Maximizing the Capacity of Edge Networks with Multicasting

Abstract

Edge networks employ local computing and caching resources to process data, thus alleviating the bandwidth pressure on backbone networks and improve users’ quality of experience. System capacity is one of the key metrics to evaluate the performance of edge networks. However, maximizing system capacity in edge scenarios faces challenges due to the dynamical user sessions and the changing content popularity. This study reports on the influence of multicast communication on the capacity of edge caching networks. When large amounts of content are requested simultaneously or over a short period, one-to-one unicast transmission will consume copious network resources due to repetitious transmission. To solve this problem, this study used the one-to-many multicast scheme to realize cooperative transmission between edge servers. First, multiple copies of the content are distributed to multiple small base stations (SBSs) based on the content’s popularity and the SBSs’ cache sizes. Second, a multicast tree is constructed by using, as the root node, the SBS that stores the content. Third, the content is transmitted along the path of the multicast tree to each end-user. Finally, a simulation platform is constructed to analyze the performance of the two transmission schemes. The results of simulating on edge caching networks show that multicast communication responds well to users’ requests even when the requested content requires sudden transmission, is highly popular or is requested often within short time. This system’s capacity has been significantly improved compared to the classical methods.

1. Introduction

In the era of the “Internet of Everything”, terminals including tablets, computers, mobile phones and diverse sensors generate massive quantities of data. Huge network resources are required to process the large data flows, which often over-burdens the current cloud computing architecture. Edge computing with the locality feature is emerging as a new computing paradigm [1]. By deploying computing and caching resources to the network edge or migrating the tasks/services from the cloud to the network edge, users receive quicker responses and experience higher quality [2].

System capacity is one of the key indicators for evaluating a network’s performance. It reflects the transmission capability of a networking system per second. Gupta and Kumar [3] were the first to present the concept of wireless network capacity $\frac{W}{L\sqrt{n}}$, where W is the transmission rate of the network channel, L is the transmission distance of the content and n is the number of nodes in the network. The subsequent works follow this approach to evaluate the metric from different perspectives such as the network architecture [4], the routing model [5] and the caching strategies [6]. The key idea behind maximizing system capacity is to shorten the content transmission distance by various ingenious conceptions. Though it is well investigated in traditional wireless networks, maximizing system capacity in edge scenarios faces challenges due to the dynamical user sessions and the changing content popularity.

Local caching is one of the main features of edge networks since it reduces the access delay of content. Recent studies propose various caching schemes to improve the edge network’s performance, such as popularity-based caching [7]. Note that the same content may be requested by many users simultaneously or over a short period, for example, the requests for breaking news or major sporting events [8]. This indicates that multicast sessions can be employed in response to users’ requests and only leave content copies at branches of a multicast tree. It thus reduces the volume of repetitive transmission links, shortens the transmission distance and improves the edge caching network’s performance. However, to our knowledge, few works focus on the joint optimization of capacity by using multicasting and caching in edge computing. Related works on multicasting in traditional wireless networks neglect the impact of content caching on the system capacity, and those on edge networks employ the unicast routing model to receive/forward content item. Hence, the joint gain in capacity with multicasting and caching is not clear in edge networks.

Considering the above, we study the capacity of edge networks with respect to the multicasting and content caching. As the number of content copies affects the caching performance and the construction of multicast tree, we first calculated the optimal number of content copies based on user request volumes and the cache sizes of SBSs. Second, we constructed an efficient multicast tree to distribute the content copies. Our contributions are summarized as follows:

• We investigated multicast communication in edge caching networks. SBS caching of the content was selected to construct the multicast tree and the content copy was held in the branch node;
• We used the term capacity as the evaluation metric and analyzed the upper and lower bounds of system capacity. The upper bound is determined by considering the full k-ary tree and the lower bound by the assumption that multicast communication happens only at the last hop.

The remainder of this study is organized as follows. In Section 2, we review the related works. Section 3 describes the system architecture. The following four sections analyze the presentation of the system capacity. Section 8 evaluates the performance of the proposed methods. Finally, conclusions are presented in Section 9.

