Joint Radio Resource Allocation and Content Caching in Heterogeneous Virtualized Wireless Networks

An efficient content caching policy at the edge of the mobile cellular network can improve the quality of services of the mobile users and reduces network congestion at the backhaul. On the other hand, the wireless network virtualization emerges as a cutting-edge technique to address the limited network capacity problem due to the exponential growth of the mobile data traffic. In addition, the wireless network virtualization can bring huge benefits such as reducing the capital expenditure (CAPEX) and the operational expenditure (OPEX) as well as improving the network capacity. In this regard, one of the key requirements to recognize the benefits of the above mentioned two technologies is to have an applicable resource allocation framework, that enables the deployment of the content caching schemes in the virtualized wireless network. In this study, we investigate a novel joint radio resource allocation and content caching problem to efficiently utilize the radio resource blocks, the transmit power, and the available cache storage at the base stations (BSs). The goal of the formulated problem presented in this paper focuses on minimizing the delays experienced by the end mobile users of mobile virtual network operators (MVNOs). We show that the formulated problem is a non-convex, mixed integer non-linear problem (MINLP), which is NP-hard and simply intractable. Therefore, we deploy the block upper-bound minimization (BSUM) algorithm to solve the formulated problem. Numerical results show that our method outperforms the existing baseline resource allocation schemes, with up to 19% performance gain in terms of network delay.


I. INTRODUCTION
Wireless Network Virtualization (WNV) has been seen as one of the exciting innovations for solving unprecedented challenges in the future wireless networks (5G) due to massive growth in mobile devices, mobile video traffic and dataoriented mobile applications [1]. Wireless network virtualization enables multiple mobile network operators (MVNOs) to share network resources such as spectrum, power, and physical infrastructure owned by the infrastructure provider The associate editor coordinating the review of this manuscript and approving it for publication was Antonino Orsino .
(InP). Thus, WNV logically disconnects the existing cellular network into two entities: infrastructure provider (InP) and mobile virtual network operators (MVNOs). Moreover, WNV facilitates to achieve higher peak rate, lower delay, lower network deploying cost, and improves network capacity [2]- [4]. In spite of the fact that network virtualization is an auspicious technology for the future wireless network, there are several challenges to be addressed. One of such is how to efficiently share network resources (i.e., wireless subchannels, power) among different MVNOs while ensuring the two main requirements: inter-isolation (i.e., no interference among different MVNOs) and intra-isolation In this work, different from the previous works, we consider the joint deployment of wireless network virtualization (WNV) and content caching at the edge of the access network to fulfill traffic demand of the future wireless network. This approach will reduce traffic congestion at the backhaul link during high-traffic hours while meeting QoS requirements of the users of MVNOs, with minimum end-toend delay. Nonetheless, in WNV, multiple MVNOs coexist on the same infrastructure and share the limited storage capacity of InP at each BS. Therefore, the network performance is severely affected by the duplication of cached contents at each BS, specially, when the InP simply divides the limited storage capacity of each BS into multiple slices, and then allocates each storage slice to each MVNO. Because of this, an efficient storage sharing and contents caching framework between MVNOs is required, jointly, with an efficient wireless resources (i.e., channels and power) allocation to upgrade the performance of the network.
In order to address the aforementioned challenges of content caching jointly with the radio resource allocation in the virtualized wireless network, we propose a unique framework for content storage sharing, content caching, subchannels and power allocation in WNV. The summary of the main contributions of this paper is given as follows: • Firstly, we formulate a joint content caching, channels and power allocation problem in the virtualized wireless network that minimizes the network delay experienced by the users of all MVNOs while downloading contents, subject to the QoS requirement of users of all MVNOs, cache storage capacity of BSs, and subchannels and power constraints of BSs. The formulated problem is a non-convex, mixed-integer nonlinear problem (MINLP), which is NP-hard due to the combinatorial properties.
• To address the proposed problem, we firstly relax the binary variables into a continuous form. Then, we transform the relaxed problem into the proximal upper bound minimization problem and apply BSUM algorithm [15], [16]. BSUM is a new powerful approach for solving nonconvex, non-smooth optimization problems, where the original complex problem is decomposed into smaller sub-problems.
