Proceedings of ICCT2013

Self-Adaptive Genetic Algorithm Based MU-MIMO Scheduling Scheme

Chengcheng Yang, Jiang Han, Yi Li, Xiaodong Xu

Beijing University of Posts and Telecommunications, Beijing 100876, China yangchengcheng@bupt.edu.cn,

Abstract: Multi-user Multiple Input Multiple Output (MU-MIMO) system is known to enhance the system capacity with low network delay. One of the biggest challenges with MU-MIMO is on the scheduling scheme which simultaneously selects multiple users to maximize the sum rate. Genetic Algorithm (GA) works perfectly as an optimal or suboptimal solution with quite low complexity to handle such problems. In this paper, we promote a MU-MIMO scheduling scheme which is based on a modified GA, namely Self-adaptive GA (SaGA). SaGA can adjust the performance of population and generation dynamically during run. Furthermore, a self-adaptive elitism strategy also makes contributions to a better behave and faster convergence beyond the traditional GA strategy. Simulation results demonstrate that the performance of SaGA based scheduling is quite approximate to the exhaustive searching while with a much lower complexity.

Keywords: MU-MIMO; Scheduling scheme; Self-adaptive Genetic Algorithm; Sum rat; Complexity

1 IntroductionDue to the promising improvement in system performance, Multi-user Multiple-input Multiple-output (MU-MIMO) is widely considered as a key technology for system capacity improvement in the modern wireless network [1]. MU-MIMO allows multiple users to be scheduled simultaneous on the same time-frequency resource to exploit the spatial multiplexing gain which is more than just one scheduled user [2]. Hence, MU-MIMO achieves higher performance and spectral efficiency than Single-User MIMO (SU-MIMO). For a practical deployment point of view, the biggest challenge for MU-MIMO is how to find an efficient scheduling scheme. Dirty Paper Coding (DPC) is an efficient strategy in MU-MIMO broadcast channels [3]. However, DPC is difficult and infeasible in practical systems due to its high computational complexity with continuous encoding and decoding.

Recently, plenty of works have been done to the optimization of schedule scheme on MU-MIMO. The optimal solution is named as Exhaustive Search (ES), which searches exhaustively through all the possible selections to maximize the solution. However, the possible selections are generally infinite and unprocurable in the practical scenario. In current research, lots of heuristic scheduling process have been

proposed to solve the above questions, which are combinatorial benefited in both complex and performance to achieve the approximate optimal solution as ES [4-6]. Among the heuristic scheduling method, bionics-based algorithm draws more attention in current research due to their better performance.

Motivated by the above job, we consider Genetic Algorithm (GA) as the optimization tool, which is an efficient option of performing the utility optimization [7]. GA simulates the genetic process in biology to achieve the approximate optimal solution with low complex and fast convergence. GA has been used to implement the maximum sum rate in MU-MIMO schedule, a GA algorithm was used to implement the maximum throughput and proportionally fair scheduling criteria in a MIMO channel in the context of orthogonal transmit spatial multiplexing (i.e. Zero-forcing Beamforming (ZFB)) in [8]. [9] demonstrates a downlink scheduling via GA for single/multi-carrier MU-MIMO system with dirty paper coding (DPC).

In this paper, we expand upon the work of [9]. In the heterogeneous network, there is a great quantity of users involved in a MU-MIMO system, traditional GA, however, cant meet the scenario with fast convergence. Hence, we propose a Self-adaptive Genetic algorithm-based (SaGA) based MU-MIMO downlink scheduling scheme to maximize the sum rate with faster convergence. There are some new features in our SaGA based scheme. Firstly, to improve the convergence, we propose the self-adaptive rate of crossover and mutation for each generation. Secondly, in order to optimize the results continually, we propose a self-adaptive elitism strategy and an elite population structure. Compared with the traditional GA, ES and Greedy Search (Greedy) [10], the advantage of system improvement, complexity and convergence is demonstrated for the proposed Self Adaptive GA based scheduling scheme in downlink MU-MIMO system.

The remainder of this paper is organized as follows. Section II shows the downlink system model of MU-MIMO and formulates the optimal sum rate scheduling problem. The proposed SaGA-based scheduling algorithm is described in details in Section III. Section IV shows the simulation results of system performance and the complexity of different schemes. Finally, conclusion is given in Section V.

____________________________________978-1-4799-0077-0/13/$31.00 2013 IEEE

Proceedings of ICCT2013

2 System Model and Problem Formulation We consider a downlink MU-MIMO system, which consists of a base station with several antennas and active users in the system, while each users equipment has only one receive antenna. The channel gain between any given channel is modeled as an independent identically distributed circularly symmetric complex Gaussian process with unit variance, the channel gains are approximately constant for the entire interval since the time of the channel is much longer than the scheduling and transmission interval, the channel state information (CSI) is assumed to be well known [9].

