Modeling and analysis of applying adaptive modulation ...

Modeling and analysis of applying adaptive modulation codingin wireless multicast and broadcast systems

Yu-Cheng Liang • Ching-Chun Chou •

Hung-Yu Wei

Published online: 31 May 2011

� Springer Science+Business Media, LLC 2011

Abstract Traditional wireless communications only uti-

lize fixed-rate multicast and broadcast. In other words, only

the most robust modulation and coding scheme can be

applied for data transmission. Such a scheme fails to suf-

ficiently exploit the potential gains of multicast and

broadcast, resulting in bandwidth waste. To overcome such

a problem, investigating the rate adaptation of multicast

and broadcast wireless systems is the primary task. Unlike

the traditional wireless systems, this paper presents an

analytical model with rate adaptation for both multicast and

broadcast. Adaptive modulation and coding are applied to

achieve rate adaptation. We construct a stochastic model

by using Finite State Markov chains for the multicast

broadcast system modeling. The model’s outputs are

shown to approximate to the results of our system level

simulations. The model derives the performance of rate

adaptation in multicast and broadcast. With the deduced

modeling results, we can predict the system throughput

providing the channel states, and the modulation and

coding schemes variations.

Keywords Adaptive modulation and coding �Markov chain model � Multicast � Broadcast

1 Introduction

Multicast and broadcast have been widely utilized in

numerous systems for many years. The applications of

multicast are especially ubiquitous in radio and television

programs. Multicast and broadcast services (MBS) are

extensively studied and widely used because of its excellent

radio resource efficiency, which results from its one-source-

multiple-receiver transmission [1, 2]. Since bandwidth is a

finite resource, developing an efficient delivery mechanism

to save bandwidth resources is the key to provide high

quality multimedia services. MBS is a core research for

utilizing the finite bandwidth resource [3]. Applying adap-

tive modulation and coding (AMC) to the MBS is a prom-

ising technique to improve the efficiency of the next

generation wireless systems [4–7].

The conventional MBS fails to exploit the potential of

adaptive wireless transmission. The data packets are dis-

tributed over the air with a fixed rate, and only a specific

modulation and coding scheme (MCS) is applied

throughout the transmission. To increase transmission

robustness, the fixed transmission rate is mostly the lowest

of all the available MCSs. Because the MBS source has to

consider the receiver with the worst channel condition. In

contrast, if all receivers are in their excellent channel

conditions, the MCS should be dynamically adjusted to the

optimum MBS transmission rate. At the same time, the

MCS should also satisfy the robustness requirement in

addition to increasing the transmission rate. In other words,

the high-rate MCS should be able to serve receivers with

the worst channel state. AMC scheme adjusts the MCS

according to all receivers’ channel conditions. With higher-

rate MCS, the MBS data delivery can also be accelerated.

Note that the AMC implementation requires feedback

channels, namely, receivers must send channel state reports

back to the source through feedback channels.

AMC for unicast transmission is thoroughly studied in

many wireless systems. She et al. [8, 9] present cross

layer designs on video streaming to improve video quality.

The feedback channel provides dynamic video coding and

Y.-C. Liang � C.-C. Chou � H.-Y. Wei (&)

Department of Electrical Engineering, National Taiwan

University, Taipei, Taiwan

e-mail: [email protected]

123

Wireless Netw (2011) 17:1373–1386

DOI 10.1007/s11276-011-0354-7

rate selection to improve video quality. Deb et al. [10]

conduct an integrated scheme that applies layered video

coding and AMC for video transmission. Chi et al. [11]

propose a video transmission method using AMC. This

method dynamically adjusts the MCS according to the

channel states within each group of pictures (GOP).

Chen et al. propose MCS adaptation on video multicast.

Receivers are divided into different regions, and MCS is

adjusted within each region. The proposed algorithm is

analyzed [12, 13]. Fatih et al. [14] also proposes rate

adaptation multicast, using dynamic FEC based on the

queue and the channel status report. Xiong et al. propose

flow control mechanism for the wireless multi-rate multi-

cast. The incoming rate and buffering status is feedback to

the upstream PID (Proportional Integral and Derivative)

controller to adjust the multicast rate [15, 16]. Zhou et al.

consider the tradeoff between video delay and distortion of

real-time wireless video transmission. The DAWVS

scheduling algorithm is proposed and analyzed [17]. Zhou

et al. also propose a graph based authentication method to

maximize the quality of video security within the delay

[18]. Though these works all apply AMC or similar flow

control mechanism to improve the multimedia services, a

complete mathematical model to exploit AMC’s perfor-

mance is not yet supplied. Additionally, Markov Chain

models have been applied to evaluate the performance of

rate-adaptive unicast transmission for years [19–21], but

the performance of rate-adaptive multicast and broadcast

service have not taken account of the channel conditions of

all multicast receivers and robust multicast requirement.

Fen et al. propose a cooperative multicast using two-phase

transmission with AMC. However, the throughput is ana-

lyzed based on the summation of received signal strength

and the corresponding data rate. The advantages of AMC

on MAC layers still await further investigations [22].

This paper focuses on using Markov Chains to analyti-

cally model the MBS system with AMC. A simplified

model is provided to predict the system throughput, cap-

turing the channel states and MCS variations. Tradition-

ally, Kronecker product [20, 21] is applied to model the

MBS system’s behavior. The channel state of each inde-

pendent receivers is multiplied with a Kronecker product to

generate the system’s throughput matrix. But constructing

the whole system state matrix to predict the system

throughput is unnecessary. Since the MBS system’s

throughput is only determined by the BS with the channel

states and MCS of the MBS group, it is possible to reduce

the computing and storage complexity for a naive product

expansion. This is the major reason to propose a new model

for the MBS system with AMC. With the varying channel

states and MCS, the channel state matrices for the Kro-

necker model are required to be frequently updated.

Therefore, the Kronecker product of the MBS group should

be computed frequently. Once the model can be reduced

and simplified, the computation and transmission delay of

the MBS system is then reduced, and moreover, the system

performance is improved.

To the author’s best knowledge, this work is the first

paper presenting a comprehensive mathematical model for

AMC under MBS. We propose a finite state Markov chain

(FSMC) model for the multicast and broadcast service with

AMC to efficiently evaluate the possible performance of

AMC under MBS. We assume that perfect channel state

information and channel error rate are available here. With

the proposed model, any given channel error probability or

error pattern can be completely fit into the modeling.

Moreover, the system throughput becomes predictable

under the given initial channel states and the error proba-

bility. To sum up, this could be the most comprehensive

work consisting of both modeling and analysis of MBS

with AMC ever presented.