• In the simulation section, firstly we present the total network delay experienced by the users of all MVNOs under the proposed solution approach based on BSUM method. Then, the performance of our proposed algorithm is compared with other baseline schemes: Equal cache storage (ECS) allocation, Equal power EP) allocation, Equal channel (EC) allocation, and Equal channel + Equal power + Equal cache storage (EC+EP+ECS) allocation. We observe that the proposed algorithm attains a significant performance gain: likely 19.75%, 10.06%, 5.35%, and 4.28% in comparison with (EC+EP+ECS), ECS, EP, and EC allocation scheme, respectively. Lastly, we show that our proposed algorithm converges quickly to the optimal.
The remainder of this paper is arranged according to the following: In Section II, we summarize related works. The system model and content caching in the heterogeneous virtualized wireless networks framework are presented in Section III. Section IV demonstrates the proposed joint resource allocation and content caching problem in the heterogeneous virtualized wireless networks. The overview of BSUM algorithm and the solution approach based on the BSUM method to solve the proposed problem are introduced in Sections V and VI, respectively. Section VII discusses the numerical results of our proposed solution approach. Finally, Section VIII concludes the paper.

II. RELATED WORKS
We can categorize the existing works into two groups: (i) communication resources (i.e., subchannels, power) allocation, and (ii) content caching in virtualized wireless network.
(i) Communication resources allocation in virtualized wireless network: In [17], the authors introduced a joint subchannels and power allocation in WNV. The work of [18] considered user clustering and non-orthogonal multiple access (NOMA) based joint channels and power allocation in WNV. A Lyapunov optimization based hierarchical wireless resource management in virtualized wireless networks was presented in [19]. However, in the above mentioned works, the authors considered a single cell scenario that undermines inter-tier (inter-cell) interference and that can make it easy to solve the optimization problem. In [20], the authors proposed VOLUME 8, 2020 a stochastic optimization based cells selection and resource allocation framework in virtualized wireless networks. The work of [21] introduced an efficient power allocation algorithm and interference management in multi-cell virtualized wireless network. Nonetheless, in this work, the authors have omitted subchannels allocation. A joint user association and allocation of resources in WNV and a stochastic game-based wireless spectrum allocation was proposed in [22] and [23]. Furthermore, some of the research works studied spectrum sharing from an InP to the MVNOs. As an example, in [24], the resource allocation in WNV was studied by using an opportunistic spectrum sharing framework. A bankruptcy game-based dynamic spectrum sharing among MVNOs was proposed in [25]. However, in all of these research works, individual users were not considered for spectrum sharing among MVNOs.
Moreover, a few existing works proposed resource allocation in WNV from an economic perspective rather than considering the technological aspects. In our previous work [8], we introduced Generalized Kelly Mechanism (GKM) based hierarchical resource allocation in the virtualized wireless networks. In this work, an InP firstly allocates its wireless resources among MVNOs based on an auction mechanism, and then each MVNO efficiently shares the obtained wireless resource from the InP to its mobile users. Similarly, VCG (Vickrey-Clarke-Groves) auction mechanism based power allocation and spectrum sharing by using a sequential auction in the wireless network virtualization were proposed in [26] and [27]. In [28], the authors proposed stochastic resource allocation in WNV. In their work, they tried to maximize the profit of the mobile network operator (i.e., the profit of the infrastructure provider (InP)). Furthermore, a cyclic three-sided matching game based resource allocation in the WVN was introduced in [29]. All of these works aimed at maximizing the profits of the MVNOs.
(ii) Content caching in the virtualized wireless network: Although there are several existing works which focus on improving the performance of content caching, i.e., to diminish the transmission delay, and increase the cache hit ratio at the edge of the wireless network, only a few works considered efficient content caching in virtualized wireless network [30]- [32]. However, signal-to-noise ratio (SNR) based content placing at the base station will not be efficient because content caching decision at each BS will be long-term scale, but SNR is changing over a short period. In [33], the authors have proposed a deep learning based cache buying strategies for the MVNOs where each MVNO buys cache storage from the InP in order to maximize their own users' experience. The authors in [34] proposed software-defined networking based resource allocation and content caching problem in the virtualized network where they tried to maximize the utility of the mobile virtual network operators (MVNOs). In [35], the authors considered the optimal caching strategy in the virtualized wireless networks, and the work in [36] introduced the joint resource allocation in content caching for wireless network virtualization with the objective of minimizing the content request rejection rate. However, the authors did not consider the frequency reuse between the BSs, i.e., no consideration for inter-tier interference, backhaul delay, and power allocation.
