The JPRA Algorithm

The JPRA algorithm is mainly composed of the link proportional power allocation (LPPA) scheme in the first phase and the evolutionary computing rate assignment (ECRA) method in the second phase.

3.3.1 The LPPA Scheme

The link proportional power allocation (LPPA) scheme estimates the required transmission power for soft handoff user h, p^∗_h(r); then it distributes p^∗_h(r) to all serving base stations in D_h under the constraint of maximum link power to each user by base station b ∈ D_h, ep_b. And p_b,h(r) is proportional to the link quality between the serving base station b and the soft handoff user h [8]. If the required transmission power of one link reaches to the constraint of maximum link power, LPPA will compensate the required power through other links. The LPPA scheme is an iterative method to distribute p^∗_h(r) to all serving base stations so that the required signal quality can be satisfied. The design tries to accomplish power balance between cells in the CDMA cellular system with mixed-size cells. Besides, it is noteworthy that due to the constraint of the maximum link power, there exists a forced termination situation for the soft handoff because the soft handoff user cannot obtain required signal quality even though all active links are allocated with maximum link power. If the soft handoff is forced to terminate, p_b,h(r) of each link i in the active set D_h are reset to zero.

The LPPA scheme and the proof of the convergence has detailed in the Chapter 2. The required transmission power of each active link, pb,h(r), b ∈ Dh, for all soft handoff users with all kinds of service rates can be obtained through the LPPA scheme.

3.3.2 The ECRA Method

The ECRA method performs the rate assignment for multirate soft handoff users. It formulates the rate assignment issue as a constrained optimization problem with an objec-tive to maximize the total throughput of multirate soft handoffs such that the total power allocated to soft handoffs in cell b would be constrained by a maximum value, denoted by Be_b, and eB_b < P_b^T. Note that the total power budget for each base station is limited. When there are a larger number of multirate soft handoff users being managed, the computation time would become a major concern for system operators. In this chapter an evolutionary computing algorithm [40], which is a promising intelligent technique to effectively search a global optimal solution, is adopted. Assume there is N_d soft handoff users with data ser-vices. If there are N_S kinds of data service rates, the searching complexity is (N_S+ 1)^Nd by using exhaustive method, in which 1 means zero service rate for suspending transmission.

For example, if N_d is 10 and N_S is 4, there are nearly 10⁷ searching complexity. This is far beyond the reasonable computation time for the system’s requirement. In order to reduce the complexity of exhaustive search, the evolutionary computing technique [40] is applied.

The evolutionary computing technique can represent the service rate of each user as a chromosome in a population, in which each population is the possible solution with a collection of chromosomes for all handoff data users. For N_S kinds of data service rates, each rate r is encoded into blog₂(N_S+ 1)c binary digits, denoted by x, and the decoder function for x is denoted by s(x). Thus, for soft handoff data user h with service rate r, its corresponding required total transmission power is p^∗_h(s(xh)), in which the transmission power from active link h is p_b,h(s(x_h)) by the LPPA scheme.

Assume there are N_v number of the voice users in each cell. Also, r_v is the service rate of soft handoff voice users, and p_b,h(r_v) is the corresponding allocation power from active link b to soft handoff user h. The ECRA method is to find an optimal rate assignment vector (decision vector) of N_d soft handoffs, x^∗ = [x^∗₁, x^∗₂, · · · , x^∗_Nd], for maximizing the objective function O(x), which is defined to be the total throughput of soft handoff data users, given

by

where N_b is the number of base stations in the system. And

γ_h(s(x_h)) ≥ γ^∗(s(x_h)) , ∀ h. (3.11)

Because of these constraints, some decision vectors may be out of the feasible domain. A violation function, which is proportional to the square of violation, is used to rank violated constraints of the decision vector [40]. The values of the constraint violation function indicate how far the solutions deviate from the feasible region. This constrained violation function is defined as

where Z_b, and H_h are the Heaveside operators [40], i.e. Z_b(·) = 1 whenever the constraint in (3.10) is violated, and Z_b(·) = 0 otherwise. The evolutionary computing method is a more advanced genetic algorithm, which uses stochastic searches through simulating natural genetic processes of living organisms, including selection, mutation, and crossover, to solve difficult optimization problem in real-world. Based on the formulation of constrained opti-mization problem, the optimal decision vector, x^∗, can be found by maximizing the objective function, O(x). The ECRA method is described in the following. Noticeably, the allocation power for the soft handoffs are corresponding to the ones obtained by the LPPA scheme.

[The ECRA method]

Step 1: [Initialize]

• Set crossover rate p_c, mutation rate p_u, and maximum number of generations T .

• Initialize generation t = 1, optimal objective value O^∗ = 0, and optimal decision vector x^∗ to be a zero pattern.

• Generate K_P populations that are randomly selected decision vectors x_k = [x^k₁, · · · , x^k_Nd], 1 ≤ k ≤ K_P.

Step 2: [Constraint tournament selection]

• Choose K_P tournament pairs randomly among all populations.

• Calculate the violation function X(x) in (3.12) for each competitive pair, and determine one winner, which owns a smaller value of the violation function.

• Replace each population x_k with the winner population of each competitive pair, thus form K_P new populations.

Step 3: [Variable point crossover]

• Choose K_P/2 crossover pairs from adjacent population x_k and x_k+1, where k is odd.

• Generate a random number c in [ 0, 1 ] for each chromosome in each crossover pair.

• For the chromosome with c < p_c, generate the crossover point randomly in [1, blog₂(N_S+ 1)c], and make the crossover operation within this crossover chromosome.

Step 4: [Uniform mutation]

• Generate a random number u in [ 0, 1 ] for every bit in each population, and mutate bits whenever u < p_u.

Step 5: [Calculate the objective function of resulting new population]

• Calculate the violation function value for each population.

• Find feasible population { xf } with zero violation among KP populations.

• IF { xf } is not empty set, THEN Calculate objective function value { O(xf) }.

− IF max { O(x_f) } > O^∗, THEN

Set O^∗ = max{O(x_f)} and optimal decision vector x^∗ = arg_xf max{ O(x_f) }.

ELSE Goto Step 6.

ELSE Goto Step 6.

RM

Rµ Start

Figure 3.3: The mixed-size cellular model (ρ = 1/2) with an example of mobility trajectory.

Step 6: [Check the stop criterion]

• IF t < T , THEN Set t = t + 1, and Goto Step 2.

ELSE DONE. ¥