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Computing offloading of mobile devices (MDs) through cloud is a greatly effective way to solve the problem of local resource constraints. However, cloud servers are usually located far away from MDs leading to a long response time. To this end, edge cloud servers (ECSs) provide a shorter response time due to being closer to MDs. In this paper, we propose a computing offloading game for MDs and ECSs. We prove the existence of a Stackelberg equilibrium in the game. In addition, we propose two algorithms, F-SGA and C-SGA, for delay-sensitive and compute-intensive applications, respectively. Moreover, the response time is reduced by F-SGA, which makes decisions quickly. An optimal decision is obtained by C-SGA, which achieves the equilibrium. Both algorithms above proposed can adjust the computing resource and utility of system users according to parameters control in computing offloading. The simulation results show that the game significantly saves the computing resources and response time of both the MD and the ECSs during the computing offloading process.

The popularity of the Internet of Things (IoT) allows people to enjoy the convenience of the Internet in most scenarios of daily life. Especially for mobile devices (MDs), network services provide convenience and functional possibilities. However, functional and computationally intensive applications consume a large amount of energy and computing time on MDs, such as augmented reality [

Edge cloud computing (ECC) is promising for mobile computing offloading, which also considers a promoter of 5G mobile networks because they are located near the edge of the network [

Edge cloud network.

In this paper, we consider the equilibrium problem during the computing offloading process, which not only considers the needs of MDs, but also considers the maximum benefits of service providers. For this reason, we propose a strategy for computing offloading by computing the equilibrium between a MD and ECSs. Liu

Realistically considering the real offloading scenario, we formulate the interaction of the MD and the ECSs during the computing offloading process as a Stackelberg game.

We prove the existence of equilibrium in the Stackelberg game. Furthermore, we propose the C-SGA for computing the equilibrium. And we also design the F-SGA for delay-sensitive applications, which greatly reduces response time.

We verify the performance of the proposed algorithms via the simulation experiments. The results show that the proposed algorithms increase the efficiency of computing offloading. In addition, we performed a detailed analysis of the performance of the strategy for the changes in the values of the different parameters.

The rest of the paper is organized as follows. We present the related work in Section

Most previous works have been done on computing offloading [

Proposition of MAUI [

Some of previous work conducted extensive research on game theory. There has been some research on the game that can be used in computing offloading [

As the primary deployment method for MEC [

We assume that computation offloaded profile is

Therefore, the cost of the MD for performing computing offloading is determined by the computing time, the transmission time, and the payment for the ECSs, i.e.,

ECSs usually have their own computations to compute as shown in Figure

The profit of the ECS

The utility of each ECS

The strategy among the MD and the ECSs is formulated as a Stackelberg game. Our strategy has two steps as shown in Figure

Two-step Stackelberg game.

We assume that the MD gives a first initial payment; then the optimal decision of each ECS

Next, we obtain the optimal strategy by maximizing the utility of the MD after receiving the decision of the ECS, i.e.,

As discussed above, the MD determines the payment profile of computation offloading to the ECSs, while the ECSs provide corresponding amount of computation. We model the problem as a two-step Stackelberg game among the MD and the ECSs based on the noncooperative game theory. Explicitly, as shown in Figure

In this work, we denote by

A Nash equilibrium of the Stackelberg game is a strategy profile

where

In this part, we will analyze the equilibrium in the Stackelberg game that we proposed and prove the existence of Nash equilibrium in the Stackelberg game. In order to confirm the existence of Nash equilibrium in our Stackelberg game, we have made the following proofs. We first discuss the existence and uniqueness of Nash equilibrium of the Stackelberg game.

The strategy profile set of the ECSs is nonempty, convex, and compact.

First, according to the characteristics of ECSs, we calculate the first-order derivative of

The ECS

Obviously, if the MD gives a

When

For the optimal strategies of ECSs, the MD has a unique optimal strategy.

The first-order partial derivative of

From (

With Lemmas

A unique Nash equilibrium exists among the MD and the ECSs in our proposed Stackelberg game.

In this section, the process of strategy is shown in Figure

The game process of our Stackelberg game.

We propose F-SGA to quickly achieve equilibrium as shown in Algorithm

As discussed before, we set two points, denoted as

The algorithm proposed in Algorithm

In contrast, C-SGA only needs the ECSs to compare the the bound of payment, which not only greatly reduces the computation of the ECSs, but also obtains more optimized payment. Therefore, C-SGA is suitable for situations where the ECSs have less computation resources. The process of C-SGA is summarized as follows.

The MD first sends the initial value of payment. With the strategy

After receiving the results, the MD calculates the utility function

Then the MD and the ECSs continue to perform procedures (1) and (2) until

The proposed Algorithm

Since

We use extensive simulations to verify our proposed strategy and algorithms. We set up 3 ECSs as the initial default settings for our simulations. The total computation

In this part, we analyze the changes in the utility of MD as a function of different conditions. First we analyze the relationship between the utility of a MD and the revenue of the ECSs. Figure

The utility of the MD with different unit reward.

Then we analyze how the payment of a MD affects the utility of the MD. Figure

The utility of the MD with different payment.

Figure

The utility of the ECSs with different unit reward.

Figure

The utility of the ECSs with different revenue.

In order to evaluate how the unit revenue impacts computation for offloading, we set three different unit revenues for simulation. We set

Payment of MD.

Figure

Offloading quantity.

We are interested in the impact of

The impact of computational accuracy.

Figure

Response time of F-SGA and C-SGA.

In this paper, we proposed a game for the computing offloading between a MD and ECSs. We provided a Stackelberg game theoretic analysis and proved the existence of the equilibrium in the Stackelberg game. Furthermore, we proposed two algorithms for different scenarios and provided the upper and lower boundary of the payment. Moreover, multiple optimization results were obtained by regulating model parameters. The simulation results showed that the game is effective in improving the utility of both the MD and the ECSs.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grants no. 61602155, no. 61871430, no. U1604155, and no. 61370221 and in part by Henan Science and Technology Innovation Project under Grant no. 174100510010 and in part by the industry university research project of Henan Province under Grant No. 172107000005 and in part by the basic research projects in the University of Henan Province under Grant No. 19zx010.