In this work, we consider the significant requirements that were mostly overlooked in the prior works, and jointly combine the communication and caching schemes in the heterogeneous virtualized wireless network to formulate an optimization problem. Moreover, in order to solve our proposed optimization problem, we introduce the BSUM algorithm. BSUM is a powerful framework for solving big data, nonconvex and non-smooth optimization problem. Furthermore, BSUM allows to effectively decompose the initial optimization problem into multiple smaller-subproblems, which can be solved disjointly and in parallel. In this work, we assume that the mobile users of MVNOs are associated to the specific BS in prior, and orthogonal frequency division multiple access (OFDMA) based wireless access protocol is used among the users of a BS to avoid intra-cell interference. We define y bn mu ∈ {0, 1} as a channel assignment variable which indicates whether or not channel n of BS b is assigned to user u of MVNO m.
Then, the received signal to interference plus noise ratio (SINR) of user u of MVNO m on channel n of BS b can be expressed as follows: where h bn mu , p bn mu are the channel gain and the allocated power to the user u of MVNO m who is assigned to channel n of BS b, respectively, and is the interference on mobile user u of MVNO m who is assigned to channel n of BS b, and σ 2 is the noise power spectral. Therefore, we can calculate the data rate of user u To calculate the data rate of users of all MVNOs at each BS, we consider the following constraints: • Transmit Power Constraint: The transmit power allocated to the users of all MVNOs at each base station must be less than the maximum transmit power of that base station. Therefore, we can write the transmit power constraint as follows: C1 : where P max b is the maximum transmit power of BS b ∈ B.
• QoS Constraint: The InP guarantees pre-agreement with MVNOs by the minimum data rate requirement constraint as: where R min m is the minimum rate requirement of MVNO m ∈ M.
• Resource Scheduling Constraint: According to OFDMA technique for channel access, a channel is assigned to at most one mobile user in each BS. Therefore, we have the following constraint: C3 : On the other hand, the communication cost (i.e., delay) is very high to access contents from the content server located at the core network (CN). In order to reduce the communication delay and save the backhaul link capacity, in this work, we assumed that the InP deploys and manages the cache storage at each BS with proactive caching [37], where cache storage is shared and used by all MVNOs. Let C b denote the maximum storage capacity of the cache repository deployed at BS b ∈ B. Without loss of generality, different BSs have different storage capacity. Here, the mobile users of MVNOs are willing to download contents in the following content catalog S = {1, 2, . . . , S}. We consider that all of the contents in the content list have same size (i.e., L 1 = . . . = L s = . . . = L S ). This is feasible using advanced coding technique, by applying which all of the contents can be divided into the same block length [1]. In the coverage area of BS b ∈ B, each content has a different value of popularity which depends on the probability of it being requested by the users of MVNOs. In this work, we consider the content requests by the users of MVNOs (i.e., content popularity) follow the Zipf-like distribution [38], where the probability of requesting content s ∈ {1, 2, . . . , S} by the users is given by: , and 0 ≤ λ ≤ 1 is the exponent characterizing the Zipf distribution. When λ = 0, all of the contents have the same popularity. Otherwise, when λ = 0, VOLUME 8, 2020 the contents have different popularity. Furthermore, the contents are ranked in order of their popularity. Here, we consider that BSs cache the contents in the order of their popularity starting from the highest popularity. Now, we define another new variable, q bs mu ∈ {0, 1} which represents whether user u ∈ U of MVNO m ∈ M requests the content s ∈ S at BS b or not, i.e., Then, let the new variable z bs m be the cache decision variable, which represents whether or not content s is cached at BS b and shared to MVNO m as In particular, each BS caches the popular contents among the mobile users of MVNOs in advance for possible future requests [2], [3]. Moreover, mobile users of different MVNOs can share the cached contents at each BS's content storage, so that the same content will not be cached at each BS to serve the mobile users from different MVNOs. However, user mobility analysis in joint radio resource allocation and content caching is outside the scope of this paper and will be subjected to our future research. Therefore, we can express the above constraint in mathematical form as follows: Moreover, the maximum content storage capacity constraint at each BS m can be expressed as follows:

IV. PROBLEM FORMULATION
Next, we derive the joint radio resource allocation and content caching problem in the heterogeneous virtualized wireless networks which aims to minimize the overall network delay. The network delay consists of two main terms: the wireless transmission delay, and backhaul delay. Firstly, we introduce wireless transmission delay of user u ∈ U in MVNO m ∈ M to download content s ∈ S when associating with BS b ∈ B.
