Ensemble Learning-Based Edge Caching Strategies for Internet of Vehicles: Outage and Finite SNR Analysis

In this paper, an ensemble learning-driven edge caching (ELDEC) strategy and a meta-based ensemble learning-driven edge caching (MELDEC) strategy are proposed for content popularity prediction and cache content placement in Internet-of-Vehicles (IoV) networks. Specifically, the proposed MELDEC and ELDEC strategies incorporate meta learning and ensemble learning for enhanced content popularity prediction in IoV networks. Closed-form outage probability and finite signal-to-noise ratio (SNR) diversity gain expressions are also derived to establish the relationship between the proposed edge caching strategies and the wireless performance of IoV networks. When compared against benchmark schemes, the proposed MELDEC and ELDEC strategies achieve near-optimal cache hit rates, outage probability, and finite SNR diversity gain under imperfect channel state information (CSI) estimation. We also show that the outage probability decay rate in the IoV network depends on the number of base stations and roadside units, and it is independent of the content popularity prediction of the MELDEC strategy, ELDEC strategy, and benchmark schemes. The performance analysis demonstrates that the proposed MELDEC and ELDEC strategies are promising solutions towards achieving reliable content access in IoV networks.


I. INTRODUCTION
R ECENT interests to support connected vehicles (CVs), road side units (RSUs), and cellular base stations (BSs), as part of next-generation intelligent transportation systems (ITSs) have necessitated a transition towards Internet-of-Vehicles (IoV) networks [1], [2], [3]. Such IoV networks are envisioned to be able to meet the mission-critical requirements of ITS applications. However, one challenge in realizing IoV networks lies in the fact that the intensity of information exchanged in ITS applications ultimately results in significant backhaul overheads [4]. Apart from backhaul overheads, developing frameworks for accurate outage and finite signal-to-noise (SNR) characterization and identification of performance bounds in IoV networks remain an open research problem due to vehicular mobility, wireless channel environment, and the accurate modeling of RSU and BS placements for vehicular communications [5].
To reduce backhaul overheads, one can turn towards the concept of edge caching from the multi-access edge computing (MEC) paradigm [6], [7], [8]. The main idea is to employ edge caching in IoV networks to reduce overall backhaul overheads by proactively storing, i.e., caching, popular contents at MEC servers located near CVs [4], e.g., cellular BSs or RSUs. In this aspect, the adoption of edge caching in IoV networks has been widely studied in the literature, e.g., [4], [5], with studies investigating the application of deep learning for edge caching and the performance of edge caching in IoV networks.

A. RELATED WORKS 1) DEEP LEARNING-BASED EDGE CACHING IN IOV NETWORKS
In the literature, studies employing deep learning-driven schemes for content popularity prediction and placement in edge caching have been noted in IoV networks. For example, the authors in [9] proposed an edge caching scheme based on cross-entropy to determine popular contents that should be cached at RSUs. An analysis showed that the proposed edge caching scheme achieves higher cache hit rates and lower latency than other benchmark non-learning schemes. In [10], deep reinforcement learning and particle swarm schemes are employed as part of an edge caching and resource allocation framework in IoV networks. Likewise, the authors in [11] propose a cooperative deep reinforcement learning-based edge caching scheme, while [12] proposed a convolutional neural network-based edge caching scheme to predict the popularity of infotainment contents.
It is evident from [9], [10], [11], [12] that the focus has largely been on enhancing the content popularity prediction accuracy. Yet, the proposed schemes can be computationally complex to implement in practice. In turn, this may lead to longer execution time and unnecessary backhaul overheads. Furthermore, the performance characterization of deep learning-based edge caching schemes to identify performance bounds in IoV networks remains unaddressed.

2) PERFORMANCE CHARACTERIZATION OF CACHE-ENABLED IOV NETWORKS
As noted earlier, the performance of cache-enabled wireless networks has been widely investigated in the literature. In [13], [14], the successful content delivery probability was analyzed for different cooperative schemes in cellular networks where closed-form expressions for coverage probability were provided and evaluated to identify performance bounds. Likewise, the content retrieval delay in cooperative cellular networks was analyzed in [15] while [16] characterized the successful content delivery performance for backhaul-limited cache-enabled heterogeneous networks. In [17], the successful content delivery probability was analyzed for reinforcement learning-based probabilistic caching with a priori knowledge of content popularity. Similar successful content delivery probability analysis was also conducted in [18] for online content popularity prediction techniques. In [19], an outage probability analysis was also noted for deep learning-based edge caching in device-todevice networks.
For IoV networks, the outage probability and content retrieval delay for in-vehicle caching was investigated in [20]. Similar studies on coverage probability analysis have also been noted in [4] and [5]. Apart from outage and coverage probability, finite SNR analysis is another useful tool to aid in the performance characterization of wireless networks. In particular, finite SNR analysis enables one to evaluate the finite SNR diversity gain of wireless networks [21], [22], [23], [24]. The finite SNR diversity gain of wireless networks effectively quantifies the outage probability decay rate for a given SNR [23]. With finite SNR analysis, one can uncover outage probability behaviors which are noticeable at only finite SNR regimes, e.g., SNRs below 30dBm. As most wireless networks operate at finite SNR regimes [24], finite SNR analysis has been shown to be useful in providing accurate outage probability characterizations. For instance, finite SNR diversity gain has been employed in unmanned aerial vehicle networks to determine interference-limited scenarios [21], [24].