During the scheduling process, 0 {1,2, , }S K , }, is denoted as a set of all active users, and S denotes the possible set of scheduled users. Thus, there is | |S N , which means one eNodeB can schedule at most N users simultaneously in a MU-MIMO system. Here, N denotes the maximum number of support schedule user.

Define the aggregate channel matrix of all K users as:

1 2[ , , , ]T T T T

KH h h h , ]T T]K, (1)

And the aggregate precoding matrix of all K users as:

1 2[ , , , ]KW w w w , ]K, (2) where the corresponding precoding matrix W as:

1( )T TW H H HH (3) For Zero Forcing Beamforming (ZFB), the precoding vector kw is selected to satisfy the zero-interference condition: 0j kh w for all j k and 1 ( , )j k K

Hence, the received signal of scheduled k th user can be expressed as:

k k k k k j k j j kj Sj k

y p h w x p h w x z

(4)

where kp and kx is the transmit power and transmitted data symbol of the k th user, kz is the additive white Gaussian noise with 2{ }Tk k nE z z .

As for the set S of scheduled users, the aggregate channel matrix and the corresponding precoding matrix are:

1 2( ) [ , , , ]T T T T

k SH S h h h , ]T T]k S, (5)

1

( ) ( )( ) ( ( ) ( ) )T T

W S H SH S H S H S

. (6)

According to Shannon theorem, the sum rate of k th users as (7), The scheduling and resource allocation optimization problem is equal to maximize the total sum rate of the scheduled users as R:

2

2 2 2

2 2

| |log (1 )

| |

log (1 )

k k jk

j k jj Sj k

k

p h wR

p h w

p

(7)

1max: 0

( )

max log(1 )k k k

k S

kk S

k

p p P k S

R S R

pn

(8)

where the constraint conditions are:

s.t. S1: | | ,S N (9)

S2: 1 max| |k kk S

p P

(10)

Here we define 2 * 1

,

1 1|| || [( ( ) ( ) ) ]k k k kw H s H s

and

( )k kp

, kp is the optimum power allocation for each scheduled users. The optimum power allocation can be obtained with water-filing algorithm shown in [11].

Note that the constraint S1 means that at most N users could be selected simultaneously on one scheduling time. Constraint S2 means that power of one eNodeB is fulfilled and limited. Therefore, (8) can be concluded as a combinational optimization problem.

How to select the scheduling set S is the optimal solution. For MU-MIMO system, taken user selection into account, there are Q kinds of possible combinations, which are given as:

1 1

!( )!( )!

N N

i i

K KQi i K i

(11)

In current research, ES is quite complex due to the required number of function evaluations and huge search area. Current wireless and hardware systems cannot support a complete ES efficiently. Therefore, the objective of this paper is to propose a low complexity suboptimal optimization algorithm which can achieve the approximate solution, we consider to use the SaGA strategy in MU-MIMO scheduling.

3 SaGA-based Solution

3.1 General Aspects Genetic Algorithm (GA) is the adaptive swarm searching algorithm premised on the evolutionary ideas of natural genetic and selection. Usually GA is used to solve the optimization problem to approach the approximate optimal solutions with comparatively low complexity and fast convergence compares to the other algorithms. The basic principle of GA is to simulate processes in natural system necessary for evolution. Based on the characteristic of problem space, GA defines a population of individuals that undergo the

Proceedings of ICCT2013

genetic process such as selection, crossover and mutation operations. Furthermore, we added modification and elitism strategy which can improve the performance of GA highly. A fitness function is used to evaluate individuals.

In this paper, we will give the prominence to the creative self-adaptive characteristics [12], [13] of the proposed algorithm, such as the self-adaptive rates of crossover and mutation and the self-adaptive elitism strategy. The process of SaGA will be introduced in the following parts.

3.2 Coding and Initialization In [9], binary bits are used to indicate the scheduling in MU-MIMO system with DPC. In our approach, this method will be used and extended in scheduling strategies, individuals are coded into one-dimension-binary chromosomes with K bits in representation which users are scheduled, with 1 denoting a scheduled user and 0 denoting an unscheduled user. The number of scheduled users should be less than N . Practically, the scale of population can be defined as

pN , which means the number of individuals in one population. The number of generations can be predetermined as

gN , which means the whole process repeat for a total of

gN generations.