2 System model and assumptions

2.1 System overview

Our system includes a multicast group formed by one BS

and several MSs, which is the standard setting of an MBS

system. We focus on the effects of utilizing AMC to pro-

vide a rate-adaptive MBS. Modeling the channel quality

requires a solid mathematical formulation. As there are

limited numbers of applicable MCS within the system, we

separate the channel states into several intervals. Each

corresponds to an adequate MCS. If the MCS can be

applied to the current channel state intervals, it is an

appropriate MCS selection; by contrast, improper MCS

leads to low throughput and high error rate of transmission.

A specific MCS under a fixed channel state interval results

in fixed transmission rate.

To further simplify the idea, we assume that a specific

number of packets can be transmitted under the given MCS

and channel states. This technique relates the transmission

rate directly to the channel states, and thus avoids intro-

ducing extra variables or functions for the connection

between the actual transmission rate and the channel states.

Such assumptions can be referred in [21], and it is used in

unicast AMC. Similar technique, using a transmission rate to

represent the channel states, is also applied in multicast

system [23]. In our works, we apply this technique to mul-

ticast and broadcast system with AMC. That is, we assume a

specific number of packets can be transmitted within one

time slot in the multicast AMC system (Table 1).

In this paper, the channel quality is considered to be

divided into C ? 1 interval, and each can be represented

by a channel state (0–C) which remains unchanged during

1374 Wireless Netw (2011) 17:1373–1386

123

every time slot. We also assume that when the channel

state is ‘x’, the receiver can receive maximally x packets

transmitted from the base station (BS). These x packets are

transmitted in a single time slot at a relatively low error

rate. On the other hand, if the base station transmits more

than x packets at a time, every packet’s error rate will rise

significantly. To ensure that every receiver receives all the

packets with a low error rate, we make the base station

transmit packets according to the lowest channel state of all

the receivers in the multicast group. We denote this

channel state as the group state. The corresponding trans-

mission rate is deemed as the suitable transmission data

rate for the whole multicast group. For example, state 0 is

BPSK with 1/2 code rate, state 1, 2, 3, 4, 5, 6 are QPSK

with 1/2 code rate, QPSK with 3/4, 16-QAM with 1/2,

16-QAM with 3/4, 64-QAM with 2/3, and 64-QAM with

3/4 code rate. Assume that we have three different users A,

B, and C. Suppose that A is at state 3, B state 2, and C state

4. In this case, the lowest state is 2, and the group state is

also 2. Therefore, BS transmits two packets to all the

receivers in the group, and all of them receive the packets

at the lowest error rate. Suppose we know the probability of

every channel state at time slot d, and the channel state

transfer matrix T, then we can estimate the system per-

formance at the time slot d ? 1 using d and T. These

parameters are defined in the Table 2.

Markov chain model is a popular analytical tool in

capturing the wireless channel states variations. With such

characteristics, it is most suitable for modeling MBS with

AMC. However, most previous works focuses on applying

Markov chain on unicast AMC. Therefore, we applied the

Markov chain model to the MBS with AMC.

Definition 1 Let T, and Ti denote the group’s transfer

matrix and the ith MSs’ channel state transfer matrix,

respectively. The element of Ti is txyi . txy

i represents the

probability that the ith MS’s channel jumps from state x to

state y. We can write down each MS’s channel state

transfer matrix like Ti. Combining all MSs’ channel state

transfer matrices, we can form the group channel state

transfer matrix T.

Ti �

ti00 ti

01 � � � ti0C

ti10 ti

11 � � � ti1C

..

. ... . .

. ...

tiC0 ti

C1 � � � tiCC

0BBB@

1CCCA

T �

T1 0 � � � 00 T2 � � � 0

..

. ... . .

. ...

0 0 � � � Tn

0BBB@

1CCCA

ð1Þ

The formulation of the Ti is under the presumption that

the transition probability matrix is a given variable. The

elements of each Ti and T are environmental variables

provided for the AMC analysis. The other variables,

including the following (group) channel states matrices and

(group) error rates, are also given variables.

Definition 2 Let Md and Mdi denote the group channel-

state and the ith MS’s channel state probability at time slot

d, respectively. Both Md and Mdi are row vectors. The

element of Mdi is mij

d. It represents the probability of the ith

MS’s channel state being j at time slot d. Mdi is composed

of mijd.

mdij ¼ Prfthe ith MS0s channel state is j at time slot dg ð2Þ

Mdi � md

i0 � � � mdij � � � md

iC

� �1�ðCþ1Þ ð3Þ

If we exactly know that the ith MS’s channel state is in j at

time slot d, the vector will consist of ‘‘0’’s except for the

only one ‘‘1’’ at the jth component. That is, the channel

state of a specific MS are considered static within the time

period. The channel state remains fixed within the time

Table 1 MCS and corresponding channel state in WiMAX [24]

State Modulation Coding rate

0 BPSK 1/2

1 QPSK 1/2

2 QPSK 3/4

3 16-QAM 1/2

4 16-QAM 3/4

5 64-QAM 2/3

6 64-QAM 3/4

Table 2 Notations used in

Sect. 3Notations Descriptions

n Number of channels (receivers), n C 0

C The best channel quality state, C C 0

err = [errij] Packet error rate matrix. errij means channel state at i,and the BS transmits packets with rate j

T Group’s transfer matrix

Ti The ith MS’s channel state transfer matrix

p Steady state channel state matrix

Wireless Netw (2011) 17:1373–1386 1375

123

interval. This is the assumption made in the Sect. 2.1.

Finally, the all-MS’s channel state could be obtained as

follows:

Md � Md1 Md

2 � � � Mdn

� �1�nðCþ1Þ ð4Þ

Note that the MSs channel states and the group channel

states are given variables. The formulation of M and T can

capture the channel states information of each MS. For

example, the channel states may vary according to time or

position variation. However, what we need in analyzing the

performance of the AMC in MBS is only the system

throughput. Therefore, an abstract model should be

constructed to simplify the discussion.

Next, let us focus on a special case, Steady Channel

State. Steady channel state indicates the long term behavior

of the system. In this case, short term channel states vari-

ation and changes are not studied.

2.2 Steady channel state

In the case of Steady Channel State, we investigate into the

long term behavior of the system. The transition probability

of channel state is determined by matrix T. If the system is

in the steady state, the channel state probability distribution

at any time slot will be the same. The channel state tran-

sition matrix T and Ti can be used to deduce the steady

state probability. The steady state probability pi can be

obtained by solving the equation set pi = pi Ti andP

j=1C

pij = 1. The solution indicates the long term channel state

probability distribution. We can use the steady state dis-

tribution pi to evaluate the system throughput.

