Minimizing Delay and Power Consumption at the Edge

1.1. The Main Results Presented in This Paper

1.2. Related Work

2.1. Summary of Notation and Symbols and Abbreviations

• $t\ge 0$ is the real-valued time variable.
• DP is the task dispatching platform that transfers tasks from the end users to the servers.
• $S_i$ denotes a server that receives tasks assigned by the DP, as well as “locally generated tasks”, e.g., from its local owner or user or as part of its operating system.
• $\Lambda >0$ is the rate of arrival of external tasks to the DP.
• ${\lambda }_i$ is the rate of arrival of locally generated tasks to $S_i$.
• ${\mu }_i>0$ is the average service rate for tasks at the server $S_i$. Thus, the average service time per task at $S_i$ is ${\displaystyle \frac{1}{{\mu }_i}}$.
• We define ${\rho }_i={\displaystyle \frac{{\lambda }_i}{{\mu }_i}}$, $\lambda ={\sum }_{i=1}^n{\lambda }_i$, and $\mu ={\sum }_{i=1}^n{\mu }_i$.
• $p_i$, $0\le p_i\le 1$, is the probability that, when $S_i$ completes the current task that it is executing, it requests to receive a task from the DP.
• $a_i$, $0\le a_i\le 1$, is the probability that the DP accepts $S_i$’s request when the DP’s input queue is non-empty.
• $C_i=p_i{\dot{a}}_i$ is the probability that when $S_i$ asks for a new task from the DP, it receives it provided that a new task is available at the DP.
• $y\left(t\right)\ge 0$ is the non-negative integer-valued length of the queue of externally arriving tasks waiting at the Dispatching Platform (DP) at time t.
• $y_i\left(t\right)\ge 0$ is the integer-valued total number (queue length) of all the tasks that are in the queue at $S_i$ at time t.
• k is a particular value of $y\left(t\right)$.
• $k_i$ is a particular value of $y_i\left(t\right)$, and we define the vectors as follows:

  $$\begin{array}{ccc}& & Y\left(t\right)=(y\left(t\right),y_1\left(t\right),\dots ,y_n\left(t\right)),\hfill \\ & & K=(k,k_1,\dots ,k_n).\hfill \end{array}$$
• The following vectors are related to $K=(k,k_1,\dots ,k_n)$, where $k\ge 0,k_i\ge 0$:

  $$\begin{array}{ccc}& & K^{-0}=(k-1,k_1,\dots ,k_n)ifk>0,\hfill \\ & & K^{+0}=(k+1,k_1,\dots ,k_n),\hfill \\ & & K^{-i}=(k,k_1,\dots ,k_i-1,\dots ,k_n)if{k}_i>0,\hfill \\ & & K^{+i}=(k,k_1,\dots ,k_i+1,\dots ,k_n).\hfill \end{array}$$

  (2)
• ${\Phi }_i$ is the fraction of external user tasks that the DP allocates to $S_i$.
• ${\Phi }_i^{+}$ is the fraction of external user tasks that the DP allocates to $S_i$ to minimize the average task response time of the edge system.
• ${\Phi }_i^{*}$ is the fraction of external user tasks that the DP allocates to $S_i$ to minimize the average energy consumption per external task assigned to the edge system.
• $X_i={\lambda }_i+{\Phi }_i\Lambda$ is the total arrival rate of tasks to server $S_i$, i.e., the load of $S_i$.
• $X_{i1}$ is the upper bound for the linear approximation of the power consumption of $S_i$, and $X_{i1}<{\mu }_i$
• $q_i={\displaystyle \frac{{\lambda }_i+{\Phi }_i\Lambda }{{\mu }_i}}$ is the utilization rate of server $S_i$. If $q_i<0$, it can be interpreted as the probability that $S_i$ is busy processing tasks.
• $R_{DP}$ is the average response time at the DP for externally arriving tasks.
• $R_S$ is the average response time of all tasks at the n servers.
• ${\pi }_{i0}$ is the power consumption of server $S_i$ when the server is idle, i.e., when $X_i=0$.
• ${\pi }_{iM}$ is the maximum power consumption of server $S_i$. It is attained when $X_i$ is just under the value ${\mu }_i$.
• ${\alpha }_i>0$ is the approximate linear increase in power consumption of $S_i$ as a function of the load $X_i$.
• ${\pi }_i\left(X_i\right)={\pi }_{i0}+{\alpha }_i{X}_i$ is the approximate power consumption of $S_i$ when its load is $X_i$, for $X_i<{\mu }_i$.
• ${\pi }_i^{\prime }$ is the derivative of ${\pi }_i\left(X_i\right)$ with respect to ${\Phi }_i$.
• ${\pi }_i^{″}$ is the second derivative of ${\pi }_i\left(X_i\right)$ with respect to ${\Phi }_i$.
• E is the average energy consumption of the externally arriving tasks that are assigned by the DP to the different servers, and $E={\sum }_{i=1}^n{\Phi }_i{\pi }_i\left(X_i\right){\mu }_i^{-1}$.

Allocating Incoming Tasks to Minimize the Average Additional Energy Consumed by the Servers