The wireless transmission delay of user u of MVNO m on channel n of BS b to download content s can be expressed as follows: where L s is the size of content s. From (12), we observe that the transmission delay experienced by user u of MVNO m to download the content from the base station b depends on the size of the content, and signal to interference plus noise ratio (SINR). The second component of the network delay is the backhaul delay. We add the backhaul delay experienced by the user u in MVNO m associated with the BS b as H b m,u when downloading the content from the content server. In case of wired backhaul, the characterizing components of the backhaul delay are: a) the average link distance, b) the average traffic load, and c) the average number of SBSs associated with a single small cell gateway interface. In this work, we follow the backhaul delay model as in [39].
Therefore, the network delay experienced by user u of MVNO m on channel n of MBS b to download content s can be expressed as follows: .
Considering the limitation of bandwidth efficiency, transmission power restriction, and the storage capacity constraint, the joint resource allocation and content caching in multicells virtualized wireless networks problem to minimize the total delay experienced by all MVNOs users can be formulated as follows: s.t.
where the objective function described in (14) is a nonconvex function because of the inter-cell interference, and the decision variables (i.e., y, p and z) are coupling in the objective function. Moreover, the constraints mentioned in (15)- (22) are non-linear and which are the mixture of continuous and binary variables, i.e., y, p, z. As a conclusion, the above optimization problem is a non-convex mixed integer nonlinear problem (MINLP), which is NP-hard and takes exponential time complexity to find the optimal solution. Therefore, we deploy BSUM algorithm to solve our proposed problem.

V. OVERVIEW OF BSUM ALGORITHM
BSUM, a distributed algorithm, can be used for parallel computing. The use of BSUM in centralized algorithms results in advantages for both solution speed and problem decomposability. Hence, the following function is considered as a block-structured optimization problem for the introduction of the standard form of BSUM [40]: a 1 , a 2 , . . . , a x ), where D := D 1 × D 2 × · · · × D X , f (·) is the continuous function, and X denotes the set of index. When x = 1, 2, . . . , X , D x is considered as a close convex set and a x as a block of variables. If we apply block coordinate descent (BCD) in every iteration i, it will optimize a single block of variable by solving the following problem: where However, both (23) and (24) are challenging to solve. The hardest condition will happen when the function in (23) is non-convex and BCD cannot guarantee convergence for every time. So, by applying BSUM at a given feasible point e ∈ D, the proximal upper-bound function g(a x , e) of f (a x , e −x ) can be introduced. There are three common schemes used for choosing the proximal upper-bound function: linear upperbound, quadratic upper-bound, and Jensen's upper-bound. Here, the selected proximal upper-bound function g(a x , e) has to satisfy the following assumption. ; r), e x + r x ∈ D x . Assumption 1(i) and 1(ii) guarantee that the objective function f is globally upper-bounded by the proximal upperbound function g. Then, Assumption 1(iii) ensures the existence of first-order derivative satisfying the fact that g(a x , e) takes negative steps towards the gradient of the objective function f (a x , e −x ) in the direction of r. For ease of presentation, the following proximal upper bound function is defined by adding quadratic penalty term to the objective function: where θ is the positive parameter, i.e., θ > 0. The BSUM algorithm addresses the proximal upper-bound function in (25) at each iteration i along with the following update: The detailed procedures of the BSUM algorithm is presented in Algorithm 1. The BSUM can be viewed as the generalized form of BCD which successively updates the primary variable blocks to maximize the upper-bound function of the original objective function. It can be used to solve separable convex optimization problems with linear coupling constraints that are smooth or non-smooth. Specifically, each block of variables is iteratively updated to minimize the upper-bound proximal function before convergence to both a coordinate-wise minimum and a stationary solution. If a block of variables reaches the minimum point a * x = a (i+1) x , a stationary solution is assumed to be a coordinate-wise minimum. In other words, the whole point vector cannot find a better minimum direction at stationary points [40], [41]. Based on [40], we have the following remark:

Remark 1 (Convergence of BSUM Algorithm): In order to converge to the -optimal solution, BSUM algorithm takes at most O(log(1/ )) iterations. In other words, it is a sub-linear convergence.
