ÿ ::: ÿ tkÿ1†dtkÿ1:::dt2dt1: …20†

To accommodate a new/handoff call, an MS switches from a microcell to a macrocell if no channel is available in the microcell. Let M be the period between when the MS

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the macrocell (see Fig. 3). Suppose that the MS visits k microcells during M. Then, the density function of Mis

r…k†M…M† ˆ  vo vo‡ vh   Z M mˆ0 Z Mÿm t2ˆ0    Z Mÿmÿt2ÿ...ÿtkÿ2 tkÿ1ˆ0 rm…m†  fm…t2† . . . fm…tkÿ1†fm…Mÿ mÿ t2ÿ . . . ÿ tkÿ1† dtkÿ1. . . dt2dm ‡ vh vo‡ vh   fM…k†…M†: …21† The first term of the righthand side in (21) represents the case that the MS switches from a microcell to a macrocell when a new call arrives. The second term of the righthand side in (21) represents the case that the MS switches from a microcell to a macrocell when a handoff call arrives.

Now, we describe a random walk model to compute when the MS leaves a macro cell. Our model considers a regular microcell/macrocell overlay structure shown in Fig. 2a. A macrocell is referred to as an n-layer macrocell if it overlays with N ˆ 3n2ÿ 3n ‡ 1 microcells. Fig. 2a shows

4-layer macrocells. The microcell at the center of the macrocell is called the layer 0 microcell. The microcells that surround layer x ÿ 1 microcells are called layer x microcells. There are 6x microcells in layer x except exactly one microcell which is in layer 0. An n-layer macrocell overlays microcells from layer 0 to layer n ÿ 1. The microcells that surround the layer n ÿ 1 microcells are referred to as boundary neighbors, which are outside of the macrocell.

We assume that an MS resides in a microcell for a period ti, then moves to one of its neighbors with the same

probability (i.e., with probability 1/6; see Fig. 4). For an MS in conversation, we derive the number k of moving steps (a step represents an MS movement from a microcell to another) from the beginning of the conversation until the MS moves out of the macrocell. A random walk model with 6…n ÿ 1† absorbing states in a two-dimensional hexagonal plan [5] can be used to study when an MS moves out of an n-layer macrocell boundary. In this model, a state repre-sents the microcell where the MS resides. In order to compute the number of microcells visited before the MS leaves a macrocell, whenever the MS moves out of the macrocell, the random walk enters an absorbing state. A

potential problem of this model is that the number of states increases rapidly as n increases.

To reduce the computation complexity, we modify the two-dimensional random walk model based on our previous work [1]. According to the equal routing probability assump-tion, we classify the microcells in a macrocell into several microcell types. A microcell type is of the form hx; yi, where

00x00 0 indicates that the microcell is in layer x and y  0

represents the y ‡ 1st type in layer x. Microcells of the same type have the same traffic flow pattern because they are at the symmetrical positions on the hexagonal macrocell. Based on the derivation in [1], Fig. 5 illustrates the type of microcell for 6-layer macrocells. For example, the microcell in layer 0 is of the type h0; 0i. The six microcells in layer 1 are grouped together and assigned to the same type h1; 0i. A layer 2 microcell may have two or three neighbors in layer 3 and is assigned to type h2; 1i or h2; 0i, respectively.

In the modified random walk model for an n-layer macrocell, a state …x; y† represents that the MS is in one of the microcells of type hx; yi, where 0  x < n and 0  y  x ÿ 1. The state …n; j† represents that the MS moves out of the macrocell from the state …n ÿ 1; j†, where 0  j < n ÿ 1. For 0  x < n and 0  y  x ÿ 1, states …x; y† are transient and, for 0  j < n ÿ 1, states …n; j† are absorbing. The total number S…n† of states for an n-layer macrocell random walk is S…n† ˆn…n‡1†2 and the computation complexity is significantly reduced. Details of the modified random walk and derivation of its steady state probabilities are given in [1] and the results are summarized as follows: The transition matrix of this random walk is an S…n†  S…n† matrix P ˆ …p…x;y†;…x0;y0††. An element p…k†…x;y†;…x0;y0† in P…k† is the

probability that the random walk moves from state …x; y† to state …x0; y0† with exact k steps. For 0  j < n ÿ 1, define

pk;…x;y†;…n;j†as

pk;…x;y†;…n;j†ˆ

p…x;y†;…n;j†; for k ˆ 1;

p…k†…x;y†;…n;j†ÿ p…kÿ1†…x;y†;…n;j†; for k > 1: 8

< :

Then, pk;…x;y†;…n;j† is the probability that an MS initially

resides at a hx; yi microcell, moves into a hn ÿ 1; ji microcell at the k ÿ 1st step and, then, moves out of the macrocell at the kth step.