3.3 Breeding Process An effective GA representation as well as the meaningful fitness evaluation [9] is the key to the success of GA applications. In this paper, we define the sum rate to be the fitness of each individual.

As shown in Figure 1, during each generation, the populations evolve through a breeding process consists of 4 steps, namely: selection, self-adaptive crossover, self-adaptive mutation, modification, whats more, we promote a self-adaptive elitism strategy to improve the performance.

Population

Selection

Self-adaptive crossover

Modification

Parent 1 Parent 2

Child 1 Child 2

Child 1' Child 2'

Modification

Child 1'' Child 2''

Sa-Mutation Sa-Mutation

Figure 1 Flowchart of breeding process

1) Selection

A pair of parent individuals is selected in selection operation. The selection rule is based on roulette Wheel Selection Method (RWS), defining the possibility kG of chromosomes to be selected as:

1

( )( =( )pk

k Njj

Fit Gp GFit G

(12)

Note that the selected individuals are still in the population. The higher fitness, the more possible of the chromosome will be selected repeatedly.

2) Crossover

One-point in [9] of crossover and rate cp is used to do crossover operation. The selected parents swap the bits after a random point to produce two children chromosomes which inherit partial characteristics from the parents.

3) Mutation

After the above process, the populations join in the mutation operation with the mutation rate mp (should be very small to stabilize system). Creating a random

[0,1]r for every bit in the chromosome, if mr p reverse the bit. Figure 2 gives an example of crossover and mutation.

1

0 01 0 1 1 00

0 01 0 0 11

1

1 1 000 01 0

0 11

Crossover pointParent 1 Offspring 1

1 0 000 11 0

0 01 0 1 1 101 00 0

Crossover Mutation

Parent 2 Offspring 2

Figure 2 Crossover and mutation for single-carrier

MU-MIMO Scheduling scheme with DPC

4) Self-adaptive crossover and mutation

Rate of crossover and mutation influence the behavior of GA. However, in order to achieve a better solution,

mp and cp should be self-adaptive with the progress to keep diversity and convergence. The worse individual should get a higher cp and mp to produce new individuals quickly. On the other hand, the elite individuals should have a low 2cp and 2mp to keep the best performance of population, and enter into the next generation instead of breeding. The performance of GA improve highly than the traditional.

Formulas of self-adaptive modulation of cp and

mp are:

'1 2 max

1max

2 max

( )( ) ,{

c cc avg

avgc

p p f fp f ff fp

k f f

(13)

Proceedings of ICCT2013

1 2 max1

max

1 max

( )( ) ,{

m mm avg

avgm

m

p p f fp f ff fp

p f f

(14)

Where maxf is the maximum fitness in population,

avgf is the average fitness in each generation, 'f is

the higher fitness of the selected parents, f is the fitness before mutation. In this paper, we define 1 2 10.9, 0.6, 0.1c c mp p p and

2 0.001mp .

5) Modification

After crossover and mutation, the offspring chromosomes may unexpectedly violate the constraint S1. Therefore, it is necessary to make some modifications to keep the sum 1 in each individual less than N . In the wrong individual, 1s should be replaced randomly to be 0s until the sum is N . 6) Self-adaptive Elitism Strategy

Elitism strategy can keep the sustainable growth with generation [14]. Self-adaptive Elitism Strategy is put forward to keep the best individuals as elite in each generation with self-adaptive 2cp and 2mp . The elites enter into next generation instead of joining in breeding process, the elitism structure is as Figure 3 shows. The elites are chosen from the following update rules:

If the best individual of current population is better than elite, elitism will be replaced by the best individual;

Otherwise, the elite remain unchanged.

110100100100100101011100

...101010010001001001001010

Npindividual

Elites

Figure 3 Elite-population structure

4 Simulation Results and Complexity Analysis

4.1 Simulation Results In this section, we demonstrate the performance of the proposed SaGA in implementing the maximum sum rate and resource allocation of scheduling scheme which uses DPC and ZF for single-cell and single-carrier system. In our simulation, we consider a system with

8K active users and 4N transmit antennas, the channel between eNodeB and users includes pathloss, shadow fading and Rayleigh fading, more details are based on 3GPP LTE criterions, the parameters of

channel model are based on 3GPP SCM criterions in Table I 2max ( ) /SNR P PL D is denoted the signal noise ratio (SNR). The performances of SaGA are compared with the other scheduling strategies, such as traditional GA, ES and Greedy.

