For the conventional exhaustive-searching optimization algorithm, it has to try all possible channel allocations and runs the iterative optimization algorithm in Fig. 4.1 in each channel allocation. If there are N VOCs and d requested data types, the exhaustive-searching optimization algorithm will have dN possible channel allocations. In addition, in each channel allocation, it has to run the iterative optimization algorithm once for each data type. Thus, the conventional exhaustive-searching optimization algorithm is an O(d × dN) computational procedure.
The computational complexity of the DPRA approach is a function of the number of stages and states. For the DPRA algorithm, there are N stages and d states in each stage. Since the ith VOC will be excluded from each data type subset at the ith stage, we can use the idea discussed in the fourth complexity-reduction technique of the BBRA scheme to reduce the repetitious calculations in each state at the ith stage. For this reason, DPRA method requires to run the iterative optimization algorithm for at most d times at each stage and d × N times towards the whole algorithm, so O(d × N) is the upper bound of its computational procedure.
The complexity of the BBRA method is hard to estimate directly. It is highly dependent on the bounding value and techniques. We use computer simulations to estimate the approximate complexity.
Fig. 4.1 compares the complexities of DPRA and BBRA schemes with the computa-tional complexity evaluated by their processing times. The normalized noise power level σ2 is assumed to be 0.01 and different numbers of data types with the same normalized sum rate are compared.
For the downlink scenario, Figs. 5.10 and 5.11 indicate that the complexity of the DPRA scheme increases with the number of VOCs, as has been expected. The com-plexity, however, is much lower than the upper bound. Besides, these results reveal an interesting fact that the average complexity of the case with 64 VOCs is higher than
that of the case with 128 VOCs. This is because there are much more VOCs with high GNRs in the 128-VOC case. A better VOC can support larger data rate, so less VOCs are needed.
Figs. 5.10 and 5.11 indicate that the complexities of both DPRA and BBRA schemes increase with the number of data types. For the DPRA algorithm, the degradation with respect to the optimal performance is an increasing function of the number of data types as is shown in Fig. 5.3. Using the DPRA solution as its bounding function, the complexity of BBRA algorithm thus increases very rapidly when the number of data types increases.
With regard to the uplink scenario, the complexities of DPRA and BBRA schemes in Figs. 5.12 and 5.13 increase with the number of data types as the downlink scenario.
The major difference is that for the uplink scenario, the complexity of BBRA approach with more VOCs is higher than it with fewer VOCs. The cause of it is that the GNR of the VOC is the same to each data type. When there are more important VOCs in each data type subset, the more compare is needed to be calculated.
From the results, DPRA method has much reduced the computational complexity compared to the full-searching algorithm and guarantees that the complexity is not over the upper bound. Thus, DPRA algorithm is very suitable to be a practical algorithm for its low computational and hardware requirements. Moreover, BBRA approach also shows an acceptable computational complexity to find the optimal solution within less normalized sum rate.
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102 103 104 105 106
The normalized sum rate
Average processing times of the mono−rate iterative algorithm
BBRA scheme (d=5) BBRA scheme (d=10) BBRA scheme (d=15) DPRA scheme (d=5) DPRA scheme (d=10) DPRA scheme (d=15)
Figure 5.10: Average complexities of BBRA and DPRA schemes in a 64-VOCs downlink scenario.
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101 102 103 104 105
The normalized sum rate
Average processing times of the mono−rate iterative algorithm
BBRA scheme (d=5) BBRA scheme (d=10) BBRA scheme (d=15) DPRA scheme (d=5) DPRA scheme (d=10) DPRA scheme (d=15)
Figure 5.11: Average complexities of BBRA and DPRA schemes in a 128-VOCs downlink scenario.
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102 103 104 105
The normalized sum rate
Average processing times of the mono−rate iterative algorithm
BBRA scheme (d=3) BBRA scheme (d=4) BBRA scheme (d=5) DPRA scheme (d=3) DPRA scheme (d=4) DPRA scheme (d=5)
Figure 5.12: Average complexities of BBRA and DPRA schemes in a 10-VOCs uplink scenario.
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101 102 103 104 105 106
The normalized sum rate
Average processing times of the mono−rate iterative algorithm
BBRA scheme (d=3) BBRA scheme (d=4) BBRA scheme (d=5) DPRA scheme (d=3) DPRA scheme (d=4) DPRA scheme (d=5)
Figure 5.13: Average complexities of BBRA and DPRA schemes in a 15-VOCs uplink scenario.