The -optimal solution can be determined as a as the desired solution. VOLUME 8, 2020 For analyzing the complexity of the BSUM Algorithm 1, we use the complexity analysis described in [42] and make the following remark: Assumption 1 (i, ii) and Remark 1, with flexible update rules, BSUM algorithm has sub-linear rate of convergence. Therefore, as described and proved in [42], the Algorithm 1 has sub-linear iteration complexity O(1/i), where i is the index of iteration.

VI. SOLUTION APPROACH FOR JOINT RADIO RESOURCE ALLOCATION AND CONTENT CACHING
To address our proposed problem, firstly, we relax the channel assignment variable y b,n m,u and the cache decision variable z bs m in constraints (21) and (22) into the continuous form, i.e., 0 ≤ y b,n m,u ≤ 1 and 0 ≤ z bs m ≤ 1. Then, the above optimization problem (14) can be rewritten into the following form: z bs m ∈ [0, 1], ∀b ∈ B, ∀m ∈ M, ∀s ∈ S.
Likewise, we can rewrite the optimization problem in (27)  The problem in (27), equivalent to (31), is non-convex after relaxing the binary variables into the continuous form, and in general, is challenging to solve. Even though, as an example, the subproblem of optimizing the variable y while the other variables such as p and z are fixed is still non-convex because of intercell (or inter-tier) interference. Therefore, we cannot directly use the traditional block coordinated descent (BCD) algorithm [43] because it cannot guarantee the convergence when the problem is non-convex. Consequently, we use the BSUM algorithm, which is a different method from the exact BCD. The BSUM algorithm guarantees convergence to the set of stationary points of the non-convex problem [41].
At each iteration i, ∀x ∈ X i , where X is the indexes set, we define the proximal upper-bound function Q x of the objective function described in (31). At this point, we add the quadratic penalty term to the objective function in (31) to keep the proximal upper-bound function convex as where µ x is the positive penalty parameter, i.e., µ x > 0. (32) is the proximal upper-bound function of the objective function in (31) for the vector of variableỹ, and it can be deployed for other vectors for variablesp, andz, respectively. Additionally, the proximal upper-bound function is convex as a result of its penalty term, µ x 2 (y x −ỹ) 2 . Strictly speaking, with respect to y x , p x , and z x , the proximal upperbound function has the unique minimizers vectorỹ,p, andz at each iteration i, which are examined as the solution of the previous iteration (i − 1). Then, the solution in each iteration (i + 1) can be updated by solving the following subproblems: Moreover, the subproblems in (33)- (35) can be solved by using our proposed block successive upper-bound minimization (BSUM) algorithm for joint resource allocation and content caching in WNV as shown in Algorithm 2.

VII. SIMULATION RESULTS
In this section, we assess the performance of our proposed BSUM algorithm based joint resource allocation and content caching in the heterogeneous virtualized wireless networks.

Algorithm 2 BSUM Algorithm for Joint Radio Resource
Allocation and Content Caching in WNV 1: Initialization: Set i = 0, > 0, and find initial feasible solutions (y (0) , p (0) , z (0) ); 2: repeat 3: Choose index set X ; 4: Find the solution for channel allocation problem at each BS, y (i+1) x , by solving the following subproblem; 5: Find the solution for power allocation problem at each BS, p i = i + 1; 11: Then, set y as the desired solutions. 36770 VOLUME 8, 2020

A. SIMULATION SETUP
In this network scenario, the InP deploys a single macro base station (MBS) having the coverage radius of 500 m. Then, 3 SBSs with the radius of 100 m are randomly deployed by the InP within the coverage area of the MBS. Moreover, the InP installs the cache storage at the base stations, where the maximum cache storage capacity of each base station can be varied from 15% to 50% of the total content size. From above, it is clear that the total cache storage capacity of the BS is always smaller than the total content size. In this network setup, 3 MVNOs are sharing the infrastructure provided by the InP, where each MVNO is subscribed by the number of users U ∈ [5,10]. Here, every user of each MVNO is associated with BS that has the strongest wireless signal strength. Then, mobile users of MVNOs access the list of 1000 contents (i.e., S = 1000), where the size of each content is 15 KB. In our simulation, we also consider that each BS is operating at the maximum system bandwidth 10 MHz. Therefore, according to LTE standard, the maximum available subchannels at each BS is 50, where each subchannel has the system bandwidth 15 kHz. Furthermore, the maximum transmit power of the MBS and each SBS are 48 dBm and 38 dBm, respectively, and the thermal noise density of the system is -174 dBm/Hz. Moreover, the small scale fading model, i.e., Rayleigh fading model is considered. The details of the parameters used in our simulation are presented in Table 2.

B. NUMERICAL RESULTS
In this subsection, we mainly focus on the performance gain of the proposed algorithm. Moreover, we compare the performance of our proposed solution approach with the existing baseline schemes. In this work, we consider the following 4 baseline schemes to compare our work with.
• Equal channel + Equal power + Equal cache storage (EC+EP+ECS) allocation: In this scheme, the channels and the transmit power of BSs are equally allocated to the users of all MVNOs. Moreover, the total cache storage capacity of each BS is also equally partitioned amongst MVNOs and the cache decision is independently made for each MVNO. Furthermore, the contents will not be shared amongst different MVNOs to serve their mobile users.
• Equal cache storage (ECS) allocation: Under equal cache storage (ECS) allocation scheme, the cache storage of each BS is equally shared amongst the associated MVNOs. Furthermore, contents sharing amongst MVNOs to serve their mobile users is also not possible. Then, channels and the transmit power of each BS are allocated to the users of MVNOs by using our proposed algorithm.
• Equal power (EP) allocation: In EP allocation scheme, the transmit power of each BS is equally allocated to its associated users of MVNOs.
• Equal channel (EC) allocation: The channels of each BS are equally allocated to its associated users of all MVNOs. Fig. 3 demonstrates the total delay of the network under different number of users. Here, all BSs are operating at the maximum bandwidth 10 MHz (i.e., maximum available channels at each BS is 50) and the maximum content storage capacity of each BS is 50% of the total contents size. From Fig.3, we observe that the total delay increases with the increase in the number of users of MVNOs. Furthermore, we observe that the total network delay is 523.344 seconds (EC+EP+ECS), 467.345 seconds (ECS), 445.23 seconds (EP), 440 seconds (EC), and 420 seconds (Proposed), for 30 users in the network. We clearly observe that our proposed algorithm outperforms other baseline schemes. From there, we can conclude that equal sharing of radio resources and cache storage of the BSs among the users of MVNOs is not efficient. Fig. 4 demonstrates the network delay experienced by the users of all MVNOs under different number of available channels (i.e., N ∈ [50, 100, 150, 200] ). Concurrently, the cache storage capacity of each BS is 50% of the total contents size. As an example, we recognize from Fig. 4 that  the total delay when the number of available channels is 50 is 850.57 seconds (EC+EP+ECS), 760 seconds (ECS), 748 seconds (EP), 745 seconds (EC), and 726.23 seconds (Proposed). Thus, our proposed algorithm achieves the lowest network delay compared with other baseline schemes. Moreover, we present the delay experienced by the users of all MVNOs for different cache storage capacities of the BSs (i.e., C b ∈ [20%, 50%]) in Fig. 5. Meanwhile, the maximum available bandwidth at each BS is 10 MHz (i.e., the total available channel at each BS is 50). In Fig. 5, we observe that the delay is the lowest under our proposed scheme. Therefore, similar to the previous results, our proposed algorithm outperforms other baseline schemes. Fig. 6 demonstrates the number of channels assigned to each MVNO's users under our proposed algorithm. From Fig. 6, we observe that the number of channels allocated to the users of MVNO-1 is 70, MVNO-2 is 87, and MVNO-3 is 42, respectively. Here, the number of channels assigned to each MVNO's users depends on the pre-agreement between the InP and MVNO, as characterized in (5). It is clear that the BSs need to allocate more channels to the users of MVNO with the higher QoS requirements. Similar to the scenario of channels allocation, the transmit power allocated to the users of MVNOs is shown in Fig. 7. From Fig. 7, we observe that the transmit power of the BSs allocated to the users of MVNO-1, MVNO-2 and MVNO-3 are 50.34 dBm, 73.14 dBm, and   38.52 dBm, respectively. Therefore, it is clear that higher transmit power is allocated to the users of MVNO with the higher QoS requirements.
As illustrated in Fig. 8, with the number of users, the achievable data rate of each MVNO increases. It is clearly seen that the achievable data rate of MVNO-2 is the highest among all MVNOs. Statistically, the data rates of MVNO-1, MVNO-2 and MVNO-3 are 4.7 Mbps, 5.2 Mbps and 3.9 Mbps, respectively. The reason is that each BS allocates more channels and power to the users of MVNO-2 following the initial agreement between InP and MVNOs. In Fig. 9, we compare the performance of the proposed algorithm  with the other two schemes: 1) Equal power sharing among MVNOs, and 2) Equal channel sharing among MVNOs. In the equal power sharing among MVNOs, the transmit power the BS is equally shared among MVNOs without considering the pre-agreement (i.e., without considering the QoS requirements of the MVNO), as in (5). It is same as the equal channel sharing scheme, where the channels are equally allocated among MVNOs. In the above aforementioned schemes, the MVNO with the large number of users will be allocated equal proportion of channels and power of the BS compared with the MVNO who has the few number of users. In other words, fairness is not guarantee among MVNOs. To this end, from Fig. 9, we observe that the total delay when the number of users in the network is 30 is 516 seconds (Equal power sharing among MVNOs), 504 seconds (Equal channel sharing among MVNOs), and 420 seconds (Proposed). Therefore, when compared with other existing schemes, our proposed algorithm achieves the lowest delay.
In Fig. 10, we present the normalized cache hits at the BSs based on the content requests made by the users of MVNOs. The requested contents, if not cached at the BS (cache misses), need to be retrieved from the content server located at the core network (CN). We observe that the normalized cache hits increase with the Zipf parameter. This is intuitive as well, as the algorithm caches the contents having high user demand, the cache hit rates will increase correspondingly.  In Fig. 11, we demonstrates the delay experienced by the users of MVNOs under caching (i.e., the popular contents are cached at the storage repository of the BSs to serve users of all MVNOs) and no-caching scenarios. In the no-caching scenario, an InP does not deploy content storage capacity at the BSs. Therefore, the contents requested by the users of MVNOs are downloaded directly from the content server located at the core network. Therefore, the backhaul delay will be present for each user of MVNOs when the contents are downloaded from the content server. As an example, in Fig. 11, the total network delay experienced by the users of all MVNOs is 634.31 seconds (not caching), and 420 seconds (caching) for 30 users in the network. According to the result, we can conclude that deploying the cache storage and caching the popular contents at the BSs can reduce the network delay experienced by the users of MVNOs.
Finally, in Fig. 12, the convergence rate of the proposed algorithm is shown. We also compare the proposed algorithm's convergence rate with different values of penalty parameter (i.e., different µ x ). When µ x = 1, the algorithm converges within 30 iterations. However, the proposed algorithm converges in fewer than 40 iterations when we set the value of the parameter as 10 (i.e., µ x = 10). According to the above results, we observe that the value of the penalty parameter can effect on both the convergence rate and the value of the function in (31). VOLUME 8, 2020

VIII. CONCLUSION
In this paper, we have formulated a problem of joint radio resource allocation and content caching in the heterogeneous virtualized wireless networks. We have shownthat the problem formulated is a non-convex. Therefore, by adding a proximal term, we transformed the proposed problem into the convex form and then solved the transformed problem using a successive block upper-bound minimization algorithm (BSUM). Simulation results have reflected that the total content downloading delay experienced by the users of MVNOs with our solution approach is lesser than that provided by the other baseline schemes. Finally, we introduced the convergence of our proposed algorithm, where it is clear that our proposed algorithm converges within a few iterations. In future, we will consider joint communication and computation resource allocation in the virtualized wireless networks.