2. Background
2.4 Quantitative Measure of Fairness
Fairness consideration is very important in the field of radio resource management.
We should show the how fair resource allocation is among all users numerically. How-ever, there are many fairness measures proposed to indicate the fairness level of re-source allocation. We describe those fairness indices as follows. Define xithe resource quantity allocated to user i, and n the number of total users.
Variance
Coef f icient of V ariation (COV ) = V ariance
Mean (2.8)
As above, all the indices can represent the degree of fairness of resource allo-cation among users. Nevertheless, in [19], we desire that the fairness index has the following properties.
(a) Population size independence: The index should be suitable for infinite or finite users. The above four indices all satisfy the requirement.
(b) Scale and metric independence: We expect that the indices are indepen-dent of scale. In other words, if users are allocated ten times quantity of resource simultaneously, the indices should be the same values. However, the variance index does not meet the goal.
(c) Boundedness: In addition to population size independence and scale in-dependence, the indices are desired to be bounded between 0 and 1. Therefore, we can judge the policy of resource allocation fair according whether the fairness index approaches 1 or not. The COV (coefficient of variation) is not bounded. Its value distributes from 0 to infinity.
(d) Continuity: The continuous index can respond the little change in allo-cation way. The min-max ratio index only take the users with best resource and with worst resource into account. Therefore, if the allocation changes among medium users, the min-max ratio index does not change.
To sum up, we observe the behaviors of the indices. We find that Jain’s fairness index satisfies all the desired properties. Consequently, we adopt the index as one of our simulation performance metric.
Tab. 2.2: OFDMA downlink subcarriers allocation
Parameters Value
Number of DC carriers 1
Number of Guard Carriers: Left, Right 173,172
Number of Used Carriers 1702
Nused 1702
Total Number of Carriers 2048
NvarLocP ilots 142
Number of Fixed-location Pilots 32
Number of Variable-Location Pilots which 8
coincide with Fixed-Location Pilots
Total Number of Pilots 166
Number of data carriers 1536
Nsubchannels 32
Nsubcarriers 48
Number of data carriers per subchannel 48
BasicFixedLocationPilots {0,39,261,330,342,351,522,636,645,651,708,726, 756,792,849,855,918,1017,1143,1155,1158,1185, 1206,1260,1407,1419,1428,1461,1530,1545,1572,
1701}
{P ermutationBase0} {3,18,2,8,16,10,11,15,26,22,6,9,27,20,25,1,29 7,21,5,28,31,23,17,4,24,0,13,12,19,14,30}
Tab. 2.3: OFDMA uplink subcarriers allocation
Parameters Value
Number of DC carriers 1
Nused 1696
Number of Guard Carriers: Left, Right 176,175
Nsubchannels 32
Nsubcarriers 53
Number of data carriers per subchannel 48
{P ermutationBase0} {3,18,2,8,16,10,11,15,26,22,6,9,27,20,25,1, 29,7,21,5,28,31,23,17,4,24,0,13,12,19,14,
30}
Tab. 2.4: MMDS band frequency spacing (fs/BW = 8/7) OFDMA(NF F T = 2048)
T
g(µ
s)
BW(MHz) ∆f (kHz) T
b(µ
s) T
b/32 T
b/16 T
b/8 T
b/4
1.5
36431194
2337
1374
23149
13298
233.0 1
6089597
1318
2337
1374
23149
136.0 3
238298
239
1318
2337
1374
2312.0 6
3956149
134
239
1318
2337
1324.0 13
112874
232
134
239
1318
23Throughput and Fairness Enhancement for OFDMA Broadband Wireless Access
Systems Using the Maximum C/I Scheduling
In this chapter, we demonstrate that the simple maximum carrier to interference ratio (C/I) scheduling can both enhance system throughput and maintain fairness perfor-mances for the orthogonal frequency division multiple access (OFDMA) system. The maximum C/I scheduling has long been recognized as an effective method to enhance throughput, but it is viewed as an unfair scheduling policy in the the single carrier code division multiple access (CDMA) system. We reassess the fairness performance of the maximum C/I scheduling in the context of the multi-carrier OFDMA system.
Through analysis and simulations, we find that the maximum C/I scheduling is indeed an fair scheduling for OFDMA systems. Thus, with respect to the OFDMA system, we develop a maximum C/I scheduling based resource allocation algorithm. Our re-sults show that the fairness of the maximum C/I scheduling in OFDMA systems is comparable to that of the proportional fair scheduling scheme. In short, we conclude that in the OFDMA system, the maximum C/I scheduling not only can maximize system throughput, but simultaneously maintain very good fairness performance.
3.1 Introduction
With the growing demand for high data rate communication, orthogonal frequency division multiple access (OFDMA) is becoming an important technology. OFDMA has been used in many broadband wireless systems, such as the IEEE 802.16a wireless metropolitan area network (WMAN) [13] [20]. This chapter investigates the benefits of OFDMA systems from both frequency diversity and multiuser diversity perspec-tives. Frequency diversity inherently exists in OFDMA systems, while multiuser diversity can be achieved by adopting scheduling algorithms. Although both diver-sity gains can enhance the system throughput, the challenging issue here is how to select a scheduling algorithm that can achieve high system throughput and maintain the fairness among users simultaneously.
OFDMA is not only a modulation scheme but also a multiple access technology.
In an OFDMA system, each user is allocated a set of orthogonal subcarriers. In addition to overcoming the inter-symbol interference (ISI), an OFDMA system can also mitigate the multiple access interference (MAI) due to the orthogonality among subcarriers. Moreover, it can result in frequency diversity benefit with interleaving and channel coding. To further take advantage of frequency diversity [21–25], many adaptive resource allocation techniques were suggested from a view point of subcarrier power allocation [12] [26].
The goal of this chapter is to investigate the OFDMA system from another resource allocation viewpoint, i.e, scheduling algorithms. Wireless scheduling tech-niques are developed to exploit the multiuser diversity. In a multiuser wireless system, different users may have different channel responses in a time varying wireless chan-nel. Thus, a channel may be viewed as a bad channel, but may be viewed as a good channel by other users. Consequently, if the system can first pick a user with the best channel quality among a group of users to serve in each channel , the system capacity
can be improved significantly. We call this capacity improvement as the multiuser diversity gain. Clearly, for providing delay-tolerant data services, wireless scheduling is an inevitable technique to exploit multiuser diversity which inherently exists in the multiuser system.
Many scheduling algorithms have been developed for the single carrier time division multiple access (TDMA) or code division multiple access (CDMA) systems [14–18]. First, the maximum C/I scheduling scheme allocates the channel to the user that has the best channel condition [14]. This scheduling algorithm can fully exploit multiuser diversity at the expense of sacrificing the fairness performance for other users. Second, the round-robin scheduling approach allocates resource to each user periodically, which can provide the best fairness performance, but has lowest throughput because it does not take the channel information into account. Third, the proportional fair scheduling algorithm [15] was proposed to use the ratio of the short-term channel response to the long-short-term channel condition of each user to allocate the resource. Last, the exponential rule scheduling method [16–18] further considers the service delay of each user. If the user has waited for a long period of time, this user will be allocated a channel with a higher priority. These wireless scheduling algorithms were only evaluated in the single carrier wireless system. To our knowledge, how these resource management algorithms perform in the multi-carrier OFDMA system is an open issue.
There were a lot of dynamic radio resource management technologies in OFDM based multicarrier systems discussed in the literature. In traditional wired dis-crete multitone asymmetric digital subscriber lines (ADSL), a resource management method named water-filling power allocation [27] is popularly used. Therefore, a lot of papers [21] [23] [24] adopted this rule to solve the optimization problem that to maximize the system capacity under the total power constraint.
In this chapter, we first assess the fairness performance of the maximum C/I
scheduling in the multi-carrier OFDMA system. The maximum C/I scheduling has long been recognized as an effective method to enhance throughput, but it is also viewed as an unfair scheduling policy in the the single carrier CDMA system. Through analysis and simulations, we will find that the maximum C/I scheduling is indeed an fair scheduling for OFDMA systems. Thus, with respect to the OFDMA system, we develop a maximum C/I scheduling based resource allocation algorithm. We will show that the fairness of the maximum C/I scheduling in OFDMA systems is comparable to that of the proportional fair scheduling scheme. Hence, we conclude that the simple maximum C/I scheduling can enhance both system throughput and fairness performances for the OFDMA system.
The rest of this chapter is organized as follows. Section 3.2 introduces the channel models for an OFDMA based IEEE 802.16a system. Section 3.3 formulates this problem. In Section 3.4, we analyze the system throughput performance with the maximum C/I scheduling algorithm in the multicarrier systems. Section 3.5 introduces the current two resource allocation strategies. Simulation results are given in Section 3.6. We give our concluding remarks in Section 3.7.
3.2 Channel Models for the IEEE 802.16a System
We will introduce more complicated but practical channel models specified in the IEEE 802.16a WMAN standard [28]. There are six typical Stanford University In-terim (SUI) channel models for three types of terrains. These SUI channels are used for the fixed broadband wireless applications (BWA) in the multichannel multipoint distributed service (MMDS) band. We will use the two SUI channel models, SUI-1 and SUI-5, in our simulations. Parameters in the two SUI channels are summarized in Tables 3.1 and 3.2.
SUI-1 channel model is for low mobility with small delay spread, which is close
Tab. 3.1: SUI-1 Channel
Tap 1 Tap 2 Tap 3 Units
Delay 0 0.4 0.8 µs
power (omni. ant) 0 -15 -20 dB
power (30◦ antenna) 0 -21 -32 dB
K Factor 18 0 0
Maximum Doppler frequency 0.4 0.4 0.4 Hz
Tab. 3.2: SUI-5 Channel
Tap 1 Tap 2 Tap 3 Units
Delay 0 5 10 µs
power (omni. ant) 0 -5 -10 dB
power (30◦ antenna) 0 -11 -22 dB
K Factor 0 0 0
Maximum Doppler frequency 2 2 2 Hz
to Rician fading. On the other hand, SUI-5 is close to Rayleigh fading channel and it is exposed severe multipath fading effect. Moreover, the channel response value “1”
is defined to be the state that received signal-to-noise ratio (SNR) can be satisfied. If the value is above 1, the channel is in good condition. In our simulation, we evaluate the system capacity by using the QPSK with coding rate 1/2 case [13]. The channel response “1” corresponds to the received SNR 9.4 dB. The receiver with SNR values in Table 3.3 can achieve BER less than 10−6.
Tab. 3.3: Receiver SNR and Eb/N0 assumptions Modulation coding rate Receiver SNR
QPSK 1/2 9.4
QPSK 3/4 11.2
16QAM 1/2 16.4
16QAM 3/4 18.2
64QAM 2/3 22.7
64QAM 3/4 24.4
3.3 Problem Description
3.3.1 Two-state Random Channel Matrix
A simple channel model is adopted to describe the impact of the number of subchan-nels when using the maximum C/I scheduling algorithm. We assume that there are N users requiring the same data rate. Each subchannel has two states: good and bad [29]. Good state means that the channel could bear 1 + δ times of the required rate rate, while the bad state means that the channel only could transmit 1 − δ times of the normal data rate. We also assume that the channel condition on which each user observed is independent. In other words, the same channel may be viewed as a good channel for a user, but a bad one for others.
As described above, an arbitrary subchannel may have different states for
different users. Consequently, we can form a channel matrix H :
where hn,m represents the m−th subchannel condition to the n−th user. For example, h3,2 means the response of subchannel 2 observed by user 3. Each element can be in two states, good or bad with equal probability 12. This model will be used for only analysis.
3.3.2 Problem Formulation
For simplicity, we adopt the two-state channel model to analyze both the throughput and fairness performance of multicarrier systems. We first assume that each user uses just one subchannel. Next, we will calculate the probability that all users are allocated with good subchannels. This is an issue of permutation and combination in mathematics. As the numbers of users and subchannels increase, the process of permutation and combination calculation becomes very complicated. Therefore, we propose a systematic analytical approach. Owing to too many possibilities in permutation and combination as the numbers of users and subchannels increase, we will apply the Inclusion-Exclusion Principle to analyze system performance.
We define a permutation matrix P, which contains all permutations of 1,2,3,...,N.
Take N=3 as an example.
P =
where P is an N × N! matrix. This matrix will be used to permutate all conditions that all users have observed good channel conditions. The value x of each entry in the i − th row of the permutation matrix P represents the entry located at the i − th row and the x − th column of the channel matrix H in a good condition. In other words, the x − th subchannel is in a good condition for the i − th user. For example, if the second column vector of P, [1, 3, 2]T, this means that h11, h23 and h32 are in the good state. Then the channel matrix becomes
H =
where the elements labelled ”∨” in the i − th row and the j − th column in channel matrix H mean that the j − th subchannel is in a good state for the i − th user. The elements labelled ”f ree” mean that the subchannel conditions responding to some users can be either good or bad. Thus, all users can use good subchannel without conflicts. Consider both the second and fourth columns of P in (3.2), i.e. [1, 3, 2]T and [2, 3, 1]T, simultaneously. Then the channel matrix becomes
H =
Since there are at least N good subchannels in different rows and different columns, each user can have a good subchannel for transmissions. In the following, we introduce a systematic approach to analyze the impact of the maximum C/I scheduling algo-rithm in the multicarrier systems by applying the Inclusion-exclusion principle [30].
3.4 Analysis
3.4.1 Inclusion-Exclusion Principle
Our goal is to calculate all conditions that all users can use good subchannels. Con-sider N users and N subchannels. We count the number of cases that the good subchannels can distribute in N different rows and different columns in the channel matrix HN ×N. First, we will use the parameter, permutation matrix P . Any combi-nations of the columns in matrix P corresponds to a channel matrix H. It is possible that different combinations of columns in P map to the same channel matrices H.
Our objective is to calculate the number of matrices H in which all users can find a good subchannel without conflicts. For example,
H =
represents a case that all users can have good subchannels without conflicts. By contrast,
represents a case that users 1 and 2 compete for subchannel 1. Next, we apply the Inclusion-Exclusion Principle to calculate the number of all users having good channels.
Lemma To calculate the size of A1S A2S
. . .S
An, calculate the sizes of all possible intersections of sets from {A1, A2, . . . , An}, and then add the results obtained
by intersecting an odd number of the sets and then subtract the results obtained by intersecting an even number of the sets [30].
For example, if we will calculate the number of multiples of 2 and 3 from 1 to 100, we will first count the number of multiples of 2, then we add the number of multiples of 3; and finally we subtract the number of multiples of 6.
We define F (k) as the number of matrices H for selecting k columns from the permutation matrix P . For an even number of k, F (k) is denoted as Fe(k), whereas for an odd number of k, F (k) is represented by Fo(k). Note that k is ranged from 1 to N! and P is an N × N! matrix. By applying the Inclusion-Exclusion Principle, we can calculate the number of the non-conflict conditions as
X
k=1,3,...,N !−1
Fo(k) − X
k=2,4,..,N !
Fe(k) (3.7)
For example, if N = 3, then the permutation matrix P
P =
For Fo(1), there are six (C13!) selections, which corresponds to the case that {1}, {2}, {3}, {4}, {5} and {6} columns in permutation matrix P are selected individually. In this case, each Fo(1) corresponds to 26 channel matrices H because there are six free elements in H. (see (3.3) as an example). For Fe(2), there are C23! combinations, which means that we choose {1,2}, {1,3},{1,4},...{4,5},{4,6} and {5,6} columns from the permutation matrix P . Fe(2) may be either 23 (e.g. {1,4}) or 24 (e.g. {2,4}).
When N increases, the permutation and combination conditions becomes huge. The systematic approach based on (3.7) can solve the complex permutation and combi-nation problems.
3.4.2 Fairness Index
According to [31] [19], we define a fairness index F in the multiuser systems as follows:
F =
where ri is the transmission data rate of the i − th user, and N is the number of total users. For F = 1, it is the fairest condition between users, and it is not fair as F < 1.
For example, if there are two users transmitting data, one is transmit at 1.2 times of the required data rate, and the other transmit at 0.8 times of the required data rate.
Then the fairness index F is about 0.96. If the transmission data rate of one user is 1.5, and the other is 0.5, the fairness index is 0.8. Thus the former example is fairer than the latter.
We will illustrate that a random assignment method cannot easily achieve high value of the fairness index. We illustrate this point as follows. We generate a set of random variables. Each random variable represents the resource allocated to each user. We assume the random variables are uniformly distributed in the interval (0,1). Fig. 3.1 shows the cumulative distribution function (CDF) of the value of the fairness index. From Fig. 3.1, we find that the more the users, the harder the fairness is achieved. When there are 8 users, the probability that the fairness index is larger than 0.8 is 38%. However, if 32 users exist in the system, the probability that the fairness index is greater than 0.8 is smaller than 20%. From this example, we know that a random assignment approach can not easily achieve the fairness index higher than 0.8 or 0.9. Thus, it is not trivial to design a resource allocation scheme achieving the fairness index higher than 0.9.
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Value of fairness index
CDF
8 users 32 users 56 users 80 users
Fig. 3.1: Distribution of fairness index value for a random assignment
3.4.3 Observation
By applying the Inclusion-Exclusion Principle, we can systematically calculate prob-ability that all users can use good subchannels when N is not too large. We list the results in Table 3.4.
Tab. 3.4: Analytical results of non-conflict condition Number of
From Table 3.4, we observe that the probability of the non-conflict condition (i.e., all users can use good subchannels.) increases with the number of users (sub-channels) increasing apparently. By increasing the number of subchannels and users, we find that system throughput performance can be improved even without other complicated scheduling algorithms, such as proportional fairness scheduling or even exponential rule scheduling algorithms.
Furthermore, we observe the effect of the number of subcarriers on the fair-ness when the maximum carrier-to-interference scheduling algorithm is used in the multicarrier systems.
Due to the complexity, we obtain the numerical results by programming when N is larger than 5. We find that we can further achieve good fairness performance between users by efficiently exploiting both multiuser diversity and frequency diver-sity. For the case of N = 7 in Fig. 3.2, one can find that with 90% probability, all
5 10 15 20 25 30 0
0.5 0.65 0.85 0.9 0.93 0.96 0.99 0.995 0.999 0.9999
number of subcarriers(users)
probability that all users use good subcarrier
Fig. 3.2: Probability of the non-conflict condition with the varying number of users and subchannels.
the seven users can have the good subchannels and the fairness index F = 1.
Figure 3.3 shows the effect of increasing number of the subcarriers on fairness performance with different numbers of users in a two-state random channel model. We can easily see that the more the number of subcarriers, the better the system fairness performance. However, as the number of users increases, the required number of subcarriers to provide satisfying fairness performance increases. Observing Fig. 3.3, if we require the system fairness index has to be larger than 0.9 when there are
24 users in the system, we should divide available total bandwidth into at least 22 subcarriers.
3.5 Resource Allocation Strategies
Besides some scheduling algorithms mentioned in Section 2.2, we will describe some other resource allocation approaches mathematically in the multicarrier OFDMA sys-tem.
3.5.1 Dynamic Power Allocation
The dynamic power allocation is commonly used in traditional wired discrete multi-tone (DMT) [27] systems, such as ADSL. We allocate power in different multi-tones with different channel condition. The goal is to maximize the system capacity. Conse-quently, this issue becomes an optimization problem under total power constraint.
We describe this problem by the following equations.
maxPn,m
5 10 15 20 25 30 0
0.1 0.3 0.5 0.6 0.7 0.8 0.9 0.95 0.98 0.99
number of subcarriers
number of subcarriers