Multi-objective edge server placement using the whale optimization algorithm and game theory

Due to the users’ mobility, new online applications, as well as low processing power and limited energy of smart devices, traditional cloud computing models could not provide new required services. Cloud service providers improve the quality of their services by moving some servers to the edge of the network and closer to mobile users. Considering the moving nature of users and the heterogeneous service demands in different areas, the optimal placement of servers plays an important role in increasing the quality of service provided to users. However, because of the large number of servers, finding the optimal location of these resources is a serious challenge. In the proposed method of this paper (MES-WG), in the first step, the geographical area of server deployment is divided into smaller sub-regions to reduce the complexity of the problem. Then, by using the WOA algorithm the search agent finds the optimal location of the servers. In the next step, a neural network is used for the local placement of all servers in each area. Finally, game theory is deployed for the convergence of resource placement in all sub-regions. The experimental results show that the proposed method reduces the network latency by 33.5% and also improves the load balance on servers by 28.2%, compared to some of the state-of-the-art methods.


Introduction
Cloud computing is a popular model to provide various services for its users.Users can easily connect to the servers in cloud data centers and receive their services (Asghari et al. 2020(Asghari et al. , 2021)).With the rapid growth of new communication technologies and smart mobile equipment, new applications, and the mobility of users, traditional cloud models could not meet the new needs in this area.These needs include online games, multimedia applications, virtual education, e-commerce, etc., which must be done with tolerable delay.In response to this need, cloud service providers transfer their resources to the vicinity of their users and the edge of the network.Users can access these resources through cellular base stations (CBS) and Wi-Fi networks.
Proper placement of mobile edge servers (MES) (Huang et al. 2021a, b) has an important role in the efficiency of these resources and services delivered to users.The service demand is not the same in different areas of the network.The volume of services requested by users is high in some places and less in others.Therefore, the proper server placement must be done considering the rate of demand in each area; otherwise, there will be a possibility of overloading resources in some places while underloading in other areas.Users' mobile smart equipment has batteries with limited energy and certain processing power.For this reason, and in most cases, user applications and data are offloaded on mobile edge servers (Kumar et al. 2013).Improper placement of resources can lead to poor service quality delivered to mobile users.Server placement must be a function of service requests and data exchanged in each area.Some researchers have conducted studies and introduced methods for placing mobile servers.Considering that proper placement of resources is an NP-hard problem, some optimization methods have been used in this field.Some of these research include PSO-based (Liang et al. 2022), GA-based (Li et al. 2020), clusteringbased (Lee et al. 2019), MIP-based (Premsankar et al. 2018), and learning-based (Asghari and Sohrabi 2022a).However, there are still important challenges.The extensive area of the network and the high number of servers that must be placed have caused finding the best server locations to face a serious challenge.Some researchers have used clustering techniques (Lee et al. 2019) to reduce the dimensions of the problem.However, still, there are some problems with clustering algorithms, such as falling into local optima or finding the optimal number of clusters.
Most of the research conducted in this field considers the resource deployment area to be flat.These methods will have scalability problems as the area dimensions become larger.The proposed method reduces the complexity of the problem by dividing the area into some sub-regions, using the BOA algorithm.
Also, most of the previous methods have used algorithms with discrete nature and predefined locations for server placement and only these locations can be selected.The proposed multi-objective and multi-agent algorithm searches the entire area with no predefined limitations and uses a continuous approach.In other words, instead of determining predefined candidate locations, the entire area is searched to find the optimal location for each server.For this reason, the WOA algorithm is used because of its continuous nature, powerful operators, and its local and global search features.
To converge the solutions in each sub-region, the MLP neural network model is used to avoid falling the problem into the local optimum, and finally, game theory is used for the global convergence of all regions until reaching an optimal global solution.Latency reduction and better load balancing are the major objectives of the proposed method.
The contributions of the proposed method are as follows: • Dividing the resource deployment area into smaller zones to reduce the complexity of the problem.• Using the WOA algorithm to find the optimal placement of each server.• Using the neural network to get the best local solutions in each sub-region.• Applying game theory to find the optimal servers locations in the whole area.
Other sections of the paper are organized as follows.In Sect.2, a literature review has been conducted on cloud resource placement.Section 3 is devoted to the basics and concepts of the research.The proposed model is described in Sect. 4. Section 5 is related to the evaluation of the proposed method and results.And finally, the conclusion and future research suggestions are explained in Sect.6.

