Open Access

A Novel Method for Improving Fairness over Multiaccess Channels

EURASIP Journal on Wireless Communications and Networking20102010:395763

DOI: 10.1155/2010/395763

Received: 7 June 2010

Accepted: 29 November 2010

Published: 8 December 2010

Abstract

It is known that the orthogonal multiple access (OMA) guarantees for homogeneous networks, where all users have almost the same received power, a higher degree of fairness (in rate) than that provided by successive interference cancellation (SIC). The situation changes in heterogeneous networks, where the received powers are very disparate, and SIC becomes superior to OMA. In this paper, we propose to partition the network into (almost) homogeneous subnetworks such that the users within each subnetwork employ OMA, and SIC is utilized across subnetworks. The newly proposed scheme is equivalent to partition the users into ordered groups. The main contribution is a practical algorithm for finding the ordered partition that maximizes the minimum rate. We also give a geometrical interpretation for the rate-vector yield by our algorithm. Experimental results show that the proposed strategy leads to a good tradeoff between fairness and the asymptotic multiuser efficiency.

1. Introduction and Preliminaries

Rate allocation in multiuser communication systems is an important task which should consider simultaneously the fairness and the spectral efficiency. This paper is focused on fairness of multiple-access (MA) schemes working under maximum spectral efficiency evaluated in terms of sum rate. The state of art is the method recently introduced in [1]. However, the main drawback of this algorithm is a significant decrease of the asymptotic multiuser efficiency (AME) [24]. We propose a new strategy that combines the strengths of two different MA schemes such that to guarantee a good tradeoff between fairness and AME.

1.1. System Model

Consider a single-antenna Gaussian MA channel with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq1_HTML.gif users transmitting to the base station (BS). The system model can be written as [1, Example  1]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ1_HTML.gif
(1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq2_HTML.gif is the received signal, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq3_HTML.gif models the fading channel from the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq4_HTML.gif th user to the BS, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq5_HTML.gif is the symbol transmitted by the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq6_HTML.gif th user. The additive noise https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq7_HTML.gif is assumed to be white circular Gaussian with variance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq8_HTML.gif for each real and imaginary component. Under the hypothesis that the transmitting powers of the users are constrained such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq10_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ2_HTML.gif
(2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ3_HTML.gif
(3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq11_HTML.gif is the rate of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq12_HTML.gif th user and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq13_HTML.gif . The interested reader can find in [5, Chapter 6] a comprehensive discussion on the significance of (2) and (3). The following two methods can be applied to achieve equality in (2):

  1. (1)

    OMA: orthogonal multiple-access with degrees of freedom (DOF) allocated proportional to users' received powers;

     
  2. (2)

    SIC: successive interference cancellation.

     

We refer to [5, Chapter 6] for more details on OMA, SIC, and the definition of DOF.

It is also pointed out in [5] that, whenever the received power is almost the same for all users, that is, the network is homogeneous, OMA guarantees a higher degree of fairness (in rate) than that provided by SIC. The situation changes in heterogeneous networks, where the received powers are very disparate: if the decoding is performed in the decreasing order of the received powers, then SIC becomes superior to OMA. However, the SIC systems have drawbacks which do not exist for OMA. Because the signals received from the users are estimated and subtracted from the composite signal one after the other, the inaccurate estimation for the current user makes the next users decoded unreliably. This deficiency becomes more severe when the number of users increases. In fact, it is known that SIC works well only when a specific disparity of the powers is enforced (see, e.g., [6, 7] and Chapter 5 in [8]).

To measure the fairness and the performance, we employ two criteria that have been used frequently in the past. For instance, it is customary to evaluate the fairness with the following max-min criterion: a rate vector is called max-min fair (MMF) if and only if an increase in the rate of one user results in the decrease in the rate of one or more users who have smaller or equal rates [1, 9]. Additionally, we consider the AME. Note that AME quantifies the loss of performance when the interferer users are present and the background noise vanishes [24]. More precisely, AME is a measure of degradation in bit error rate because of the presence of multiple-access interference in a white Gaussian channel.

1.2. Basics of the New Method

Our approach exploits the beneficial aspects of both OMA and SIC. Because we do not aim to improve fairness by sacrificing the throughput, we assume that (2) is satisfied with equality.

