Advanced Wireless Networks - 4G Technologies

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and

where δi* is the minimum required SINR for user i, ri is the minimum rate limit for i. The problem presented in Equations (12.68)–(12.70) can be reduced to a system of linear equations. If the constraints Equations (12.69)–(12.70) cannot be achieved, then the problem is called infeasible. In this case either some user should be dropped from this link or some of the constraints should be relaxed. At the optimal solution, all QoS constraints are met with equality. Also, the optimal power vector is the one that achieves all rate constraints with equality. So, the optimum rate vector is R* = [r1, . . . , rQ ]T. The corresponding power vector can be obtained by solving the QoS equation. This is a system of linear equations in power. From Equation (12.69) we have

In order to obtain a nonnegative solution of Equation (12.77), the following condition should hold:

ρ(rH) < 1

where ρ(A) is the spectral radius of matrix A.

12.5.2 Maximum throughput power control (MTPC)

This algorithm has been suggested in Chawla and Qiu [85]. The algorithm is based on the maximization of the total throughput in a cellular system. There is no need to generate all solutions in this method. Since the gain links and the interference of other users are

needed to calculate the transmitted power of each user, the MTPC algorithm is a centralized algorithm. The throughput of user i can be approximated when M-QAM modulation is used by

Ti = + log2( i )

where Ti is the throughput of user i, is a constant, and i is the CIR of user i, and it is given in general by

where Q = number of mobile stations, M = number of base stations and Gk j = channel gain between mobile station j and base-station k. The total throughput T is given by

where Q is the number of users.

Now the problem can be defined as follows: given the link gains Gi j of the users, what is the power vector P = [P1, P2, . . . , PQ ] which maximizes the total throughput? Since the first term in Equation (12.77) is constant and the logarithmic function is an increasing function, then maximizing the multiplicative term

Q

i

i=1

will lead to maximizing the total throughput T . The problem considered in Chawla and Qiu [85] is

where = { P| Pmin ≤ Pi ≤ Pmax, i = 1, . . . , Q} Q . The MTPC algorithm to solve Equation (12.77a) is given by

where Gi j is the channel gain between user j and base station i. Without loss of generality user i is assumed to be assigned to base station i. Starting from any initial vector P(0) , the iteration specified by Equation(12.78) converges to a unique point P* , which achieves the global maximum [85].

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12.5.3 Statistically distributed multirate power control (SDMPC)

A distributed solution of the optimization problem given by Equations (12.68)–(12.70) is proposed for one cell case in Morikawa et al. [80]. It is assumed that every user has two states, active ON or passive OFF. The transition probabilities of the ith user from idle to active state at any packet slot is υi , and from active to idle state is ζi . The durations of the active and idle periods are geometrically distributed with a mean of 1/ζi and 1/υi (in packet slots), respectively. The optimization problem Equations (12.68)–(12.70) is slightly modified as

Find

One parameter has been added to original optimization problem which is the indicator function β j (t). The indicator function is equal to one if the jth user is currently active,

and zero otherwise. It is assumed in Morikawa et al. [80] that the random process ˆ (t) has

β

Markovian property since geometric distribution is memoryless over the duration of traffic. The centralized solution (if the system is feasible) is given by

The main idea behind the SDMPC algorithm is to estimate the other users’ information part.

Q

Therefore the term ( j=1 β j (t)γ j ) is estimated. The Markovian property of the random process β j (t) has been exploited to obtain a good estimate of the other users’ information part. The SDMPC algorithm is given by

The estimated parameter ˆ (t) has been derived in Morikawa et al. [80] for two cases: (i)

β

there is no ‘collision’ at t; and (ii) a ‘collision’ occurs at t. There are at least three drawbacks of this algorithm. First of all in the cellular CDMA system there is a control channel always

So far, the RRM problem was formulated as a single objective (SO) optimization problem considering the other parameters as constraints. Solving the objectives (1)–(4) at the same time as using MO optimization technique, leads to a more general solution than the conventional methods. In this section we discuss an MO optimization method to solve the RRM problem. In subsequent subsections we will discuss some radio resource scheduler algorithms based on MO optimization. The field is very wide and many different algorithms and methods can be derived based on MO optimization. One formulation of the RRS optimization problem can be defined as:

where O P is the outage probability. The outage probability is defined as the probability that a user cannot achieve at least the minimum required QoS. We can see that the O P reflects the fairness situation in the communication system. The minus sign associated with the sum of the rate function in Equation (12.90) refers to the maximization process of the total utility functions.