Definition 3 Let pij denote the probability of the ith MS’s

channel at state j in steady state. Then we can write down

the steady state matrix pij as follows.

p �

p10 p11 � � � p1C

p20 p21 � � � p2C

..

. ... . .

. ...

pn0 pn1 � � � pnC

0BBB@

1CCCA ð5Þ

2.3 Transmission error rate

The transmission error rate is state dependent. That is, the

combination of the channel state and the MCS determines

the transmission error rate. Two cases are provided to

depict the concept. First, the channel is at a good state, and

the BS adopts low-data-rate modulation and coding. Then

it is obvious that the error rate will be small. However, the

channel utilization is relatively low. Second, the channel is

at a bad state, and the BS uses high-data-rate modulation

and coding. The error rate will be large in this case. To

model this phenomenon, we need to define the error rate

matrix. We denote errxy as the transmission error rate,

given the current channel state being in x, and the lowest

channel state at this time slot being y. In other words, the

channel state is at x, the group rate is at y, and the BS

transmits the data with error rate errxy.

Definition 4 Let err denote the error rate matrix

err �

err00 err01 � � � err0C

err10 err11 � � � err1C

..

. ... . .

. ...

errC0 errC1 � � � errCC

0BBB@

1CCCA ð6Þ

Here we arrange all denotations into three tables. These

notations will be used in the following chapters. Variables

in Table 2 are applied across the whole work (Sect. 3).

Variables in Table 3 are used for stationary behavior

analysis, Steady Channel State (Sect. 3.1). Finally,

variables in Table 4 are for transient cases, Transient

with Initial State Probability (Sect. 3.2) and Predict

Throughput Based on Previous Throughput (Sect. 3.3).

3 Model analysis

In the following sections, we will analyze different situa-

tions, including (1) Steady Channel State, (2) Transient

with Initial Channel State Probability, and (3) Predict

Throughput Based on Previous Throughput. Steady

Table 3 Notations used in

stationary analysis (Sect. 3.1)Notations Descriptions

si State of the ith MS’s channel

s State of the group. s ¼ mini si

Ptx(ctx) The probability that the BS transmits ctx packets during

one time slot Ptx(ctx) = Pr{s = ctx }

Prx(crx) The probability that all receivers successfully receive crx packets during one time slot

Perr(cerr) The probability that the BS transmits with cerr packets failure during one time slot

ctx Number of transmitted packets during one time slot

crx Number of successfully received packets during one time slot

cerr Number of packets failed to be received during one time slot

1376 Wireless Netw (2011) 17:1373–1386

123

channel state is the system’s long term behavior. This

specific channel state can be induced by the transition

probability matrix. Therefore, the steady channel state

probability distribution is much more important than the

channel state transfer matrix. The Transient with Initial

Channel State Probability is the transient behavior of the

system. It addresses the problem of predicting the next-

time-slot system state. For example, we can predict the

system’s performance at time d ? 1 given the state in time

d. The last one, Predict Throughput Based on Previous

Throughput, is another type of system’s transient behavior.

In this case, we assume that the data throughput at time d is

known. The data throughput is determined by the base

station’s transmission rate. This rate is limited by the group

rate of all MSs. While we know all MSs’ channel state in

the Transient with Initial Channel State Probability case,

now we only have the group rate of the system. We predict

the system performance at time d ? 1 knowing only the

group rate at time d. Under these three cases, we deduced

the general and special case results of the system. The

special case includes zero error rate, errxy = 0.

3.1 Steady channel state

In this section, we will focus on steady channel state. For a

FSMC with all states communicative, the system will

surely enter a steady state. After the system has run a long

period of time, the system will reach the steady state. From

Definition 3, the channel state probability distribution at

any time slot will be the same under steady state. The

distribution can be presented as:

p �

p10 p11 � � � p1C

p20 p21 � � � p2C

..

. ... . .

. ...

pn0 pn1 � � � pnC

0BBB@

1CCCA

Definition 5 We denote two parameters as follows. They

can simplify the equations later.

UðctxÞ ¼Yn

i¼1

XC

j¼ctx

pij ð7Þ

WðctxÞ ¼Yn

i¼1

XC

j¼ctx

pijð1� errjctxÞ

" #ð8Þ

Theorem 1 The probability that the group state is at ctx

could be presented as:

PtxðctxÞ ¼ Prfs ¼ ctxg ¼Yn

i¼1

XC

j¼ctx

pij �Yn

i¼1

XC

j¼ctxþ1

pij

¼ UðctxÞ � Uðctx þ 1Þ ð9Þ

Referring to the notation tables Table 3, the state of the

group is denoted as s.

Proof We can deduce from Definition 3 that the proba-

bility of the ith MS’s channel being at state ctx is

Prfsi ¼ ctxg ¼ pictx1� i� n ð10Þ

Prfsi� ctxg ¼XC

j¼ctx

Prfsi ¼ jg ¼XC

j¼ctx

pij ð11Þ

The probability of the group state being ctx in one time

slot is

Prfs ¼ ctxg ¼Yn

i¼1

Prfsi� ctxg �Yn

i¼1

Prfsi� ctx þ 1g

¼Yn

i¼1

XC

j¼ctx

pij �Yn

i¼1

XC

j¼ctxþ1

pij

ð12Þ

h

According to our transmission policy, base station

transmits packets according to the lowest MS’s channel

state. The transmission rate ctx, which is the number of

packets transmitted during one time slot, should be equal

to the group channel state s. That is, ctx = s. However,

even with the same ctx, the combinations of MSs’

Table 4 Notations used in

transient analysis

(Sects. 3.2, 3.3)

Notations Descriptions

sid?1 State of the ith MS’s channel at time slot d ? 1

sd?1State of the group at time slot. d ? 1 sdþ1 ¼ mini sdþ1

i

Ptxd?1(ctx

d?1) The probability that the BS transmits ctxd?1 packets during time slot d ? 1.

