Research Article  Open Access
Jiajie Ren, Demin Li, Lei Zhang, Guanglin Zhang, "Scaling Performance Analysis and Optimization Based on the Node Spatial Distribution in Mobile ContentCentric Networks", Wireless Communications and Mobile Computing, vol. 2021, Article ID 8880015, 17 pages, 2021. https://doi.org/10.1155/2021/8880015
Scaling Performance Analysis and Optimization Based on the Node Spatial Distribution in Mobile ContentCentric Networks
Abstract
Contentcentric networks (CCNs) have become a promising technology for relieving the increasing wireless traffic demands. In this paper, we explore the scaling performance of mobile contentcentric networks based on the nonuniform spatial distribution of nodes, where each node moves around its own home point and requests the desired content according to a Zipf distribution. We assume each mobile node is equipped with a finite local cache, which is applied to cache contents following a static cache allocation scheme. According to the nonuniform spatial distribution of cacheenabled nodes, we introduce two kinds of clustered models, i.e., the clustered grid model and the clustered random model. In each clustered model, we analyze throughput and delay performance when the number of nodes goes infinity by means of the proposed cellpartition scheduling scheme and the distributed multihop routing scheme. We show that the node mobility degree and the clustering behavior play the fundamental roles in the aforementioned asymptotic performance. Finally, we study the optimal cache allocation problem in the two kinds of clustered models. Our findings provide a guidance for developing the optimal caching scheme. We further perform the numerical simulations to validate the theoretical scaling laws.
1. Introduction
During recent years, wireless traffic is undergoing explosively increase due to the subscribersâ€™ enormous data demands (such as video streaming). Contentcentric networks (CCNs) [1] have emerged as a promising solution to deal with the increasing traffic, which shifts the traditional hostoriented communication pattern to the novel contentoriented communication pattern. In CCNs, nodes or user terminals are allowed to cache and forward contents based upon their names rather than the host addresses. This enables users request desired contents by local communications, without communicating with backhaul links to the core networks, which also reduces the delivery time of desired contents. In this context, as the number of users continually grows, the scaling performance of contentcentric networks has attracted research interests, which is important to help us understand the scalability of CCNs.
In the pioneer work of Gupta and Kumar [2], they first study the scaling behavior of largescale wireless ad hoc networks. In a static unit network consisting of randomly distributed nodes, Gupta and Kumar [2] shows the asymptotic throughput of each node scales as (Given two nonnegative functions and : means there exists a constant such that for large enough; if ; means both and ; means ; and means .), which indicates the poor scalability of wireless networks as the number of nodes increases. In [3], Franceschetti et al. apply the percolation theory to improve the asymptotic performance of wireless networks and pernode throughput is achieved as . Grossglauser and Tse [4] first take the node mobility into consideration, and they propose a twohop relaying policy to obtain the constant pernode throughput, which costs a vast transmission delay. Subsequently, a series of researches focus on the scaling laws of various wireless ad hoc networks. Talak et al. [5] investigate the broadcast capacity and transmission delay in highly mobile wireless networks. In [6], Lin et al. investigate the optimal throughputdelay tradeoff under the i.i.d mobility model for mobile ad hoc networks. Jia et al. [7] introduce the correlated mobility into the analysis of throughput and delay. They find that strong correlation of node mobility results in poor asymptotic performance. The nature of different mobility models [6â€“9] affects the network performance remarkably. In addition, there are several technics to improve the performance of wireless networks, such as directional antennas [10â€“12], infrastructure support [13â€“15], secrecy analysis [16, 17], and reinforcement learning [18, 19].