Table I System Parameters

System parameters Cellular layout Antenna configuration Cell radius Spectrum allocation Channel number Pathloss(dB) Shadowing standard deviation Shadowing correlation distance

Hexagonal grid, 1 cell site 1 Tx/eNodeB, 1 Rx/user

500mD 1MHz bandwidth at 2 GHz

1 10( ) 128.1 37.6log ( )PL d d km

8dB 50m

Rayleigh channel -174 dBm/Hz

GA parameters Size of population Number of generation Crossover rate Mutation rate

50

100

0.60.005

p

g

c

m

NNpp

Figure 4 describes the performance of the maximum sum rate versus the K of active users at SNR=10dB and SNR=20dB. It is shown that the performance of SaGA achieve approximate 94%-98% of the sum rate compare with ES, while save much more calculation complex. Greedy is easy to fall into the local optimal solution with the larger K . Specially, it conspicuously reveals the superiority of SaGA with the increasing number K is better than the traditional GA, since SaGA-based scheduling illustrates the advantageous effect of multiuser diversity based on its self-adaptive performance. As for SNR value, it is obvious that high SNR makes higher sum rate for all scheduling schemes.

8 10 12 14 165

10

15

20

25

30

35

40

45

50

Number of users

Sum

rat

e [b

it/s/

Hz]

ESSaGAGAGreedy

SNR=10dB

SNR=20dB

1.24%

Figure 4 Sum rate versus number of users at SNR=10dB and 20dB.

Figure 5 describes the convergence versus the number of active users K at SNR=10dB for SaGA and GA. Defining a uniformization rate of convergence (URC) to describe the convergence performance for system with different K [15]. There are three comparisons for SaGA and GA. First, the URC of SaGA is bigger than that of GA with same K . Second, the gap of URC

Proceedings of ICCT2013

between SaGA and GA diminish gradually with K increasing. Third, URC diminishes with K increasing in a non-line relationship. Hence, SaGA demonstrates its advance of convergence with self-adaptation.

8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of active users

Uni

form

izat

ion

rate

of c

onve

rgen

ce

SaGAGA

28.9% promotion

16.5% promotion

Figure 5 Uniformization rate of convergence versus number of users

for SaGA and GA.

Figure 6 describes the sum rate versus generations. The generation increases, both SaGA and GA will achieve the convergence at a certain point of generation. The promotional efficiency demonstrates what the curve shows.

10 20 30 40 50 60 70 80 90 1005

10

15

20

25

30

35

40

45

50

Generation

Sum

Rat

e bi

ts/s

/Hz

SaGA (K=8)SaGA (K=16)GA (K=8)GA (K=16)

Figure 6 sum rate versus generation of SaGA and GA

4.2 Complexity Analysis Usually complexity is mainly considered by the required number of floating-point calculations (FLOPs) [16]. For example, m n complex-valued matrix m nA C , multiplying an m n matrix by n p matrix requires mnp complex additions and mnp complex multiplications, for a total of 8mnp flops. In this paper, ignoring the details in calculation of breeding process, the complexity of SaGA and comparing algorithms mainly lie in the fitness calculations [17]. The complexity for once fitness evaluation involving with matrix inverse and multiplication can be given as

2( )O N . SaGA calculates p gN N times. Thus, the

complexity of SaGA is 2( )p gO N N N , where SaGA is equal to GA. As for Greedy, the calculation step is based on the beyond step, no iteration is needed, thus,

2( )O N for Greedy. Since ES has the same calculation method as SaGA, the complexity of ES is

2 2( ( )) ( )N

N

i

KO N O N K

i2( 2 N(( 2 . In conclusion, the

complexity of SaGA is much lower than ES but bigger than Greedy. The comprehensive performance of SaGA is approximately best on performance with an acceptable complexity.

5 Conclusions In this paper, we propose a SaGA-base scheduling scheme and resource allocation for MU-MIMO system with implementation scenarios, where multiple users are scheduled and served simultaneously by the multiple transmit antennas under zero-forcing beamforming and water-filling power allocation. The object is to achieve the maximum sun rate of system. In allusion to the huge number of active users in the current system, there are some innovations for the proposed algorithm, including the self-adaptive rates of crossover and mutation; a self-adaptive elitism strategy based on population structure with a certain rate of crossover and mutation. Matlab simulation results demonstrate that SaGA-based scheme achieving a better optimal solution then the pervious GA and approaching to ES with lower complexity. In conclusion, a SaGA with higher performance and lower complexity costs is realizable in current system. In the future, scheduling scheme will be considered furthermore on QoS, energy efficiency, mixture with heterogeneous network and so on.

Acknowledgements This work was supported by NSFC Project (No. 61001116, 61027003), International Scientific and Technological Cooperation Program (No.2010DFA11060).

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