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Figure 6.8. Illustration of crossover operation in genetic algorithm.
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in the intermediate population. The fractional part of G*,i/ G indicates the a probability that an additional copy of that chromosome is placed in the intermediate population. For example, a chromosome with G*,i/ G = 1.36 a places one copy in the intermediate generation and receives a 0.36 chance of placing a second copy. The selection process continues until all the Np slots in the intermediate generation have been lled up. In this way, we see that chromosomes that are tter will be allowed more chance to propagate into the next population. Step 3: Breeding. The breeding process consists of two stages: the crossover and the mutation. Randomly select a pair of chromosomes in the intermediate population and recombine the two parents into two offspring according to crossover and mutation rules. The crossover operation is characterized by a crossover probability pc. For every selected pairs of parents, there is pc probability of performing crossover operation. By crossover, a randomly selected crossover point (between 1 and K) is selected for the pair of chromosomes. The two parents are split respectively in the crossover point selected, and the two offspring are obtained by crossing the fragments of the two parents as illustrated in Figure 6.8. For every bit in the chromosomes of the offspring, there is a pm (mutation rate) chance of toggling the bit. This is called the mutation operation. Throughout the remaining of this subsection, we shall dynamically adjust the mutation rate on the basis of the spread of the tness in the current population. Speci cally, the mutation rate of current generation is given by pm = 1 b1 + b 2s G G (6.34)
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where sG is the standard derivation, G is the mean of the tness of the current population (before selection), and b1 and b2 are two constants.5 These two processes introduce randomness into the intermediate generation so that the new population will be a combination of the best chromosome in the current population as well as some new random elements.
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Here, we set b1 = 1.2 and b2 = 10 for the maximal throughput and proportional fair utility functions.
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Step 4: Termination. Replace the original population with the new population and repeat Steps 2 and 3 until the number of iterations reaches Ng. When forming a new population, it is ensured that the tness chromosome in the current population will be saved and inserted into the next population, and all members of the next population are checked against the constraint Skak nT. If any chromosome violates this constraint, 0 is inserted into a randomly selected bit position in the violating chromosome until the constraint is satis ed. The computation complexity of the genetic algorithm is bounded by Ng Np function evaluations. As will be illustrated in the next section, this represents enormous computational saving compared with the optimal algorithm.
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In this section, we shall discuss the performance of the capacity-optimized cross-layer scheduling algorithms for multiuser systems with multiple antennas at the base station. We rst consider the system performance (network capacity) based on two commonly used system utilities: the total throughput and the proportional fairness. For each of the two system utilities, we compare the performance of the optimal cross-layer scheduling with respect to the greedy scheduling as well as the genetic-based scheduling. To highlight the contribution of multiuser selection diversity, we also compare the performance with respect to the round-robin scheduler, where the K users share the channel evenly in time irrespective of their channel matrices at every fading block. In addition, we shall discuss the effect of nT on the overall network capacity. Next, we shall discuss the performance of the coverage-optimized crosslayer scheduling. Similarly, we compare the coverage performance of the optimal algorithm versus the conventional round-robin, greedy scheduling as well as genetic-based scheduling. Finally, we shall discuss the tradeoff between the capacity and coverage of multiuser systems. Note that in the simulations, each data point consists of 5000 realizations of channel fading. Channel fading of the K users are generated in accordance with an independent complex Gaussian distribution with unit variance. We assume 0 dB antenna gain in the transmit and receive antennas and 2 GHz carrier frequency. The data rate is expressed in terms of bits per second per hertz, and network coverage is expressed in terms of cell radius (m). 6.6.1 System Capacity Performance
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Figure 6.9a d illustrates the scheduling performance of the maximal throughput scheduler with respect to nT and SNR. Mobile users are assumed to be homogeneous in terms of path loss and transmit power constraint. We observe
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