2. Related Work

Many factors impact system capacity. In this study, we classify the related works from the following perspectives.

2.1. Influence of Routing Scheme on the Capacity

The system capacity provides a reliable index to reflect the network performance. The papers [3,9] first proposed the computability of network capacity. The authors designed an optimal scheduling transmission scheme and established the upper limit of wireless network capacity. The authors in [10] further studied the influence of routing and scheduling algorithms on the capacity of ad hoc networks [11]. The authors used “shortest-path like” selection strategies to incorporate the path-length requirement [12]. Liu et al. [4] concluded that the gain in capacity is substantial even by minimally increasing the number of cell stations. A similar result is given in [13], where the transmission efficiency of device-to-device communication is improved with user-assisted cellular connections. In addition, the authors in [5] employed the scale-free characteristic to formulate network capacity. They rewired a small number of the links connecting to the central nodes to improve the transmission efficiency.

The aforementioned works give general solutions for computing system capacity based on the unicast routing mode. Conversely, the multicast transmission scheme has shown superiority in saving energy and improving delays [14,15,16]. Li et al. [17,18] studied the upper and lower bounds of multicast capacity in wireless networks. Their results indicate that the number of nodes in each multicast session layer plays a big role in system capacity. Furthermore, the authors of [19] showed that the achievable transmission rate per multicast session is bounded by the number of multicast flows in large-scale wireless networks. In addition, the authors of [20] studied the scaling law of capacity and delay for broadcast routing in highly mobile networks. Recent work [21] considered the capacity constraints for enabling workload sharing and load balancing between servers.

2.2. Influence of Caching Scheme on the Capacity

Recent studies show that a pre-caching scheme positively influences the network’s performance [22]. The authors of [23] analyzed the impacts of caching schemes on the wireless content distribution networks and two schemes were considered. The first is the nearest caching node scheme that responds to a user request by caching the content at the nearest node. The second is the transparent en route caching scheme that responds to a user request via an intermediate node on the route path caching the content. The authors of [24] investigated the capacity of an information-centric network when the cached content’s lifetime is limited. Liu et al. [25,26] studied the scaling law of a cache-enabled wireless network and concluded that, even if the number of nodes increases, the system’s capacity could remain unchanged under certain cache constraints. In the paper [27], the authors took the multicast routing and content caching into account. They concluded that the use of multicast transmission technology can significantly save bandwidth if each task request is processed from a local caching and computing site.

2.3. Multicasting in Edge Networks

The authors of [28] investigated the task offloading with multicast in vehicular edge networks. The study [29] studied the multicast-aware resource allocation for edge computing, and computing and caching are jointly optimized in multicast scenarios. Furthermore, the authors of [30] proposed a cooperative multicast-aware caching scheme to improve the delivery delay. The transmission bandwidth gain was analyzed in [27], where the local caching and computing policy are jointly considered subject to multicast transmission constraints.

2.4. Gap in the Existing Literature

The aforementioned works employ multicasting to improve the performance of edge networks. However, to the best of our knowledge, few works investigate the relationship between multicasting and the system capacity in edge networks. Furthermore, because of the dynamics in user sessions and content popularity, it is difficult to compute the content transmission distance. The work [27] inspired us to study the joint influence of multicast and caching on the system capacity, where the content is popular and requested by many users. This provides the basis for deploying multicast sessions in edge caching networks.

3. System Architecture

3.1. Network Model

This paper mainly studies the effect of multicasting on the capacity of edge caching networks. n SBSs are deployed in the network with a Poisson distribution intensity of ${\lambda }_e$, and these nodes function as relays for multicast transmission. Each node has ${\lambda }_u\pi R^2$ users in the coverage range, where ${\lambda }_u$ is the user distribution density and R denotes the communication range of edge servers. In addition, the entire network contains m independent content, and each SBS stores S. As shown in Figure 1, when multiple users covered by several SBSs request the same content simultaneously, the SBS caching the content is selected to transmit data to adjacent SBSs using multicasting, and distant SBSs obtain the requested content via the relay (red lines in the figure); in addition, if none of SBSs cache the content, the first SBS receiving the request forwards users’ demands to the BS. The BS then multicasts the content to the requesting users (black lines in the figure).