Defining the objectives and the constraints is the first step. Selecting the proper MO optimization method to solve the problem is the second step. Then the (weakly) Pareto optimal set of solutions is generated, where every solution is optimal in a different sense. Finally, the decision maker selects the optimum solution from the optimal set which best achieves the required specifications. In this section we discuss a framework to use the MO optimization techniques in RRM.

12.5.7 Multiobjective distributed power and rate control (MODPRC)

The algorithm is based on minimizing a multi-objective definition of an error function. In this algorithm we defined three objectives: (1) minimize the transmitted power; (2) achieve at least the minimum CIR, which is defined at the minimum data rate; and (3) achieve the maximum CIR, which is defined at maximum data rate. An optimized solution can be obtained using an MO optimization.

The derivations of the algorithms are based on a VSL-CDMA communication system. After the dispreading process at the receiver, the SINR is

where δi (t) is the SINR of user i at t, Rs is the fixed chip rate (= 3.84 Mb/s for UMTS), Ri (t) is the data rate for user i at t, and i (t) is the CIR of user i at t. In wireless and digital communication, it is well known that the BER is a decreasing function in the SINR. In case of coherent binary PSK, the BER can be approximated by (when the interference is assumed Gaussian)

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For example, if the BER should not be more than 10−4 then the target SINR is obtained from Equation (12.93) as δT ≥ 8.3 dB. In the case of fixed data rate power control there is one target CIR that corresponds to the target SINR, because we have only one spreading factor value. In a case of multirate services there are different target CIR values corresponding to the target SINR. From Equation (12.92) it is clear that, in case of constant target SINR, maximizing CIR leads to maximizing data rate as follows:

Trying to achieve the maximum CIR for all users will end up in high outage probability. If there is a reasonable dropping algorithm then only one or few users will be supported [84]. To reduce the outage probability, we will define the target CIR at the minimum transmitted rate as

Also we will define the maximum CIR which is defined at the maximum transmitted rate as

The target SINR, the minimum/maximum CIR, and the minimum/maximum data rate are time-dependent, but we dropped the time symbol (t) for simplicity. In UMTS specifications the power is updated on slot-by-slot basis. The data rate is updated on a frame-by-frame basis. To generalize the analysis, we use the same time symbol for power and rate.

To increase the fairness, the users should achieve at least the minimum target CIR, which corresponds to the minimum transmitted rate (e.g. 15 kb/s in UMTS). The multirate power control problem is defined as: given the target SINR vector δ = [δ1T, δ2T, . . . , δTQ ] , the minimum requested data rate vector Rmin = [R1,min, R2,min, . . . , RQ,min] , and without loss of generality, assuming the maximum allowed data rate Rmax to be the same for all users, find the optimum power vector P = [P1, P2, . . . , PQ ] and the optimum rate vector R = [R1, R2, . . . , RQ ] that minimize the following cost function

N is the optimization time window, γ is a real-valued constant adaptation factor. The notation ( ) is used for transposed. The error ei (t) has been defined according to the weighted metrics method

advantages of joining the weighting metrics method with the least square formula of

is the multiobjective function, 1 = [1, 1, . . . , 1] , and the tradeoff factors are real numbers, 0 ≤ λ1 ≤ 1, and λ2 = 1 − λ1.

Necessary conditions for solving the problem Equation (12.109) are

where O(P) = [∂ O/∂ P1, ∂ O/∂ P2, · · · , ∂ O/∂ PQ ] is the gradient of O. Substituting the CIR expression into Equation (12.109) we obtain

J =i

To maximize the reward functions, Equation (12.111), we find the power vector P which satisfies Equation (12.110). Since the obtained equations are nonlinear, it will be very complicated to get an analytical solution. An iterative solution for k = 1, . . . , Q will be formulated (we will drop the iteration argument t for simplicity)

∂ O = ∂ Pk