Analysis on Cache-enabled Wireless Heterogeneous Networks
Caching the popular multimedia content is a promising way to unleash the ultimate potential of wireless networks. In this paper, we contribute to proposing and analyzing the cache-based content delivery in a three-tier heterogeneous network (HetNet), where base stations (BSs), relays and device-to-device (D2D) pairs are included. We advocate to proactively cache the popular contents in the relays and parts of the users with caching ability when the network is off-peak. The cached contents can be reused for frequent access to offload the cellular network traffic. The node locations are first modeled as mutually independent Poisson Point Processes (PPPs) and the corresponding content access protocol is developed. The average ergodic rate and outage probability in the downlink are then analyzed theoretically. We further derive the throughput and the delay based on the multiclass processor-sharing queue model and the continuous-time Markov process. According to the critical condition of the steady state in the HetNet, the maximum traffic load and the global throughput gain are investigated. Moreover, impacts of some key network characteristics, e.g., the heterogeneity of multimedia contents, node densities and the limited caching capacities, on the system performance are elaborated to provide a valuable insight.
Caching, HetNets, content popularity, Poisson Point Processes, Markov process.
The total mobile data traffic of 2020 will increase 1000 times compared with the 2010 traffic level . Despite the deployment of the fourth generation Long Term Evolution (LTE) and LTE-Advanced systems, the rapidly increasing wireless data demands overwhelms the throughput increase that the wireless network could afford. Various innovative throughput-increasing methods have been investigated to tackle the ever-growing wireless data challenge, such as the heterogeneous network (HetNet)  and the cache-enabled content-centric network [4, 5, 6]. The state of the art is elaborated in the perspective of the two aspects respectively in the following.
HetNets, bringing the network closer to users: One widely regarded as the cornerstone technology is denser node deployment, including macro base station (BS), micro BS, pico BS, femto BS and relays. Such a HetNet decreases the distance between BSs/relays and users, and thus increases the area spectral efficiency, yielding the increase of network capacity [7, 8, 9]. However, the exponential growth in traffic also requires the high-speed backhaul for the connection of different type of BSs/relays and content servers [10, 11].
Cache-enabled content-centric networks, bringing the content closer to users: It has been shown that of the wireless traffic is from multimedia contents, e.g., videos . Meanwhile, the multimedia contents are not accessed with the same frequency. Only a small fraction () of “popular” contents are consumed by the majority of the users, and the less popular contents are requested by a much smaller number of users . Moreover, following the uncannily accurate Moore’s law, a tremendous amount of computing and storage capacity is held by the intelligent terminal devices and networks. As such, the popular contents can be cached in BSs, relays and devices, bringing the content closer to users. It allows users to access to the cache-enabled nodes and reduces the duplicate content transmissions, mitigating the over-the-air traffic .
Therefore, taking advantage of the caching capability within the wireless HetNet, the content diversity and network diversity can be exploited to relieve the burden of the fast growing traffic [4, 14, 15].
I-a Related Work
The role of the caching technology in the fifth generation (5G) wireless network is demonstrated in [14, 13]. Urs Niesen et al. investigate a large wireless caching network with the hierarchical tree structure of transmissions, and scaling results on the capacity region are derived. An arbitrary traffic matrix and cooperative transmissions over arbitrarily long links are assumed .  introduces the distributed caching at the macro BSs to improve the network capacity and reduce the video stalling. The authors of  advocate to set up relays with caching ability in the cellular network to reduce the access delay. The content placement scheme has received significant attention, e.g.,  proposes a novel coded caching scheme to improve both the local and the global caching gain.  considers the scenario where a user in the overlapping coverage area can connect to any of the stations covering it. The optimal caching strategy maximizing the caching hit ratio is formulated by solving the Geographic Caching Problem. Optimal request routing and content caching are investigated in  to minimize the average content access delay. In , the energy consumption is minimized by appropriately pre-caching popular contents.  studies how to disseminate the content via cellular caching and Wi-Fi sharing to trade off the dissemination delay and the energy cost. Based on the content popularity, the cache-based multimedia content delivery scheme is proposed and analyzed in . Terminal users can share the received content via opportunistic local connectivity to offload the traffic of cellular links in .  exploits redundancy of user requests and the storage capacity of terminal devices via dividing the cell into virtual square grids.