Literature review
Proper server placement increases the efficiency of network services.For this reason, many studies have been conducted in this field.Some researchers have used evolutionary and meta-heuristic algorithms on this subject.For example, in Kasi et al. (2020), using genetic and hillclimbing algorithms, a method of server placement has been introduced to reduce the access delay and load balancing of resources.A genetic algorithm is used to determine the value of cloud server locations.In the next step, a placement state is determined by applying the hill-climbing algorithm.A state is equivalent to the connection between a location and an edge server.Then, the value of these states is determined.And finally, the simulated annealing algorithm is used for more exploration of the algorithm.In another similar work (Ma 2021), using particle swarm optimization, genetic algorithm, and the simple additive weighting method, a model of server placement has been introduced to balance the user's workload on mobile servers.Genetic algorithms and particle swarm optimization algorithms have been used for service offloading and optimization of resource placement strategy, respectively.In another research (Wang et al. 2022), using genetic and gray wolf algorithms, a method of server placement has been introduced.The server placement model is deployed for Wi-Fi sites.The gray wolf algorithm is used to accelerate finding a solution.Also, in Li and Wang (2018), using the improved PSO algorithm, a new server placement method has been introduced to reduce both the energy consumption of resources and the access delay to servers.Based on this method, cloud servers are considered particles of the PSO algorithm.PSO algorithm is used in another paper (Li et al. 2021a, b).The objectives of this paper are to reduce the access delay and maximize the profit of the servers.The mapping between servers and their locations is considered.Then, in the next iterations, this mapping is improved so that the best allocation of resources to locations is reached.Finding the best server location is done using the local and global features of the PSO algorithm.Also, other similar methods have been performed on this subject (Li et al. 2021a;Zhang et al. 2021a, b;Xu et al. 2020;Shen et al. 2021;Chen et al 2020).
Another category of server placement methods is based on machine learning models.In Kasi et al. (2021), a method of server placement using multi-agent reinforcement learning is introduced with the objectives of reducing resource access delay and balancing workload on cloud servers.Each of the multi-objective learning agents tries to reduce the resource access delay and better workload distribution on edge servers.In Yin et al. (2016), a study on CRP with the objectives of reducing costs and increasing fault tolerance has been proposed.The proposed algorithm using the clustering technic finds the optimal locations of the servers.It then creates a map between these optimal locations and the actual placement on the physical network.This algorithm can find unforeseen locations in the entire network.Analysis of the results shows that this method saves the cost by 45%.Another similar work using capacitated location-allocation (Lahderanta et al. 2021) has been considered for better load balance of servers and reliability.This method tries to balance a load of servers by reducing the distance between servers and access points.The k-means algorithm and a new objective function are used in this algorithm.The proposed method is used in both MEC and Fog computing environments.
In Lu (2020), using the max-min algorithm, authors presented a new edge server placement method.In the proposed algorithm, their algorithm maximized workloads on network resources, considering the unforeseen failures of servers.Another research has been introduced by Meng et al. (2019) to reduce both service and operational costs in the optimal placement of mobile servers and access points.For this purpose, the authors have introduced a local search algorithm called SPAC.The proposed model of the paper is based on an undirected acyclic graph in which the nodes of the graph represent the locations of the servers.The experiments show that this method reduces operational and service costs.Also, similar methods have been conducted in other papers (Yin et al. 2016;Lee et al. 2019;Mohan et al. 2018;Li et al. 2021a, b).
Green cloud computing is the designing of energy-efficient and environmentally friendly architectures, in a way that reduces the negative impact on the environment.Abbasi-Khazaei and Rezvani (2022) introduced a new multi-objective virtual machine placement, namely carbonefficient VM placement to joint minimize energy costs and scheduling, using the modified memetic algorithm.The experiments showed that the proposed method can reduce SLA violations.Khosravi et al. (2013) proposed a new virtual machine placement heuristic to increase environmental sustainability for distributed servers with different carbon footprint rates.Experimental results show that the proposed heuristic can save up to 45% carbon footprint in the ecosystem compared to some similar algorithms while keeping the SLA violation.In Jabbari et al. (2022) and using a new queueing model, the authors proposed a method of green edge server placement to reduce the power consumption of the servers in the edge environment.The problem formulation considers the queueing delay through load assignments on servers for the offloaded tasks.Their proposed algorithm also is equipped with SDN technology for better load balancing on servers.In Nadalizadeh and Momtazpour (2022), a new virtual machine placement algorithm has been proposed to reduce the energy cost of servers.The authors introduced a new cost metric to select the most suitable servers in the data center by considering the green energy availability and cost, avoiding server fragmentation.The experiments show that the proposed method saves energy costs while maintaining the quality of services.In Khamari et al. (2022), a new edge server placement has been proposed for energy efficiency.The proposed method deploys integer linear programming to address the tradeoff between energy, latency, and cost while considering edge servers' processing power and the volume of the vehicle's traffic on the road.This model reduces energy consumption by decreasing the number of edge servers while maintaining the communication latency and preventing servers' overloading.
A review of papers on server placement methods shows that most of the methods consider the geographical area of resource deployment to be flat.Because of the size of the area and the number of servers, this model cannot be scalable.Also, some methods operate in a coarse-grained manner and do not pay special attention to each resource.The proposed hierarchical algorithm searches to place servers with a global, local, and single view for obtaining the best mapping between servers and the locations.