Ptxd?1(ctx

d?1) = Pr{sd?1 = ctxd?1 }

Prxd?1(crx

d?1) The probability that all receivers successfully receive crxd?1

packets during time slot d ? 1

Perrd?1(cerr

d?1) The probability that the BS transmits with cerrd?1 packets fail

during time slot d ? 1

ctxd?1 Number of transmitted packets during time slot d ? 1

crxd?1 Number of successfully received packets during time slot d ? 1

cerrd?1 Number of packets failed to be received during time slot d ? 1

Wireless Netw (2011) 17:1373–1386 1377

123

channel states may be different. These combinations may

result in the same s and ctx. Let us go through two

examples for detailed explanations. Assume

n = 3, C = 5, and the first MS’s channel is at state 5;

the second is at state 1, and the third is at state 5.

Therefore the lowest state is 1, ctx = s = 1. We may

construct another case: the first MS’s channel is at state

1; the second is at state 1 and the third is at state 1. In

this different case, the lowest state is also 1, ctx = s = 1.

However, the combination of each MS’s channel state is

different. Thus, for a given ctx, there are many combi-

nations of channel states that lead to the same group

state, s, so as to the throughput of data transmission

from base station, ctx. Therefore, all group rate and

group channel state combinations have distinct error rate.

First, we denote N(s) as the number of channels at state

s. Therefore, N(s) can be viewed as a function. A spe-

cific channel states s is input, and the number of MSs’

which are at channel state s are output.

Definition 6 Let N(s) denote the number of channels at

state s. Therefore, N(s) = n0 means that exactly n0 MSs’

channels are at state s, and the other n - n0 MSs’ channels

are at state better than s

Definition 7 Let e(ctx) denote the error rate under given

group channel state ctx. That is, the domain is the group

state, so ctx is a nonnegative integer with 0 B ctx B C. The

codomain is the error rate, which is a nonnegative real

number with 0 B err(ctx) B 1.

errðctxÞ � E½error rate j s ¼ ctx�E½success rate j s ¼ ctx� ¼ 1� errðctxÞ

ð13Þ

E½success rate j s ¼ ctx� ¼Xn

n0¼1

E½success rate j NðsÞ ¼ n0;

s ¼ ctx� � PrfNðsÞ ¼ n0 j s ¼ ctxgð14Þ

To evaluate the system’s throughput, what we need is

the probability of successful transmission. We can

deduce it from err(ctx). From Definition 6, N(s) is the

number of MSs’ channels at state s. Therefore we can

deduce the probability and the rate of success

transmission. To compute 1 - err(ctx), we list all the

terms in Table 5. We can derive 1 - err(ctx) by

summing these terms.

Then Prx(crx), Perr(cerr) can be represented by Ptx(ctx)

and err(ctx)

PrxðcrxÞ ¼XC

ctx¼crx

PtxðctxÞctx

crx

� �1� errðctxÞ½ �crx errðctxÞ½ �ctx�crx

ð16Þ

PerrðcerrÞ¼XC

ctx¼cerr

PtxðctxÞctx

cerr

� �1�errðctxÞ½ �ctx�cerr errðctxÞ½ �cerr

ð17Þ

where Prx(crx) is the probability that all receivers suc-

cessfully receive crx packets during one time slot, and

Perr(cerr) is the probability that the BS transmits cerr erro-

neous packets during one time slot.

Lemma 1

XC

crx¼0

crxPrxðcrxÞ ¼XC

ctx¼0

ctxPtxðctxÞ 1� errðctxÞ½ � ð18Þ

Proof

XC

crx¼0

crxPrxðcrxÞ¼XC

crx¼0

crx

XC

ctx¼crx

PtxðctxÞctx

crx

� �

� 1�errðctxÞ½ �crx errðctxÞ½ �ctx�crx

¼XC

ctx¼0

ctxPtxðctxÞ½1

�errðctxÞ�Xctx

crx¼1

ctx�1

crx�1

� �1�errðctxÞ½ �crx�1 errðctxÞ½ �ctx�crx

For the latter part of the above expressions, we expand it

as a polynomial using err(ctx), and we can derive the

coefficient of each power of err(ctx). The coefficient of

[err(ctx)]p is as follows

Xctx

crx¼ctx�p

ctx � 1

crx � 1

� �crx � 1

p� ctx þ crx

� �ð�1Þp�ctxþcrx

¼ctx � 1

p

� � Xctx

crx¼ctx�p

p

ctx � crx

� �ð1Þctx�crxð�1Þp�ctxþcrx

¼ctx � 1

p

� �½1þ ð�1Þ�p ¼

1 if p ¼ 0

0 if p 6¼ 0

�

XC

crx¼0

crxPrxðcrxÞ ¼XC

ctx¼0

ctxPtxðctxÞ 1� errðctxÞ½ �

h

1� errðctxÞ ¼Qn

i¼1

PCj¼ctx

pijð1� errjctxÞ

h i�Qn

i¼1

PCj¼ctþ1 pijð1� errjctx

Þh i

Qni¼1

PCj¼ctx

pij �Qn

i¼1

PCj¼ctxþ1 pij

¼ WðctxÞ �Wðctx þ 1ÞUðctxÞ � Uðctx þ 1Þ

ð15Þ

1378 Wireless Netw (2011) 17:1373–1386

123

Finally, the expectation value of transmitted, received,

and erroneous packets from the BS, E[ctx], E[crx] and

E[cerr], can be derived from Lemma 1 and Eq. 15. E[ctx]:

the average number of transmitted packets from the BS to

all MSs E[crx]: the average number of successfully

received packets at any MS E[cerr]: the average number of

packets failed to be transmitted during one time slot at any

MS

E½ctx� ¼XC

ctx¼0

ctxPtxðctxÞ ¼XC

ctx¼1

Yn

i¼1

XC

j¼ctx

pij ¼XC

ctx¼1

UðctxÞ

ð19Þ

E½crx� ¼XC

crx¼0

crxPrxðcrxÞ ¼XC

ctx¼0

ctxPtxðctxÞ½1� errðctxÞ�

¼XC

ctx¼0

ctx½WðctxÞ �Wðctx þ 1Þ�

ð20Þ

E½cerr� ¼XC

cerr¼0

cerrPerrðcerrÞ ¼ E½ctx� � E½crx� ð21Þ

If the system’s transmission error can be ignored (which

indicates error-free transmission, errxy = 0), the

expressions can be simplified as follows

E½ctx� ¼XC

ctx¼1

UðctxÞ; E½crx� ¼XC

crx¼1

UðcrxÞ; E½cerr� ¼ 0

ð22Þ

3.2 Transient with initial channel state probability

In this section, we turn our attention to the transient

behavior. We know the information at time slot d, and we

use it to predict the system performance at time slot d ? 1.

What we have is the channel state transfer matrix, the

initial channel state probability, and the error rate matrix.