B. MAIN MOTIVATIONS AND CONTRIBUTIONS
To address the above challenges, we propose an ensemble learning-driven edge caching (ELDEC) strategy that incorporates ensemble learning and a meta-based ensemble learning-driven edge caching (MELDEC) strategy which combines both meta and ensemble learning for edge caching in IoV networks.
Ensemble learning allows for the outputs of weaker learning models to be combined to enhance prediction performance [25], [26], while meta learning enables one to optimally combine the outputs of several weaker learning models [26]. By leveraging on meta and ensemble learning, simple deep learning models can be used for content popularity prediction in IoV networks. Specifically for the MELDEC strategy, meta learning is employed to optimally prioritize better performing deep learning models within an ensemble of base deep learning models for enhanced content popularity prediction. It is also worth noting that the amalgamation of meta learning with ensemble learning has not been widely investigated for content popularity prediction in edge caching. This is in contrast to the techniques noted in [9], [10], [11], [12], which have focused on enhancing content popularity prediction at the expense of potentially higher computational complexity.
A comprehensive outage and finite SNR analysis framework is also presented for the performance characterization of the MELDEC strategy, ELDEC strategy, and considered benchmark schemes in the IoV network. Specifically, outage probability and finite SNR diversity gain expressions are obtained in closed-form for the MELDEC strategy, ELDEC strategy, and benchmark schemes by extending the power series approaches proposed in [21] and [27] for IoV networks. The power series approach enables complicated functions to be expressed as a power series, leading to a rigorous analysis of the interplay between CV mobility, stochastic geometry, outage probability, and finite SNR diversity gain. To the best of our knowledge, such a performance characterization framework for deep learning-based edge caching schemes in IoV networks is unavailable in the literature.
Therefore, the main contributions of this paper are summarized as follows: • This paper proposes MELDEC and ELDEC for content popularity prediction and cache content placement in IoV networks, which to the best of our knowledge, has not been investigated in the literature. • Closed-form outage probability and finite SNR diversity gain expressions are derived for the proposed MELDEC strategy, ELDEC strategy, and benchmark schemes to characterize and identify performance bounds in the IoV network. • We demonstrate that the proposed MELDEC and ELDEC strategies achieve near-optimal cache hit rates, outage probability, and finite SNR diversity gain when compared against the benchmark deep learning-based technique in [28] and non-learning based benchmark schemes under imperfect channel state information (CSI) estimation. • When ideal CSI estimation is attained, we demonstrate through finite SNR analysis that the content popularity prediction accuracy of the MELDEC strategy, ELDEC strategy, and benchmark schemes plays a limited role in the overall outage probability decay. Instead, the outage probability decay is greatly influenced by the number of cellular BSs and RSUs, and the severity of small-scale fading in the IoV network. The remainder of this paper is organized as follows. The proposed system model is introduced in Section II, with the MELDEC and ELDEC strategies presented in Section III. The closed-form outage probability and energy efficiency expressions are derived Section IV. Finally, the results of the numerical analysis are presented in Section IV, with the conclusions and future research directions of the paper discussed in Section VI. A summary of key mathematical notations used in this paper is also provided in Table 1.

II. SYSTEM MODEL
In this work, we consider an IoV network comprising M RSU RSUs and M BS cellular BSs (Fig. 1). Specifically, we are interested in analyzing downlink cellular transmissions between the typical CV and the infrastructure when edge caching is adopted in the IoV network. 1 As part of the analysis, we assume that downlink cellular transmissions are impaired by imperfect CSI due to Doppler shift 1. Enhancing the performance of the IoV network through cache-enabled vehicle-to-vehicle links is being investigated as an extensions of this work.  from the mobility of CVs [29], [30] and small-scale fading effects [31], [32]. Furthermore, network coordination within the IoV network is assumed to occur either among RSUs or BSs through horizontal handovers, or between RSUs and BSs through vertical handovers [33].

A. STOCHASTIC GEOMETRY MODEL
To model the spatial locations of the RSUs and cellular BSs, the binomial point process (BPP) model is employed in this study. The BPP model is used to model the uniform distribution of distances in a finite-sized cell for a fixed number of nodes [34], [35]. In the context of IoV networks, the number of RSUs, cellular BSs and CVs operating in a geographical area is generally known in practice. Therefore, the BPP model is more suitable than Poisson point processes (PPPs) to model the distance distribution of nodes in an IoV network.
Let the typical CV associated to the RSUs or cellular BSs be at the origin O with a communication radius of r a . Then, the spatial location of the RSUs and cellular BSs are assumed to be uniformly distributed in a disc centered at the CV with radius r a and angle [0, 2π). The Euclidean distance between BS-i and the CV is denoted as d i , while the Euclidean distance between RSU-j and the CV is denoted as d j . From [34, eq. (3)], [27], the probability density function