In light of the asymptotic analysis of traditional wireless ad hoc networks, the throughput and delay scaling behaviors of largescale cacheenabled contentcentric networks have also received wide attention in recent years. In a static square network, Gitzenis et al. [20] formulate a joint optimization problem for content replication and transmission. They derive the minimum link capacity by utilizing a Zipf popularity law. In [21], by assuming that the content cached time in a node is finite, Azimdoost et al. investigate the throughput and delay of contentcentric networks for the static grid network model and the static random network model, respectively. Jeon et al. [22] study the pernode throughput for wireless static devicetodevice networks based on a decentralized caching scheme. In [23], Mahdian et al. analyze the scaling laws for pure static contentcentric networks and heterogeneous contentcentric networks, respectively. The authors further consider the optimal cache strategy for two kinds of networks. Zhang et al. [24] investigate the capacity of static hybrid contentcentric wireless networks. In [25, 26], the authors study how the content popularity impact the network throughput. Contrast to the static networks, [27, 28] investigate the optimal throughputdelay tradeoff for mobile contentcentric networks. Do et al. in [27] adopt a decoupling approach to achieve the optimal caching allocation in the hybrid mobile contentcentric networks. In [29], Alfano et al. investigate the throughput and delay performance of mobile contentcentric networks with limited cache space. They find that the stronger mobility degree of nodes results in poor network performance. In [30], Luo et al. introduce fast and slow mobility models into mobile contentcentric networks according to different time scales. Then, the authors analyze the asymptotic performance with an arbitrary content popularity distribution.
The distribution of all nodes in the aforementioned literatures is uniform regardless of the traditional ad hoc networks or the contentcentric networks. However, the node distribution or the mobility degree is nonuniform in the real world. For example, the density of nodes around hot spots or home points is relatively intensive. The mobile terminals are more likely to move around the airports or the tourist attractions. Motivated by the above considerations, the researchers begin to investigate the nonuniform wireless traffic [31â€“34]. Alfano et al. [31] first analyze the upper bounds to the pernode throughput of inhomogeneous static wireless networks, where the nodes are distributed following a shotnoise cox process. In [32, 33], Garetto et al. investigate the scaling behaviors of heterogeneous mobile ad hoc networks under the assumption that each node moves around a home point. The interesting results in [32, 33] show that the mobility degree directly affects the data exchange among the nodes. In [34], the authors study multicast capacity of heterogeneous sensor networks, jointly analyzing sensor energy efficiency. In contrast with [31â€“34], Zheng et al. [35] study the asymptotic performance of inhomogeneous static informationcentric networks with infrastructure support. However, the mobility of nodes is of great importance in lots of application scenarios like vehicular ad hoc networks. It is meaningful to understand the impact of mobility on the scaling performance in the contentcentric networks where node spatial distribution is nonuniform.
To this end, we investigate the scaling laws of mobile contentcentric networks where node distribution is nonuniform in this paper. Instead of assuming each node moving the network area uniformly and independently, we consider that all nodes move around their corresponding home points, and they spend more time in the proximity of their home points than the network boundary. We assume each mobile node is equipped with a finite local cache and is able to store contents by using a static cache allocation scheme. Each cacheenabled node requests the desired content based on a Zipf popularity distribution. Moreover, we introduce two clustered models based on the spatial distribution of cacheenabled nodes, i.e., the clustered grid model and clustered random model. In each clustered model, we analyze the asymptotic performance according to the proposed scheduling and routing schemes. Finally, we study the optimal cache allocation problem of the two clustered models, respectively. We also perform the numerical simulations to validate the theoretical scaling laws. The main contributions of this paper are summarized as follows: (i)Firstly, we construct a novel system model in the contentcentric networks where each cacheenabled node moves around its corresponding home point. Based on the deployment of home point, we formalize the spatial distribution of nodes into the two clustered model(ii)Secondly, we devise a cellpartitionbased TDMA scheduling scheme to maximize the concurrent transmissions and develop a distributed multihop routing scheme. On this basis, we derive the asymptotic performance of the two kinds of clustered models and further establish the closed form of the throughputdelay tradeoff(iii)Thirdly, we design the optimal cache allocation and investigate optimal throughput and delay performance by utilizing Lagrangian relaxation method under the assumption of a Zipf content popularity distribution. Moreover, massive numerical simulations are conducted to validate aforementioned theoretical results
The rest of the paper is organized as follows. In Section 2, we describe the system models and outline some definitions. In Section 3, we analyze the throughput and delay performance in both clustered grid model and clustered random model. In Section 4, we introduce the optimal cache allocation. Finally, we conclude the paper in Section 5.