The state transfer function is defined in Definition 1. T and

Ti denote the group’s transfer matrix and the ith MS’s

channel state transfer matrix, respectively. The element of

Ti is txyi . txy

i represents the probability that the ith MS’s

channel jumps from state x to state y.

Ti �

ti00 ti

01 � � � ti0C

ti10 ti

11 � � � ti1C

..

. ... . .

. ...

tiC0 ti

C1 � � � tiCC

0BBB@

1CCCA

T �

T1 0 � � � 00 T2 � � � 0

..

. ... . .

. ...

0 0 � � � Tn

0BBB@

1CCCA

We apply the definition of initial channel state probability

distribution at time slot d from Definition 2. Md and Mdi

denote the group channel state and the ith MS’s channel state

at time slot d, respectively. The element of Mdi is mij

d which is

mdij ¼ Prfthe ith MS0s channel is in state j at time slot dg

Mdi � md

i0 � � � mdij � � � md

iC

� �1�ðCþ1Þ

Md � Md1 Md

2 � � � Mdn

� �1�nðCþ1Þ

Theorem 2 The probability of the base station

transmitting ctxd?1 packets during the next time slot, which

is time slot d ? 1, is as follows:

Pdþ1t cdþ1

tx

� �¼Yn

i¼1

XC

j¼cdþ1tx

mdþ1ij �

Yn

i¼1

XC

j¼cdþ1tx þ1

mdþ1ij

¼ U cdþ1tx

� �� U cdþ1

tx þ 1� �

ð23Þ

We introduce two notations to simplify the calculation

process.

Definition 8

U cdþ1tx

� �¼Yn

i¼1

XC

j¼cdþ1tx

mdþ1ij

W cdþ1tx

� �¼Yn

i¼1

XC

j¼cdþ1tx

mdþ1ij 1� errjcdþ1

tx

� �24

35

ð24Þ

Proof According to Definitions 1 and 2, we can directly

derive mijd?1, which is the probability of each MS’s channel

state at time slot d ? 1.

Table 5 PrfNðctxÞ ¼ n0 j s ¼ ctxg and corresponding avg. packet

success rate

n0 PrfNðctxÞ ¼ n0 j s ¼ ctxg E½success rate j NðctxÞ¼ n0; s ¼ ctx�

nQn

i¼1pictx

PtxðctxÞ(1 - errc_txctx)

n

n -

1

Qn

i¼1pictx

px1y1px1ctx

PtxðctxÞ(1 - ec_txctx)

n-1(1 - erry_1c_tx)

for

x1 ¼ 1; . . .; ny1 ¼ ct þ 1; . . .;C

..

. ... ..

.

n -

k

Qn

i¼1pictx

Qk

m¼1

pxmympxmctx

PtxðctxÞð1� errctxctx

Þn�k Qki¼1

ð1� erryictxÞ

where

xm ¼ 1; . . .; nym ¼ ctx þ 1; . . .;Cbuta 6¼ b) xa 6¼ xb

..

. ... ..

.

1 ... ..

.

Wireless Netw (2011) 17:1373–1386 1379

123

Mdþ1 ¼ Mdþ11 � � � Mdþ1

n

� �¼Md � T ð25Þ

Pr sdþ1 ¼ cdþ1tx

¼ mdþ1

icdþ1tx

Pr sdþ1� cdþ1tx

¼XC

j¼cdþ1tx

Pr sdþ1 ¼ j

¼XC

j¼cdþ1tx

mdþ1ij

ð26Þ

The probability of the base station transmitting ctxd?1

packets at time slot d ? 1 is

Pdþ1t ðcdþ1

tx Þ ¼Yn

i¼1

Pr sdþ1� cdþ1tx

�Yn

i¼1

Pr sdþ1� cdþ1tx þ 1

¼Yn

i¼1

XC

j¼cdþ1tx

mdþ1ij �

Yn

i¼1

XC

j¼cdþ1tx þ1

mdþ1ij

ð27Þ

h

The form of 1 - err(ctxd?1) is the same as Eq. 15.

However, pij is replaced by mijd?1.

Prx(crxd?1) and Perr(cerr

d?1), which are the probability of the

BS receiving crxd?1 or losing cerr

d?1, can be expressed as

follows

Pdþ1rx cdþ1

rx

� �¼

XC

cdþ1tx ¼cdþ1

rx

Pdþ1tx cdþ1

tx

� � cdþ1tx

cdþ1rx

!

� 1� errðcdþ1tx Þ

� �cdþ1rx err cdþ1

tx

� �� �cdþ1tx �cdþ1

rx

Pdþ1err cdþ1

err

� �¼

XC

cdþ1tx ¼cdþ1

err

Pdþ1tx cdþ1

tx

� � cdþ1tx

cdþ1err

!

� 1� errðcdþ1tx Þ

� �cdþ1tx �cdþ1

err err cdþ1tx

� �� �cdþ1err

Finally, using Lemma 1, the expectation value of the

transmitted, received, and erroneous packets from BS,

E[ctxd?1], E[crx

d?1] and E[cerrd?1], can be deduced.

E cdþ1tx

� �¼XC

cdþ1tx ¼0

cdþ1tx Pdþ1

tx ðcdþ1tx Þ ¼

XC

cdþ1tx ¼1

Yn

i¼1

XC

j¼cdþ1tx

mdþ1ij

¼XC

cdþ1tx ¼1

U cdþ1tx

� �

ð29Þ

E cdþ1rx

� �¼XC

cdþ1rx ¼0

cdþ1rx Pdþ1

rx cdþ1rx

� �

¼XC

cdþ1tx ¼0

cdþ1tx Ptx cdþ1

tx

� �1� err cdþ1

tx

� �� �

¼XC

cdþ1tx ¼0

cdþ1tx Wðcdþ1

tx Þ �Wðcdþ1tx þ 1Þ

� �

ð30Þ

E cdþ1err

� �¼XC

cdþ1err ¼0

cdþ1err Pdþ1

err cdþ1err

� �¼ E cdþ1

tx

� �� E cdþ1

rx

� �

ð31Þ

Suppose errxy = 0, which the system has error-free

channels,

E cdþ1tx

� �¼XC

cdþ1tx ¼1

U cdþ1tx

� �E cdþ1

rx

� �¼XC

cdþ1rx ¼1

U cdþ1rx

� �

E cdþ1err

� �¼ 0 ð32Þ

3.3 Predict throughput based on previous throughput

Predict Throughput Based on Previous Throughput means

that the given the throughput of previous time slots, the

throughput of current time slot is predicted. At first sight,

the whole situation is similar to Transient with Initial State

Probability. However, there exist essential differences.