3.2. Multicast Tree

In contrast to the one-to-one unicast transmission mode, multicasting follows the one-to-many transmission mode. A separate link for each source-destination pair need not be established and the content copy and distribution only happens at the branches of the multicast tree.

During cooperative transmission, the relays form a multicast tree with the SBS where the content is located at the root node, and the leaf nodes are the SBSs near users that have requested the same content. In the multicast tree, if the source node or relay node has multiple child nodes, it can send content to all child nodes concurrently. Therefore, multicast can use fewer hops to achieve the same purpose, which, compared to unicast transmission, greatly reduces the transmission distance. In other words, the advantage of multicasting is that the content is only transmitted once to multiple receivers on the repeated path of multiple links. This greatly reduces the average transmission distance during dense link transmission, which indirectly affects the system capacity. In the edge network, SBSs have limited storage and computing capabilities and most tasks/services rely on cooperation between SBSs. This indicates that there are many cross-links in the network. Using multicasting to build a multicast tree for content transmission can reduce the transmission delay, optimize the network performance and increase the network capacity.

We next analyze the generalized representation of system capacity and calculate the average transmission distance in different cases.

4. Generalized Representation of Network Capacity

The classical work concludes that, when the number of nodes remains constant, the system capacity is inversely proportional to the transmission distance, that is, the smaller the transmission distance is, the larger the capacity is [31].

Let $L_i\left(C_i\right)$ represent the transmission distance of the ith content with $C_i$ copies on average, and $p_i$ denote its requested probability (The content popularity can follow the Zipf distribution as most related works did [26,32,33,34], or a predicating model can be used [35]). We obtain

$$L_m=min\left(\sum _{i=1}^m{L}_i\left(C_i\right)p_i\right).$$

(1)

s.t.

$$\left\{\begin{array}{c}{C}_i\le n\hfill \\ {\displaystyle \sum _{i=1}^m}{C}_i\le ns\hfill \end{array}\right.$$

(2)

The first constraint imposes the limit that the maximum number of copies of each content item in the network cannot be bigger than the total number of SBSs, that is, to ensure that each content item has at most one copy in each SBS. The second constraint limits the total number of content copies, and these cannot exceed the total cache size. Therefore, in addition to content popularity, capacity is mainly dependent on the transmission distance $L_i\left(C_i\right)$, which includes two parts. The first is the cooperative transmission distance $\overline{L_i\left(C_i\right)}$ and the other is the length of the last hop, that is, the distance between the SBS and the user.

Note that each SBS has a cache size s and the total size of the content is m; the response probability that the SBS caches the requested content is therefore $\frac{s}{m}$, and the probability of SBS collaboration is $\frac{m-s}{m}$ [36]. Then the transmission distance $L_m$ of the content can be presented as:

$$\begin{array}{cc}{L}_m\hfill & ={\displaystyle \sum _{i=1}^m}{L}_i\left(C_i\right)p_i\hfill \\ \hfill & =\frac{m-s}{m}{\displaystyle \sum _{i=1}^m}\overline{L\left(C_i\right)}{p}_i+0\left(s/m\right)p_i.\hfill \\ \hfill & =\frac{m-s}{m}{\displaystyle \sum _{i=1}^m}\overline{L_i\left(C_i\right)}{p}_i\hfill \end{array}$$

(3)

where we ignore the length of the last hop because it is within the communication range R.