In a homogeneous dual band GSM network, an MS may be in any microcell of the overlaid macrocell with the same probability when it switches from the microcell to the macrocell during a phone call. On the other hand, the MS may enter a new macrocell through different types of microcells (and, thus, with different probabilities) during the conversation. Let q…nÿ1;j† be the probability that an MS

enters the macrocell through a hn ÿ 1; ji microcell at the first step and ~q…nÿ1;j†be the probability that an MS moves out of the

macrocell through a hn ÿ 1; ji microcell at the last step. Then, we have: ~q…nÿ1;j†ˆ Xnÿ2 yˆ0 X1 kˆ1

q…nÿ1;y†pk;…nÿ1;y†;…n;j† for 0  j < n ÿ 1 …22† Fig. 4. The routing pattern.

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Xnÿ2 jˆ0

~q…nÿ1;j†ˆ 1: …23†

Equation (22) indicates the situation where an MS moves out of the macrocell. Each term of the righthand side of (22) represents the probability that the MS initially resides at a hn ÿ 1; yi microcell and leaves the macrocell from a hn ÿ 1; ji microcell at the kth step. Equation (23) ensures that all probabilities sum up to 1. For a 6-layer macrocell, we use M2ˆ 200 truncated terms in (22) to approximate the infinite

summations. Computation indicates that the error for the pk;…nÿ1;y†;…n;j† summation truncation is within 10ÿ4.

Now, we consider the situation where an MS passes through the boundary between two macrocells. For n ˆ 2,

q…1;0†ˆ ~q…1;0†ˆ 1:

For n > 2, the following equations hold:

q…nÿ1;0†ˆ23~q…nÿ1;0†‡12~q…nÿ1;1† …24†

q…nÿ1;1†ˆ13~q…nÿ1;0†‡12~q…nÿ1;nÿ2† …25†

q…nÿ1;j†ˆ12~q…nÿ1;nÿjÿ1†‡12~q…nÿ1;nÿj† for 2  j < n ÿ 1: …26†

We use the 6-layer macrocell configuration in Fig. 5 as an example to explain (24). The same reasoning applies to (25)

and (26). Consider a h5; 0i microcell A. An MS entering microcell A comes from one of three surrounding boundary neighbors (microcell B, C or D). The probability that an MS at B moves out of the macrocell is ~q…5;0†. Since B has three

boundary neighbors, the probability that the MS moves from B to A is ~q…5;0†=3. Similarly, the MS moves from C to A

with probability ~q…5;0†=3. For 0 < j < n ÿ 1, microcell h5; ji

only has two boundary neighbors. Thus, the MS moves from microcell D to microcell A with probability ~q…5;1†=2.

Therefore, the righthand side of (24) indicates the prob-ability that the MS moves to A from either B, C, or D. Following the same reasoning, (25) applies to microcell h5; 1i and (26) applies to microcells h5; 2i, h5; 3i, and h5; 4i.

For n ˆ 3, (22)-(24) and (25) are used to compute q…2;0†

and q…2;1†. For n > 3 and 0  j < n ÿ 1, we compute q…nÿ1;j†

by using (22)-(26). For a 6-layer macrocell, we have q…5;0†ˆ

27:27% and q…5;j† ˆ 18:18% for 1  j  4.

From (20), the MS residence time density function for an n-layer macrocell is fM…tM† ˆ X1 kˆ1 Xnÿ2 yˆ0 Xnÿ2 jˆ0

q…nÿ1;y†pk;…nÿ1;y†;…n;j†fM…k†…tM† for n > 1:

…27† From (20) and the Laplace transform convolution rule (P1.2.2 in [19]), the Laplace transform for fM…k†…t† is

(8)

fM…k†…s† ˆ 

f m…s†

k

and from (27), the Laplace transform for fM…t† is

f M…s† ˆ X1 kˆ1 Xnÿ2 yˆ0 Xnÿ2 jˆ0 q…nÿ1;y†pk;…nÿ1;y†;…n;j†  f m…s† k for n > 1: …28† From (21), the density function rM…M† for Mis

rM…M† ˆ X1 kˆ1 Xnÿ1 xˆ0 Xxÿ1 yˆ0 X nÿ2 jˆ0 …x;y† N pk;…x;y†;…n;j†r…k†M…M†   for n > 1; …29† where hx;yiis the number of microcells of type hx; yi:

hx;yiˆ 1; for hx; yi ˆ h0; 0i;6; otherwise:



By using the Laplace transform convolution rule and from (7), (18), (19), and (21), we have

r…k†M …s† ˆ oÿ …oÿ sh†fm…s† s…o‡ h†   f m…s† kÿ1 : For n > 1 and from (29), the Laplace transform for rM…t† is

r M…s† ˆ X1 kˆ1 Xnÿ1 xˆ0 Xxÿ1 yˆ0 Xnÿ2 jˆ0 …x;y† N pk;…x;y†;…n;j†   ( ) ( oÿ …oÿ sh†fm…s† s…o‡ h†   f m…s† kÿ1) : …30†

Let tON and tOH be the channel occupation times of a

macrocell for a new call (an overflow new/handoff call from a microcell) and a handoff call, respectively. From (13), (30), (14), and (28) E‰tONŠ ˆ1  1 ÿ r M…†  and E‰tOHŠ ˆ1  1 ÿ f M…†  : …31†

Equations (28), (30), and (31) are computed with truncated terms. The Appendix shows how to select the number of truncated terms so that the errors for (28) and (30) can be controlled within a small predefined value.

4 T

HE

M

ACROCELL

M

ODEL

This section describes the macrocell model. The iterative algorithm for the macrocell model is similar to Algorithm 1. Algorithm 2 (Macrocell).

Input parameters

. N ˆ 3n2ÿ 3n ‡ 1: the number of microcells in an

n-layer macrocell (given).

. C: the number of channels in a macrocell (given). . 1=: the expected call holding time (given).

. vo: the overflow new call arrival rate (computed by

using (18), where pois computed by Algorithm 1).

. vh: the overflow handoff call arrival rate (computed

by using (19), where po and h are computed by

Algorithm 1). . f

M…s†: the Laplace transform of the tM distribution

(derived by using (28)). . r

M…s†: the Laplace transform of the density function

for M(derived by using (30)).

Output measures

. H: the handoff call arrival rate to a macrocell.

. pO: the new call blocking (handoff call forced

termination) probability to a macrocell.

. pnc;M: the incompletion probability of a call (either

blocked or forced to terminate) in the macrocell. Step 1. Select an initial value for H.

Step 2a. The load onto a macrocell is computed by using (32) and (33):

 ˆ …vo‡ vh†E‰tONŠ ‡ HE‰tOHŠ:

Step 2b. The probability pOis computed by using the Erlang

loss equation:

pOˆ B…; C†:

Step 3. H;old H.

Step 4. Compute H by using (12), (28), and (30):

H ˆ…1 ÿ pO†r 

M…†…vo‡ vh†

1 ÿ …1 ÿ pO†f…† :

Step 5. If jHÿ H;oldj > H (where  is a predefined

value), then go to Step 2. Otherwise, go to Step 6. Step 6. The values for H and pOconverge. Compute pnc;M

by using (17), (28), and (30): pnc;M ˆ 1 ‡ …1 ÿ pO†‰r  M…† ÿ fM…†Š  pO 1 ÿ …1 ÿ pO†fM…† :

Step 7. Compute the call incompletion probability as pNCˆ pncpnc;M:

5 P

ERFORMANCE OF THE

D

UAL

B

AND

PCS

N

ETWORK

We assume that the MS residence times in a microcell have a Gamma distribution with the shape parameter . As we mentioned before, the Gamma distribution is selected because it can be shaped to represent many distributions, as well as measured data that cannot be characterized by a particular distribution.

Fig. 6 shows the effect of the variation (or standard deviation) for the residence time distribution on pNC. Note

that, for Gamma distributions with the same mean value 1=, the standard deviation  ˆ 1

p increases as

de-creases. By the Chebyshev's Inequality, the probability that the residence times are out of the range ‰1= ÿ 5=3; 1= ‡ 5=3Š is smaller than 36 percent for all  values. For

(9)

example, if  ˆ 6=, then 5=3 ˆ 10= and the probability that the residence time exceeds 11= is smaller than 36 percent.

For a macrocell overlaying 61 microcells (N ˆ 61) with 50 channels (C ˆ 50), and the number of channels in each microcell is 10 (c ˆ 10), we observe the following:

. pNC increases as  decreases.