In the case of Transient with Initial State Probability,

the initial channel states of each MS, the transition matri-

ces, and the error rate matrices are provided. What we

deduce is the transient behavior of MBS system channel

states. In the case of Predict Throughput Based on Previ-

ous Throughput, we only know the system throughput of

the previous time slot. The system throughput is deter-

mined by the lowest channel state (group states) of the

MBS users. Therefore we are using the channel states to

predict the throughput of current time slot. It means that the

std at previous time slot is given, but the combination of

each MS’s channel state is not. We only know the lowest

channel states in this case, while the case in Transient with

Initial State Probability is provide with the channel states

combination. Hence, these two situations are two different

problems.

1� errðcdþ1tx Þ ¼

Qni¼1

PCj¼cdþ1

txmdþ1

ij 1� errjcdþ1tx

� �h i�Qn

i¼1

PCj¼cdþ1

tx þ1 mdþ1ij 1� errjcdþ1

tx

� �h iQn

i¼1

PCj¼cdþ1

txmdþ1

ij �Qn

i¼1

PCj¼cdþ1

tx þ1 mdþ1ij

¼W cdþ1

tx

� ��W cdþ1

tx þ 1� �

U cdþ1tx

� �� U cdþ1

tx þ 1� �

ð28Þ

1380 Wireless Netw (2011) 17:1373–1386

123

Now we have the information of the channel state

transfer matrix and the MS’s channel state probability at

time slot d. The definition of transfer matrix T is in Defi-

nition 1. Thus, txyi represents the probability that the ith

MS’s channel jumps from state x to state y. However, all

the equations solved throughout this section require only

txyi . The T and Ti are not required for the deduction process.

The last information we have is the channel state

probability at time slot d. We use notations which are

similar to the state probability in the Steady Channel State.

Definition 9 Let pijd denote the probability of the ith MS’s

channel at state j at time slot d.

From Theorem 1, we know that the BS transmitting cdtx

packets at time slot d is

Pdtxðcd

txÞ ¼Yn

i¼1

XC

j¼cdtx

pij �Yn

i¼1

XC

j¼cdtxþ1

pij ð33Þ

We denote two notations for simplicity.

Definition 10

Because many combinations can result in the group state

being equal to sd, we express the successful transmission

probability conditioned on the number of MSs in state sd at

time slot d. Thus, we need to use Definition 6 for N(sd).

Now we can compute the conditional probability with the

given MSs’ channel state at time d?1. The expectation

value for the number of successfully received packets at the

BS is

E cdþ1rx j sd ¼ cd

tx

� �¼Xn

n0¼1

E cdþ1rx j NðsdÞ ¼ n0; sd ¼ cd

tx

� �

� Pr ðcdtxÞ ¼ n0 j sd ¼ cd

tx

ð36Þ

The computation details of Eq. 36 is provided in the

‘‘Appendix’’. The derivation of each equation term is listed

in Table 6. After summing these terms up, we can obtain

E cdþ1tx j sd¼ cd

tx

� �;E cdþ1

rx j sd¼ cdtx

� �, and E cdþ1

err j sd¼�

cdtx�.

They are the number of transmitted, successfully received,

and erroneous packets at the BS.

E cdþ1tx j sd ¼ cd

tx

� �¼XC

cdþ1tx ¼1

U cdtx; c

dþ1tx

� �ð37Þ

E cdþ1rx j sd ¼ cd

tx

� �¼XC

cdþ1tx ¼1

cdþ1t W cd

tx; cdþ1tx

� ��W cd

tx; cdþ1tx þ 1

� �� �

ð38Þ

E cdþ1err j sd ¼ cd

tx

� �¼ E cdþ1

t j sd ¼ cdtx

� �� E cdþ1

rx j sd ¼ cdtx

� �ð39Þ

We can observe that the difference between

E cdþ1tx j

�sd

t ¼ cdt � and E cdþ1

rx j sdt ¼ cd

t

� �is

1� err cdþ1tx

� �¼

W cdtx; c

dþ1tx

� ��W cd

tx; cdþ1tx þ 1

� �

U cdtx; c

dþ1tx

� �� U cd

tx; cdþ1tx þ 1

� � ð40Þ

Now consider the special case of user with error-free

channel, errxy = 0,

E cdþ1tx j sd ¼ cd

tx

� �¼XC

cdþ1tx ¼1

U cdtx; c

dþ1tx

� �ð41Þ

E cdþ1rx j sd ¼ cd

tx

� �¼XC

cdþ1rx ¼1

U cdtx; c

dþ1rx

� �ð42Þ

E cdþ1err j sd ¼ cd

tx

� �¼ 0 ð43Þ

4 Performance evaluation

In this part, we construct a software platform for simulation

provided with all parameters related to our FSMC model.

We attempt to use the simulation to evaluate the system

performance. The anticipated results from our model are

compared with these simulation results to verify our

model’s accuracy.

Note that the goal of this paper is to examine the the-

oretical performance of applying AMC to MBS, so we

have to verify if our model captures the performance of

AMC. The channel states variation is captured using the

formulation of M and T in Definition 2, and the mobility,

time/space variance, etc are all included in the formulation.

We use the lowest channel states, the group states, as the

MCS to be applied. As a result, the group states now can

represent the throughput of the system, and we then can

examine our model’s validity by verifying if the throughput

corresponds to the simulation results.

U cdtx; c

dþ1tx

� �¼Qn

i¼1

PCj¼cd

txpd

ij

PCk¼cdþ1

txtijk �

Qni¼1

PCj¼cd

txþ1 pdij

PCk¼cdþ1

txtijk

Pdtxðcd

txÞð34Þ

W cdtx; c

dþ1tx

� �¼Qn

i¼1

PCj¼cd

txpd

ij

PCk¼cdþ1

txtijk 1� errkcdþ1

tx

� ��Qn

i¼1

PCj¼cd

txþ1 pdij

PCk¼cdþ1

txtijk 1� errkcdþ1

tx

� �

Pdtxðcd

txÞð35Þ

Wireless Netw (2011) 17:1373–1386 1381

123

4.1 Simulation methodology

The system’s performance is evaluated in the following

simulations. First, we derive the theoretical throughputs

with the proposed system model. Second, a system level

simulation is performed to simulate the varying channel

states and MCS settings. Considering the MBS system

throughput, we can evaluate the model’s effectiveness by

examining the throughput discrepancy between the theo-

retical and simulation results. Once the simulation results

approximate the theoretical results, we can confirm that the

proposed model accurately captures the throughput of the

MBS system using AMC.

