Relative Efficiency Performance

I Performance of BBRA and DPRA algorithms

As only the GNRs ai affect the performance we assume, without loss of generality, that σij = σ, ∀ i, j. We normalize the bandwidth of each sub-carrier (channel) such that W = 1 and set the normalized noise power lever σ² to be 1. We also normalize the Rayleigh-distributed channel gains ||hij||² such that E[|hij|²] = 1 and assume that channels are independently faded. These two normalization assumptions effectively imply E[aij] = 0 dB. Since the channel capacity or the normalized rate is a function of the product pijaij, the simulation results shown in Figs. 3–6 are scalable in the sense that a higher (lower) E[aij] needs a proportionally lower (higher) minimum required power. The normalization of the channel bandwidth W has a similar purpose in interpreting the normalized data

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rates Ri which now have the unit of bits/sec/Hz. We also have the normalized sum rate asPd

j=1Rj. Various normalized rate distributions with the same sum rate are examined.

Let JDP and JBB be the total required transmit power determined by the DPRA and BBRA algorithms, respectively, and define the relative efficiency (RE) of the former algorithm by

η = 1 −E[JDP] − E[JBB]

E[JBB] (3.11)

Since the power-rate allocation is solely determined by OMPA once the subcarrier as-signment is fixed, we say two algorithms give the same solution if both suggest the same subcarrier allocation. Fig. 3.2 plots the probability that the DPRA algorithm converges to the optimal solution for several cases (N = 64, 128 d = 5, 10, 15). Since the BBRA algorithm is guaranteed to give the optimal solution, this probability is equal to Pr[JDP = JBB]. It is found that when d  N (say d/N < 0.1) the probability that the DPRA algorithm yields the optimal solution is greater than 0.9 if the sum rate is less than 8 bits/sec/Hz. Although for other cases under investigation, this probability is smaller than 0.9, Fig. 3.3, which plots the RE of the DPRA algorithm, indicates that the corresponding solutions still lie very close to the optimal one. It is clear that the DPRA algorithm is capable of offering a near-optimal solution that even in the worst case (64 channels, 15 users and a normalized required sum rate of 20) it achieves a RE as high as 99.82%.

In general, the larger the number of the users d or the sum rate is, the less efficient the DPRA algorithm becomes. Such a behavior is consistent with the fact that, when d increases but N is fixed, a correct channel selection at each level become less likely so is the probability of obtaining the optimal channel allocation. On the other hand, the assumption of independent fading of channels implies that the probability of having

“good” channels increases as N increases, and the probability of correct or good decision at each level increases as well. Hence, for a fixed d, the probability of obtaining the optimal or near optimal solution is an increasing function of N and so is the RE (η).

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Step 1: (Initialization) Set t = 1 and let the SCS

and the assigned channel set (ACS) for user j be C_t^s(j) = {1, 2, · · · , N } and C_t^a(j) = ∅, respectively.

Step 2: (Find the most demanded channel) Compute the sum rate R^s_i ^def= Pd

j=1rij for each channel, where r_ij is obtained by applying the OMPA algorithm for each SCS C_t^s(j), compute

Ψ(i) = { j | 1 ≤ j ≤ d, rij 6= 0}, C_A =Sd

j=1C_t^a(j), Ch = { i | 1 ≤ i ≤ N, |Ψ(i)| ≥ 2}

If |Ch| 6= ∅ and ` = maxi∈ChR<sup>s</sup><sub>i</sub>, then the `th channel is re-indexed as channel t (i.e., µ(`) = t)

and assign this channel to user k if k = maxjr`j. Go to Step 4 if |Ch| = ∅.

Step 3: (Updating) The ACS for user k and the SCSs are updated by C_t^a(k) ← C_t^a(k) ∪ {`},

C_t^s(j) ← C_t^s(j) \ {`}, ∀ j 6= k, respectively.

The tree level index is updated by t ← t + 1.

If t < N , go to Step 2;

otherwise, the sorting process is completed.

Step 4: (Sorting the less demanded channels) If C_t^s(j)T C_t^a(j) = ∅, ∀ j, go to Step 5;

otherwise, for all j, C_t^s(j)T C_t^a(j) 6= ∅, modify the corresponding SCS by C_t^s(j) ← C_t^s(j) \ {jm}, where j_m = arg max_i∈Ct^s_(j)T C_t^a(j)r_ij,

and go to Step 2.

Step 5: (Sorting the remaining channels) The order (numbering) of the channels in the set {i|1 ≤ i ≤ N, i 6∈ CA} is determined by the maximum GNR criterion used in the DPRA algorithm.