We replace L with $L_m$ in the formula $C=\frac{W}{L\sqrt{n}}$ and obtain

$$\begin{array}{c}\hfill C=\frac{Wm}{(m-s){\displaystyle \sum _{i=1}^m}\overline{L_i\left(C_i\right)}{p}_i\sqrt{n}}.\end{array}$$

(4)

Before we analyze the average distance in the multicasting transmission, the wireless interference is firstly discussed. Let $P_t$ represent the transmission power of SBSs and $V\left(t\right)$ denote the set of SBSs transmitting signals at time t. For content to be transmitted from $i\in V\left(t\right)$ to $j\in V\left(t\right)$, the following path loss model is used

$$\frac{\frac{P_t}{D_{i,j}^{\alpha }}}{N_0+{\sum }_{\phi \in V\left(t\right),\phi \ne i}\frac{P_t}{D_{\phi ,j}^{\alpha }}}\ge \sigma ,$$

where $\sigma$ denotes the minimal signal-to-interference-plus-noise ratio, $D_{i,j}^{\alpha }$ is the distance between nodes i and j, $N_0$ is white noise and $\alpha$ denotes the path-loss exponent, which describes how the signal strength decays with distance and is generally greater than two.

5. Multicast Transmission Distance

From the aforementioned analysis, the system capacity can be represented if the average transmission distance of the multicast tree can be formulated. However, it is difficult to calculate the average transmission distance since the content may be duplicated at any branch, and the distribution of branches is uncertain. To make the study more tractable, the upper and lower bound on the transmission distance are analyzed by employing the multicast tree feature. The first is the last-hop multicasting, i.e., the transmission adopts the unicast mode in the former hops and the multicast mode only occurs in the last hop as shown in Figure 2a. The second is the complete k-ary multicast tree, i.e., each branch node has k children.

We here employ Figure 2 to illustrate the difference in the two communication models. In Figure 2a, the root node needs to transmit two times for the same content if the last-hop multicasting is adopted, the number of transmitting time is only one if the complete k-ary multicasting technology is used in Figure 2b, whereas the number is four if we use the unicast communication.

The average transmission distance in the two cases can now be analyzed.

5.1. Upper Bound on the Transmission Distance

Lemma 1. 

In last-hop multicasting, the average distance from one SBS without caching the requested content to other SBSs can be calculated as

$$\begin{array}{c}\hfill \overline{L\left(C_i\right)}=\frac{Q_1}{\left(m-s\right)C_i}\end{array}$$

(5)

where $Q_1$ denotes $\frac{s\left(n-1\right)}{2\sqrt{{\lambda }_e}{\lambda }_u\pi R^2}$.

The proof of Lemma 1 can be found in Appendix A. From Lemma 1, we obtain Theorem 1 as follows.

Theorem 1. 

In last-hop multicast, the system capacity can be represented as

$$C=\frac{Wm}{Q_1\sqrt{n}{\displaystyle \sum _{i=1}^m}\frac{p_i}{C_i}}.$$

(6)

Proof of Theorem 1. 

The process of proving this by replacing $\overline{L\left(C_i\right)}$ with Equation (5) in Equation (4) is highly intuitive.    □

5.2. Lower Bound on the Transmission Distance

Lemma 2. 

In the complete k-ary multicast tree, the average distance from one SBS without caching the requested content to other SBSs is

$$\overline{L\left(C_i\right)}=\frac{Q_2}{\left(m-s\right)C_i}$$

(7)

where $Q_2$ denotes $\frac{sh\left(h-1\right)\left(1-k\right)}{4\left(1-k^h\right)\sqrt{{\lambda }_e}{\lambda }_u\pi R^2}$.

The proof of Lemma 2 can be found in Appendix B.

Theorem 2. 

In the complete k-ary multicast, the system capacity can be represented as

$$C=\frac{4Wm}{Q_2\sqrt{n}{\displaystyle \sum _{i=1}^m}\frac{p_i}{C_i}}.$$

(8)

Proof of Theorem 2. 

The proof is almost identical to that of Theorem 1; it is omitted here for concision.    □

6. Optimal Number of Copies

As shown above, two factors impact on the transmission distance. The first is the number of cooperative SBSs and the second is the number of content copies. They are correlated since the number of copies relies on the multicast tree structure and the number of cooperative nodes is decided by the copy distribution method. This study investigates the first factor with the multicast tree structure, that is, the number of cooperating SBSs will not exceed the number of nodes in the tree assuming that we obtain the copy numbers. The second factor is now analyzed.