Research basis
In this section, the basics used in the proposed method, including the whale optimization algorithm and game theory, are described.

WOA algorithm
WOA (Mirjalili and Lewis 2016) is an intelligence-based algorithm based on the hunting behavior of humpback whales.They identify the location of the prey and surround it.At the beginning of the hunt, they release bubbles around their prey to confuse it.This behavior is called the bubble-net feeding method.This algorithm has been used to solve some problems related to mobile cloud networks.For example, in Huang et al. (2021a, b) WOA is used for computational offloading in the mobile edge computing environment.Since the best position of the whale is not known at the beginning of the hunt, it considers its current position as the best position toward proximity to the prey.
When a whale with the best position is detected near the prey, the other whales update their next position accordingly.Equations 1 and 2 determine this updated position (Mirjalili and Lewis 2016).
where t represents the current iteration, C and A are the coefficient vectors, X Ã t ð Þ is the position vector of the best solution, and XðtÞ is the current position vector.Coefficients A and C are calculated using Eqs. 3 and 4 (Mirjalili and Lewis 2016).
where a decreases linearly from 2 to zero during the iteration of the algorithm and r [[0,1].The exploitation phase of the algorithm is as follows: (a) Shrinking encircling: It is obtained by reducing the values of a, according to Eq. 2.
(b) Spiral updating: Calculates the distance between the prey and the whale.Equation 5shows this movement (Mirjalili and Lewis 2016).
where b is constant and l is a random number between , where D i expresses the distance between ith whale and the prey.To select the spiral or shrinking encircling mechanism, a 50% probability number is defined as follows (Mirjalili and Lewis 2016): where p is a random number between [0,1] with a uniform distribution.In the exploration phase, -1 \ A \ 1 is used to force the agents to behave more randomly.Here, the whales randomly move toward each other.Equations 7 and 8 indicate the exploration phase (Mirjalili and Lewis 2016).
where X rand is a random solution or a whale that is selected from the current population.
In short, the algorithm starts with an initial random population of solutions.In each iteration, an agent changes its position randomly or according to the best answer obtained so far.The parameter a decreases from 2 to zero for the exploration and exploitation phases, respectively.When |A| [ 1, random search agent is selected.While choosing the best solution and updating the position of search agents, |A| \ 1 is considered.And finally, the value of p gives the ability to switch between spiral or circular movements.