.  has a significant effect on pNCfor  > 1= and has

almost no effect on pNC for  < 1= (note that a

Gamma distribution is exponential when  ˆ 1=). . pNC increases as  increases.

The call incompletion probability of the system increases as the standard deviation (variance) of the MS microcell residence time distribution decreases. For the same ex-pected microcell residence times, more long and short residence times will be observed as the standard deviation increases. More long residence times implies that more new

calls will complete without any handoff actions. Thus, smaller call incompletion probability is observed.

Figs. 7 and 8 illustrate the effect of the macrocell size. Four scenarios are considered: a macrocell of size N ˆ 19 with the channel number C ˆ 16 (C=N ˆ 0:8421), N ˆ 37 with C ˆ 31 (C=N ˆ 0:8378), N ˆ 61 with C ˆ 51 (C=N ˆ 0:8361), and N ˆ 91 with C ˆ 76 (C=N ˆ 0:8352). We observe the following:

. For a fixed C=N value, the performance of a large macrocell is better than that of a small macrocell. . For a fixed C=N value, the benefit of a large

macrocell is significant for a large  value or a small  value.

. For a fixed C=N value, the benefit of a large macrocell becomes insignificant for heavy traffic. Queuing theory indicates that systems with more common resource pool perform better than systems with isolated resources. Thus, the utilization of radio channels is more efficient for a large macrocell. Our study indicates that

Fig. 6. The effect of the standard deviation for the microcell residence time distribution (N ˆ 61; C ˆ 50; c ˆ 10). (a)  ˆ 0:5. (b)  ˆ . (c)  ˆ 2. (d)  ˆ 4.

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large macrocells have good performance if pNC< 2%.

About 7:05 offered load per microcell can be served by a macrocell with N ˆ 91 microcells and about 6:65 offered load per microcell can be served by a macrocell with N ˆ 19 microcells when pNCˆ 2%.

A single band PCS network is a special case of the dual band PCS network where N ˆ 1. Table 1 compares the performance of the single band system and the dual band system. In Table 1, N ˆ 61 and C ˆ 50 for the dual band system. The table indicates that the dual band architecture significantly outperforms the single band system. For example, for  ˆ 1=…6†;  ˆ ; oˆ 7, pNCˆ 1:413% for

the dual band system with the total channels number N  c ‡ C ˆ 660. For the single band system, pncˆ 2:62%,

with the total channel number 61  13 ˆ 793 or pncˆ 1:37%

with the total channel number 61  14 ˆ 854.

6 C

ONCLUSIONS AND

F

UTURE

E

XTENSIONS

We studied the performance of dual band GSM networks by assuming that a call is first served by a microcell. If no channel is available in the microcell, then the call is served

Fig. 7. The effect of the size of a macrocell (c ˆ 10). (a)  ˆ 0:5,  ˆ 6=. (b)  ˆ 0:5,  ˆ 1=. (c)  ˆ 2,  ˆ 6=. (d)  ˆ 2,  ˆ 1=.

Fig. 8. The effect of the size of a macrocell (c ˆ 10) for large traffic.  ˆ 2,  ˆ 6=.

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by the overlaid macrocell. The call is blocked (or forced to terminate) if no channel is available in the macrocell. We observed the following:

. The call incompletion probability of the system increases as the standard deviation of the MS microcell residence time distribution decreases. Furthermore, the call incompletion probability of the system increases as the user mobility increases, though the forced termination probability is de-creased. These results are consistent with those in the single band system [13].

. For the same channel budget, the call incompletion probability decreases as the number N of microcells covered by a macrocell increases. However, the size of a macrocell cannot be arbitrarily increased due to power consumption of the radio system. The trade-off between the size of N and the power consump-tion deserves further study.

This paper assumed that the switching of a call from a microcell to a macrocell is based on the channel availability of the microcell (the model in [7] followed the same assump-tion). However, a call may be switched according to the mobility of a mobile user as suggested in [8]. If a mobile user moves too fast, the calls may be served by macrocells to reduce the number of handoffs. Our model can be easily modified to accommodate this condition as follows:

. Suppose that the user's speed can be measured. For example, we define the user moves too fast if 5 microcells have been visited during a fixed period of three minutes. Compute the probability that a mobile user is too fast.

. In Fig. 2b, 1 ÿ of the handoff/new call traffic (i.e., the portion of the call traffic generated by the slow

moving users) is directed to the microcell models as before and of the traffic (i.e., the portion of the traffic generated by the fast moving users) is directed to the overlaying macrocell.

