A Comparative Study of Modified BBO Variants and Other Metaheuristics for Optimal Power Allocation in Wireless Sensor Networks - Advances in Heuristic Signal Processing and Applications - page 90

Fig. 5.2 Linear migration
model—the relationship of
fitness of islands (number of
species), emigration rate μ
and immigration rate λ

We now consider the probability P S that the island contains exactly S species.
The number of species will change from time t to time t + Δt as follows:

− λ S Δt − μ S Δt) + P S − 1 λ S − 1 Δt + P S + 1 μ S + 1 Δt, (5.31)
which states that the number of species on the island in one time step is based on the
total number of current species on the island, the new immigrants, and the number
of species which leave during the time period. We assume here that Δt is small
enough so that the probability of more than one immigration or emigration can be
ignored. In order to have S species at time t

P S (t + Δt) = P S (t)( 1

+ Δt , one of the following conditions

must hold:

• There were S species at time t , and no immigration or emigration occurred be-

tween t and t + Δt ;

• One species immigrated onto an island already occupied by S − 1 species at

time t .

• One species emigrated from an island occupied by S + 1 species at time t .

The limit of ( 5.31 )as Δt → 0 is given by Eq. ( 5.32 ):

⎧
⎨

− (λ S + μ S )P S + μ S + 1 P S + 1 if S = 0,

P S =

− (λ S + μ S )P S + λ S − 1 P S − 1 + μ S + 1 P S + 1 if 1 ≤ S ≤ S max − 1, (5.32)

⎩

− (λ S + μ S )P S + λ S − 1 P S − 1 if S = S max .

Equation ( 5.32 ) can be arranged into a single matrix form:

⎡
⎣

⎤
⎦

− (λ 0 + μ 0 ) μ 1 0 ... 0

⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦

. . .

P 0
P . . .
P n

P P . . .
P n

λ 0 − (λ 1 + μ 1 ) 2 ...

. . . . . .

. . . . . . . . .

.

. . .

. . .

λ n − 2 − (λ n − 1 + μ n − 1 ) μ n

0 ... 0 λ n − 1 − (λ n + μ n )

(5.33)