Table 3.4: The channel-sorting algorithm

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I Performance of Representative Sub-optimal Algorithms

Although many RA schemes have been proposed, they assume different scenarios and costs. Those dealing with RA problems similar to (4) often follow a three-step procedure [5],[6]. (S1) Resource allocation–determine the resource (number of channels) to be given to each user based on some criterion. (S2) Subcarrier assignment–decide which subcarrier should serve which user. (S3) Local optimization–each user computes the optimal power allocation according to its channel set and rate requirement. The average GNR criterion, i.e., the BABS (bandwidth assignment based on GNR) algorithm [5], is perhaps the simplest and most popular choice for use in (S1). Such an approach treats the channel of concern as a flat-faded wideband channel when determining the number of subcarriers an user is entitled to possess. It simplifies the resource allocation procedure by ignoring the selectivity of a wideband channel but is very likely to exclude the optimal solution from further consideration. Many methods were proposed to obtain the water-filling solution for (S3) with various degrees of precision. The main difference lies in (S2).

The first approach called the amplitude craving greedy (ACG) algorithm [5] sequen-tially assigns the subcarriers to the user with the largest GNR unless the channel number quota determined in (S1) has been exceeded. An alternate method called rate craving greedy (RCG) algorithm [5] finds the water-filling rate level for all users, assuming they have been given all subcarriers. The subcarriers are then assigned to the one with the highest achievable rate unless its channel number quota is exceeded. The original ACG and RCG algorithms can not be used to solve (4) as they are designed for discrete-rate constraints. Moreover, they use an approximate instead of exact water-filling solution.

For the purpose of fair comparison, we modify both algorithms by using the rate-power equation (2) and the OMPA solution. The resulting algorithms are henceforth referred to as the modified ACG (MACG) and RCG (MRCG) algorithms, respectively.

We assume GNR =20 dB for all subcarriers and Rj = 5 bits/sec/Hz for all j. Besides

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the independent-fading channel model, for N = 128 we also consider the ITU Vehicular A model [11] which has been adopted by UMTS and WiMax forum as one of the reference channel models. The RE performance shown in Fig. 3.4 indicates that our DPRA algorithm does outperform both MACG and MRCG algorithms. It yields a near-optimal solution that even in the worst case (N = 64, d = 14), achieves a 99.92% RE while the the two modified algorithms give 91.74% and 92.92% efficiencies. Due to the fixed rate requirement, a larger N results in better performance for all suboptimal approaches. The increase of d, on the other hand, leads to reduced efficiency but DPRA is much more robust in the sense of maintaining almost constance RE for different d, N and channel conditions. The optimal allocation probabilities for these two suboptimal algorithms are not presented for they are simply too small.

3.4.2 Complexity Evaluation

The computing complexity of various RA methods is dominated the number of calls to the single-user (mono-rate) water-filling algorithm whether it is the OMPA or any other algorithm that computes the power or rate associated with each channel assigned to an user. An exhaustive search requires O(N d^N) single-tone power- or rate-level computing operations [4] or O(d · d^N) calls of OMPA.

I Average complexities of the proposed algorithms

For the DPRA algorithm, at most d²× N calls of OMPA is needed. But the complexity of BBRA method is difficult to analyze directly for it depends on the channel order and the initial upper bound value. By using computer simulation, we estimate the average complexities, measured in terms of number of calls of the OMPA algorithm, of the DPRA and BBRA algorithms and present the results in Figs. 3.5-3.6. We assume that the normalized GNR is 0 dB and examine the required complexity for different numbers of users with the same normalized sum rate. A few observations on the last two figures can be made. First, because of Guidelines 4 and 5, the complexity of DPRA

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algorithm is limited to at most dN + 2d OMPA calls. Second, although the complexities of both DPRA and BBRA algorithms increase with the number of users d, the latter is much more sensitive to this parameter. Finally, the average complexity of the BBRA algorithm is higher when there are 64 channels than if there are 128 channels. The reason for this interesting fact is that there are more good channels when N = 128 and, for a fixed sum rate requirement, as good channels tend to support higher data rates, fewer channels are needed and early terminations due to Guideline 2 and Guideline 4 occur more often.

I Complexities of Other Sub-optimal Algorithms

Since the complexity of water-filling is a function of the channel number involved, a more precise and fair comparison is counting the number of rate(power)-evaluation iterations, i.e. the [MR2]-[MR4] loop of the OMPA algorithm; see Appendix A. For BBRA or DPRA algorithms, the iteration number in every call is upper-bounded by log₂N as bisection search is in place. The complexity of the DPRA algorithms can be reduced by using Guidelines 4, 5 and the iteration number is thus upper-bounded by (dN + 2d) log₂N .

Fig. 3.7 shows the average complexity performance of our algorithms and the MACG, MRCG algorithms for N = 64 and 128. The system and channel parameter values used here are the same as those used in Fig. 3.4. As expected, the performance in correlated fading is worse than that in independent fading and the computation complexity of all algorithms are far less than the DRRA upper-bound, (dN + 2d) log₂N . Moreover, BBRA requires the highest average complexity, followed by DPRA, MRCG and the MACG algorithms. The complexity of the DPRA algorithm is about twice that of the RCG based algorithm but is far less than that of the BBRA algorithm. The DPRA algorithm, as mentioned before, yields near-optimal performance and is robust against the variations of the numbers of users and subcarriers.

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°

Figure 3.1: A complete search tree for multiuser channel allocation. For the DPRA algorithm, only one child node survives at each level.