Chapter 6
Simulation Results
In this chapter we revisit the two application scenarios discussed in Chapter 2 and examine the numerical performance of our algorithms when applied to solve the radio resource allocation problems arisen in these two operation scenarios. To demonstrate the usefulness of the proposed algorithms and see how they perform with realistic QoS constraints, we consider four distinct services whose rate requirements are given by
Table 6.1: Transmission rate requirements for video, audio, voice and data services Service Data rate
Video 128 kbps Audio 56 kbps Voice 9.6 kbps Data No specific
Two independent multimedia sources whose respective probabilities of generating different services are listed in the following table are assumed in our simulation.
Table 6.2: Statistical characterizations of two independent multimedia sources.
Video Audio Voice Data
Source 1 0.25 0.25 0.25 0.25
Source 2 0.125 0.125 0.5 0.25
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6.1 Resource Allocation for an OFDMA Downlink System
The first application example we consider is an OFDMA system which has N = 64 or 128 VOCs. A similar system can be found in IEEE 802.16e. The normalized required rates for video, audio and voice are calculated by dividing the required data rates of Table 6.1 by the sub-carrier frequency spacing. As for the data service, since there is no specific QoS constraint, the normalized rate for data is assumed to be uniformly distributed in [0, 5]. The system parameters used in simulation are given in Table 6.3 below. We first examine the performance of DPRA approach which gives suboptimal
Table 6.3: Simulation parameters of the OFDMA system Sub-carrier frequency spacing (W ) 10.94 kHz Number of sub-carriers (N ) 64, 128 Number of data types (d ) 3, 5, 8, 10
Noise power level 2 0.01
Normalized required rate for video 0.872 Normalized required rate for audio 5.09 Normalized required rate for voice 11.64 Normalized required rate for data 0~5
performance. Since the probabilities of generating video and audio services in Source 1 are higher than those of Source 2, the expected normalized sum rate for Source 1 should be higher than that of Source 2. Thus, in Figs. 6.1 and 6.2, the probability of achieving the optimum allocation and the performance loss for Source 1 are inferior to those of Source 2.
Figs. 5.2 and 5.3 indicate that the performance loss increases with the number of data types or the normalized sum rate. In this case, the normalized sum rate is proportional to the number of data types, so the performance loss in Fig. 6.2 degrade much fast than that in Fig. 5.3, when the number of data types increases. Note that the performance
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loss of the DPRA method is very low: even if the number of data types is as large as 10, the performance loss is still maintained to within 1%.
On the other hand, Fig. 6.3 shows that the BBRA approach requires very high computational complexity when the number of data types becomes larger than 5. By contrast, the complexity of the DPRA scheme increases very slowly; it is still reasonably affordable even when the number of data types is 10. We conclude that for the downlink OFDMA system, the DPRA algorithm provides a simple and efficient solution that offer near optimum solution (< 1%) with very little complexity.
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The number of data types
Probability of achieving the optimum
DPRA scheme (Case1,N=64) DPRA scheme (Case1,N=128) DPRA scheme (Case2,N=64) DPRA scheme (Case2,N=128)
Figure 6.1: The probability that the DPRA method yields the optimum performance in an OFDMA downlink system.
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0.05 0.1 0.15 0.2 0.25
The number of data types
Average loss from the optimum (%)
DPRA scheme (Case1,N=64) DPRA scheme (Case1,N=128) DPRA scheme (Case2,N=64) DPRA scheme (Case2,N=128)
Figure 6.2: The average performance degradation (with respect to the optimum perfor-mance) of the DPRA algorithm in an OFDMA downlink system.
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The number of data types
Average processing times of the mono−rate iterative algorithm
BBRA scheme (Case1,N=64) BBRA scheme (Case1,N=128) BBRA scheme (Case2,N=64) BBRA scheme (Case2,N=128) DPRA scheme (Case1,N=64) DPRA scheme (Case1,N=128) DPRA scheme (Case2,N=64) DPRA scheme (Case2,N=128)
Figure 6.3: The average complexity of the BBRA and DPRA schemes when used in an OFDMA downlink system.