A game theoretical distributed approach for opportunistic caching strategy

Abstract

Wireless network virtualization (WNV) has been a hot topic recently, which can provide the customized service to construct virtual networks for various requirements quickly and flexibly by resource sharing. However, most of the traffic is generated by duplicate downloads for a few popular contents. Meanwhile, due to the isolation in the existing WNV, these popular contents cannot be shared among various requests, and thus the congestion and overhead would be caused in the remote data center. Therefore, in order to decline the congestion and overhead from the remote data center, the caching problem is addressed in this paper, which aims to overhear and cache content opportunistically to provide the quicker response for the future requests by in-network nodes. To well understand the impact of content caching, we formulate this issue as a Markov chain, and analyze the content response ratio according to the steady-state mathematically. Furthermore, we define the content caching strategy decision as an optimization problem, and use potential game to decide caching strategy based on Nash Equilibrium in a distributed manner, denoted as Game based Opportunistic Caching Strategy (GOCS). The experimental results show the effectiveness of GOCS in terms of average response hops, average cache hit ratio, data center load reduction ratio, and average resource consumption.

Appendices

Appendix 1

Proof

Equation (15) can be rewritten as follows.

$$\begin{aligned} E({H}) = \frac{\sum \limits _{\forall i \in {\mathcal {V}}}\sum \limits _{\forall c_k \in {\mathcal {O}}}R_{k}\sum \limits _{x=0}^{H_i}xG_{i,k}(x)}{NK} \end{aligned}$$

(28)

In order to analyze that (28) is continuous and bounded, we give Theorem 1 as follows.

Theorem 1

If a function is continuous in the closed interval [a, b], the function must be bounded in closed interval [a, b] [31].

Because variable$p_{i,k}$is continuous and bounded in [0, 1], according to Eqs. (4–7), $P_{mn}$is a linear function of$p_{i,k}$. Therefore, we can conclude that$P_{mn}$is a continuous and bounded function when$p_{i,k} \in [0,1]$. Next, as$P_{mn}$is a variable for$\pi _{i}(n)$in Eq. (8), we can also derive that$\pi _{i}(n)$is a continuous and bounded function.Similarly, since$G_{i,k}(x)$is affected by$g_{i,k}$in Eqs. (12) and (20), it is also a continuous and bounded function when$p_{i,k} \in [0,1]$. Finally, according to expression (28), we can conclude thatE(H) is a continuous and bounded function with$p_{i,k}$.

Appendix 2

Proof

We define the potential function of ${\mathcal {G}}$ as

$$\begin{aligned} \begin{aligned} \Phi _i({s_{i}}, {s_{-i}}) = {\mathcal {H}}_i({s_{i}}, {s_{-i}})\\ \end{aligned} \end{aligned}$$

(29)

$\square$

According to Definition 2, the ith player $(\forall i \in {\mathcal {N}})$ changes its strategy from ${s_{i}}$ to ${s_{i}^{\prime }}$ (${s_{i}}, {s_{i}^{\prime }} \in {\mathcal {S}}_i$). The variation of cost function of the ith player is given as follows.

$$\begin{aligned} \begin{aligned} \Delta {\mathcal {H}}_i&= {\mathcal {H}}_i({s_{i}}, {s_{-i}})-{\mathcal {H}}_i({s_{i}^{\prime }}, {s_{-i}})\\&=\frac{\sum \limits _{\forall c_k \in {\mathcal {O}}}\left\{ E({H_{i, k}(s_i,s_{-i})})-E({H_{i, k}(s_{i}^{\prime },s_{-i})})\right\} }{K}. \end{aligned} \end{aligned}$$

(30)

Simultaneously, the variation of potential function is given as:

$$\begin{aligned} \begin{aligned} \Delta \Phi _i&= \Phi (s_i,s_{-i})-\Phi (s_{i}^{\prime },s_{-i})\\&=\frac{\sum \limits _{\forall c_k \in {\mathcal {O}}}\left\{ E({H_{i, k}(s_i,s_{-i})})-E({H_{i, k}(s_{i}^{\prime },s_{-i})})\right\} }{K} \\&\quad + \frac{\sum \limits _{f \ne i, \forall f \in {\mathcal {N}}}\sum \limits _{\forall c_k \in {\mathcal {O}}}\left\{ E({H_{f, k}(s_i,s_{-i})})-E({H_{f, k}(s_{i}^{\prime },s_{-i})})\right\} }{(N-1)K}\\&= \Delta {\mathcal {H}}_{i}+\Delta {\mathcal {H}}_{-i}. \end{aligned} \end{aligned}$$

(31)

The ith player changes its strategy from $({s_i},{s_{-i}})$ to $({s_{i}^{\prime }},{s_{-i}})$, while the other players do not change their strategies unilaterally. As a result, $E({H_{f, k}(s_i,s_{-i})}) = E({H_{f, k}(s_{i}^{\prime },s_{-i})})$, and $\Delta {\mathcal {H}}_{-i} = 0$. Hence, we can conclude that ${\mathcal {H}}_{i}({s_{i}}, {s_{-i}}) -{\mathcal {H}}_{i}({s_{i}^{\prime }}, {s_{-i}})= \Phi ({s_{i}}, {s_{-i}}) -\Phi ({s_{i}^{\prime }}, {s_{-i}})$. Therefore, the game ${\mathcal {G}} = [{\mathcal {N}}, \{{\mathcal {S}}_{i}\}_{i \in {\mathcal {N}}}, \{{\mathcal {H}}_{i}\}_{i \in {\mathcal {N}}}]$ is an exact potential game, and Eq. (23) is a potential function.