A

PPENDIX

S

ELECTING THE

N

UMBER OF

T

RUNCATED

T

ERMS FOR

E‰t

ONS

Š

AND

E‰t

OHS

Š

The summations of infinite series in (30) and (28) can be accurately approximated by the summations of finite series as follows: Note that 0 < f

M…† < 1. For (30), let s ˆ ,

X ˆoÿ …oÿ h†fm…†

…o‡ h† > 0

and note that

0  Y ˆXnÿ1 xˆ0 Xxÿ1 yˆ0 Xnÿ2 jˆ0 …x;y† N pk;…x;y†;…n;j† 1:

Consider a positive value  << 1 and an integer K1. We

have K1 log  X‰1 ÿ fm…†Š  log‰f m…†Š   ) X1 kˆK1‡1 X  f m…† kÿ1   ) X1 kˆK1‡1 XY  f m…† kÿ1  : Similarly, for an integer K2

TABLE 1

Comparison of the Single Band System and the Dual Band System (N ˆ 61, C ˆ 50, and c ˆ 10 for the Dual Band System

(12)

K2 logf‰1 ÿ f  m…†Šg log‰f m…†Š   ÿ 1 ) X1 kˆK2‡1 Xnÿ2 yˆ0 Xnÿ2 jˆ0 q…nÿ1;y†pk;…nÿ1;y†;…n;j†  f m…† k  : Let K ˆ max…K1; K2†, then the error introduced by the

following approximations for E‰tONŠ and E‰tOHŠ is bounded

by =: E‰tONŠ ˆ1ÿ1 XK kˆ1 Xnÿ1 xˆ0 Xxÿ1 yˆ0 Xnÿ2 jˆ0 …x;y† N pk;…x;y†;…n;j† " # ( ( oÿ …oÿ h†fm…† …o‡ h†   f m…† kÿ1)) : …32† and E‰tOHŠ ˆ1 1 ÿ XK kˆ1 Xnÿ2 yˆ0 Xnÿ2 jˆ0 q…nÿ1;y†pk;…nÿ1;y†;…n;j†  f m…† k " # : …33†

R

EFERENCES

[1] I.F. Akyildiz, Y.-B. Lin, W.-R. Lai, and R.-J. Chen, ªA New Model for Random Walks in PCS Networks,º Submitted for publication, Technical Report CSIE-PCS2000-1; see http://liny.csie.nctu. edu.tw, 1999.

[2] R. Beraldi, S. Marano, and C. Mastroianni, ªPerformance of a Reversible Hierarchical Cellular System,º Int'l J. Wireless Informa-tion Networks, vol. 4, no. 1, pp. 43-54, 1997.

[3] Y. Fang, I. Chlamtac, and Y.-B. Lin, ªChannel Occupancy Times and Handoff Rate for Mobile Computing and PCS Networks,º IEEE Trans. Computers, vol. 47, no. 6, pp. 679-692, June 1998. [4] Y. Fang, I. Chlamtac, and Y.-B. Lin, ªModeling PCS Networks

under General Call Holding Time and Cell Residence Time Distributions,º IEEE/ACM Trans. Networking, vol. 5, no. 6, pp. 893-906, 1998.

[5] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I. John Wiley & Sons, 1966.

[6] G.J. Foschini, B. Gopinath, and Z. Miljanic, ªChannel Cost of Mobility,º IEEE Trans. Vehicular Technology, vol. 42, no. 4, pp. 414-424, Nov. 1993.

[7] L.R. Hu and S.S. Rappaport, ªMicro-Cellular Communication Systems with Hierarchical Macrocell Overlays: Traffic Perfor-mance Models and Analysis,º Proc. WINLAB Workshop, pp. 143-174, 1993.

[8] C.L. I?, L.J. Greenstein, and R.D. Gitlin, ªA Microcell/Macrocell Cellular Architecture for Low- and High-Mobility Wireless Users,º IEEE J. Selected Areas in Comm., vol. 11, no. 6, pp. 885-891, 1993.

[9] N.L. Johnson, Continuous Univariate Distributions-1. John Wiley & Sons, 1970.

[10] I. Katzela and M. Naghshineh, ªChannel Assignment Schemes for Cellular Mobile Telecommunication Systems: A Comprehensive Survey,º IEEE Personal Comm., vol. 3, no. 3, pp. 10-31, June 1996. [11] F.P. Kelly, Reversibility and Stochastic Networks. John Wiley & Sons,

1979.