B. MOBILITY MODEL
In the IoV network, the CV is associated with the cellular BS or the RSU with the strongest instantaneous signal-tointerference-plus-noise ratio (SINR) and CSI is estimated at the CV over a feedback period T [29], [30]. We denote h x , x ∈ {i, j} as the channel gain between node-x and the CV, h x as the estimated channel gain, e x as the CSI estimation error, and 0 < < 1 as the correlation coefficient between h x and h x+1 . Using Jakes' statistical model, is defined as [29], [30], [36], [37]: where J 0 is the zero-order Bessel function of the first kind [38, eq. (9.1.10)], f d = vf c c is the maximum Doppler frequency, f c is the carrier frequency, v is the vehicle speed in m/s, and c = 3 × 10 8 m/s. Then, we employ the first-order Gauss-Markov process to model h x such that [29], [30], [36], [37].

C. CHANNEL MODEL AND INSTANTANEOUS SINR
At the CV, the instantaneous SINR of cellular transmissions from BS-i (γ i ) or RSU-j (γ j ) is given as [29]: where l is the pathloss exponent and x ∈ {i, j}. We assume |ĥ x | 2 follows the Gamma distribution with average received powerP t 2 and Nakagami-m fading param- L 0 η is the normalised transmit power, P t is the transmit power, L 0 is the pathloss at 1 m reference distance [39], η = −174 + 10 log 10 (B w ) is the noise power in dBm, and B w is the bandwidth. Likewise, we assume |e Such an approach allows a multitude of small-scale fading effects to be modeled for both |ĥ x | 2 and |e x | 2 , e.g., Rayleigh fading [29], [30].

D. FILE POPULARITY MODEL
A finite library of K total files is assumed in the IoV network, where each file has a size of K size . CVs in the IoV network transmit file requests to both RSUs and cellular BSs such that the file requests are first transmitted to the RSUs, and if the requested file is unavailable at the RSU, then the file request is serviced by cellular BSs.
To model file popularity and placement in the IoV network, we assume that RSUs are equipped with MEC servers that have a cache size of K cache , and cellular BSs are equipped with file servers hosting the library of K total files such that 1 ≤ K cache ≤ K total [19]. Furthermore, it is assumed that the popularity of each file is unknown since file request patterns are dynamically changing in practice. However, for the sake of analysis, the popularity of the k-th file is assumed to follow a Zipf distribution and the request probability of file-k, 1 ≤ k ≤ K total is given as where z is the file popularity factor [13].
As the popularity of each file is unknown, the main aim of this paper is to propose an ELDEC strategy for IoV networks to enable file popularity prediction and caching at RSUs.

III. ENSEMBLE LEARNING-BASED EDGE CACHING STRATEGIES
In this section, detailed descriptions pertaining to the proposed ELDEC strategy, MELDEC strategy, and benchmark schemes are provided.

A. PROPOSED ELDEC STRATEGY
The proposed ELDEC strategy incorporates ensemble learning to predict the popularity of all K total files based on file request statistics from CVs in the IoV network. In ensemble learning, multiple base learning models are trained and the resultant outputs are amalgamated for improved prediction accuracy, e.g., [26]. On its own, such base learning models may produce lower prediction accuracy. However, if orchestrated properly, a set of base learning models can be applied to achieve greater prediction or classification accuracy [25]. One tradeoff to such an approach is the need to train additional base learning models, which can be computationally intensive and may lead to longer training time. However, in the context of IoV networks, this can be mitigated by distributing the training of base learning models across an IoV network to RSUs and cellular BSs with sufficient processing capability.
Therefore, we apply ensemble learning for edge caching by training an ensemble of K model deep learning neural network (DNN) models using the file request statistics. Thereafter, the prediction output from each DNN model is combined into a single prediction output for all K total files. Files with high predicted popularity are then cached at RSUs to improve the overall performance of the IoV network. Such a technique has also been adopted in similar studies, e.g., [26], [28].

1) DATASET GENERATION
In the IoV network, all file indices requested by CVs are tracked by RSUs and are periodically uploaded for content  popularity prediction over a wired backhaul link to a cloudbased deep learning platform. 2 The deep learning platform maintains a content demand table (X) with M dataset rows by aggregating all file indices which have been requested by CVs at each RSU. For the m-th row (X m ), i.e., the m-th sample or m-th row in X, the deep learning platform tabulates the vector X m = [x 0 , . . . , x k , . . . , x K total −1 ] from all RSUs to obtain the instantaneous file access pattern across the IoV network, where x k denotes the total number of times file-k was requested, before y m = X m is obtained as the ground truth label of X m (Fig. 2). 3,4 After populating X, M train training samples and M test testing samples are randomly selected from X for training and testing such that X m is used as the input to each DNN model. The objective behind such an arrangement is to predict the cache content score of file-k for content popularity prediction, where 0 ≤ k ≤ K total − 1.