Game theory
A normal game contains a limited set of strategies S 1 Â S 2 . ..:S n and the set of the real values utility function u 1 Â u 2 . ..:u n , where S i and u i are the strategy set and utility function of player i, respectively.Each member of S 1 Â S 2 . ..:S n is known as a strategy profile.The set of all strategy profiles forms the game state space.For the strategy profile s, s i is the strategy of player i, and s Ài represents the n-1 strategy vector of other players.And finally, the value of u i s ð Þ is the payoff of player i.In a symmetric game, u i s ð Þ depends on the strategy of player i and other players (Fundenberg and Tirole 1991).If a game G contains n players with different strategies and utility functionsG ¼ ð S i f g; u i f gÞ, then p 1 ; p 2 ; . ..p n is the set of the probability distribution over the set of strategies.Also p 1 ; p 2 ; . ..p n is Nash equilibrium, if for each player i, choosing the strategy s i leads to maximizing the payoff of that player, assuming that other players have chosen their strategy with the probability distribution p j i 6 ¼ j ð Þ from their possible strategies.When a game leads to a Nash equilibrium, no player will get more payoff by changing the strategy, and the game will end (Papadimitriou and Roughgarden 2008).A dynamic game is a type of game in which players act sequentially or repeatedly.In this model, all players know the rules of the game and every player tries to maximize their payoffs with the knowledge of their opponents.Players may have to choose more than one action and might observe other players' past choices.In cooperative games, actions are taken by groups of agents, and payoffs are given to them.

Network model
In edge server placement, the two main entities are the cellular base station (CBS) and mobile edge server (MES).In the proposed model, B ¼ b 1 ; b 2 ; . ..bN and S ¼ s 1 ; s 2 ; . ..sM are the set of base stations and servers, respectively.Each MES is covered by one or more CBSs.Cloud users connect to servers through CBS. Figure 1 shows the network model.Due to a large number of servers, which makes finding their optimal locations complicated, the resource deployment area in the proposed model is divided into some sub-areas.A server belongs to a base station that is within its range.

123
Considering that the objectives of the proposed method include reducing latency and better load balancing on the servers, therefore, the latency and load balance models are defined in the following.

Latency
Latency refers to the time difference between the server response and the service request.This time, especially in online applications, must be tolerable; otherwise, these services will not be efficient.Latency is a function of the distance between CBS and MES, as well as the bandwidth of the communication network and the amount of data sent or received.In most studies, this time is only considered the distance between each CBS and MES, without considering the user workloads and the bandwidth of the communication network.In the proposed method, latency is calculated using Eq. 9 (Gibbon and Little 1996).
where Pd and Td are propagation and transmission delays, respectively.The propagation delay of a signal depends on the length of the transmission line between server S and base station C and the propagation speed, which is shown in Eq. 10.The distance between each CBS and MES is computed using the Euclidean distance.The propagation speed depends on the propagation environment, which is proportional to the speed of light.
The data transmission delay is related to the data volume and the transmission line's bandwidth, which is shown in Eq. 11.
where Data À size and BW are the size of the data sent or received and the bandwidth of the communication network, respectively.