[12] L. Kleinrock, Queueing Systems: Vol. IÐTheory. New York: Wiley, 1976.

[13] Y.-B. Lin, ªPerformance Modeling for Mobile Telephone Net-works,º IEEE Network Magazine, vol. 11, no. 6, pp. 63-68, Nov./ Dec. 1997.

[14] Y.-B. Lin, L.-F. Chang, and A. Noerpal, ªModeling Hierarchical Microcell/Macrocell PCS Architecture,º Proc. IEEE Int'l Conf. Comm., 1995.

[15] Y.-B. Lin, S. Mohan, and A. Noerpel, ªQueueing Priority Channel Assignment Strategies for Handoff and Initial Access for a PCS Network,º IEEE Trans. Vehicular Technology, vol. 43, no. 3, pp. 704-712, 1994.

[16] M. Mouly and M.-B. Pautet, The GSM System for Mobile Communications. 1992.

[17] S. Tekinary and B. Jabbari, ªA Measurement Based Prioritization Scheme for Handovers in Cellular and Microcellular Networks,º IEEE J. Selected Areas in Comm., pp. 1,343-1,350, Oct. 1992. [18] L.-C. Wang, G.L. StuÈber, and C.-T. Lea, ªArchitecture Design,

Frequency Planning, and Performance Analysis for a Microcell/ Macrocell Overlaying System,º IEEE Trans. Vehicular Technology, vol. 46, no. 4, pp. 836-848, 1997.

[19] E.J. Watson, Laplace Transforms and Applications. Birkhauserk, 1981.

[20] M.D. Yacoub, Foundations of Mobile Radio Engineering. CRC Press, 1993.

[21] C.H. Yoon and K. Un, ªPerformance of Personal Portable Radio Telephone Systems with and without Guard Channels,º IEEE J. Selected Areas in Comm., vol. 11, no. 6, pp. 911-917, Aug. 1993

Yi-Bing Lin (S'80-M'96-SM'96) received his BSEE degree from National Cheng Kung Uni-versity in 1983 and his PhD degree in computer science from the University of Washington in 1990. From 1990 to 1995, he was with the Applied Research Area at Bell Communications Research (Bellcore), Morristown, New Jersey. In 1995, he was appointed a professor in the Department of Computer Science and Informa-tion Engineering (CSIE), NaInforma-tional Chiao Tung University (NCTU). In 1996, he was appointed deputy director of the Microelectronics and Information Systems Research Center, NCTU. During 1997-1999, he was elected chairman of CSIE, NCTU. His current research interests include design and analysis of personal communica-tions services network, mobile computing, distributed simulation, and performance modeling.

Dr. Lin is an associate editor of IEEE Network, an editor of the IEEE Journal of Selected Areas in Communications: Wireless Series, IEEE Personal Communications Magazine, Computer Networks, ACM Mobile Computing and Communication Review, International Journal of Communications Systems, ACM/Baltzer Wireless Networks, Computer Simulation Modeling and Analysis, Journal of Information Science and Engineering, a columnist for ACM Simulation Digest, and guest editor for the ACM/Baltzer MONET special issue on personal communications, IEEE Transactions on Computers special issue on mobile computing, and IEEE Communications Magazine special issue on active, program-mable, and mobile code networking. Dr. Lin received the 1997 Outstanding Research Award from the National Science Council, Republic of China (ROC), and the Outstanding Youth Electrical Engineer Award from CIEE, ROC.

Wei-Ru Lai received her BSEE degree and PhD degrees from the Department of Computer Science and Information Engineering, National Chiao Tung University in 1991 and 1999, respectively. In 1999, she was appointed an assistant professor and was elected chairman of the Department of Information Management, Chin-Min College. Her current research interests include design and analysis of personal com-munications services network.

Rong-Jaye Chen received his BS (1977) in mathematics from National Tsing Hua University and his PhD (1987) in computer science from the University of Wisconsin-Madison. He is currently a professor and chairman of Computer Science and Information Engineering Depart-ment at National Chiao Tung University. His research interests include cryptography and security, personal communication service, algo-rithm design, and theory of computation.

數據

Fig. 2. Homogeneous microcell/macrocell layout and channel assignment procedure. (a) Microcell/macrocell configuration
Fig. 3. The timing diagram.
Fig. 5. Type assignments for 6-layer macrocells.
Fig. 6 shows the effect of the variation (or standard deviation) for the residence time distribution on p NC
+3

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