2) ENSEMBLE ARCHITECTURE
An ensemble of K model DNN models are trained with the main goal of predicting the total cache content score for each file (Fig. 3). Together, the K model DNN models make up an ensemble architecture and each DNN model comprises an input layer, hidden layers, and an output layer. We employ fully connected layers in the DNN model, with a rectified linear unit, i.e., ReLU, used as the activation function for some of the hidden layers.
2. Examples of such platforms include Microsoft Azure and Google Compute Engine.
3. It is worth noting that the DNN models used in this work only require file index information for content popularity prediction. Future works will investigate utilizing additional information, e.g., SINR, CV mobility and, content lifespan [40], together with distributed learning-based or cooperative-based methods, e.g., [28], [41], [42] to enhance the accuracy of content popularity prediction. 4. In other words, X m indicates the total number of requests for file-k in the IoV network, which is obtained from all RSUs. Using the training and testing samples, each DNN model is trained to minimize the categorical cross-entropy loss function. Finally, softmax [43] is used as the activation function for the output layer and the Adam optimiser is employed for weight update. A description of the DNN model design is provided in Table 2, where a dropout fraction rate of 0.1 is used.

3) OFFLINE TRAINING AND DEPLOYMENT
The K model DNN models in the ELDEC strategy are trained offline and validated online using the cloud-based deep learning platform with M train samples and M test samples, respectively. Specifically, the vectorf is obtained during online validation using M test test samples for content popularity prediction. Starting with the s-th test sample for the ν-th DNN model, an output vectorȳ s,ν = [ȳ s,ν,0 , . . . ,ȳ s,ν,K total −1 ] with K total elements is obtained, whereȳ s,ν,k is the cache content score of file-k for the s-th test sample that is predicted with the ν-th DNN model. 5 Then, the vectorf is obtained as: where The vectorf is used to determine the total cache content score of each file based on the ensemble learning of the K model DNN models. As files with larger cache content scores are more likely to be requested, the top K cache files with the highest cache content scores inf are cached at RSUs, with the corresponding file index populated in a vectorf cache . A summary of the offline training and deployment process for the ELDEC strategy is presented in Algorithm 1 and illustrated in Fig. 3.

4) ESTIMATED CACHE HIT RATE
Based onf cache , the estimated cache hit rate of the ELDEC strategy P hit,EL is obtained as [44], [45]: 5. It is important to note that since M test testing samples are randomly selected from X, the s-th test sample is also a vector which is tabulated as where the total number of requests in the M test testing samples is computed in the denominator of (4), and 1(s) = x k if the k-th file requested in the s-th sample is inf cache .
After calculating P hit,EL and caching the K cache files at RSUs, the content demand table (X) is emptied and repopulated with new file index information aggregated from all RSUs until M dataset samples are collected. 6 Offline training and content popularity prediction are then conducted with the ensemble of K model DNN models, and P hit,EL recalculated, before the updated K cache files are cached at RSUs. In this way, the ELDEC strategy continues to predict popular contents which are relevant in the near future even under time-varying content request statistics. 7

B. PROPOSED MELDEC STRATEGY
It is worth noting that the ELDEC strategy considers all DNN models equally when computing the predicted total cache content score of each file, i.e.,f. Therefore, we propose the MELDEC strategy that incorporates meta learning and ensemble learning to further improve content popularity prediction accuracy. The main idea behind the MELDEC strategy is to employ a meta model within a meta-based ensemble learning architecture to aid in prioritizing better performing learners from within an ensemble of base learners such thatf more accurately reflects actual content popularity.

1) META-BASED ENSEMBLE LEARNING ARCHITECTURE
An ensemble of K model,meta DNN models are trained for content popularity prediction using the design described in 6. The time taken to collect M dataset samples depends on the number of RSUs and CVs in the IoV network due to the frequency of file requests. Thus, popular content prediction and distribution can be conducted more frequently in a larger IoV network compared to a smaller IoV network.
7. It is worth noting that P hit,EL can be further improved by caching a mix of contents that are popular or have large content diversity in the ELDEC strategy, as was done in [13], which will investigated in future extensions of this paper.   Table 2. The output from each of the K model,meta DNN models are then aggregated as inputs into a meta model which comprises an input layer, hidden layers, and an output layer. The meta model employs fully connected layers, with ReLU used in the hidden layers. The resulting meta-based ensemble learning architecture comprises K model,meta DNN models as the first-level learner and the meta model as the second-level learner.
Using the training and testing samples, the meta-based ensemble learning architecture is trained to minimize mean squared error. Finally, sigmoid [43] is used as the activation function for the output layer and the Adam optimiser is employed for weight update. The proposed meta model design is described in Table 3.

2) OFFLINE TRAINING AND DEPLOYMENT
We adopt similar offline training and deployment steps seen in the ELDEC strategy for the MELDEC strategy. In particular, the K model,meta DNN models and the meta model in the MELDEC strategy are trained offline with M train samples and validated online with M test samples using the cloudbased deep learning platform. For each of the K model,meta DNN models, the output and input layers are aggregated and used as the inputs to the meta model and meta-based ensemble learning architecture, respectively. The meta-based ensemble learning architecture is then used for offline training, beforef andf cache are computed with M test samples. The top K cache files with the highest cache content scores inf are then selected to be cached at RSUs, with the corresponding file index populated inf cache . It is worth noting that the estimated cache hit (P hit,MEL ) for the MELDEC strategy is obtained using (4). A summary of the offline training and deployment process for the MELDEC strategy is presented in Algorithm 2 and illustrated in Fig. 4.