We use MATLAB as the simulation platform, and

assume that the number of data packets exceeding the

channel capacity every time to rule out any idle channel.

During every time slot, the transmitter (BS) transmits the

packets in the front end of the queue, and the packets can

either be received or not. In our simulations, we collect

data samples no less than 104 time slots to compute their

average value.

4.2 Numerical settings and results

As the goal is to verify the throughput prediction’s effec-

tiveness, what we look for is the comparison between the

simulation and theoretical results. Therefore, four levels of

channel states are assumed. That is, C = 3, and the channel

states ranges from 0 to 3. We categorize our parameter

settings into groups according to the previous sections

Table 6 PrfNðcdtxÞ ¼ n0 j sd ¼ cd

txg and the corresponding avg. packet throughput with different n0

n0 PrfNðcdtxÞ ¼ n0 j sd ¼ cd

txg E½cdþ1r x j Nðcd

txÞ ¼ n0; sd ¼ cdtx�

nQn

i¼1pd

icdtx

Pdtxðcd

txÞPC

cdþ1tx ¼1

cdþ1tx

Qni¼1

PCj¼cdþ1

tx

ticd

txj1� errjcdþ1

tx

� �"

�Qn

i¼1

PCj¼cdþ1

tx þ1

ticd

tx j1� errjcdþ1

tx

� �#

n - 1 summation over any combination of x1,y1

Qn

i¼1pd

icdtx

pdx1y1

pd

x1cdtx

Pdtxðcd

txÞ

PCcdþ1

tx ¼1

cdþ1tx

Qni¼1

PCj¼cdþ1

tx

ticd

txj1� errjcdþ1

tx

� �A

"

�Qn

i¼1

PCj¼cdþ1

tx þ1

ticd

tx j1� errjcdþ1

tx

� �B

where

x1 ¼ 1; . . .; ny1 ¼ cd

tx þ 1; . . .;C

where

A ¼PC

l¼cdþ1tx

tx1y1 l

1�errlcdþ1

tx

� �

PC

l¼cdþ1tx

tx1

cdtx

l1�err

jcdþ1tx

� �

B ¼PC

l¼cdþ1txþ1

tx1y1 l

1�errlcdþ1

tx

� �

PC

l¼cdþ1txþ1

tx1

cdtx

l1�err

jcdþ1tx

� �

..

. ... ..

.

n - k summation over any combination of xm,ym

Qn

i¼1pd

icdtx

Qk

m¼1

pdxmym

pd

xm cdtx

Pdt ðcd

txÞ

PCcdþ1

t ¼1

cdþ1tx

Qni¼1

PCj¼cdþ1

tx

ticd

txj1� errjcdþ1

tx

� �Qkm¼1 G

"

�Qn

i¼1

PCj¼cdþ1

tx þ1

ticd

txj1� errjcdþ1

tx

� �Qkm¼1 H

#

where

xm ¼ 1; � � � ; nym ¼ cd

tx þ 1; � � � ;Cbuta 6¼ b) xa 6¼ xb

where

G ¼PC

l¼cdþ1tx

txmyml

1�errlcdþ1

tx

� �

PC

l¼cdþ1tx

txm

cdt

l1�err

lcdþ1tx

� � ;

H ¼PC

l¼cdþ1txþ1

txmyml

1�errlcdþ1

tx

� �

PC

l¼cdþ1txþ1

txm

cdt

l1�err

lcdþ1tx

� �

..

. ... ..

.

1 ... ..

.

1382 Wireless Netw (2011) 17:1373–1386

123

titles. We then verify these cases: Steady Channel State,

Predict Throughput Based on Previous Throughput, Pre-

dict Throughput Based on Previous Throughput. For a

Markov chain with finite and communicative states, there

will exist one and only one steady state. As our model has

limited numbers of states and they are all communicative,

the steady channel state will surely exist, regardless of the

initial states. If long-term system behavior is to be pre-

dicted, the initial channel states of the system could be

arbitrarily determined. The initial state probability is listed

below. Besides, the error rate probability matrices should

be determined. We randomly construct 3 sets of error

probability matrices with elow = 10% error rate,

emid = 20%, and ehigh = 30%.

The state transfer matrices are randomly determined

using different average values. State transfer matrix T1

with 50% probability to stay within the same channel

states, and 25% probability of transition to higher or lower

channel states. This is to investigate the system behavior of

a relatively static channel. On one hand, state transfer

matrix T2 is with 75% of probability staying at the first two

levels of channel states, to simulate a more interfered

channel state. On the other hand, state transfer matrix T3

has 75% probability staying at the last two levels of

channel states, to simulate a channel in excellent condition.

The initial state probability matrix is given by InitProb,

which is arbitrarily determined with a 60% ‘‘dominant

initial state’’ probability for each level of channel states.

The ’dominant initial state’ refers to the initial channel

state that is most likely at that state. For example, if C = 3

and the initial state probability is (0.1, 0.1, 0.6, 0.2), then

the dominant initial state is 2, with the largest probability

of 0.6. The initial state probability matrix InitProb is for-

mulated to consider all the cases of different initial states. It

is obvious that the higher the dominant initial state is, the

higher the system throughput. Therefore the InitProb

should possess dominant initial state from channel state 0

to 3. The current state probability p is also formulated in

the same way, with different dominant state inclining for

medium, high, and low channel quality.

(1) Steady Channel State: In this section, we investigate

the case of steady channel state. We first study the

case with non-constant error rate. The number of

users and the channel states are set to n = 3 and

C = 3, as mentioned in previous section. That is,

there are 3 users, and the channel states range from 0

to 3. Each user adopts a distinct initial state: all-low,

all-high, and rand. The all-low initial states indicates

that the initial states of all users are set to state 0, and

the all-high 3. The rand initial states are randomly

generated with mean state=2. The state transfer

matrices are T1, T2, and T3. The error rate is emid,

which is 20% of error rate probability. Figure 1

shows the result of steady channel state with

nonconstant error rate. The dotted line is the expected

value of successfully received packets and error

packets. The other solid lines are simulation results.

Different initial states are applied. As shown in the

Fig. 1, the simulation performance approaches to the

expected value with time for both successful and error

packets. The different initial state settings have no

effects on the long-term results, which prove that our

model is adequate. The simulation results will

converge to the analytical expected value after a

time interval of 104 time slots.