2. System Models and Definitions
In this paper, we study a mobile contentcentric network composed of mobile nodes. We assume that nodes move over a square region of area with the wrapcondition, to eliminate border effects. Note that, under aforementioned assumption, we adopt an extended network model that the node density over the square area remains constant as the number of mobile nodes increases. In the following subsections, we first describe the mobility model, content request model, and interference model. Then, we give some important definitions and notations used in this paper.
2.1. Mobility Model
In this paper, we adopt a bidimensional i.i.d mobility model for each mobile node, and time is divided into slots of equal duration. At the beginning of each slot, every node moves to a new location, which is independent of other nodes, and stays in the new location for the remaining duration of the slot. Let denote the location of node at time , and denotes the distance between node and node .
To characterize the spatial distribution of nodes, we assume that each node is associated with a home point , which is uniformly and independently selected over the square region . We consider that a node moves independently around its home point following a general ergodic process, which can be described by a rotationally invariant spatial distribution . Here, denotes the Euclidean distance between the mobile node and its corresponding home point.
We further assume that is an arbitrary nonincreasing function that decays as a power law of exponent , i.e., . We take function and consider the following normalized probability density function (PDF) over the whole mobility area to avoid convergence problems in proximity of the home point.
Exponent reflects the mobility degree of nodes, that is, the probability that each node moving to one point of the network area and stay at that point at a given time slot is not uniform. According to the PDF , it indicates that each node moves in a limited region, differing from the global mobility. In addition, the probability for the node moving in the proximity of its corresponding home point is larger than the probability that the node moves to the relatively far network area.
However, exponent just describes the individual mobility behavior and cannot reflect the mobile node density over the network area at a time slot. In real mobile world, the mobile node density is tightly related with the number of home points. Mobile nodes are more likely to move around the hotspots [36] or social spots [37]. For example, the density of mobile terminals (e.g., phones and ipads) is relatively dense in the office buildings or the tourist attractions while the density of mobile terminals in the suburb is reversely sparse due to the less home points. That is, the number of home points will affect the spatial distribution of mobile nodes. Motivated by this fact, clustering behavior has been found in [32, 33, 38] based on the longterm observations. In our work, we introduce the clustered model combined with the distribution of home points. First, we assume clusters and each cluster has a middle point (the center of the cluster), which are distributed over the area independently. Then, each home point randomly chooses one of the clusters with equal opportunity. Finally, the home points of the same cluster are belonged to a disk of radius centered at the cluster middle point.
Considering the asymptotic performance, we assume that , and the cluster density over the whole network area . The average distance between two cluster middle points is . For the ease of analysis, we do not consider the cluster overlapping behavior. Hence, the cluster radius should be satisfied by .
2.2. Content Request Model
In the mobile contentcentric networks, there are distinct content objects of same size, where . We assume each mobile node is equipped with an equalsized local cache, which can store content objects ( is a positive constant). For the problem to be not trivial, we assume , that is, each node has to decide which kinds of content to cache. We refer to a mobile node requesting a desired content at any time slot as the requseter of content . We call a mobile node carrying a content in its local cache the holder of content .
In this paper, a caching scheme consists of two phases: content placement and content retrieve [22]. In the content placement phase, each node randomly and independently chooses contents to be stored in its local cache. Let denote the set of nodes that cache content in their local caches, where . Thus, the probability that the content is cached by a mobile node in its local cache is . In order to achieve a feasible cache allocation, the total cache constraint should be satisfied.