C. BENCHMARK AND OPTIMAL SCHEMES
We employ the deep learning edge caching (DLEC) strategy in [28] and non learning-based Least Frequently Used (LFU), Least Recently Used (LRU) as the benchmark schemes [44], [45] to benchmark the performance of the proposed MELDEC and ELDEC strategies in the IoV network. We also employ the oracle algorithm as the optimal scheme in this work [44], [45] 8,9 In particular, the LFU and LRU schemes respectively removes the least frequently used and least recently used file from the RSUs when the MEC server cache is full. For the oracle algorithm, the popularity of all K total files is assumed to be known a priori [45], i.e., the oracle algorithm is the optimal scheme to determine file popularity. Using (4), we obtain P hit,DL , P hit,LFU , P hit,LRU , and P hit,Oracle as the estimated cache hit rate for DLEC in [28], LFU, LRU, and the oracle schemes, respectively. The estimated cache hit rate P hit,u , u ∈ {EL, MEL, DL, LFU, LRU, Oracle} enables closed-form outage probability and finite SNR diversity gain expressions to be derived for the proposed MELDEC strategy, ELDEC strategy, and benchmark schemes, which will be further discussed in subsequent sections.

IV. PERFORMANCE ANALYSIS OF THE IOV NETWORK
In this section, we derive the closed-form outage probability and finite signal-to-noise ratio (SNR) diversity gain expressions in the IoV network.

A. OUTAGE PROBABILITY ANALYSIS
Let τ be the latency to download the requested file from RSUs or cellular BSs to the CV, and γ th = 2 K size τ Bw − 1 be the 8. We choose the work in [28] as one of the benchmark schemes due to the relatively low computational complexity, which makes it suitable for deployment in a practical IoV network. 9. The approaches in [9], [10], [11], [12] may be computationally complex to implement in practice. However, future works will investigate these techniques for benchmarking. threshold [13, eq. (7)]. Then, the outage probability (P out ) of the IoV network is given in the following theorem.
Theorem 1: The outage probability of the CV in the IoV network is: where u ∈ {EL, MEL, DL, LFU, LRU, Oracle}, α x (p) = The proof is given in Appendix A. The function α x (p) represents the power series expansion of the cumulative distribution function (CDF) for the random variable (RV) |ĥ x | 2 , x ∈ {i, j}, while E{Z q x } is the q-th moment of the RV Z [46, Table II]. Also, the function x (p, q) accounts for the impact of stochastic geometry on outage probability in the IoV network and x (p, q) accounts for the impact of CSI estimation errors on outage probability due to the speed of the CV v and feedback period T. In the next corollary, we show that the closed-form outage probability expression in (5) is convergent.
Corollary 1: The closed-form outage probability expression in (5) has a convergence radius of ∞.
Proof: The proof is given in Appendix B. At highP t regimes, the asymptotic outage probability (P out,∞ ) when BS-i and RSU-j of the CV is given in the following corollary.
Corollary 2: The asymptotic outage probability of the CV in the IoV network is: Proof: From (5), limP t →∞ (P t ) q−m x −p = 0 when q < m x + p for x ∈ {i, j}. For q = m x + p, limP t →∞ (P t ) q−m x −p = 1. Therefore, evaluating (5) with q = m x +p leads to the final expression in (6). This completes the proof.
Remark 1: It is observed through (6) that the random location of the typical CV, i.e., stochastic geometry, in the IoV network does not affect P u out,∞ . Instead, the estimated cache hit rate, small-scale fading in the IoV network, and CSI estimation errors due to mobility play a larger role in determining P u out,∞ regardless of the underlying edge caching strategies employed.
When the feedback period (T) is sufficiently long or when the vehicle speed (v) is adequately low, then one may obtain the ideal CSI estimation, i.e., ≈ 1. In such cases, the outage probability P u out,Ideal CSI of the CV can be approximated from (5) as shown in the following corollary.
Corollary 3: The outage probability of the CV when ideal CSI estimation is obtained in the IoV network is: Proof: We begin by noting that x (p, q) ≈ 1 when q = 0, while x (p, q) ≈ 0 when q > 0. Therefore, P u out,Ideal CSI can be approximated from (5) with q = 0 when ≈ 1. Finally, the final expression in (7) is obtained after some algebraic simplifications. This completes the proof.
Using (5), (6), and (7), one can evaluate P u out , P u out,∞ , and P u out,Ideal CSI , in closed-form, respectively, after obtaining the estimated cache hit rate of the proposed MELDEC strategy, ELDEC strategy, or any of the benchmark schemes, i.e., P hit,u where u ∈ {EL, MEL, DL, LFU, LRU, Oracle}. Additionally, one will be able to evaluate any edge caching strategy with the closed-form expressions in (5), (6), and (7), as long as the estimated cache hit rate is obtainable.

