Decentralized Hierarchical Coded Caching Over Heterogeneous Wireless Networks with Multi-level Popularity Content

Abstract

Decentralized hierarchical coded caching is studied with two layers of caches which users receive their demands through intermediate helpers from a main server. Coded caching in hierarchical model reduces rates in both layers and this scheme has been shown to be order-optimal when all contents are uniformly popular and there is no tension between the rates in each of two layers. We consider two-layer coded caching over heterogeneous wireless networks which can reduce the rates more effectively in both layers. The contents in our work are multi-level, based on the different levels of popularity. We use two different approaches in our hierarchical model: when the coded caching is provided within each layer and when it is provided across multiple layers. We consider combination of these two approaches and then optimize the proportion of using each approach to minimize the rates of the first layer and the second layer. We show that the rates in our scheme are much lower than the traditional hierarchical model. Moreover, common-memory method is introduced in two-layer network and is compared with the other schemes. It can be observed that the rates of common-memory method are lower than the previous hierarchical model but are higher than our proposed multi-level scheme.

Appendices

Appendices

1.1 Appendix 1: Approximation of Rate Formula

The rate of decentralized multi-level coded caching is as follows [18]

$$\begin{aligned} R(M)= \,& {} max \left\{ \sum _{l=1}^L R_l(M),0 \right\} \nonumber \\= & {} max \left\{ \sum _{l=1}^L U_l\left( \frac{N_l}{\lambda _lM} -1\right) \left( 1-\left( 1-\frac{\lambda _lM}{N_l}\right) ^{ K}\right) ,0 \right\} . \end{aligned}$$

(14)

Optimization of the above formula is complex because there are L terms in power of K which is the number of access points. Considering that K is usually a large number, we derive an accurate approximation for the rate. We study two different cases.

1.2 A. $\lambda _lM<N_l$

When the memory size of level l is less than the number of files in level l, we have

$$\begin{aligned} 0<\lambda _lM<N_l, \end{aligned}$$

so

$$\begin{aligned} 0<\left( 1-\frac{\lambda _lM}{N_l}\right) <1, \end{aligned}$$

and when K is large

$$\begin{aligned} \left( 1-\frac{\lambda _lM}{N_l}\right) ^K \approx 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \left( 1-\left( 1-\frac{\lambda _lM}{N_l}\right) ^K\right) \approx 1. \end{aligned}$$

1.3 B. $\lambda _lM>N_l$

In this case, the memory size of level l is more than the number of files in level l. Hence, all files are saved in the cache of users in the placement phase and the transmission rate for this level is zero, so

$$\begin{aligned} R_l(M) =0. \end{aligned}$$

According to parts A and B, we conclude that in some levels, we have $R_l(M) =0$ and in other levels, we have $(1-(1-\frac{\lambda _lM}{N_l})^K)\approx 1$. So Eq. (14) turns into the following expression

$$\begin{aligned} R(M)= \,& {} max \left\{ \sum _{l=1}^L R_l(M),0 \right\} \nonumber \\= & {} max \left\{ \sum _{l=1}^L U_l\left( \frac{N_l}{\lambda _lM} -1\right) ,0 \right\} . \end{aligned}$$

(15)

It is important to note that when we set $(1-(1-\frac{\lambda _lM}{N_l})^K) =1$, in fact we consider the worse state or the largest amount of rate so this approximation is an appropriate method to minimize the rates. We use this approximation in all our calculations.

1.4 Appendix 2: Optimization of $\alpha$ and $\beta$ in the Multi-level Method

By approximating (8) and (9) with the approximation explained in “Appendix 1”, we have

$$\begin{aligned} R_1^M(\alpha ,\beta )= \,& {} max \left\{ \alpha k_2\left( \frac{\alpha N}{M_1}-1\right) , 0 \right\} \nonumber \\&+ \, max \left\{ (1-\alpha )\sum _{l=1}^{L} U'_l \left( \frac{(1-\alpha )N_l}{(1-\beta )\lambda _{l}M_2}-1\right) , 0\right\} , \end{aligned}$$

(16)

and

$$\begin{aligned} R_2^M(\alpha ,\beta )= \,& {} max \left\{ \alpha \sum _{l=1}^{L} U'_l \left( \frac{\alpha N_l}{\beta \lambda _l M_2}-1\right) , 0\right\} \nonumber \\&+ \, max \left\{ (1-\alpha )\sum _{l=1}^{L} U'_l \left( \frac{(1-\alpha )N_l}{(1-\beta )\lambda _{l}M_2}-1\right) , 0\right\} . \end{aligned}$$

(17)

Starting with $R_2^M$ by setting $\frac{\partial R_2^M(\alpha ,\beta )}{\partial \alpha } =0$ or $\frac{\partial R_2^M(\alpha ,\beta )}{\partial \beta } =0,$ hence we conclude that $\alpha =\beta$. By setting $\alpha =\beta$ and minimizing $R_1^M$, we have

$$\begin{aligned} \alpha ^*=\beta ^*=\frac{ M_1\left( \sum _{l=1}^{L} U'_l\left( \frac{N_l}{\lambda _{l} M_2}-1\right) +k_2\right) }{2 k_2 N}. \end{aligned}$$

(18)

1.5 Appendix 3: Optimization of $\alpha$ and $\beta$ in the Common-Memory Method

If (11) and (12) are approximated by the approximation in “Appendix 1”, we have

$$\begin{aligned} R_1^{c-m}(\alpha ,\beta )= \,& {} max \left\{ \alpha k_2 \left( \frac{\alpha N}{M_1}-1\right) , 0\right\} \nonumber \\&+\, max \left\{ (1-\alpha )\lfloor \frac{k_2}{q} \rfloor \left( \frac{(1-\alpha )N}{(1-\beta )M_2}-1\right) , 0\right\} , \end{aligned}$$

(19)

and

$$\begin{aligned} R_2^{c-m}(\alpha ,\beta )= \,& {} max \left\{ \alpha \lfloor \frac{k_2}{q} \rfloor \left( \frac{\alpha N}{\beta M_2}-1\right) , 0\right\} \nonumber \\&+\, max \left\{ (1-\alpha )\lfloor \frac{k_2}{q} \rfloor \left( \frac{(1-\alpha )N}{(1-\beta )M_2}-1\right) , 0\right\} . \end{aligned}$$

(20)

To minimize these formulas, we start with $R_2^{c-m}(\alpha ,\beta )$. Setting $\frac{\partial R_2^{c-m}(\alpha ,\beta )}{\partial \alpha } =0$ or $\frac{\partial R_2^{c-m}(\alpha ,\beta )}{\partial \beta } =0,$ results in $\alpha =\beta$. Then we set $\alpha =\beta$ in $R_1^{c-m}(\alpha , \beta )$ and optimize it, so

$$\begin{aligned} \alpha ^*=\beta ^*=\frac{M_1 \left( \lfloor \frac{k_2}{q} \rfloor \left( \frac{N}{M_2}-1\right) +k_2\right) }{ 2 k_2 N}. \end{aligned}$$

(21)