6.1. Optimal Number of Copies in the Last Hop Multicast

For convenient calculation, the optimization objective for Equation (6) becomes

$$min\left(\sum _{i=1}^m\frac{Q_1{p}_i}{C_i\left(m-s\right)}\right).$$

(9)

s.t.

$$\left\{\begin{array}{c}{C}_i\le n\hfill \\ {\displaystyle \sum _{i=1}^m}{C}_i\le ns\hfill \end{array}\right.$$

Theorem 3. 

In the proposed multicast cooperation model, the optimal solution of Equation (9) is

$$C_i=\left\{\begin{array}{cc}0& p_i\le 0\\ \sqrt{\frac{Q_1{p}_i}{\beta \left(m-s\right)}}& 0<p_i<\frac{n^2\beta \left(m-s\right)}{Q_1}.\\ n& p_i\ge \frac{n^2\beta \left(m-s\right)}{Q_1}\end{array}\right.$$

(10)

The proof of Theorem 3 is placed in Appendix C.

6.2. Optimal Number of Copies in the Complete K-Ary Multicast

In the complete k-ary multicast, the optimization objective becomes

$$min{\displaystyle \sum _{i=1}^m}\frac{Q_2{p}_i}{\left(m-s\right)C_i}.$$

(11)

s.t.

$$\left\{\begin{array}{c}{C}_i\le n\hfill \\ {\displaystyle \sum _{i=1}^m}{C}_i\le nm\hfill \end{array}\right.$$

(12)

Theorem 4. 

In the proposed multicast cooperation model, the optimal solution of Equation (11) is

$$C_i=\left\{\begin{array}{cc}0;& p_i\le 0\\ \sqrt{\frac{Q_2{p}_i}{\beta \left(m-s\right)}};& 0<p_i<\frac{n^2\beta \left(m-s\right)}{Q_2}.\\ n;& p_i\ge \frac{n^2\beta \left(m-s\right)}{Q_2}\end{array}\right.$$

(13)

The proof of Theorem 4 is placed in Appendix D.

Theorems 3 and 4 reflect the relationship between the number of copies and the content’s popularity. When the number of nodes, the number of files and the cache size remain constant, the number of the ith content copies is mainly decided by its popularity. The more popular the content, the bigger the copy number. The next step is to distribute these copies throughout the edge caching network.

7. Multicast Network Capacity

7.1. Content Caching Algorithm

The aforementioned confirms that an efficient content caching algorithm is also important for improving the system’s performance. This is mainly because this study adopts the multicast transmission scheme and the collaborative distance between the SBS transferring the requested caching content to the SBS that received the request plays a key role in system capacity. An intuition is to place one copy in each subarea of the network to minimize the transmission distance. This is the proportional regional caching (PRC) algorithm. Specifically, first divide the SBS in the entire network into $C_i$ subareas (lines 3–5), and then store each copy in the $C_i$ subareas (lines 6–11). After that, the number of copies for the i-th content is reduced (line 12) as shown in Algorithm 1. When one copy is cached in one subarea, random caching is performed for the remaining copies.

Algorithm 1 The proportional regional caching algorithm

1: The nodes are distributed according to the Poisson point process and the intensity is ${\lambda }_e$. The users are distributed according to the Poisson point process and the intensity is ${\lambda }_u$. 2: for $i=1$ to m do 3:    All SBSs are divided into $C_i$ subareas 4:    Select $C_i-1$ cell randomly, and the number of SBSs in these cells is ceil $\left(n/C_i\right)$ 5:    Divide the remaining SBSs into the last cell 6:    for $t=1$ to $C_i$ do 7:      Randomly select an SBS j from cell t 8:      while the cache size $s_j$ is equal to 0 do 9:         Select another SBS randomly from the cell node 10:      end while 11:      Store content i into $SB{S}_j$ 12:      $c_i\leftarrow c_i-1$ 13:    end for 14: end for

7.2. Capacity

After we acquire the presentation of the optimal copy number in the two multicast schemes, the multicast capacity of the edge caching network can be presented as follows.

Theorem 5. 

The lower limitation on the system capacity with the last-hop multicast can be calculated by

$$\begin{array}{c}C=\frac{Wm}{Q_1\sqrt{n}{\displaystyle \sum _{i=1}^m}\frac{p_i}{C_i}}.\hfill \end{array}$$

(14)

where

$$C_i=\left\{\begin{array}{cc}0;& p_i\le 0\\ \sqrt{\frac{Q_1{p}_i}{\beta \left(m-s\right)}};& 0<p_i<\frac{n^2\beta \left(m-s\right)}{Q_1}\\ n;& p_i\ge \frac{n^2\beta \left(m-s\right)}{Q_1}\end{array}\right.$$

Correspondingly, Theorem 6 presents the system capacity in the complete k-array multicast.

Theorem 6. 

The upper limitation on the system capacity with the complete k-array multicast can be calculated by

$$C=\frac{4Wm}{Q_2\sqrt{n}{\displaystyle \sum _{i=1}^m}\frac{p_i}{C_i}},$$

(15)

where

$$C_i=\left\{\begin{array}{cc}0;& p_i\le 0\\ \sqrt{\frac{Q_2{p}_i}{\beta \left(m-s\right)}};& 0<p_i<\frac{n^2\beta \left(m-s\right)}{Q_2}\\ n;& p_i\ge \frac{n^2\beta \left(m-s\right)}{Q_2}\end{array}\right.$$

7.3. Scalability Analysis

From Theorems 5 and 6, it is clear to see that the system capacity is mainly relevant to the number of SBSs n if other parameters remain unchanged. Therefore, the system capacity tends to zero if n approaches to infinity. This result is consistent with the conclusion of [3]. However, the declining trend is slower than that of [3] because of the caching number of $C_i$ in edge scenarios.

8. Simulation Results and Performance Evaluation

8.1. Simulation Scenarios

This study converted the content transmission mode from unicasting to multicasting. This change to the transmission mode accelerated the edge network’s response to multi-user content requests, especially for highly popular breaking content. Multicast transmission alleviates the problems of network congestion and resource shortage, etc. To illustrate the superiority of the proposed scheme, this study uses the classical multicast work on network capacity [19], as a benchmark, and the paper [25], one recent but representative study on content caching networks to make a performance evaluation. In addition, we use the real data-set MovieLens as the data source to make the study as realistic as possible. This data-set contains a total of 100,000 rating records [37]. The scores of each file were summed and the normalized value was used as the file’s requested probability. Based on the normalized requested probability, the top 5000 files are selected as samples.

Table 1. Initial settings of Parameters.

Simulation area	100 × 100
Number of SBSs	100
Number of users	200
${\lambda }_e$	10
${\lambda }_u$	20
Number of files	5000
Buffer size of SBSs	50
Communication range	20

shows the initial values of some key parameters used in the simulation. The following comparative tests were based on these experimental values.

8.2. Simulation Results

Figure 3 plots the networks topology and the construction process of the multicast tree. In Figure 3a, one hundred SBSs and two hundred users are distributed according to the Poisson distribution intensity ${\lambda }_u$ and ${\lambda }_e$, respectively. Figure 3b,c show the proportional regional caching scheme, where the copy number 7 is used as an example. Figure 3d illustrates the construction process of the multicast. When one user (green dot) makes a content request, the adjacent SBS starts to request cooperation from the nearest SBS. When the SBS (small blue circle) of the content cache is found, the multicast transmission structure is established. The content is stored at the root node and starts being transmitted to the adjacent SBS with each branch. Then, if the requesting SBS is found, the transmission is stopped. Otherwise, the SBS that has received the content as the relay SBS continues to search for the most adjacent SBS and looks downstream for the transmission target until the requesting SBS is found, and the transmission process stops. Here, to facilitate viewing, the fork number of the multicast tree is taken as