Load balancing
Load balancing on servers increases their efficiency and provides more suitable services to mobile users (Asghari and Sohrabi 2021).This avoids overloading resources, while some other resources are under-loaded.In the proposed method of the paper, the standard deviation metric is used to determine the load balancing of resources, which is shown in Eq. 12 (Asghari and Sohrabi 2021).
where u j and b u k are the utilization rate of server j and the average utilization rate of all servers in zone k, respectively.Finally, n k is the total number of servers in zone k.In multi-objective problems, Pareto-based, like NSGA-III users (Asghari and Sohrabi 2022b) and weighted average methods are used to determine the value of the solutions.In our proposed method, when the specified weight vector is close to one extreme, e.g., (0.1, 0.9) and (0.2, 0.8), the NSGA model shows a little bit better performance, but in other cases, the weighted sum method is superior to NSGA, so for more control over the weights of the objectives, we used this technic in our proposed method (Chen and Li 2022).
The proposed method uses the weighted average model shown in Eq. 13.
where w 1 and w 2 are the weights of the objectives that are normalized and can be adjusted.Although these parameters are adjustable, in all experiments, these parameters are assumed equal to 0.5.

Proposed method
In this section, the details of the proposed three-step algorithm are described.Based on our model, placing each server is independently followed by the WOA algorithm.Then, in the second step, using a neural network, the local placement of each sub-area is performed, and finally, in the third step, the global server placement is obtained by utilization of the game theory model.
Step one: The geographical area of resource deployment is divided into k sub-areas to reduce the complexity of the problem.Then, some servers are randomly assigned to each of these regions.Based on the WOA algorithm, each CBS and MES are considered as prey and whale, respectively.This step aims to place each server in its best position in the domain of CBSs, considering the objectives of the algorithm.Then, in the next iterations, the new position of the servers is determined using Eq.14.
where X s; b; t þ 1 ð Þis the position of the server s relative to the base station b at time t ? 1 and X Ã s; b; t ð Þ is the best position of the servers relative to the base station b at time t.D is determined by Eq. 15.
Also, A ¼ 2:a:r À a; C ¼ 2:r, where a decreases linearly from 2 to 0 during the iterations of the algorithm and r 2 ½0; 1.
Two main parameters of meta-heuristic algorithms are exploitation and exploration.Exploration involves the algorithm looking for new solutions in new areas, whereas exploitation is using existing solutions and improving them to increase their fitness.
In the exploitation phase of the proposed model, the new position of a server relative to the covered base station is determined using Eq.16.Here, the new position of a server or Xðs; b; t þ 1Þ is updated considering its best previous global position, i.e., X Ã ðs; b; tÞ.
where b is constant and l is a random number between j determines the distance between the ith base station and the covered server.To choose random or greedy models in changing the position of a server, a 50% probability selection is defined as follows: To avoid falling into the local optima, and in the exploration phase of the proposed method, the new location of the servers or X s; b; t þ 1 ð Þis determined with more random behavior to discover the new search space.This displacement corresponds to a random position or X rand in the search space.
Equation 18 is used to increase exploration and more random behavior of the algorithm to find new locations.
Our proposed method comprises three layers or steps, including personal, local, and global server placement strategy.
Algorithm 1: In this step, and to reduce the complexity of finding a global solution, the geographical area of resource deployment area is divided into some sub-regions.In lines 2 to 4, some servers are randomly assigned in these areas and their fitness is calculated by Eq. 13.The optimal location of each server will be updated in the next iterations using Eqs.15, 18, and 16 according to operators of Wall's algorithm described earlier.According to Eq. 13, If the new location of each server is better than its previous position, its optimal location, i.e., S Ã , is updated.At this stage, the algorithm focuses on each server independently.Convergence between these independent servers will follow in the second phase.This step is considered the local coarse-grain server placement method.Algorithm demonstrates the first step of the proposed method.
Algorithm 2: As mentioned in the previous step, the optimal location of each server is found independently and regardless of the location of other resources.In the second phase and for local convergence of the optimal placement of all resources that belong to each area, some servers are moved from their personal optimal places to get the local optimal locations, considering all resources in that region.This is done by using MLP neural network model (Villarrubia et al. 2018).The network input is the geographical coordinates of servers that belong to each region.Also, the output of the network is the average value of the objective function that should be minimized.Then, the learning process determines the exact location of local resources to maximize their efficiency in each area.The details of this step are shown in Algorithm 2. The used MLP neural network is designed with three layers with some inputs and one output.The connections between the layers are fully connected.The error function is the mean square error (MSE), the activation function model is sigmoid, and the learning model is feed-forward.The number of network inputs is equal to the number of resources used in each subregion.This step is considered the local fine-grain server placement method.
Multi-objective edge server placement using the whale optimization algorithm and game theory 123 Algorithm 3: In the second step of the proposed method, the neural network is used to find the optimal location of servers in each area.However, to avoid falling into local optima, it is necessary to converge all sub-regions to find a global solution in the entire area.For this purpose, game theory is used.In the proposed method, the number of players is equal to the number of sub-regions.Each solution vector of a region is defined as a strategy for that player.Players find the global solution and minimize the fitness function in Eq. 13.The type of game is dynamic and cooperative.The game terminates when the players' strategy change does not lead to more payoff for them, and the global placement of resources is obtained according to the objectives of the proposed method.Algorithm demonstrates the third step of the proposed method.Here, each area will contain one player who will act according to their local placement strategies.All regions try to do their best in resource placement.However, to reach the global optimum, each region chooses its best strategy that improves the performance of the entire network.When the change of strategy does not improve this performance, the game balance has occurred and the final placement of servers is obtained in the entire area.
Figure 2 shows the proposed three-level model.As this figure shows, in server-level mode, each server independently and regardless of other resources tries to find its suitable location.This helps to get the approximate location of the servers according to Algorithm 1.At the local level for the convergence of all the servers of a region, the multi-layer perceptron network (MLP) is used based on Algorithm 2. In this case, the position of each server is received from the previous level as the input of the neural network.The output of the network is the fitness function of the problem (Eq.13), which must be minimized.The learning process tries to find the best location for the servers in each region, which is updated according to the efficiency of the entire sub-area.Finally, at the global level and by using the game theory model in Algorithm 3, the final location of the servers is determined and the optimal solution is obtained.

Evaluations and discussion
In this section, the proposed method is evaluated and compared with some similar and state-of-the-art methods to determine its efficiency in different test scenarios.The dataset, resource descriptions, and compared algorithms are described.Finally, the efficiency of the proposed method is validated by several experiments.

Experiments settings
To conduct the evaluations, real scenarios have been used based on the information from the MCI Telecommunication network of Tehran, the capital of Iran.Tehran is a modern city with an area of 750 square kilometers and a population of about 9 million people, including mountainous and flat areas.Commercial and political centers are mainly located in the city center.During the day, over three million people living in the suburbs come to this city for work and education.The number of service requests varies in different areas and at different times.Proper server placement will play an important role in increasing the quality of services.Figure 3 shows part of the CBS map of this city (OpenCelliD 2022).The geographical locations of the base stations and their average number of online users are shown in Table 1.MATLAB v9.6.0R2019a software has been used on a computer with IntelÒ core (TM) i7 2.5 GHz CPU and 16 GB of main memory to perform experiments.The used servers have processing power from Fig. 2 The three-step model of the proposed method Multi-objective edge server placement using the whale optimization algorithm and game theory 123 10 to 50 GHZ.Tasks are generated and offloaded on servers with the Poisson distribution, according to the number of users of each CBS.Table 2 shows the characteristics of the resources that are used in experiments.The resources consist of CPU, RAM, Disk, and communication network bandwidth.
The calibration of an optimization algorithm's parameters is one of the fundamental problems in evolutionary and meta-heuristic algorithms.These parameters control selection, population size, mutation, and other important variables.It is necessary to tune these parameters for a wide range of situations.Some classic methods are used to calibrate the parameters of these algorithms.In this study, we use a numerical calibration and relevance estimation (CRE) method to choose and calibrate the parameters that aim for the validity of our proposed method (Nannen and  123 Eiben 2006).We can get relevant metrics of calibration difficulty from the CRE approach in two different ways: numerically, as Shannon entropy, and visually, as percentiles.Selecting and calibrating the pertinent parameters of the proposed method only requires visual inspection and manual addition or removal of parameters.

Compared algorithms
Three similar and new algorithms have been used to compare with the proposed method.The first algorithm uses the multi-agent reinforcement learning (MARL) (Kasi et al. 2021) technique.In this method, agents try to learn the dynamics of the environment and optimal resource placement with the objectives of reducing latency and better load balance of resources.The reasons for using this algorithm are the similarity in objectives and the use of the learning model.Our proposed algorithm uses the neural network algorithm as an important learning technique.So, for a fair comparison, this algorithm has been chosen.The second algorithm, PACK (Lahderanta et al. 2021), with the objectives of load balance and reliability, uses capacitated location-allocation.The authors have used the K-means clustering algorithm and a new heuristic for server placement at the edge of the network.Our proposed algorithm divides the area into some sub-regions to reduce the complexity of the problem.PACK algorithm like our model clusters the servers into some groups.The second reason for choosing this algorithm is the similarity in the objectives.So, for these reasons, this algorithm has been selected to compare with the proposed method.Finally, BGWGS (Wang et al. 2022) uses genetic and gray wolf optimization algorithms with the objectives of cost reduction and more load balancing.The genetic algorithm has been used to find servers' optimal locations.The gray wolf algorithm accelerates the algorithm.Considering that the proposed method, similar to BGWGS, has deployed a new meta-heuristic and population-based algorithm, and also because of the similarity in the objectives, this algorithm has been used to compare with the proposed method.

Simulation results and discussion
Finding the location of servers in a shorter time determines the superiority of the algorithm.In the server placement problem, as the number of servers increases, potential solutions exponentially increase.Partitioning the area and using the proposed method will reduce the complexity of the problem and place resources in a shorter time.Table 3 shows the convergence time of the proposed method and other algorithms to find the optimal solutions for different numbers of servers.As this table shows, except in the first scenario, the proposed method has found the optimal locations of the servers in less time.When the number of servers increases, the proposed method shows better performance.
The service error metric has been used to show the proposed method's efficiency.While a requested service can be run on edge servers, cloud data centers will not be requested.However, there is no guarantee that network edge servers can always provide the service required by users.Here, user requests will be forced to use traditional cloud servers.In this case and because of the long distance between cloud data centers and mobile users, the resource access delay will increase.The proper placement of servers reduces the rate of server access errors and the number of requests to cloud data centers.The proposed method uses this criterion as a resource access error and tries to reduce it.Table 4 indicates the error rate of the proposed method and its comparison with other algorithms.As the result, because of the more appropriate placement of resources, the proposed method has a lower error rate than other algorithms.
Reducing the server's access delay is one of the main objectives of cloud service providers.An inappropriate server placement can increase the probability of overloading the servers in one area, while some others underloaded in other regions.Due to a large number of servers, finding their optimal locations is a big challenge.The proposed method, using the partitioning approach and its three-step features, reduces the average network latency.In most similar methods, the distance between servers and CBSs is considered a measure of resource access delay (Wang et al. 2019) and communication network bandwidth and volume of exchanged packets are not considered.The proposed method of this paper considers these parameters and presents a real resource access delay model.Figure 4 shows the network average latency in the proposed method and other algorithms, based on the number of servers used.As expected, network latency increases as the number of resources decreases.Two major reasons cause this problem.First, when the number of servers reduces, the average distance of users to the resources increases and Multi-objective edge server placement using the whale optimization algorithm and game theory 123 consequently affects the average network delay.The second reason is that user requests are sent to fewer servers, which can occupy the bandwidth of the communication network.Dividing the resource deployment area into smaller sub-regions will reduce the complexity of the problem, and, as a result, more suitable server locations will be obtained.Increasing the number of sub-regions causes the neural network to face fewer inputs in each subregion, and a more appropriate convergence occurs in finding the server's locations.However, the increase of sub-regions increases the number of players in game theory to reach the global solution.For this reason, there is a tradeoff between these two issues.As Fig. 4 shows, latency gradually decreases as the number of sub-areas increases up to eight sub-areas.But when the number of sub-regions reaches 16, latency increases, and therefore, dividing the area into eight sub-regions is the optimal point.Better load balancing of servers is another goal of the proposed method.Improving the load balance on servers will increase the quality of services, reduces resource access errors, and increases their availability.As mentioned in the previous sections, the proposed method uses the standard deviation metric to determine the load balance on the servers.Figure 5 shows this issue in the proposed method and other algorithms.Based on the proposed method, the resource deployment area is divided into 2 to 16 regions.As this figure shows, the load balance on servers gradually improves as the number of clusters increases to 8.But by increasing the number of regions to 16, the load balance of the servers decreases.This is because of the tradeoff between neural networks and game theory.For this reason, the optimal number of sub-areas is equal to 8.
Finally, another objective of the proposed method is to reduce the number of used resources while maintaining the network's performance.If placing resources can be done with fewer servers, it will save the user cost and energy consumption of the servers.Table 5 shows the efficiency of the proposed method and other algorithms.As this table shows, by reducing the number of servers to 395, the efficiency of all algorithms is still maintained.But by

123
reducing the number of servers to 385, the proposed method still maintains its efficiency.However, with the further reduction of servers, the efficiency of all algorithms gradually decreases.But the efficiency of the proposed method reduces at a lower rate.

Performance measures of the proposed model
Proper server placement makes the servers to be used efficiently.Otherwise, there will be a possibility of server overloading and underloading.When a server cannot respond to the requested services, failure occurs, and the service migrates to traditional cloud data centers.The proper prediction of failures avoids the migration rate and  Multi-objective edge server placement using the whale optimization algorithm and game theory 123 improves the performance of the network.We use this measure as an efficiency criterion.
The performance of the proposed method is evaluated based on precision, recall, and F-measure metrics.The evaluation criterion of the proposed method is the correct server selection to avoid task migration.1000 jobs are randomly selected from the datasets, and according to Table 6, the success or failure of the proposed method is measured.
According to the values in Table 6, the performance of the proposed method can be calculated as follows: According to these criteria, the proposed method has high accuracy because of the correct server placement, which leads to accurate convergence in global resource placement.

Conclusion
When the number of mobile edge servers increases, finding the best location for them becomes more difficult and the complexity of the problem increases.This is because most of the resource placement methods use a discrete approach and also the resource placement area is considered to be integrated.To overcome these constraints, we proposed a new server placement method.In the first step of the proposed three-step algorithm, the geographical area of resource deployment was divided into several sub-areas.Then, the WOA algorithm with a continuous approach was used to find the optimal location of each server, regardless of other resources.In the second step, the neural network was deployed in each sub-region.The neural network was used for convergence in the optimal placement of all resources within each sub-region.Finally, the global deployment of all resources was obtained, by using the game theory.The simulation results showed that the proposed method reduced the network average access latency by 33.5% and also improved the load balance on servers by 28.2%, compared to some similar algorithms.As a future suggestion, we plan to extend our method in saving the energy of servers and the offloading process.
Funding The authors declare that no funds, grants, or other support was received during the preparation of this manuscript.

Fig. 3
Fig. 3 Part of the CBS map of Tehran city

Table 1
The CBS information A.Asghari et al.

Table 4
The error rate of the algorithms

Table 5
Number of used servers Number of MESs

Table 6
Confusion matrix of the proposed method