B. FINITE SNR DIVERSITY GAIN ANALYSIS
In wireless networks operating at low to moderate SNR regimes, the outage probability decay is quantified through the finite SNR diversity gain d f [22], [23]. Specifically, finite SNR diversity gain is defined as [22, eq. (5)]: where O, Pr(O), and P r are the outage event, outage probability, and average received power in the considered wireless network, respectively. Using d f , one also obtains the diversity gain at asymptotic SNRs by evaluating (8) at high SNR regimes [23], [47], [48]. To this end, we employ finite SNR analysis to investigate the outage probability decay in the IoV network. Let the finite SNR diversity gain of the CV in the IoV network be d u f , where u ∈ {EL, MEL, DL, LFU, LRU, Oracle}. Then, d u f is given in the following proposition.
Proposition 1: The finite SNR diversity gain of the CV in the IoV network is: Proof: The finite SNR diversity gain of the CV is obtained by substituting (5) into (8).
Using (9), the diversity gain of the CV, i.e., decay of P u out , can be analzyed at low-to-medium SNR regimes. Specifically, the impact of the estimated cache hit rate (P hit,u ), small-scale fading, and CSI estimation error on outage behavior in the IoV network can be observed using (9), which would otherwise not be present at high SNR regimes. To see this, the asymptotic diversity gain of the CV in the IoV network is presented in the next corollary.
Corollary 4: The asymptotic diversity gain of the CV in the IoV network is: Proof: We begin by noting that limP t →∞ d u f can be expressed as: From (6), it is seen that P u out,∞ is independent ofP t . Therefore, −P t P u out,∞ ∂ ∂P t P u out,∞ evaluates to zero, which completes the proof.
The result in Corollary 4 is due to the fact that at high SNR regimes, the instantaneous SINR γ x , x ∈ {i, j} becomes limited by interference due to CSI estimation errors as a result of mobility. Thus, increasingP t corresponds to negligible decrease in outage probability at the CV.
For scenarios where ideal CSI estimation is obtained, the closed-form finite SNR diversity gain d u f ,Ideal CSI expression is presented in the following proposition.
Proposition 2: When ideal CSI estimation is attained, the finite SNR diversity gain of the CV in the IoV network is: Proof: The finite SNR diversity gain in (12) is obtained by substituting (7) into (8).
At high SNR regimes, the asymptotic behavior of d u f ,Ideal CSI is presented in the following corollary. Corollary 5: When ideal CSI estimation is attained, the asymptotic diversity gain of the CV in the IoV network is approximated as: Proof: The proof is given in Appendix C. Remark 2: It is worth emphasizing the main reason behind adopting the BPP over the PPP in this paper lies in the fact that the number of RSUs and cellular BSs simulated in PPPs is a random variable. As such, assuming PPPs in the IoV network may not lead to a tractable outage and finite SNR analysis. This is in contrast to the BPP, which allows one to investigate the exact relationship between M RSU and M BS , and its impact on outage probability and diversity gain in the IoV network as shown in (5)- (13). Furthermore, the closed-form expressions and resulting corollaries seen in this section are possible due to the BPP assumption adopted in this paper. This is in contrast to the PPP assumption, which has been widely adopted to model the random spatial locations of traffic infrastructure in vehicular networks, e.g., [5].
Corollary 5 shows that under high SNR regimes and ideal CSI estimation, the estimated cache hit rate of the proposed MELDEC strategy, ELDEC strategy, and the benchmark schemes do not influence the diversity gain of the CV. Instead, the diversity gain is determined based on the number of cellular BSs (M BS ) and RSUs (M RSU ), and the severity of Nakagami-m fading (m x , x ∈ {i, j}) experienced in the IoV network. Using (12), (13) describes the relationship between M BS , M RSU , m x , x ∈ {i, j}, and the diversity gain under high SNR regimes and ideal CSI estimation in closed-form.

V. NUMERICAL AND SIMULATION RESULTS
In this section, numerical results pertaining to the performance of the MELDEC and ELDEC strategies in the IoV network are presented, with Monte Carlo simulations conducted with 10 5 samples. We also provide the default simulation parameters in Table 4.