Figure 4 shows the impact of the number of branches of the multicast tree on the capacity of the edge caching network. With an increase in k, the network capacity of the two schemes shows a growing trend. Multicast can directly improve the network capacity by reducing the traffic load and shortening the transmission distance. However, in the case of the full k-ary tree, when k is greater than 25, the growth trend slows down and begins to decline. Note that the rapid increase in capacity is because the cooperative transmission between SBS has changed from one-to-one unicast communication to one-to-k multicast communication, so that SBSs can simultaneously transmit content to more adjacent SBSs, reducing the transmission distance of multi-path repeated transmission; however, when the number of SBSs is kept at a certain level and is larger than its reasonable range, the overall transmission distance is limited because it is transmitted to SBSs with a longer distance, which makes the capacity growth trend worse or even lower. For the multicast tree with only the last hop, the distance reduction through multicast is very limited. When k increases, although the network capacity can be improved, the improvement speed is low. In fact, it is for this reason that only the last hop multicast tree can steadily improve the network performance with an increase in k.

The number of content copies can directly affect the transmission distance of content. Figure 5 shows that when the number of content copies increases gradually, the network capacity can grow steadily. However, as the copy number increases, the growth rate of network capacity becomes slow, and the upper and lower bounds on capacity gradually approach and tend to be stable. The reason for this trend is that when the number of content copies becomes large, the number of SBS in each subarea is small, and users in the network can obtain the requested content in the adjacent SBS. There is no need to cooperate with the remote SBS. All requests can be completed in two hops, or even one hop. In this case, all the leaf nodes are close to the root node, and the advantage of multicast becomes limited.

Figure 6 is the distribution diagram of the influence of the distribution intensity at the SBS on the network’s capacity. Generally, network capacity is improved because the distribution intensity of SBS becomes larger and the distribution is relatively dense, the distance between SBSs shortens and the capacity expands. In addition, the advantages of multicast communication are strongly apparent, and it is always superior to unicast communication. This is because multicast communication can serve multi-user requests simultaneously. Although the unicast transmission adopted in [25] does not consider the influence of SBS distribution, as the SBS distribution intensity increases, the distance between SBSs changes, therefore, capacity will also fluctuate, but generally, shows no significant improvement. The capacity change effected by the scheme proposed in this paper is always higher than that in the study [25], because as it expands, more SBSs and users serve the multicast tree, thus more rapidly improving its capacity. The capacity in [19] relies on the pair of source and destinations, therefore its optimization effect would not significantly change as the number of SBSs increases.

The parameter ${\lambda }_u$ determines the distribution density of users. A similar result is achieved in Figure 7, which verifies that the schemes proposed in this study can significantly improve the capacity with an increase in the distribution intensity of users, but there is no significant change in the papers [19,25]. This is because the proposed multicast tree structure can quickly serve more users with the increasing in ${\lambda }_u$, which is also conducive to shortening the transmission distance and further improving the capacity.

The more times the request is sent, the more energy the system consumes, so the simulation results on energy efficiency are shown in Figure 8. The scheme proposed in this paper greatly reduces energy efficiency, and it can be seen from the figure that multicast reduces by $80\%$ on average. With an increase in the K value, the energy efficiency of the system increases. However, increasing in energy efficiency within multicasting is far slower than those within the unicast. This is because the multicast transmission method can quickly respond to user requests and reduce the transmission times.

9. Conclusions

To solve the problem that traditional unicast communication schemes greatly add pressure to edge networks, this study proposes that multicasting can improve the capacity of an edge caching system. Multicast communication transmits the same content to many receivers at once, and can effectively alleviate the traffic-load pressure on the edge network because a separate link need not be established for each user. Using this feature, we constructed a cooperative transmission multicast tree, where a content node is selected as the root node. This reduced the cooperative transmission distance for content and improved the network capacity. Given the last-hop multicasting and complete k-ary multicasting, a lower and upper bound on the system capacity is proposed. The simulation results verify the superiority of the proposed scheme. In future, more effective content distribution methods and root node selection schemes will be investigated to improve the edge system performance in multicasting session scenarios.