However, in the current research of caching, the assumption of global knowledge of the stationary network topology and the node connectivity graph is critical, and the regular grid network model is too optimistic and idealistic to fully capture the randomness and complexity of node locations in the HetNet nowadays. Different from the traditional system model, lots of researches have pointed out that the node location obeys PPP instead of regular hexagonal grid in realistic HetNet [7, 24, 8, 25]. Two tiers of BS locations are modeled as independent PPPs in , where joint resource partitioning and offloading are analyzed in the HetNet.  studies the optimal node density in homogeneous and heterogeneous scenarios by modeling cellular networks with PPP. The authors in  model the node locations of the multi-tier HetNet as mutually independent PPPs, and analyze the system performance in terms of the average rate.  takes the limited backhaul into consideration to analyze the performance of the homogeneous cache-enabled small cell network, where the nodes of the small base stations are stochastically distributed. A constant service rate is assumed when the files can be found in the local cache and the downlink capacity exceeds the threshold. In , disjoint circular clusters are scattered based on the hard-core PP. Requesting users and cache-enabled users are distributed with two independently homogeneous PPPs. The requesting users obtain the content from cache-enabled users in the same cluster via the out-of-band device-to-device (D2D) in the cellular network.
Furthermore, the traditional fetching and reactive caching methods doesn’t intelligently utilize the service characteristic such as the traffic redundancy and the content popularity. Few studies considers the scenario where the radio access network (RAN) caching and the D2D caching coexist. Meanwhile, the performance of the wireless cooperative caching HetNet is not yet fully investigated. How much performance improvement actually can be reaped via the caching technology is urgent to be answered theoretically.
Towards these goals, in this paper we analyze the scheme that when the network load is off-peak, the most popular contents can be cached at the nodes via broadcasting. The BSs, relays and cache-enabled users are cooperative to transmit contents in the HetNet. The main contributions of this paper are summarized as follows:
We consider the limited caching ability of both relays and parts of the users. Popular contents are cached when the network is off-peak. Besides the cellular communication, there exists the local content sharing links from the cache-enabled user to the users. When a user triggers a request, it can be responded by BSs, relays or the cache-enabled users.
We model the node locations (BSs, relays and users) of the three-tier HetNet (BSs-users, relays-users, users with caching ability-users) as mutually independent PPPs. The content access protocol is then proposed, based on which the tier association priority is formulated.
We derive analytical expressions of the average ergodic rate and outage probability for users in different Cases. Then with the modeling of the request arrival and departure process at the service node as a multiclass processor-sharing queue, the throughput and delay of different classes are further analyzed based on the continuous-time Markov process.
We propose the steady ruler and the critical point for the HetNet to keep steady, according to which the throughput and the maximum traffic load over the entire network are then evaluated. Moreover, impacts of the cache-enabled users, content popularity and the limited storage capacity on the network performance are analyzed.
The remainder of the paper is organized as follows: In Section II, we formulate the three-tier HetNet architecture and elaborate the tier association priority based on the content access protocol. The average ergodic rate and outage probability are derived in Section III and Section IV. The performance gain in terms of the throughput and the delay are analyzed in Section V. In Section VI, numerical results are presented. Finally, we give our conclusions in Section VII.
Ii System model and Protocol description
In this section, we first model the nodes of the three-tier HetNet as mutually independent PPPs with different densities. Then the cache-enabled content access protocol is described. Afterwards, the probability of the tier association priority and the state of users are derived.
Ii-a Network Architecture
Consider a three-tier wireless HetNet consisting of a number of macro BSs, relays and users as illustrated in Fig. 1. The nodes of the -th tier ( for the users, relays and BSs, respectively) are deployed based on an independent homogeneous PPP with intensity [7, 26, 8]. Note that in the practical system there are more users than relays or BSs, so we consider in this paper. There are multimedia contents on the multimedia server, where all the contents are assumed to have the same size of bits. Each of the relays has a limited caching storage with the size of bits, but only a part (e.g., the proportion) of the users has caching ability and the corresponding size is bits, and . According to Poisson processes, the locations of the cache-enabled users are distributed as a thinning homogeneous PPP with density .
It has been observed that people are always interested in the most popular multimedia contents, where only a small portion of the contents are frequently accessed by the majority of users . The higher ranking of a multimedia content, the greater the requested probability. The popularity of the -ranked content can be modeled by the Zipf distribution as follow[12, 5, 31]
where reflects the skew of the content popularity distribution. The larger , the fewer of popular contents accounting for the majority of the requests.
Ii-B Cache-enabled Content Access Protocol
A high-capacity wired backhual solution can be used for the connection link between BSs and relays, e.g., optical fiber. When the network is at low traffic load, e.g., the traffic load in the nighttime, the most popular contents can be cached at the relays and the cache-enabled users via broadcasting . All the cache-enabled users store the same copy of the contents until the caching storage is fully occupied, and those cached in different relays are also the same.
When a user requests a multimedia content, it first checks whether the caching storage is available in its local devices. If the requested content is cached in its caching storage, the user can obtain the content immediately; otherwise, the user access the “closest” node. Here, we define the node providing the maximum received power as the “closest” node of a requesting user. The user’s received power is defined as [7, 27, 28],
where for is the transmit power of the node in the -th tier. When it means the user gets the content from a cache-enabled user via the local sharing link such as D2D  considered in this paper. denotes the path-loss exponent and is the distance between the requesting user and its closest node of the -th tier. denotes a propagation constant and is normalized as in this paper. For clarity, the association bias of the -th tier is assumed to be 1. Thus, the closest node is . As a result, there are four content access Cases in the three-tier HetNet as described below.
Case 1: The requesting user does not have caching ability, and it can successfully obtain the requested contents from the closest node (BS, relay or the cache-enabled user).
Case 2: The requesting user has caching ability, but the requested content is not cached in its local caching storage, which means all the cache-enabled users do not store the requested content. Thus, only the closest node in the relay or BS tier can respond to the request.
Case 3: The requesting user does not have caching ability, and its closest node is a cache-enabled user. However, the corresponding cache-enabled user does not cache the requested content due to the limited caching storage. So the requesting user needs to obtain the content from the closest node in other tiers, i.e., the relay or BS.
Case 4: The requesting user has caching ability, and it can obtain the requested content from its local caching storage immediately.
The HetNet without caching is considered as a baseline in this paper. In the baseline, neither users nor relays have caching ability. Therefore, BSs could not pre-broadcast the popular contents to the users and relays. And the local sharing links (D2D) among users can not work at this time. If the closest node of a user is a relay, the relay needs to fetch the content from the BS via wired backhaul firstly and then forwards it to the user; If the closest node is a BS, the BS responds the user’s request. Similarly, in the caching network, if the content is not cached in the relay, a Backhaul-needed (BH-needed) event happens and the relay needs to fetch the content from the BS via the backhaul firstly; otherwise, the Backhaul-free (BH-free) event happens and the relay can respond the request immediately without the backhaul.
Ii-C The Probability of the Tier Association Priority
As described above, the locations of users, BSs and relays are modeled as mutually independent PPPs. Therefore, the probability that there are nodes in area with radius of is given by
where and . Without loss of generality, according to Slivnyak’s theorem , we conduct analysis on assumption that there is a typical user with or without caching ability at the origin of the Euclidean area, and it is regarded as the reference user.
We first analyze the scenario where the reference user does not have caching ability with the probability of . So the probability that the distance between the reference user and its closest cache-enabled user is larger than is
Therefore, the probability density function (PDF) of the distance from the reference user to the closest cache-enabled user is given by
Similarly, the PDF of the distance from the reference user to its closest relay and BS are
respectively. As a result, the joint PDF can be given by
To derive the main conclusions in the following, we first consider the general -tier HetNet with PPPs with parameters and . Denote as the maximum received-power from the -th tier, where means that the value of the maximum received-power from the -th tier is ranked -th. We thus have the following proposition.
The probability of is
See Appendix A.
For the three-tier HetNet of this paper, the probability of is then given by
We observe from (9) that the reference user without caching ability prefers to obtain the content from the -th, -th, and -th tier in turn with probability of . Proposition 1 can be further extended to the following lemma.
The probability that the reference user without caching ability prefers to associate with the -th tier at first in the -tier HetNet is
See Appendix B.
So in the three-tier HetNet, the probability that the reference user without caching prefers to get the content from the -th tier at first is
Likewise, as to the scenario where the reference user is cache-enabled, the probability of is
Equation (12) means that the reference user with caching ability prefers to obtain the content from the -th, -th tier in turn with probability of when the requested content has not been cached in the local caching. and will be denoted as and respectively for convenience in the following. According to (11) and (12), we find that the tier association priorities are different when the user is cache-enabled or not. Users prefer to connect to the tier with higher transmit power and node density.
Ii-D The Density of the Active D2D Transmitters
In the subsection above, we have analyzed the tier association priority merely based on the geographical locations, where the detailed impacts of the limited caching space and the content popularity are not considered. Define as the Case the user may be active in. Let be the node where the user can obtain contents. Let describe whether the backhaul is needed for a user to access the content successfully. We only consider the wired backhaul between the BS and the relay, the impacts of the backhaul from the multimedia server to the BS are out of the scope of this paper. Denote as the state of the user. Probabilities of different are listed in Table I, where we rewrite as for simplification. Assign the value in the -th row -th column of Table I to the element of a matrix .
|Tier 1 (D2D)||Tier 2 (Relay)||Tier 3 (BS)||Local|
Based on Table I, the probability that a user obtains the content successfully via the D2D link is , i.e., . So the density of users to be served by D2D transmitters (TXs) is . However, the density of the cache-enabled user is , which is the maximum density of D2D TXs. Define as the density of the actually active D2D TXs. If a small fraction of users have caching ability, in the coverage of a cache-enabled user, there is at least one user to be responded via the D2D link, i.e., the density is . At this time the number of D2D links are limited by the number of cache-enabled users. All of cache-enabled users should be active as D2D TXs to satisfy the demand for the D2D link. However, if most of users are cache-enabled, some cache-enabled users may not cover any user in the corresponding coverage. The density of cache-enabled users active as D2D TXs is , which is smaller than . At this time the number of D2D links are limited by the number of the users without caching ability, and not all of cache-enabled users are active as D2D TXs. Thus, the node density of the active D2D TXs can be given by
We define as the critical point deciding whether all of cache-enabled users need to be active as D2D TXs. Let we get the critical point,
where . From (14) we observe that whether all the cache-enabled user need to be active as D2D TXs is jointly decided by the user caching ability (), the content popularity (), the transmit power (), the node density () and the path-loss exponent (). increases with the increase of and . Then equation (13) can be rewritten as
Next, we introduce another variable which decides the maximum density of the D2D TXs in the network with various . Based on the first derivative of with respect to we can get . The density of the D2D TXs increases with in the region and starts to decrease from . It implies at most D2D links can be set up in unit area.
For the other two tiers, as considered above that all the relays and BSs are fully loaded and active when , the actually active node density of the BSs and relays equal to the corresponding node density, i.e., for . Therefore, the nodes of the actually active D2D TXs, relays and BSs are scattered according to mutually independent homogeneous PPPs with the density , respectively.
Iii The Average Ergodic Rate
The average ergodic rate in the downlink is analyzed in this section. Specifically, the communication link between the relay/user and the requesting user is assumed to share the same frequency with that from the BS to the users, yielding the interference. There exist two types of interferences, namely, the inter-tier and the intra-tier interference. The full load state of the BS and relay is considered and user requests arriving at the same service node are responded one after the other in a round-robin manner . We shall note that the rate analyzed in this Section refers to that over the air, and the effect of the backhaul will be considered in Section V.
Therefore, the signal-to-interference-plus-noise ratio (SINR) of the reference user associated with the node in the -th tier is
where denotes the power of the additive noise, is the distance between the reference user and its serving node. and denote the channel power gain. Here, we consider Rayleigh fading channels with average unit power, yielding . is the distance between the reference user and its interfering nodes in the -th tier. denotes the cumulative interference from the -th tier. We define the average ergodic rate of the reference user when it communicates with the -th tier as [7, 8, 25, 26, 28],
Here, the unit of the average rate is nats/s/Hz () to simplify the analysis. The average is taken over both the channel fading distribution and the spatial PPP. The ergodic rate is first averaged on condition that the reference user is at a distance from its serving node in the -th tier. Then the rate is averaged via calculating the expectation over the distance . The metric means the average ergodic rate of a randomly chosen user associated to the -th tier.
Iii-a The Average Ergodic Rate in Case 1
Denote as the distance between the reference user and its serving node of tier . Based on the proof in , we can obtain the PDF of as follow,
Then we have the following theorem.
The average ergodic rate of the reference user associated with the -th tier () in Case 1 is
See Appendix -C.
Since the node densities are typically quite high in the HetNet, the background noise is far smaller than the interference power. The interference is dominant and the noise can often be neglected, i.e. , then the rate is further simplified to
Equations (20) and (21) reveal that when the interference is dominant, the average ergodic rate in Case 1 is independent on which tier the user connects to. Moreover, from (21) we know that, when the interference is dominant and the fraction of the cache-enabled users is small , the average ergodic rate keeps constant. On one hand, the rate keeps constant when varies in the region of such that all of the cache-enabled users should be active as D2D TXs. Higher density of D2D TXs gets the content closer to users while adding additional interference caused by the increase of the D2D pairs. On the other hand, the rate keeps constant independently on system parameters such as the transmit power and node density. This means that raising the transmit power or service node densities increases the desired signal power and the interference by the same amount, and they offset each other. However, the parameters affect the number of simultaneously active nodes in unit area. As an example, with larger and caching ability , more users can get contents via the D2D link or immediately from their local caching, yielding the change of the sum rate of the cache-enable network. However, when , the average ergodic rate increases with the increase of based on (21). It is because the distance between the user and the D2D TX is reduced with larger number of cache-enabled users, but not all the cache-enabled users need to be active as the D2D TXs at this time, breaking the balance between the desired signal power and the interference.
Iii-B The Average Ergodic Rate in Case 2
Similar to (18), the PDF of the distance between the reference user and its serving node of tier in Case 2 is
We then calculate the average ergodic rate for Case 2 as follow.
The average ergodic rate of the reference user associated with the -th tier () in Case 2 is
See Appendix -D.
When the interference is dominant, i.e., , we obtain
Compared (25) with (21), we find that the average ergodic rate of Case 2 is smaller than that of Case 1. D2D TXs bring out additionally unnecessary interference to the users in Case 2, decreasing the rate. According to , when , the average ergodic rate in Case 2 decreases with the increase of because increases with the . Furthermore, set the first derivative of with respect to to zero, we can get a critical point , which exactly is the point getting the maximum number of active D2D TXs explained in subsection II-D. The number of active D2D TXs increases monotonically with the increase of when ; otherwise, it decreases monotonically when . More active D2D TXs lead to more unnecessary interference to the users of Case 2. Therefore, the rate continues decreasing with the increase of in the region but it starts to increase from . On the whole, the rate in Case 2 decreases in the region and then increases from . Any of the network parameters such as the node density (), the content popularity (), the transmit power (), the path-loss parameter () and the caching ability () can affect the trend of the rate in Case 2.
Iii-C The Average Ergodic Rate in Case 3
Based on the definition of Case 3, we have . As described above, is the distance between the reference user and its closest cache-enabled user. Let be the distance between the reference user and its closest node in the -th tier for . Then the joint PDF of in Case 3 is
If , . The proof is derived in Appendix -E. As a result, we obtain the following theorem,
When the interference is dominant, the average ergodic rate of the reference user associated with the -th tier () in Case 3 is
See Appendix -F.
We can observe that when the interference is dominant, the average ergodic rate of the reference user in Case 3 is also independent on which tier the user connects to. Furthermore, denote as the average ergodic rate of the reference user in Case 4. shall be considered as a extremely fast speed with which the user can read out the contents from its local caching disk immediately. The higher the content popularity () and caching ability () become, the higher probability there is for users to be active in Case 4.
Iv The Outage Probability
Besides the average ergodic rate elaborated in the previous section, we will derive another important performance metric, i.e., the outage probability in this section. The outage probability can be defined as the probability that the instantaneous SINR of a randomly located user is less than a threshold . Let be the average outage probability of the reference user associated with the -th tier, which can be expressed as [7, 8, 25, 26, 28, 29],
The metric can be equivalently interpreted as the average fraction of the cell area where the receiving SINR is smaller than a specific threshold. It is also exactly the cumulative distribution function (CDF) of the SINR over the entire network. The outage probabilities of the different Cases are analyzed in the following. As to Case 1, we have the following theorem.
The average outage probability of the user connected to the -th tier () in Case 1 is
See Appendix -G.
For the special scenario where the interference is dominant, i.e., , we have
The average outage probability of the user connected to the -th tier () in Case 2 is
where and is as small as .
See Appendix -H.
When the interference is dominant, we get
When the interference is dominant, the average outage probability of the user connected to the -th tier () in Case 3 is
See Appendix -I.
From Theorems 4, 5 and 6 we see that, similar with the average egodic rate, the average outage probability is not affected by which tier the user connects to when the interference is dominant. As to users in Case 1, when the fraction of cache-enabled users is small, the interference and the desired signal power change by the same amount with the change of the transmit power or the node densities. However, the unnecessary interference triggered by D2D TXs depraves the outage probability in Case 2 and Case 3. Moreover, let be the outage probability of the user in Case 4, which is as small as 0 because of the immediate reading from the local caching disk.
V The Throughput and The Delay
We have analyzed the performance metrics from the perspective of a single user. Based on the analysis results of previous sections, the throughput of the entire network will be derived in this section. The delay and the critical condition for the network to keep steady will be elaborated.
We conduct analysis in a typical BS cell. According to the PPP model for the node locations, the average number of users in a typical BS area is . We now introduce the traffic dynamics of request arrivals and departures. Requests of users are considered as a unified event and modeled as a Poisson process with parameter requests/s, i.e., the request interarrival times are exponentially distributed random variables with mean seconds . It implies that requests of a single user is a Poisson process with parameter requests/s. The arriving requests require to access some sets of contents. Volumes of the sets are independent exponentially distributed random variables with mean contents/request, and the request interarrivals and request sets are independent . We define as the total request arrival rate and as the total traffic demand (in bits/s) in the typical BS cell.
Based on Table I, each element of matrix represents the probability that a randomly chosen user is active in the state of . Therefore, for and can represent the state of the user with the mapping . Then the density of users in the state of is . Corresponding to the consideration in Section III that service nodes are in the full load state, in this section, BSs, relays and D2D TXs without user being served are assumed to make dummy transmissions which bring interference to others as well . Consider Hz bandwidth are shared among different tiers. Let element of matrix denote the average ergodic rate of the user in the state of . is generated by
where is the indicator function and is the conversion factor between nats and bits. is the average ergodic rate analyzed in Section III and . We consider , and (for ) as just like we define the matrix even though no user is active in these states, and these virtual variables are defined to simplify the description. Due to the delay caused by the additionally wired transmission process and the limited backhaul, we assume the users can get the content with the service rate of when the backhaul is needed, which is a function of and is smaller than .
In the coverage of a D2D TX (a relay, a BS, a cache-enabled user itself), the average number of users who are in the state of is  for and , where . Up to now, we can divide users associated to a D2D TX (a relay, a BS, a cache-enabled user itself) into classes based on the -th column of the matrix . The corresponding class request arrival rate and class traffic demand are respectively and for and . Similarly, let be the number of requests in class of the queue at a D2D TX (a relay, a BS, a cache-enabled user itself) for . And let be the vector counting the number of user requests in each class. The orthogonal transmission is assumed, where user requests arriving at a service node are served one after the other in a round-robin manner with equal portion of time. We may view a service node as a processor, where 8 classes of user requests with different arrival and service rates are queueing to be served. So the request arrivals and departures at a D2D TX (a relay, a BS, a cache-enabled user itself) can be regarded as a multiclass processor-sharing queue.
Denote as the set of classes. Let the process describe the number of user requests in different classes of the queue at a D2D TX (a relay, a BS, a cache-enabled user itself) at time for . Then has discrete state space and is a continuous-time Markov process which can be generated by:
where represents the vector of whose -th element is 1 and 0 elsewhere. represents the total number of users in the queue. As to a steady network, the number of the requests leaving and arriving at the cell should be equal in the long run, i.e., the throughput is equal to the traffic demand. Define the throughput per request as the ratio of the given throughput (i.e., the traffic demand) by the mean number of user requests for a steady system . Then,
The mean number of user requests, the throughput per user request (Thr./Req.), and the delay of the -th class () at a D2D TX (a relay, a BS, a cache-enabled user itself) for are respectively given by,
where can be considered as the total traffic demand in the queue at a service node. is a critical value such that the queue will be at the steady state when . And the mean number of user requests, the Thr./Req., and the delay in the queue at a service node respectively are,
where is considered as the total traffic arrival rate of the queue at a service node.
The results can be deduced from  and the references therein, and the proof is omitted in this paper to avoid the unnecessary repetition.
Theorem 1 indicates that when the interference is dominant and , the service rates keep constant despite the change of the density of the BS/relay/D2D. It may lead to a misconception that the infrastructure can be deployed as scattered as possible. However, Proposition 2 reveals the critical condition to keep the system steady, i.e., should be satisfied when arranging the network. We call steady ruler of the network. The critical condition decides the maximum load/throughput of the system, e.g., the maximum arrival rate () of the request,
Besides, Proposition 2 points out the maximum ratio of for the network planning. Apparently, smaller densities of network infrastructures means more user connection per BS/relay/D2D, which will destruct the steady state of the queue and lead to the request congestion.
Analyzing Proposition 2 we observe that, larger content size , arrival rate and are not helpful for the improvement of the performance. Higher service rate () is important for the smooth departure of requests, yielding the performance improvement in terms of the Thr./Req. and delay. Moreover, the network performance highly depends on the number of users in each class (), which are determined by the transmission power (), node density (), content popularity (), caching ability () and association protocol. Specifically, when the user is able to obtain contents from its local caching immediately, the Thr./Req. (the delay) tends to infinity (zero) for the fact that the value of is extremely high.
Vi Numerical Results
In this section, we simulate the cache-enabled network to verify the performance of the proposed system. We obtain the results with Monte Carlo methods in a square area of , where the nodes are scattered based on independent homogeneous PPPs with intensities of . The transmit powers are dBm and 20 MHz bandwidth are shared among different tiers. We set the path-loss , total number of contents , the size of each content Mbits, the caching ability and , and the content popularity . These typical parameters do not change unless additional statements are clarified.
