Consider a fog network consisting of a pool of E cloudlets, some are active and some not, and a set u of N users that are distributed uniformly over the network area. Moreover, user-nodes (UNs)are interested in a set of A tasks, each task has required CPU cycles of k per bit of task data. In the proposed model we focus on the uplink transmission . UNs have computing tasks that arrive following Poisson distribution, hence the task data size will be assumed to follow an exponential distribution with mean $\lambda$. The task that the users have to perform is assumed to $M=k\lambda$ cycles of Cpu cycles. The Cpu capacity of each user device is $c_u$. Additionaly, each cloudlet has $c_b$ of Cpu capacity to serve serve user's offloading requests. …show more content…
\subsection{computing model} the computation of each tash $A$ of a user node can be either performed locally or offloaded to a cloudlet. the total local computing time equal \begin{equation} T_l=\frac{M}{c_u} \end{equation} each task A requested by user u and offloaded to any active cloudlet experience a total computing delay that consist of the task transmission delay $t_r$, and cloudlet computing time as follows \begin{equation} T_c=t_e+t_g+t_r \end{equation} where, $t_r=\frac{L_a}{R_u(t)}$. Moreover, $R_u(t)=\beta\log(1+\frac{P_u}{N_o})$ is the user transmission data rate, and $L_a$ is the length of the user transmission packet. assume a constant transmission rate so $t_r$ is constant, also $t_e$ and $t_g$ are independent, depending on, the following proposition characterizes $T_c$ statistically. \textbf{preposition 1. } the expected value of mean ($\mu$) and variance ($\sigma$) are given by \begin{equation} \begin{align*} \mu&=E[t_e+t_g+t_r]\\ &=E[t_e]+E[t_g]+[t_r]\\ &=\frac{1}{\lambda}+\frac{2(a+b\nu)}{\nu}+t_r \end{align*} \end{equation} and \begin{equation} \begin{align*}