Swarm Intelligence

6 Introduction

Figure 1.2: Cauchy distribution

series, and its leading-order approximation for the flight length results in a power-law

distribution defined by the following form:

L(s) |s|

−1−β

(1.10)

which is heavy tailed with infinite variance for 0 < β < 2. A simple version of the Levy

distribution can be defined as

L(s, γ, µ) =

(

q

γ

2π

exp[−

γ

2(s−µ)

]

1

(s−µ)

1

.5

, if 0 < µ < s < ∞

0, otherwise

(1.11)

where µ > 0 is minimum step and γ is the scale parameter and as s → ∞ we have the

following form:

L(s, γ, µ) ≈

r

γ

2π

1

s

1

.5

(1.12)

1.3 Pseudo-random number generation

Simulations requiring random numbers are critical in fields including communications, fi-

nancial modeling, and ...