Placement Combination between Heterogeneous Services and Heterogeneous Capacitated Servers in Edge Computing

Abstract

With the rapid increase of applications in 5G and Internet of Things, mobile edge computing (MEC) has been proposed to reduce the burden of central cloud and decrease the users request delay by deploying edge servers and edge services close to users. Due to resources constraint of edge servers and disadvantage of standalone placement optimization of edge servers or edge services, the discussion focusing on the combining optimization of server placement and service placement has engendered. At present, the placement combination is studied with strict assumption constrains, such as homogeneous service and server with unlimited resources, which are not suitable for the reality scenarios. This paper proposes Placement Combination between Heterogeneous Services and heterogeneous capacitated Servers (PCHSS). PCHSS aims to minimize the delay in computation and transmission as well as to balance resources in edge servers. Because the placement combination optimization is a NP-hard problem, we propose two solution algorithms named FHPC and IUPC. Both algorithms have a two-layer iterative optimization structure with different convergence time and result performances. FHPC can converge to a good result quickly, and IUPC can achieve better results with a relatively higher computational complexity. Then we prove that both algorithms can converge in polynomial time. Extensive simulations demonstrate the significant effectiveness of the placement combination, and our algorithms can reduce the user request delay by up to 51% compared with baseline algorithms.

Appendix: A

Let $D=\left (d_{1},d_{2},...,d_{S}\right )$ be the service placement decision space of edge servers, and in each iteration, a random edge server m randomly chooses a service placement decision from D. With Algorithm 3 iterating over the edge servers and the service placement decision space, the edge service placement policy Ser evolves as a N-dimension Markov chain, in which each dimension represents the service placement decision of each edge server. For the convenience of presentation, we analyze the scenario with 2 edge nodes, and the 2-dimension Markov chain is denoted as < ser1,ser2 >. In each iteration, one randomly selected edge server m virtually changes its current caching decision to a random service placement decision from d_s, thus there is

$$ \begin{array}{@{}rcl@{}} &&Pr(<{ser}_{1}^{\ast},{ser}_{2}>|<{ser}_{1},{ser}_{2}>) \\ &&\quad=\frac{e^{-\frac{y(<{ser}_{1}^{\ast},{ser}_{2}>)}{\varphi}}}{MS\left( e^{-\frac{y(<{ser}_{1}^{\ast},{ser}_{2}>)}{\varphi}}+e^{-\frac{y(<{ser}_{1},{ser}_{2}>)}{\varphi}}\right)} \end{array} $$

(1)

$$ \begin{array}{@{}rcl@{}} &&Pr(<{ser}_{1},{ser}_{2}>|<{ser}_{1},{ser}_{2}^{\ast}>) \\&&\quad= \frac{e^{-\frac{y(<{ser}_{1},{ser}_{2}^{\ast}>)}{\varphi}}}{MS\left( e^{-\frac{y(<{ser}_{1},{ser}_{2}^{\ast}>)}{\varphi}}+e^{-\frac{y(<{ser}_{1},{ser}_{2}>)}{\varphi}}\right)} \end{array} $$

(2)

where y(< ser₁,ser₂ >) is the objective value when the service placement policy is < ser₁,ser₂ >. In this scenario, N = 2. Let π(< ser₁,ser₂ >) denote the stationary probability distribution of service placement policy < ser₁,ser₂ >, then π(< ser₁,ser₂ >) can be derived by the fine stationary condition of the Markov chain as

$$ \begin{array}{ll} \pi(<d_{1},d_{2}>)Pr(<d_{1},d_{s}>|<d_{1},d_{1}>) =\\ \pi(<d_{1},d_{s}>)Pr(<d_{1},d_{1}>|<d_{1},d_{s}>) \end{array} $$

(3)

Substitute (1), (2) into (3), it can be derived that

$$ \begin{array}{ll} \pi(<d_{1},d_{1}>)\times \frac{e^{-\frac{y(<d_{1},d_{s}>)}{\varphi}}}{MS\left( e^{-\frac{y(<d_{1}^{\ast},d_{s}>)}{\varphi}}+e^{-\frac{y(<d_{1},d_{1}>)}{\varphi}}\right)} \\ \quad=\pi(<d_{1},d_{s}>)\times \frac{e^{-\frac{y(<d_{1},d_{1}>)}{\varphi}}}{MS\left( e^{-\frac{y(<d_{1}^{\ast},d_{s}>)}{\varphi}}+e^{-\frac{y(<d_{1},d_{1}>)}{\varphi}}\right)} \end{array} $$

(4)

It can be observed that (4) is symmetric and can be balanced if π(< d₁,d₂ >) has the form of $\pi (<d_{1},d_{1}>) = \gamma e^{-\frac {y(<d_{1},d_{2}>)}{\varphi }}$ where γ is a constant. let ℵ be the service placement policy space. To ensure ${\sum }_{<d_{1},d_{2}>\in \aleph }{\pi \left (<d_{1},d_{2}>\right )=1}$, the stationary probability distribution $\pi \left (<d_{1},d_{2}>\right )$ should be given as

$$ \pi(<d_{1},d_{2}>) = \frac{e^{-\frac{y(<d_{1},d_{2}>)}{\varphi}}}{{\sum}_{<{d_{1}^{f}},{d_{1}^{2}}>\in\aleph}{e^{-\frac{y\left( <{d_{1}^{f}},{d_{2}^{f}}>\right)}{\varphi}}}} $$

(5)

(5) can be rewritten as $\pi (<d_{1},d_{2}>) = \frac {1}{{\sum }_{<{d_{1}^{f}},{d_{1}^{2}}>\in \aleph }{e^{-\frac {y(<d_{1},d_{2}>)-y\left (<{d_{1}^{f}},{d_{2}^{f}}>\right )}{\varphi }}}}$. Let $<d_{1}^{\ast },d_{2}^{\ast }>$ be the globally optimal solution that minimizes the objective value, i.e.,$<d_{1}^{\ast },d_{2}^{\ast }>\leq any <{d_{1}^{f}},{d_{2}^{f}}>\in \aleph$. It can be concluded that $\pi (<d_{1}^{\ast },d_{2}^{\ast }>)$ increases as φ decreases, and $\pi (<d_{1}^{\ast },d_{2}^{\ast }>) = 1$ when $\varphi \rightarrow 0$.