More simulations with different error rate settings are

conducted to verify the model. Another simulation

using n = 3. C = 3 is conducted. As the long-term

behavior with different initial states is examined, now

we set the initial state to all-low. The state transfer

matrices are T1, T2, and T3. The error rate matrices

are elow; emid, and ehigh with corresponding error rate

probability 10, 20, and 30%. Figure 2 shows the

result of steady channel state system with different

levels of error rate. The relation between each level is

the same as Fig. 1. The dotted line is the expected

value, and the other curves are results from simula-

tions. Using different levels of error rate, the perfor-

mance of the simulation system constantly

approximates the expected values of our model.

(2) Transient with Initial State Probability: The transient

case with initial state probability is studied in this

section. Now n = 1. C = 3. The initial channel state

probability matrix is InitProb. The state transfer

matrix is T1. Three levels of error rates elow; emid, and

101

102

103

104

105

106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Number of time slot

Exp

ecte

d th

roug

hput

expected valuesim(all−low)sim(all−high)sim(rand)error expected valueerror sim(all−low)error sim(all−high)error sim(rand)

Fig. 1 Average number of received/error packets per time slot of

steady state with different initial state and nonconstant error rate V.S

time

Wireless Netw (2011) 17:1373–1386 1383

123

ehigh with corresponding error rate probability 10, 20,

and 30% are evaluated. Figure 3 shows the result of

Transient with Initial Channel State Probability given

non-constant error rate. Since n = 1, there is only one

receiver with C = 3. Different initial state probabil-

ities and levels of error rate are applied. Different

‘‘dominant initial states’’ are applied in this case. The

formulation is to thoroughly investigate all the

transient state probability, for which the error rate is

also a key factor for the performance. When the level

of error rate is low, the system performance is

improved. As shown in the figure, the theoretical

results approximate the simulation results.

(3) Predict Throughput Based on Previous Throughput

Now the throughput prediction is investigated in this

section. n = 3. C = 3. Besides, the current state

probability uses the matrix p as defined in Sect. 4.2.

The state transfer matrices are T1, T2, and T3. Three

levels of error rate, elow; emid, and ehigh, with corre-

sponding error rate probability 10, 20, and 30% are

also evaluated. Figure 4 is the result of Predict

Throughput Based on Previous Throughput with

non-constant error rate. In this case, we do not collect

every MS’s channel state data. Instead, we only

record the group rate, and verify whether our model

accurately predict the future performance. When the

given rate is high, the predicted throughput of the

next time slot is also increased. The simulation results

also closely approximate our model output.

5 Conclusion

In our work, we propose system performance models for

rate-adaptive multicast based FSMC. We analyze the sys-

tem performance, such as the number of average packet

count. In addition, the number of successfully received

packets and the transmission errors are both considered for

each time slot, and three distinct cases are investigated: (1)

steady channel state, (2) transient with initial state proba-

bility, and (3) predict future throughput based on the pre-

vious throughput. The prediction results of these three

cases approximate the simulation outcomes.

On the throughput analysis, we conduct our experiments

with three levels of error rates. Stationary and transient

performances are also investigated. All of these theoretical

results are found closely approximating the MATLAB

simulations. For a wireless communication system, the

101

102

103

104

105

106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Number of time slot

Exp

ecte

d th

roug

hput

expected value (e_low)sim(e_low)expected value (e_mid)sim(e_mid)expected value (e_high)sim(e_high)

Fig. 2 Expected throughput per time slot of steady state with

nonconstant error rate V.S time

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

dominant initial state

Exp

ecte

d th

roug

hput

per

tim

e sl

ot

expected value(e_low)sim(e_low)expected value(e_mid)sim(e_mid)expected value(e_high)sim(e_high)

Fig. 3 Expected throughput per time slot of transient with initial

channel state probability and nonconstant error rate V.S Different

dominant initial state

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

initial state

Exp

ecte

d th

roug

hput

per

tim

e sl

ot

expected value(e_low)sim(e_low)expected value(e_mid)sim(e_mid)expected value(e_high)sim(e_high)

Fig. 4 Expected throughput per time slot based on previous

throughput with different constant error rate V.S Different previous

throughput

1384 Wireless Netw (2011) 17:1373–1386

123

channel states, state transition probability, and the trans-

mission error rate can be measured under specific envi-

ronment. With the information, the proposed Markovian

model can precisely predict the system throughput. This

paper forms a solid model for the estimation of applying

AMC in MBS.

Acknowledgments This work was supported by National Science

Council, National Taiwan University and Intel Corporation under

Grants NSC99-2911-I-002-001, 99R70600, and 10R70500.

Appendix

PrfNðcdtxÞ ¼ n0 j sd ¼ cd

txg and the corresponding average

packet throughput with different n0.

This ‘‘Appendix’’ provides the deduction of Eq. 36, we

list every term in Table 4. Besides, we use Lemma 1 in

each row. At a first glance, this method seems very brutal.

However, making use of the symmetry in the expression

will simplify the calculation dramatically. After summing

these terms up, we can obtain E cdþ1tx j sd ¼ cd

tx

� �; E cdþ1

rx j�

sd ¼ cdtx�, and E cdþ1

err j sd ¼ cdtx

� �. They are the number of

transmitted, successfully received, and erroneous packets

at the BS.

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Author Biographies

Yu-Cheng Liang received the

B.S. degree in electrical engi-

neering from National Taiwan

University in 2009. He start Ph.D.

study in Department of Electrical

Engineering at Stanford Univer-

sity in 2010.

Ching-Chun Chou receives

Bachelor’s Degree in the Com-

puter Science and Information

Engineering Department of

National Taiwan University. He

is now a Ph.D. student in Elec-

trical Engineering Department

of National Taiwan University.

His research interest is in

mobile networking and stan-

dardization of IEEE 802.16 and

LTE-Advanced. He is a voting

member of IEEE 802.16 work-

ing group.

Hung-Yu Wei received the

B.S. degree in electrical engi-

neering from National Taiwan

University in 1999. He received

the M.S. and the Ph.D. degree in

electrical engineering from

Columbia University in 2001

and 2005, respectively. He was

a summer intern at Telcordia

Applied Research in 2000 and

2001. He was with NEC Labs

America from 2003 to 2005. He

joined Department of Electrical

Engineering at the National

Taiwan University in July 2005.

His research interests include wireless mesh networks, mobility

management in mobile Internet, sensor networks, cross-layer design

and optimization in wireless multimedia communications, and game

theoretical models for communications networks. He is a voting

member of IEEE 802.16 working group.

1386 Wireless Netw (2011) 17:1373–1386

123