As illustrated in the subfigure (a) of Fig. 2, the probability for the user to obtain contents from the D2D TX or its local caching space becomes higher with the increase of . Smaller fraction of users need to access the relay (Relay-total in the figure) or BS with more “concentrated” contents. It implies that cache-enabled network reduces the cell load of the BS tier and the relay tier. The number of users accessing content from the caching space of relays (Relay-cache in the figure) increases first and then deceases because of the traffic offloading ability of the D2D tier. In the subfigure (b), the probabilities of Case 1 and Case 4 increase with as more contents can be obtained via the D2D link or from the local caching, yielding the increase of the cache hit rate.
The theoretical estimates and simulating results of the the average ergodic rates in Case 1-3 are illustrated in Fig. 3, and they are consistent well. We obtain and with the parameters in subfigure (a) based on (14). The rate in Case 1 keeps constant when changes in the region and it starts to increase obviously from . It highly depends on whether all of the cache-enabled users are active as D2D TXs. We observe that the number of active D2D TXs increases linearly with when as all of the cache-enabled users need to be active as D2D TXs. Only a part of cache-enabled users are active as D2D TXs when and it comes to the maximum number when . As to Case 2, the rate decreases with the increase of in the region owing to the increase of D2D TXs, then it increases slightly after for the number of active D2D TXs goes down. The rates in Case 2 and Case 3 are smaller than those in Case 1. It is because the users of Case 2 and Case 3 can not obtain any benefit except for the unnecessary interference from D2D TXs. With the parameters in subfigure (b), we get and . It means the transmit power and coverage of the D2D link are limited, thus only parts of cache-enabled users are active as D2D TXs for an arbitrary . The number of active D2D TXs increases with the increase of when . Consequently, the rate of Case 1 increases but those of Case 2 decrease with the increase of in subfigure (b).
The outage probabilities of different Cases are demonstrated in Fig. 4. The outage probability decreases with the decrease of SINR target , where lower SINR target means more interference is allowed. Similar to the average rate, the outage probability of Case 1 keeps constant before goes to and then decreases obviously. Case 2 and Case 3 have higher outage probability compared with Case 1. Fig. 4 can be explained from another perspective with Fig. 5 where the CDF of the SINR are demonstrated. As an example, for SINR dB in Fig. 5, the value of the CDF of Case 2 is smaller than that of Case 3 when , while the former approximately equals to the latter when . It conforms to what is illustrated in Fig. 4. Moreover, both and are smaller than , so the CDF of SINR for Case 1 when coincides with that when in Fig. 5.
Fig. 6 compares the Thr./Req. of the cache-enabled network with that of the baseline. As in the subfigure (a), the Thr./Req. of class 1 (class 3) at the BS is higher than (approximately equals to) that in the baseline. Meanwhile, because of the strong unnecessary interference triggered by the D2D TXs, class 5 has lower Thr./Req. than that of the baseline. The other virtual classes whose Thr./Req. are zero are not illustrated in the figure. Similarly, The performance of the -th class at the relay is better than or approximately equals to that in the baseline, while class 5 and class 6 get worse. Additional process is needed for the relay to fetch the uncached contents via the wired backhaul link, so the Thr./Req. of class 4 (class 6) is smaller than that of class 3 (class 5). In Fig. 6 (c), we compare the Thr./Req. at the D2D TX with that at the BS in the baseline. The Thr./Req. of D2D TX outperforms that of BS in the baseline by 46.8%-58.1%. D2D TXs give rise to interference, yet at the same time traffic loads of BSs and relays are offloaded by D2D TXs and the caching resources. Consequently, the Thr./Req. in the queue at the relay and BS are not seriously affected by the interference, while the throughput over the entire network increases significantly because of the increase of the number of the simultaneously active nodes and the tolerable request arrival rate.