The key idea is to partition the network into (almost) homogeneous subnetworks such that the users within each subnetwork employ OMA, and SIC is utilized across subnetworks. Since OMA is applied to (almost) homogeneous subnetworks, it is likely that the degree of fairness is not deteriorated. The application of SIC to subnetworks and not directly to users allows to decrease the number of decoding stages, which potentially improves the performance.

Given that the number of users is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq14_HTML.gif , we assume that the number of subnetworks is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq15_HTML.gif . The newly proposed scheme is equivalent to partition the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq16_HTML.gif users into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq17_HTML.gif ordered groups. Note that the order matters because it corresponds to the order in which the groups are decoded. Remark for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq18_HTML.gif that the grouping method is the same with OMA. Moreover, the grouping method is identical with SIC for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq19_HTML.gif . Similarly to conventional SIC, the max-min rate achieved in this case depends on the order in which the groups are decoded. We consider the family of all ordered partitions of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq20_HTML.gif users into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq21_HTML.gif nonempty groups. Then we pick up the ordered partition for which the minimum rate is maximized, and we name it https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq22_HTML.gif (basic ordered grouping of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq23_HTML.gif users into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq24_HTML.gif groups). Conventionally, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq25_HTML.gif coincides with OMA, and we write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq26_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq27_HTML.gif .

Furthermore, one can select again from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq28_HTML.gif the ordered partition which maximizes the minimum rate. The new selection is dubbed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq29_HTML.gif . Remark that the rate vector which corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq30_HTML.gif is not necessarily the same with the max-min fair rate vector that was defined in Section 1.1. However, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq31_HTML.gif is guaranteed to be max-min fair among all possible user groupings for which the sum capacity is achieved.

We investigate how the fairness can be evaluated for OMA, SIC, and BORG. In this context we demonstrate for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq32_HTML.gif a fundamental property, which allows us to introduce a low-complexity search method for choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq33_HTML.gif from all ordered partitions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq34_HTML.gif users into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq35_HTML.gif groups.

We give also a geometrical interpretation for the rate-vector yield by our algorithm. More exactly, we point out the connections between the outcome of the proposed method and the polymatroid structure of the capacity region as it is used in multiuser information theory [1, 10, 11].

During recent years, several works have exploited the polymatroid structure for optimizing the fairness in multiaccess systems [1, 12]. The main idea is based on the fact that particular points within the sum capacity facet of the polymatroid can be achieved by successive decoding and time sharing. Then, the effort is focused on finding the time-sharing coefficients which give the fairest point. For the sake of comparison, we pick up the method from [1], which is based on time sharing, and we name it TS. It is clear that TS cannot be inferior to our method if the criterion is the fairness in the multiaccess system. But since TS is a linear combination of successive decoders with different decoding orders, it suffers from the same deficiencies like the ones mentioned earlier for SIC.

The rest of the paper is organized as follows. Section 2 contains the main contribution, where we show how a low-complexity search algorithm can be devised to find the ordered partition which maximizes the minimum rate. The geometrical interpretation of the result is included. In Section 3, the newly proposed method is compared with OMA, SIC, and TS in a simulation study which comprises four different network models. In all cases, the new strategy provides the best tradeoff between fairness and AME.

2. Fairness

2.1. Formulas for OMA and SIC

It is well known for OMA method that the degree of fairness among users is lowered when their received powers are very disparate [5]. This drawback can be easily understood from the formula which gives the rate of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq36_HTML.gif th user [5, Chapter 6]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ4_HTML.gif
(4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq37_HTML.gif . From the identity above, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq38_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq39_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq40_HTML.gif . Hence, the rates are as disparate as the received powers are, which leads to unfair rates in heterogeneous networks. For example, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq41_HTML.gif , then the minimum rate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq42_HTML.gif tends also to zero.

When SIC is applied, the fairest rate vector is obtained by decoding the users in the decreasing order of their received powers [1]. Consequently, the rate of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq43_HTML.gif th user has the expression [5]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ5_HTML.gif
(5)

With the convention that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq44_HTML.gif denotes the number of users whose received power is smaller than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq45_HTML.gif , the following inequality is readily obtained: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq46_HTML.gif . It shows that, as long as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq47_HTML.gif is not much smaller than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq48_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq49_HTML.gif does not tend to zero when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq50_HTML.gif . Hence, it is likely that SIC has a higher degree of fairness than OMA when the received powers are very disparate.