A. ESTIMATED CACHE HIT RATE AND RUN TIME ANALYSIS
We train the MELDEC strategy, ELDEC strategy, and benchmark schemes with the content demand table X using M dataset = 10 5 rows, and generate a new content demand table X eval with 10 5 rows which are populated independently from X. The evaluation of the estimated cache hit rate (P hit,u ), u ∈ {EL, MEL, DL, LFU, LRU, Oracle} for the MELDEC strategy, ELDEC strategy, and benchmark schemes is then conducted using X eval , where P hit,u is obtained with X and validated using X eval . Fig. 5 shows the comparison of P hit,u , u ∈ {EL, MEL, DL, LFU, LRU, Oracle} as cache size (K cache ) increases. It is seen that the proposed MELDEC strategy outperforms the ELDEC strategy and the benchmark schemes, and that similar P hit,EL is obtained when K model ∈ {5, 10}. It is also observed that the proposed MELDEC strategy achieves near-optimal cache hit rates, i.e., P hit,MEL is close to P hit,Oracle . Similar to the optimal oracle algorithm, the proposed MELDEC and ELDEC strategies are also able to achieve 100% P hit,EL and 100% P hit,MEL when K cache = 5000, i.e., when RSUs cache all K total files. In contrast, it is seen that the DLEC strategy achieves P hit,DL that is less than the LFU scheme but higher than the LRU scheme at low K cache , i.e., K cache < 1400. Furthermore, it is noted that the benchmark schemes exhibit a plateauing cache hit rate at high K cache , e.g., K cache > 4200 for the LFU and LRU schemes and K cache > 3200 for the DLEC strategy. Similar trends have also been observed in [44,Fig. 3] and [26,Fig. 12], and this is due to the fact that the benchmark schemes are limited by the distribution of the underlying file popularity [44]. Specifically, the LFU and LRU schemes cache contents by reacting to file requests from CVs in the IoV network. For files which are requested for the first time, a cache miss is still recorded despite K cache = K total = 5000 since the requested file has not been cached at the start. For the DLEC strategy, contents are cached based on predicted content popularity. As a result, the DLEC strategy may not cache files which were not requested during dataset collection despite K cache = K total = 5000.
A comparison of the run time for the MELDEC strategy, ELDEC strategy, and benchmark schemes is provided in Table 5. 10 We define the run time as the total time taken to run the ELDEC strategy and the benchmark schemes for content popularity prediction over M test test samples. The run times of the MELDEC strategy, ELDEC strategy, and benchmark schemes are obtained using the same Intel Core i7-1165G7 processor with multi-core processing disabled to ensure that a single core is used during simulations. 11 As with [25], [50], the MELDEC, ELDEC, and DLEC run times do not account for the training time as the training for both strategies in this paper are conducted offline. From Table 5, it is seen that the DLEC strategy exhibits a shorter run time than the MELDEC, ELDEC, LFU, and LRU schemes. The MELDEC strategy is also noted to have 50% shorter run time than the ELDEC strategy. Furthermore, we observed that the MELDEC and ELDEC strategies achieve a shorter 10. We omit the run time attained by the oracle algorithm, as it is unrealisable in practice due to the assumption that content popularity is known a priori.
11. Similar to [25], [50], the run time across the MELDEC strategy, ELDEC strategy, and benchmark schemes is chosen as the main metric for complexity analysis in this work. This is because evaluating the number of floating point operations across all schemes may not lead to a fair comparison, since both the LFU and LRU schemes are iterative in nature when compared to the ELDEC and DLEC strategies. Furthermore, popular deep learning libraries, e.g., TensorFlow, have also been optimized for deep learning implementation [25]. run time as the LFU and LRU schemes. Although MELDEC and ELDEC have longer run times than DLEC, P hit,EL and P hit,MEL are considerably higher than the benchmark schemes and is near-optimal when compared to the oracle algorithm. In practice, the run time of the MELDEC and ELDEC strategies can be reduced by utilizing parallel computing platforms such as graphic processing units (GPUs) in IoV networks. 12

B. OUTAGE PROBABILITY AND FINITE SNR ANALYSIS
The impact of K cache on outage probability is shown in Fig. 6 for the proposed MELDEC strategy, ELDEC strategy, and benchmark schemes. When K cache is small, e.g., K cache < 200, the estimated cache hit rate P hit,u , u ∈ {EL, MEL, DL, LFU, LRU, Oracle} for the MELDEC strategy, ELDEC strategy, and benchmark schemes is low due to the limited availability of cache storage at RSUs. In turn, the outage probability P u out becomes limited by the associated cellular BS link of the CV. As K cache increases, a higher P hit,u ensues due to a greater availability of cache storage at the RSUs. As a result, the outage probability P u out improves and becomes increasingly influenced by the respective estimated cache hit rate P hit,u for the MELDEC strategy, ELDEC strategy, and benchmark schemes. Specifically, this can be observed at high K cache where the outage probability of the benchmark schemes plateaus, e.g., K cache > 3000. Such an observation is due to the fact that P hit,DL , P hit,LFU , and P hit,LFU also begins to plateau when K cache > 3000, which can also be seen in Fig. 5. Likewise, a lower outage probability is attained when employing the proposed MELDEC and ELDEC strategies due to higher P hit,EL and P hit,MEL (as seen in Fig. 5). As such, Fig. 6 shows that the choice of K cache greatly influences P u out in the IoV network. In Fig. 7, the impact of transmit power (P t ) on outage probability and diversity gain is seen for low-to-high SNR regimes. It is seen in Fig. 7(a) that P LFU out < P DL out < P LRU out , as P hit,LFU > P hit,DL > P hit,LRU for K cache = 600 (Fig. 5). 12. It is worth noting that the authors in [25] have similarly proposed using GPUs to reduce the run time of deep learning-based power control in wireless networks. More importantly, it is observed that P EL out and P MEL out are close to P Oracle out for 0 dBm ≤ P t ≤ 30 dBm. Corollary 2 is also validated, as it is seen that P u out → P u out,∞ when P t → 30 dBm. In Fig. 7(b), it is observed that the diversity gains (d u f ) of the MELDEC strategy, ELDEC strategy, and benchmark schemes peak at P t = 3 dBm before gradually decreasing as P t increases. This can also be observed in Fig. 7(a), where P u out decays rapidly at low P t regimes before the decay rate becomes negligible at high P t regimes. Furthermore, Corollary 4 is validated as it is seen that d u f → 0 when P t → 30 dBm. Thus, Fig. 7 demonstrates that the proposed MELDEC and ELDEC strategies are capable of achieving near-optimal outage probability and finite SNR diversity gain in the IoV network.
In Fig. 8, the impact of P t on outage probability and diversity gain under ideal CSI estimation is seen for lowto-high SNR regimes. In particular, it is seen in Fig. 8(a) that the MELDEC strategy, ELDEC strategy, and benchmark schemes achieve similar outage probability P u out,Ideal CSI when 0 dBm ≤ P t ≤ 30 dBm, with P LFU out,Ideal CSI < P DL out,Ideal CSI < P LRU out,Ideal CSI . Furthermore, it is noted that P EL out,Ideal CSI and P MEL out,Ideal CSI are close to P Oracle out,Ideal CSI . In Fig. 8(b), it is seen that the MELDEC strategy, ELDEC strategy, and the benchmark schemes attain similar d u f ,Ideal CSI when 0 dBm ≤ P t ≤ 30 dBm. Such a trend is also noted in Fig. 8(a) where the respective decay rate of P u out,Ideal CSI for the MELDEC strategy, ELDEC strategy, and benchmark schemes remain largely similar for 0 dBm ≤ P t ≤ 30 dBm. As P t → 30 dBm, Corollary 5 is validated since it is observed that d u f ,Ideal CSI ≈ min(M RSU m j , M BS m i ). Evidently, Fig. 8 shows that the impact of the estimated cache hit rate P hit,u on outage probability and diversity gain is greatly diminished when ideal CSI estimation is attained. 13 Instead, we demonstrate through Corollary 5 that the number of cellular BSs and RSUs, along with the severity of Nakagami-m fading in the IoV network, largely determine the outage probability and diversity gain at the CV under ideal CSI estimation.