In the content retrieve phase, each holder decides whether to deliver the requested content to the corresponding requester. During the retrieve phase, each node requests its desired content independently according to a Zipf popularity distribution [39], i.e., the request probability of content is satisfied by where is the Zipfâ€™s law exponent and is a normalization constant and is given by
2.3. Interference Model
In this paper, to avoid multiuser simultaneous transmission interference, we adopt the protocol model in [2]. Moreover, we assume that the transmission range of mobile node is . If the content is transmitted from mobile node to node successfully, then the following two conditions should be held: (1)The distance between the transmitter and the receiver is no more than , i.e.,(2)Other transmitter delivering different contents at the same time slot does not interfere the receiver , i.e.,
Here, denotes a constant guard factor. denotes the location of a mobile node at the time slot . denotes the Euclidean distance between the transmitter and the receiver. We further assume that each mobile node can deliver the contents at a constant rate bits/sec.
2.4. Definitions
Definition 1. For a given scheduling and routing scheme, let be the total number of bits of the requested contents received by mobile node up to time . We define the longterm throughput of mobile node is
The average throughput over all nodes is given as
The throughput is defined as the expectation of the average throughput over all mobile nodes,
Definition 2. The delay of a content retrieve process is the moment the interest packet leaves node until the requested content arrives at node from the closest holder. For a given scheduling and routing scheme, let be the delay of the th requested content of mobile node . We define the longterm delay of mobile node is
The average delay over all nodes is given as
The delay is defined as the expectation of the average delay over all mobile nodes.
To facilitate the understanding, some important notations applied in this paper are listed in Table 1.

3. Throughput and Delay Analysis
For better understanding of content placement and retrieve process in the clustered random model, we first investigate the throughput and delay performance in the clustered grid model [33] in Section IIIA, that is, the home points are placed regularly, which can be considered as a special instance of the distribution of the home points. In the real world, the home points can be deployed regularly, for example, the base stations (BSs) and the road side units (RSUs). In general, the BSs are deployed in the center of the cells, and the RSUs are deployed along the road (i.e., one RSU every 50 meters), which can be regarded as the examples of the grid clustered model. In Section IIIB, we analyze the case of clustered random model. Specifically, when and goes to infinity, the spatial distribution tends to 0; we leave to the analysis of the network performance in this case for future work.
3.1. Clustered Grid Model
In the clustered grid model, we divide the whole network region into equalsized cells of area . The distribution of clusters should satisfy following two conditions: (i)The middle point of each cluster is located at the center of each cell, and the distance between two adjacent cluster middle points is (ii)The radii of all clusters equal to 0, i.e., the home points of the same cluster are gathered at their corresponding clusterâ€™s middle point (or the cell center).
Before introducing scheduling and routing schemes, we premise a necessary lemma that guarantees each cell has at least one mobile node at a time slot and the expected number of nodes in each cell is which guarantees the network connectivity.
Lemma 3. In the clustered grid model, assuming , for any , each cell has at least one mobile node and the expected number of nodes in each cell is in a given slot with high probability.
Proof. For an arbitrary cell, in a given slot, the probability that one node moves to any cell is , i.e., . Hence, the probability that each cell has at least one mobile node is
where the last inequality follows that for . Let and . By assuming , for any , we have . Note that when goes infinity, tends to 0; the upper bound of is 1. Then, we have
which tends to 1 when goes infinity. Hence, with high probability, each cell has at least one mobile node in a given slot.
Let denote the number of nodes in the cell , by definition:
where is an i.i.d Bernoullian random variable, and if node is in the cell ; otherwise, . Hence,
We apply Chernoff bounds and have
Then, letting , for any , we have . Thus, we have
which tends to 1 when goes infinity. Hence, we obtain the expected number of nodes in each cell is .
3.1.1. Scheduling Schemes
At a given slot, a scheduling scheme enables the contents retrieve between transmitterreceiver pairs not to be interfered. Based on the mobility model and the spatial distribution of mobile node, we adopt a cellpartitionbased TDMA scheduling scheme to avoid the multiuser simultaneous transmission interference and to maximize the number of noninterfering transmission pairs at a time slot. Figure 1(a) illustrates a general realworld node distribution case, which shows mobile terminals are relatively dense near the home point, and there are fewer mobile terminals far away from home point. Figure 1(b) illustrates home points are deployed regularly and can be regarded as a clustered grid model. Figure 1(c) shows a cellpartitionbased TDMA scheduling scheme corresponding to the time slot division, which guarantees multihop transmission in the clustered grid model.