2.2. BORG and Its Low-Complexity Implementation

Consider the following scenario: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq51_HTML.gif users are divided into nonempty groups https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq52_HTML.gif . For an arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq53_HTML.gif , we use https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq54_HTML.gif to denote the sum of the received powers for the users that belong to the group https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq55_HTML.gif . It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq57_HTML.gif .

To be in line with the previous literature, we adopt the convention that the order of the groups in the successive decoding is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq58_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq59_HTML.gif is a permutation of the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq60_HTML.gif . For simplicity, we denote by ORG the ordered partition which is given by the sequence of subsets https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq61_HTML.gif . By combining the results from (4) and (5), we get the rate of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq62_HTML.gif th user
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ6_HTML.gif
(6)

For writing the equation above more compactly, we have assumed that the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq63_HTML.gif th user belongs to the group https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq64_HTML.gif .

The naive approach for finding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq65_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq66_HTML.gif is to search among all ordered partitions of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq67_HTML.gif users into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq68_HTML.gif groups, then to compute the minimum rate in each case, and eventually to pick up the partition which maximizes the minimum rate. This leads to a huge computational burden and makes the method unpractical. We show below how the number of ordered partitions to be considered can be reduced significantly.

We need some more definitions. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq69_HTML.gif be a vector of strictly positive integers whose sum is equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq70_HTML.gif . The ordered partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq71_HTML.gif is of type https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq72_HTML.gif if for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq73_HTML.gif the cardinality of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq74_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq75_HTML.gif . Given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq77_HTML.gif , we denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq78_HTML.gif the family of all ordered partitions of type https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq79_HTML.gif . It is important to remark that for all partitions within this family we have that (i) the order of the subsets is the same and is given by the reverse order of the permutation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq80_HTML.gif ; (ii) the cardinality of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq81_HTML.gif th subset is the same, namely, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq82_HTML.gif .

Additionally, for two arbitrary subsets https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq84_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq85_HTML.gif if the received powers of all users within https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq86_HTML.gif are greater than those of the users within https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq87_HTML.gif . When the condition is not satisfied, we write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq88_HTML.gif .

Theorem 1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq89_HTML.gif . For fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq91_HTML.gif , consider all ordered partitions that belong to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq92_HTML.gif . In this class, the ordered partition that maximizes the minimum rate for a given set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq93_HTML.gif is the one which satisfies the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ7_HTML.gif
(7)

The proof is deferred to the appendix.

Now we are prepared to formalize the result which shows the decrease in computational complexity.

Corollary 2.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq94_HTML.gif , we have the following.

  1. (i)
    To select https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq95_HTML.gif by brute-force search amounts to compute the minimum rate for
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ8_HTML.gif
    (8)

    different ordered partitions.

     
  2. (ii)
    Theorem 1 allows to reduce to
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ9_HTML.gif
    (9)

    the number of ordered partitions that are considered in the evaluation process.

     
Proof.
  1. (i)

    In the case of brute-force search, it is easy to note that the rate vector must be computed for all ordered partitions of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq96_HTML.gif users into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq97_HTML.gif nonempty subsets. Hence, the number of partitions to be considered equals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq98_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq99_HTML.gif is the Stirling number of the second kind, and its closed-form expression is given by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq100_HTML.gif [13]. This proves the result in (8).

     
  2. (ii)

    From Theorem 1, we know that for all permutations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq101_HTML.gif there exists a single ordered partition of type https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq102_HTML.gif that must be considered, namely, the one which satisfies (7). For finding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq103_HTML.gif , we must evaluate a single rate vector for each vector type. This implies that the number of partitions which are investigated equals the number of ways that the integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq104_HTML.gif can be written as a sum of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq105_HTML.gif strictly positive integers. According to [13], this number is given by (9).

     

To gain more insight, let us suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq106_HTML.gif users and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq107_HTML.gif groups. Corollary 2 points out that the number of competing partitions for the selection of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq108_HTML.gif can be reduced from 55980 to 36, which implies a significant decrease of the computational complexity. However, by using the result from (9), it is easy to verify that the number of rate vectors which must be evaluated for selecting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq109_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq110_HTML.gif .

2.3. Geometrical Interpretation

We resort to the polymatroid structure of the capacity region [1, 10, 11], to give a new interpretation of BORG. The object of interest is the polyhedron defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ10_HTML.gif
(10)
where the set function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq111_HTML.gif is a mapping for all subsets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq112_HTML.gif to the positive real numbers. For the problem that we study, it is convenient to choose
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ11_HTML.gif
(11)
Then, it is a simple exercise to verify that the set function defined in (11) satisfies (i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq113_HTML.gif (normalized); (ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq114_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq115_HTML.gif (increasing); (iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq116_HTML.gif (submodular). Thus, according to the definition from [14], https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq117_HTML.gif is a polymatroid. Moreover, the hyperplane given by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq118_HTML.gif is the sum-capacity facet of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq119_HTML.gif [1]. We analyze next the points within the sum-capacity facet that correspond to OMA, SIC, and ORG. To get the point which corresponds to OMA, we rewrite the identity in (4) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ12_HTML.gif
(12)
For SIC, we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq120_HTML.gif to be an arbitrary permutation of the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq121_HTML.gif , and we assume that the users are decoded in the reverse order of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq122_HTML.gif . Therefore, the rate of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq123_HTML.gif th user is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ13_HTML.gif
(13)

The formula in (5) is easily obtained from (13) for the particular case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq124_HTML.gif is chosen such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq125_HTML.gif . More importantly, (13) shows that, for each permutation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq127_HTML.gif is a corner point of the polymatroid https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq128_HTML.gif (see [10, 14] for more details).

With the convention that the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq129_HTML.gif th user belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq130_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq131_HTML.gif is a permutation of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq132_HTML.gif , the expression in (6) is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ14_HTML.gif
(14)

Observe that, in general, the rate vector given by (14) does not correspond to a corner of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq133_HTML.gif .

To enhance intuition, we depict in Figure 1(a) the polymatroid https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq134_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq135_HTML.gif when the two users have equal received powers ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq136_HTML.gif ). Similarly, in Figure 1(b), it is shown https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq137_HTML.gif for the case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq138_HTML.gif . In both cases, the sum-capacity facet is the segment whose endpoints are the corners https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq139_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq140_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Fig1_HTML.jpg
Figure 1

The polymatroid https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq141_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq142_HTML.gif users. The points marked on the sum-capacity facet correspond to various MA methods. Here, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq143_HTML.gif means that user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq144_HTML.gif is decoded before user https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq145_HTML.gif . Two different cases are considered: (a) homogeneous network ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq146_HTML.gif ), (b) heterogeneous network ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq147_HTML.gif ).

Because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq148_HTML.gif , the number of groups for the BORG method can be either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq149_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq150_HTML.gif . Thus, we are interested in the points within the sum-capacity facet that correspond to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq151_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq152_HTML.gif , respectively. As we already know, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq153_HTML.gif is the same with OMA, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq154_HTML.gif coincides with SIC, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq155_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq156_HTML.gif . Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq157_HTML.gif is chosen by selecting between OMA and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq158_HTML.gif the one which maximizes the minimum rate. For completeness, we consider also the point TS that corresponds to the degree of fairness provided by the method from [1], which finds optimum weights for the time sharing between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq160_HTML.gif .

Note in Figure 1(a) that the OMA point is the fairest on the sum-capacity facet. In this case, it is obvious that also https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq161_HTML.gif corresponds to the fairest point. Moreover, the method from [1] gives the same weight to both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq162_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq163_HTML.gif , which makes the TS point coincide with the OMA point. The situation changes in Figure 1(b), where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq164_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq165_HTML.gif . This leads to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq166_HTML.gif . Remark also in Figure 1(b) that, even if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq167_HTML.gif is the best among OMA and SIC, the minimum of its rate vector is slightly smaller than the minimum rate for TS.

Next, we demonstrate by simulations the capabilities of various MA schemes.

3. Simulation Results

3.1. Evaluation Criteria

As it was already mentioned, the fairest rate vector is obtained by applying TS or, equivalently, by time sharing between the corner points of the sum-capacity facet. To find the fairest rate vector and also the optimal time-sharing coefficients, we have implemented in Matlab the algorithms III and IV from [1].

Let us assume that the number of runs for a specified set of experimental conditions is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq168_HTML.gif . An arbitrary method, say MET, is compared with TS by computing the Normalized Min-Rate with formula
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ15_HTML.gif
(15)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq169_HTML.gif is the minimum of the rate-vector yield by MET in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq170_HTML.gif th run. Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq171_HTML.gif is the minimum of the TS rate vector in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq172_HTML.gif th run.

The second figure of merit that we consider for evaluating the MA schemes is the AME, which is generally denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq173_HTML.gif . AME takes values in the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq174_HTML.gif and attains its maximum when OMA is utilized. Therefore, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq175_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq176_HTML.gif (see Chapter 5 in [3]).

To keep SIC inline with what we have in the corner points of the sum-capacity facet, we assume that all cancellations are perfect, and what is forwarded to the next decoder has no residual error from the already decoded users [15]. Furthermore, suppose that in each step a matched filter is used as decoder such that for computing the AME of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq177_HTML.gif th user we can apply the formula (3.123) from [3]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ16_HTML.gif
(16)

Note that the expression above takes into consideration the system model from (1). Additionally, it is assumed that the users are decoded in the decreasing order of the received powers. We emphasize that we do not use formula (7.31) from [3] because it was derived for SIC with residual errors propagated from previous steps.

It is clear that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq178_HTML.gif , we do not need to compute AME for all ordered partitions of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq179_HTML.gif users into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq180_HTML.gif subgroups but only for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq181_HTML.gif . With slight abuse of notation, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq182_HTML.gif is the ordered partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq183_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq184_HTML.gif .

More importantly, the BORG method combines the features of both OMA and SIC such that (i) the users within each group are orthogonal one to each other; (ii) the groups share the entire channel. It is evident that only the second characteristic determines the degradation of the AME. Hence, the expression of AME can be derived straightforwardly from (16) by taking into account that, when decoding the group https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq185_HTML.gif , the role of interferer is played by the groups https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq186_HTML.gif . If the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq187_HTML.gif th user belongs to the group https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq188_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ17_HTML.gif
(17)

Remark in the expression above that AME is the same for all the users within the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq189_HTML.gif -group.

It is worth mentioning here that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq190_HTML.gif does not necessarily coincide with the grouping that maximizes the AME. For example, if the received powers are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq191_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq192_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq195_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq196_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq197_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq198_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq199_HTML.gif , then the optimum AME is produced by the ordered partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq200_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq201_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq202_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq203_HTML.gif . The inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq204_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq205_HTML.gif , which shows clearly that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq206_HTML.gif cannot be the ordered partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq207_HTML.gif .

We conclude the short discussion on the second figure of merit, by noticing that, whenever an experiment is repeated https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq208_HTML.gif times, we calculate for each method MET the Average AME
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Equ18_HTML.gif
(18)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq209_HTML.gif is the AME for the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq210_HTML.gif th user in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq211_HTML.gif th run.

In the examples outlined below, the Normalized Min-Rate and the Average AME are employed to compare the performance of the following MA schemes: TS, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq212_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq214_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq215_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq216_HTML.gif . In our settings, the number of users is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq217_HTML.gif , and the number of runs for each set of experimental conditions is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq218_HTML.gif . Additionally, the power of the Gaussian noise is taken to be one ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq219_HTML.gif ). Four different network models are considered.

3.2. Examples

Model I

To quantify the degree of network heterogeneity, we consider the ratio between the power of the strongest user and the power of the weakest user: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq220_HTML.gif . The larger is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq221_HTML.gif , the more heterogeneous is the network. For a fixed value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq222_HTML.gif , we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq223_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq224_HTML.gif . The powers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq225_HTML.gif are chosen to be outcomes from a uniform distribution on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq226_HTML.gif , and the experiment is repeated https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq227_HTML.gif times. This selection guarantees that the mean power https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq228_HTML.gif is equal to 100. When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq229_HTML.gif , a single realization is considered, namely, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq230_HTML.gif .

We plot in Figure 2(a) the Normalized Min-Rate obtained for various MA schemes when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq231_HTML.gif increases from 0 dB to 30 dB. Due to the definition in (15), the graph for TS is a straight line parallel to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq232_HTML.gif -axis. Note in the same figure that the degree of fairness is very high for OMA when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq233_HTML.gif is close to 0 dB, but it decreases rapidly when the heterogeneity of the network increases. By contrast, SIC has a very low degree of fairness in homogeneous networks, but it improves with the increase of the network heterogeneity such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq234_HTML.gif  dB, SIC is clearly superior to OMA.

The beneficial effects of the newly proposed strategy can be observed for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq235_HTML.gif , which performs as well as OMA for small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq236_HTML.gif , but surpasses both OMA and SIC for large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq237_HTML.gif . More interestingly, good results are obtained not only when searching for the optimum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq238_HTML.gif , but also when the number of groups is kept fixed. Remark for the heterogeneous networks that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq239_HTML.gif performs very similarly with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq240_HTML.gif .

In Figure 2(b), we show the Average AME for the six methods which are compared. As it was already pointed out previously, OMA achieves always the maximum possible AME. We can notice from Figure 2(b) that SIC and TS yield the poorest Average AME. This drawback appears for the two schemes because the firstly decoded users receive high interference, which makes their AME close to zero. It is remarkable that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq241_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq242_HTML.gif have AME superior to that of SIC. Moreover, the performance of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq243_HTML.gif approaches the Average AME of OMA when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq244_HTML.gif increases.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Fig2_HTML.jpg
Figure 2

Experimental results for Model I: (a) Normalized Min-Rate versus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq245_HTML.gif ; (b) Average AME versus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq246_HTML.gif . The number of users is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq247_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq248_HTML.gif is expressed in dB, and for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq249_HTML.gif the reported results are obtained from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq250_HTML.gif runs. The following MA methods are compared (for each method we indicate the color and the marker symbol used in plots): TS (black left-pointing triangle), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq251_HTML.gif (blue asterisk), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq252_HTML.gif (magenta right-pointing triangle), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq253_HTML.gif (green diamond), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq254_HTML.gif (red square), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq255_HTML.gif (brown circle).

Model II

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq256_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq257_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq258_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq259_HTML.gif . We simulate a network such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq260_HTML.gif are uniformly distributed on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq261_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq262_HTML.gif are uniformly distributed on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq263_HTML.gif . The parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq264_HTML.gif controls the degree of heterogeneity of the network. When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq265_HTML.gif , all the users belong to a single cluster, and the increase of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq266_HTML.gif makes the network to consist of two disjoint clusters.

The results plotted in Figure 3 are obtained by varying the value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq267_HTML.gif from 0 to 180. For each value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq268_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq269_HTML.gif different realizations of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq270_HTML.gif are generated. In terms of fairness and AME, the performance of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq271_HTML.gif is the same with that of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq272_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq273_HTML.gif . Remark in Figure 3(a) that the graph of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq274_HTML.gif coincides with that of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq275_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq276_HTML.gif is larger than 100. A similar fact can be also observed in Figure 3(b). So, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq277_HTML.gif automatically adapts to the topology of the network.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Fig3_HTML.jpg
Figure 3

Experimental results for Model II: (a) Normalized Min-Rate versus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq278_HTML.gif ; (b) Average AME versus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq279_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq280_HTML.gif is not expressed in dB. The number of users ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq281_HTML.gif ), the number of runs ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq282_HTML.gif ), and all graphical conventions are the same like in Figure 2.

Model III

We consider again a network including two clusters. This time, the received powers of the users are generated as suggested in [16]. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq283_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq284_HTML.gif . We take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq285_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq286_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq287_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq288_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq289_HTML.gif . The distribution of the random variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq290_HTML.gif is Chi-Square with two degrees of freedom.

Remark that the heterogeneity of the network is measured by the difference https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq291_HTML.gif , which we increase from 0 to 90. The Normalized Min-Rate and the Average AME calculated for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq292_HTML.gif based on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq293_HTML.gif runs are shown in Figure 4. It is easy to observe the following outcome of the experiment. Because the Chi-Square distribution has infinite support, OMA does not provide fairness in rate allocation when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq294_HTML.gif . Due to the same reason, for all values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq295_HTML.gif , the degree of fairness yield by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq296_HTML.gif is inferior to that of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq297_HTML.gif even if, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq298_HTML.gif , the number of groups equals the "true" number of clusters. However, when comparing the Average AME, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq299_HTML.gif is ranked the second after OMA which achieves the maximum possible value.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Fig4_HTML.jpg
Figure 4

Experimental results for Model III: (a) Normalized Min-Rate versus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq300_HTML.gif ; (b) Average AME versus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq301_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq302_HTML.gif is not expressed in dB. The number of users ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq303_HTML.gif ), the number of runs ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq304_HTML.gif ), and all graphical conventions are the same like in Figure 2.

Model IV

Following the suggestion of one of the reviewers, we briefly investigate the case of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq305_HTML.gif users uniformly distributed over a two-dimensional area. For the sake of concreteness, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq306_HTML.gif be the distances from the BS to the users. According to the large-scale model, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq307_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq308_HTML.gif is the received power from the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq309_HTML.gif th user, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq310_HTML.gif is the received power from a transmitter located at distance one from the BS, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq311_HTML.gif is the distance from the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq312_HTML.gif th user to the BS, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq313_HTML.gif is the path loss exponent [17, 18].

It is widely accepted that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq314_HTML.gif for urban area cellular radio [18, Table  3.2]. In our settings, we choose the path loss exponent to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq315_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq316_HTML.gif . For two arbitrary bounds https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq317_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq318_HTML.gif with property https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq319_HTML.gif , the squared distances https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq320_HTML.gif are selected to be uniformly distributed on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq321_HTML.gif . Hence, the mean power has the expression https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq322_HTML.gif . Let us consider various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq323_HTML.gif between 1.2 and 3.0, and for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq324_HTML.gif we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq325_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq326_HTML.gif . Conventionally we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq327_HTML.gif . It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq328_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq329_HTML.gif , or equivalently all the users are located on a circle whose center coincides with the BS. Obviously, this corresponds to the case of a homogeneous network. In fact, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq330_HTML.gif , the quantity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq331_HTML.gif can be used to measure the heterogeneity of the network: the bigger is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq332_HTML.gif , the larger is the difference https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq333_HTML.gif , which makes the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq334_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq335_HTML.gif , more disparate.

In Figure 5, we plot the Normalized Min-Rate and the Average AME. They are computed for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq336_HTML.gif based on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq337_HTML.gif runs, while for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq338_HTML.gif one single realization is considered. By comparing the results within Figure 5 with those from Figure 2, we can observe that the multiaccess schemes have a similar behavior for Model IV and Model I.

As a final remark, we note that for all four network models, finding the partition which corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq339_HTML.gif is faster than applying the TS optimization strategy. In all runs, the execution time for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq340_HTML.gif was at about https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq341_HTML.gif of the execution time for TS. When the number of users is very large, the computational complexity can be decreased by searching for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq342_HTML.gif with a fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq343_HTML.gif , instead of finding the partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq344_HTML.gif . We observe from the numerical examples that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq345_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq346_HTML.gif have an acceptable level of performance.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_Fig5_HTML.jpg
Figure 5

Experimental results for Model IV: (a) Normalized Min-Rate versus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq347_HTML.gif ; (b) Average AME versus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq348_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq349_HTML.gif is not expressed in dB. The number of users ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq350_HTML.gif ), the number of runs ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq351_HTML.gif ), and all graphical conventions are the same like in Figure 2.

4. Conclusion

In this paper, we investigated how OMA and SIC can be combined to improve fairness in Gaussian wireless networks. The newly proposed method divides the network into (almost) homogeneous subnetworks such that the users within each subnetwork employ OMA, and SIC is utilized across subnetworks. Equivalently, the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq352_HTML.gif users are partitioned into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq353_HTML.gif ordered groups. The main theoretical result which we proved for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F395763/MediaObjects/13638_2010_Article_1893_IEq354_HTML.gif shows that the ordered partition which maximizes the minimum rate can be found with a low-complexity algorithm. Moreover, it was demonstrated experimentally that the user grouping strategy guarantees a good tradeoff between fairness and the asymptotic multiuser efficiency.

Declarations

Acknowledgment

This work was supported by the Academy of Finland, Project nos. 113572, 118355, 134767, and 213462.

Authors’ Affiliations

(1)
Department of Signal Processing, Tampere University of Technology

References

  1. Maddah-Ali MA, Mobasher A, Khandani AK: Fairness in multiuser systems with polymatroid capacity region. IEEE Transactions on Information Theory 2009, 55(5):2128-2138.MathSciNetView ArticleGoogle Scholar
  2. Verdu S: Optimum multiuser asymptotic efficiency. IEEE Transactions on Signal Processing 1986, 34: 890-897.MathSciNetMATHGoogle Scholar
  3. Verdu S: Multiuser Detection. Cambridge University Press, Cambridge, UK; 1998.MATHGoogle Scholar
  4. Yang B, Danilo-Lemoine F: Asymptotic multiuser efficiency of a decorrelator based successive interference cancellation DS-CDMA multiuser receiver. Proceedings of Military Communications Conference (MILCOM '06), 2006-7.Google Scholar
  5. Tse D, Viswanath P: Fundamentals of Wireless Communications. Cambridge University Press, Cambridge, UK; 2005.View ArticleMATHGoogle Scholar
  6. Viterbi AJ: Very low rate convolutional codes for maximum theoretical performance of spread-spectrum multiple-access channels. IEEE Journal on Selected Areas in Communications 1990, 8(4):641-649. 10.1109/49.54460View ArticleGoogle Scholar
  7. Warrier D, Madhow U: On the capacity of cellular CDMA with successive decoding and controlled power disparities. Proceedings of the 48th IEEE Vehicular Technology Conference (VTC '98), May 1998, Ottawa, Canada 3: 1873-1877.Google Scholar
  8. Buehrer RM: Code Division Multiple Access(CDMA). Synthesis Lectures on Communications 2006, 2: 1-192.View ArticleGoogle Scholar
  9. Bertsekas DP, Gallager RG: Data Networks. Prentice-Hall, Upper Saddle River, NJ, USA; 1987.MATHGoogle Scholar
  10. Tse DNC, Hanly SV: Multiaccess fading channels-part I: polymatroid structure, optimal resource allocation and throughput capacities. IEEE Transactions on Information Theory 1998, 44(7):2796-2815. 10.1109/18.737513MathSciNetView ArticleMATHGoogle Scholar
  11. Zhang X, Chen J, Wicker SB, Berger T: Successive coding in multiuser information theory. IEEE Transactions on Information Theory 2007, 53(6):2246-2254.MathSciNetView ArticleMATHGoogle Scholar
  12. Shum KW, Sung CW: Fair rate allocation in some Gaussian multiaccess channels. Proceedings of IEEE International Symposium on Information Theory (ISIT '06), July 2006, Seattle, Wash, USA 163-167.Google Scholar
  13. Flajolet P, Sedgewick R: Analytic Combinatorics. Cambridge University Press, Cambridge, UK; 2009.View ArticleMATHGoogle Scholar
  14. Edmonds J: Submodular functions, matroids, and certain polyhedra. Proceedings of Calgary International Conference on Combinatorial Structures and Applications, 1970, Calgary, Canada 69-87.Google Scholar
  15. Cover T, Thomas J: Elements of Information Theory. John Wiley & sons, New York, NY, USA; 2006.MATHGoogle Scholar
  16. Jagannathan K, Borst S, Whiting P, Modiano E: Scheduling of multi-antenna broadcast systems with heterogeneous users. IEEE Journal on Selected Areas in Communications 2007, 25(7):1424-1434.View ArticleGoogle Scholar
  17. Pahlavan K, Krishnamurthy P: Principles of Wireless Networks: A Unified Approach. Prentice-Hall, Upper Saddle River, NJ, USA; 2002.Google Scholar
  18. Rappaport TS: Wireless Communications: Principles and Practice. Prentice-Hall, Upper Saddle River, NJ, USA; 2002.MATHGoogle Scholar
  19. Cloud M, Drachman B: Inequalities: With Applications to Engineering. Springer, Berlin, Germany; 1998.MATHGoogle Scholar

Copyright

© S. A. Razavi and C. D. Giurcăneanu. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.