VI. CONCLUSION AND FUTURE RESEARCH DIRECTIONS
An ELDEC strategy that incorporates ensemble learning and an MELDEC strategy which combines meta and ensemble learning is proposed in this paper for content popularity prediction and cache content placement in IoV networks. Specifically, closed-form outage probability and finite SNR diversity gain expressions are derived for the performance 13. Such scenarios can occur in practice when CVs in an IoV network are stationary at traffic junctions. characterization of the MELDEC strategy, ELDEC strategy, and benchmark schemes. Through a comprehensive analysis, we show that the proposed MELDEC strategy and ELDEC strategy achieve near-optimal cache hit rates, outage probability, and finite SNR diversity gain when compared to the benchmark schemes under imperfect CSI estimation scenarios. It is also demonstrated that the finite SNR diversity gain under ideal CSI estimation is independent of the estimated cache hit rates of the MELDEC strategy, ELDEC strategy, and benchmark schemes. Instead, we reveal through finite SNR analysis that the finite SNR diversity gain under ideal CSI estimation is dependent on the number of cellular BSs and RSUs, and the severity of Nakagami-m fading in the IoV network. The near-optimal performance of the MELDEC and ELDEC strategies demonstrate both as viable solutions for deep learning-based edge caching in IoV networks.
Nevertheless, enhancing the content popularity prediction accuracy, i.e., cache hit rate, of both MELDEC and ELDEC strategies remains an open challenge. Specifically, the cache hit rates of both the MELDEC and ELDEC strategies can be potentially improved by considering other temporal factors, e.g., CV mobility and content lifespan. Thus, a redesign of the content demand table to properly consider temporal factors together with content access rates is proposed as part of future directions of this study. Additionally, the performance of content popularity prediction for both MELDEC and ELDEC strategies can be improved by incorporating elements of distributed learning and cooperative-based vehicleto-vehicle communications. Hence, enhancing the MELDEC and ELDEC strategies with distributed learning and cooperative methods while mitigating the associated high overheads and handovers is also proposed as part of future research directions of this study.

APPENDIX A PROOF OF (5)
We begin by first noting that an outage occurs in the IoV network when the CV is unable to download the requested file from the associated RSU and cellular BS. Hence, the outage probability (P out ) is defined as: where i and j are the indices of the associated cellular BS and RSU, respectively. The first term in (14) occurs if the requested file is available at RSU-j, while the second term in (14) occurs if the file is only available at BS-i. Since cellular links experience Nakagami-m fading, the outage probability conditioned on d i and d j can be written as [27], [46]: where F |ĥ x | 2 (·), x ∈ {i, j} is the CDF of |ĥ x | 2 . Thereafter, we apply the techniques in [27, Lemma 1] and [46, Th. 1] to express (15) as: Finally, averaging P out over the PDFs of d i and d j yields (5). This completes the proof.

APPENDIX B PROOF OF CONVERGENCE FOR (5)
We begin the proof by letting y x (p, q) , where x ∈ {i, j}. Then, (5) can be rewritten as: To prove the convergence of (5), we need to show that the power series expressions in (17) are convergent. Invoking the D'Alembert test, lim p→∞ |y x (p+1,q)| |y x (p,q)| is evaluated as: