Table 1 Table of parameters

${\lambda }_{\text{min}}^i$

Minimum required incoming traffic rate of the i th

application

${\lambda }_{\text{max}}^i$

Maximum required incoming traffic rate of the i th

application

λ _i

Incoming traffic rate of the i th application

S _i

Effective processor speed (instructions per second)

dedicated to the i th application in a mobile device

C _i

Instructions of i th application per time slot

$R_{\text{min}}^i$

Minimum required transmission rate of the i th application

$R_{\text{max}}^i$

Maximum required transmission rate of the i th application

R _i

Transmission rate of the i th application

γ _i

Instructions have to be processed in cloud servers i th

application

${\chi }_i^{\text{dep}}$

Utilization factor of the i th application dependent on

incoming traffic rate

${\chi }_i^{\text{indep}}$

Utilization factor of the i th application independent from

incoming traffic rate

$D_i^{\mathit{\text{th}}}$

Delay threshold of i th application traffic

D _i

Delay of i th application traffic

η _i

Mobile data offloading to clouds instructions of i th

application

T D _i

Uploading data of i th application

$R_k^{\text{max}}$

Maximum achievable transmission rate using k th

interface

H _k

Channel quality indicator of the k th interface

${\mathit{\text{Dw}}}_k^{\mathit{\text{th}}}$

Achievable guaranteed downlink delay of k th interface

D w _k

Downlink delay of k th interface

${\alpha }_k^w$

Cost coefficient of k th wireless download rate

(per instruction)

θ _k

Downlink QoS exponent of k th interface

μ _k

Download rate of k th interface

P_maint(k)

Connection power consumption of k th interface

$P_{\text{Error}}^{\mathit{\text{Th}}}$

Error rate threshold

${\beta }_j^n$

Delay between the wireless cloud and j th cloud server at

the n th time slot

$\widehat{{\beta }_j^n}$

Estimation of the ${\beta }_j^n$

$S_j^{{}^{\prime }}$

Effective processor speed of j th cloud server

${\alpha }_j^{\mathit{\text{DL}}}$

Cost coefficient of downlink traffic of j th cloud server

${\alpha }_j^{\mathit{\text{UL}}}$

Cost coefficient of uplink traffic of j th cloud server

${\alpha }_j^{\text{comp}}$

Cost coefficient cloud computation of j th cloud

P _comp

Process power consumption per processing speed unit

E ⁿ

n th moment

ε(n)

Mobile energy level at the n th time slot

B g(n)

Mobile budget fee at the n th time slot