Advances in Mobile Cloud Computing Systems

26 Advances in Mobile Cloud Computing Systems

obtain it as concave except most N nondifferentiable points by analo-

gizing as follows:

∂U

m

(y)

∂y

=(1−αx

∗

r

)

K



k=1

W

2

p

m

h

2

m

(1 −μ

k

x

r

B

rk

)

(ln2)

2

y

2

I

m

(y)(p

m

h

2

m

+ I

m

(y))λ

k

−β

K



k=1

g

2

km

σ

2

k

h

2

k

. (1.21)

Based on Equation 1.21, we know

∂

2

U

m

(y)

∂y

2

is shown as Equation 1.22.

∂

2

U

m

(y)

∂y

2

= −(1 −αx

∗

r

)

K



k=1

W

2

p

m

h

2

m

(1 −μ

k

x

r

B

rk

)

(ln2)

2

λ

k

y

3

(p

m

h

2

m

I

m

+ I

2

m

)

2

×

⎛

⎜

⎜

⎜

⎜

⎜

⎝

p

m

h

2

m

I

m

+(p

m

h

2

m

+2I

m

)

⎛

⎜

⎜

⎜

⎜

⎜

⎝

σ

2

m

−

K



k=1

g

2

km

σ

2

k

h

2

k

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎠

< 0.

(1.22)

Therefore, U

m

(y) is a concave function. And this noncooperative inter-

ference price game is a concave game. According to the formulation

Equation 1.19, this game exists at least one Nash equilibrium.

Then, from ...