We first divide each time slot into two half slots with equal length. Then, in the first half slot, we further divide the first half slot into several subslots with equal length. We partition each cell into squares with same area ( will be defined later). We call a square is active if a node in this square can transmit an interest packet or a content during the subslot. Based on the square partition, we assume that a node in a square can transmit an interest packet or a content to the node in the same square or the adjacent eight squares. We define the transmission radius of the node as . Hence, we obtain the following lemma according to the protocol model, which is essential to describe the noninterfering transmission process.
Lemma 4. Each square can be active at most every time subslots, and the achievable rate at the active square is .
Proof. According to the interference model, we consider the distance between transmitter and receiver is bounded by . If there is another simultaneously transmitting node at the same time subslot and the transmission distance between and is less than , then will cause the interference with . Hence, the area of the total interference region is bounded by . We obtain that each square has at most = interference neighbouring squares, which is a constant and independent of .
Each square gets a noninterfering transmission opportunity and becomes active in every time subslots. Since each transmitterreceiver pair can send bits in a successful transmission, the achievable transmission rate at the active square is .
From Lemma 4, we can construct subsets of regularly spaced, simultaneously transmitting squares without interference.
Next, we calculate the area of each square . From Lemma 3, we obtain the average number of nodes in each cell is . We assume there exists point in the cell that the distance between and the home point is . The mobility process of each node is independent at each slot, and the probability that the node moves to point is at each slot according to the node distribution. Hence, when , the density of nodes at point is at each slot. Applying Lemma 4 in [33], we can obtain the maximum density of achievable transmitterreceiver pairs (TR pairs) when . Thus, the density of TR pairs at any point of the network scales as =. Therefore, the area of each square is equal to , i.e., .
At a given time slot, each cell can schedule the number of concurrent noninterfering transmissions is
Since the area of each square is , we can acquire that the probability that at least one node in a subcell is a constant by applying Lemma 3.
3.1.2. Routing Schemes
Considering the mobility model of nodes and the clustering behavior, we propose a multihop transmission scheme for the content transmission between the requesterholder pairs. We assume the content request and delivery process of each node is conducted in one time slot. According to the maximum hops (derived in the following section) in the clustered grid model, the value of the one time slot can be set as to guarantee the multihop transmission within one time slot.
1: according to the TDMA scheduling phase, each time slot is divided into two half slots with equal length.
2: during the first half slot, we further divide the first half slot into several subslots with equal length. Each cell is divided into squares with the same area, (1)if there is one requested content in the same square (or the same cluster), the requester directly forwards an interest packet to the holder(2)if there is no any requesterholder pair in the same square (or the same cluster), the requester sends an interest packet to the relay node in the adjacent square at the beginning of subslot; multihop is used for all relay nodes until the interest packet reached the nearest holder. Furthermore, we assume that the first hop towards the adjacent subcell is on the horizon path (if possible) and then on the vertical path
3: during the second half slot, we continue dividing the second half slot into several subslots with equal length,
(1)if there is one requested content in the same square (or the same cluster), the requested content will be directly forwarded back to the requester . In this case, the requested content will be achieved in one hop
(2)if there is no any requesterholder pair in the same square (or the same cluster), the nearest holder receives the interest packet in the first half slot, and then, the nearest holder in the second half slot sends the requested content back to its corresponding requester in the reverse direction
Lemma 5. In the clustered grid model, for any node requesting content , the probability that an interest packet for content is satisfied by one hop is .
Proof. For an arbitrary square, the probability that there is at least one requested content within the same square or the adjacent squares of the requester is . Thus, The inequality follows that for any . If , then ; otherwise, .
In the clustered grid model, due to the transmission radius of each node is , we can derive the maximum number of hops from a content requester to the closest holder is .
Lemma 6. In clustered grid model, for any mobile node requesting content , the average number of hops needed to transmit an interest packet along the path from the content requester to the closest content holder, denoted by , is
Proof. Let denote the number of hops along the requesting path, and we have
Based on Lemma 5 and the value of , we consider three different conditions to further derive .
(1)If , then .
(2)If and , . Hence, we obtain that .
(3)If and , applying the equivalent infinitesimal, we obtain that . Hence, we get that .
This completes the proof.
Theorem 7. In the clustered grid model, the delay and throughput of each mobile node, denoted by and , are given by w.h.p.
Proof. First, we consider the delay performance. Based on the scheduling and routing scheme, each square can be active in every time subslots, where is a constant. Thus, the time spent in each hop is a constant fraction of time. In Lemma 6, we obtain the average number of hops from a requester to the closest holder for content is . In addition, the requested content forwarded back to the requester takes the same route as its corresponding interest packet in reverse direction, the total number of hops for content is . Taking the request probability for the content into consideration, we get the delay for all contents over the network as (6).
Next, we derive the throughput performance. Since there are cells over the network area, and the total number of bits that active squares in a cell can transmit is . Lemma 6 indicates the average number of hops for content is . Considering the request probability for the content , we obtain the average number of bits transmitted in the whole network is . Then, we have
The achievable throughput of each mobile node is determined by the scheduling and routing scheme. Hence, we obtain the throughput performance as (24).
Corollary 8. The throughput and delay tradeoff in the clustered grid model is given by
3.2. Clustered Random Model
In this section, we analyze the throughput and delay of each node in the clustered random model, where clusters are distributed randomly and independently in the network, that is, the home points are deployed in a random manner.
We first introduce an important lemma given in [31]. This lemma indicates the appropriate partition of the whole network area can guarantee that the average number of clusters in each cell is , which makes the multihop transmission in the clustered random model possible.
Lemma 9. Consider a set of points independently distributed over a bidimensional domain of area , with density . The domain is partitioned by regular tessellations, and denotes the tiles over the tessellations with area , . Let be the number of nodes falling within the tiles . Then, uniformly over the tessellation, is contained between and with high probability, i.e.,
This lemma can be proved by the Chernoff bound in [40]; we neglect the proof process for simplicity.
Considering Lemma 9 and the cluster density , we partition the whole network into equalsized cells with area =. Note that we choose the equalsized cell area in the clustered random model in order to make sure that nodes in each cluster has the opportunity to retrieve the content; in fact, we can choose different size area only if . Thus, we can obtain the expected number of clusters in each cell is . We assume that the transmission radius of each mobile node in the clustered random model is , which guarantees the nodes can communicate with the nodes in their neighbor cells. Since the maximum distance between the requester and the closest holder is , the maximum number of hops from a content requester to the closest holder is .
Based on the cell partition, we obtain the following lemma that guarantees the average number of nodes in every cluster is .
Lemma 10. In the clustered random model, the average number of node in every cluster is .
Proof. The proof process of this lemma is similar to that in Lemma 3 by applying he Chernoff bound.
The density of TS pairs at any point of the network scales as . We further partition each cell into squares with area , i.e., . Hence, each active cell can schedule the number of concurrent noninterfering transmissions is
Hence, a noninterference scheduling scheme in the clustered random model can be designed as following steps. First, we divide the whole network into equalsized cells with area , which makes the transmission connectivity possible. Then, we partition each cell into equalsized squares with area , which guarantees the content transmission free from interference. Figure 2 shows the cell partition and the content request process in the clustered random model. Finally, a routing scheme similar with that in the cluster grid model can be utilized to achieve the content multihop transmission at a given slot.
Lemma 11. In the clustered random model, for any node requesting content , the probability that an interest packet for content is satisfied by one hop is .
Proof. For an arbitrary square, the probability that there is at least one requested content is within the same square or the adjacent squares of the requester is . Thus, we have The inequality follows that for any . If , then . Otherwise, .
Lemma 12. In the clustered random model, for any mobile node requesting content , the average number of hops needed to transmit an interest packet along the path from the content requester to the closest content holder, denoted by , is
Proof. According to Lemma 6, we have
Based on the value of and , we consider three different conditions to further derive .
(1)If , then
(2)If and , . Hence, we obtain that =.
(3)If and , applying the equivalent infinitesimal, we obtain that . Hence, we get that =.
This completes the proof.
Theorem 13. In the clustered random model, the delay and throughput of each mobile node, denoted by and , are given by w.h.p.
Proof. The proof of delay performance is similar to that of Theorem 7, and we do not repeat it for simplicity.
We analyze the throughput performance of each node. Based on the cell partition of the network, we can obtain the total number of cells is . Considering the scheduling scheme and routing scheme, the total number of bits that active square in a cell can transmit is . Moreover, Lemma 12 indicates the average number of hops for content is . Since the request probability of the content is , the average number of bits transmitted in the whole network can be calculated as . The rest proof is similar to the Theorem 7; we neglect it here for simplicity.
From Theorem 13, we can conclude that the difference of throughput and delay performance between the clustered random model and the clustered grid model is the logarithm factor and the average transmission hops.
Corollary 14. The throughput and delay tradeoff in the clustered random model is given by which differs from Corollary 8 by a factor .
4. Optimal Cache Allocation
In this section, we analyze the optimal throughput and delay performance of the mobile contentcentric network with respect to optimal cache allocation strategy. To achieve the optimal goal, we need to select appropriate based on the cache constraints for the clustered grid model and clustered random model, respectively.
4.1. Clustered Grid Model
From Corollary 8, it indicates that minimizing the transmission delay is equivalent to maximizing the throughput performance. In order to achieve the minimum delay, we formulate the following optimization problem: where is given by
The first constraint in (34) comes from the feasible cache allocation in (2), and the second constraint guarantees that for different types of contents, there is at most one copy content in each square. Note that the second derivatives of the objective function in (34) is always positive, which can be considered as a strictly convex optimization problem. Hence, we apply Lagrangian relaxation method to find the unique optimal solution.
Based on the Zipf distribution law, it shows that decreases as increases and so is . For the convenient and tractable analysis, we first define three sets , , and according to the size of , respectively. That is,
(i)let be the set of contents such that
(ii)let be the set of contents such that
(iii)let be the set of contents such that
Next, we take the Lagrangian multiplier for the first constraint in (34) and combine with the second constraint; the necessary conditions for the minimal are given as
When , we obtain
By adding up in (37) for , we have
Combined (37) and (38), we obtain
where . Therefore, we obtain the optimal number of content , denoted by (In this paper, we apply symbol to denote the optimal value.), i.e.,
By substituting (40) into (34), we achieve the minimum delay, denoted by .
Lemma 15. In the clustered grid model, for , the values of and is given as follows
Proof. By applying the condition , we derive that
When , we obtain that . Since , according to (40), we have . Hence, it follows that
From the fact that is the smallest index in set such that , we decrease by one, which will result in . Hence, we obtain that
Combining (44) and (45), for , we obtain the approximation value of scales as
Similarly, we can derive the approximation value of . We know that is the smallest index in set such that , which result in
decreasing by one, we have , which leads to
Therefore, combining (47) and (48), we obtain the approximation value of scales as
The value of and is determined by the different value of . Therefore, we have following three conditions.
(1)For : We can simplify (46) as
which results in
According to (43), can be obtained by
(2)For : We assume , and (49) can be simplified as
which follows that
resulting in . This contradicts . Thus, we obtain .
Assuming , we can simplify (46) as
by using (43), we obtain .
(3)For : Assuming , we can simplify (49) as
which results in
This contradicts . Hence, we have .
By simplifying (46), we have
resulting in .
Theorem 16. In the clustered grid model, letting , according to the Zipf distribution, the optimal delay and throughput are given by w.h.p.
Proof. We first prove the optimal delay based on the values of and . Then, the optimal throughput can be easily derived by the tradeoff relation given by (26). Substituting the value of into (41), we have
We divide the RHS of (60) into three terms, which are denoted by ,, and , respectively. Then, we have following five conditions according to the values of , , and .
(1)For : We have