The Game Model Description

Published Date: 02 Nov 2017

Abstract

Cloud computing is a hot research point, for it can provide as many resources as one needing. Usually, it consists of a large number of ordinary servers which run all the time. Due to the high-available characteristic, the cloud computing consumes lots of energy which needs to be economized. So green clouding computing becomes one of the most important aspects in cloud computing research. In this paper, resource allocation is studied based on cooperative differential game model to maximize the payoff of the resource utilization and minimize the energy consumption. Lastly, optimal allocation strategy is analyzed using Shapley theorem.

Keywords: resource utilization, energy consumption, cloud computing, game theory

1 Introduction

The cloud computing platform is usually composed by two parts which are end user part and cloud server part [10, 11]. End users usually store their resources in the cloud server part and use special algorithms retrieving the data which they need. The cloud server part works as a large pool of resources and is composed by lots of ordinary servers. There are three kinds of services that clouding computing can provide, namely Software as a Service (SaaS), Platform as a Service (PaaS) and Infrastructure as a Service (IaaS) [12-14]. In the SaaS model, cloud servers install and operate software in the cloud part and end users access and get service without installing any software, such as virtual game or virtual desktop. In the PaaS model, cloud servers offer a computing platform and/or solution stack usually including operating system, programming language, database and so on. End users can develop and run their software sources on a cloud platform without building their own the underlying hardware and software layers. In the IaaS model, cloud servers offer computers as physical or more often as some virtual machines which are being used raw storage, firewalls or load balancers.

In [15], the authors claimed in data centered network only 20-30% power consumption is useful, and the additional 70-80% is wasted due to over-provisioned idle resources. Cloud computing is one type of data centered networks, and servers in the system need high-available in order to coping the uncertain serving requests. Then green cloud computing becomes a hot research point which aims reducing the consumption of energy while maintains the excellent performance.

Some researches focused on the green cloud computing network infrastructure [5-7]. In [5], the authors concluded that there are four classes of solution can be used in network infrastructure to achieve green networking, namely resource consolidation, virtualization, selective connectedness and proportional computing. In [6] the authors proposed a model-driven engineering approach in order to optimizing the cost of cloud auto-scaling infrastructure in configuration, energy consumption, and operating aspects to reduce emissions resulting from idle resources to create greener computing environments. In [7] the authors built virtualization security assurance architecture, called CyberGuarder. The CyberGuarder can address several key security problems of the green cloud computing context and provides three kinds of services; namely, a virtual machine security service, a virtual network security service and a policy based trust management service. Application layer algorithms are another way to achieve green cloud computing [8, 16-17]. In [8] the authors used a neural network based predictor to build a green scheduling algorithm to achieve energy savings. The predictor can predict future load requirements based on collected historical requirements. Using this information, they could decide whether or not the servers needed being running. A generic gossip protocol GRMP-Q was proposed in [16], which can minimize the power consumption via server consolidation and can also satisfy an uncertain load pattern. A new framework was presented in [17] which provided efficient green enhancements within a scalable cloud computing platform.

Some researches paid their attention on the performance of green cloud computing [9, 18]. In [9] the authors analyzed energy balancing in processing switching and transmission, data storage and data transport. They set three cases, called storage as a service, software as a service, and processing as a service, to evaluate the energy consumption. In [9] the authors investigated the relationship between cloud, mobile, social, green computing and IT strategy, enterprise architecture of the financial services organizations

In this paper, based on cooperative differential game model, we just pay attention to the resource allocation in cloud computing in order achieve low energy and bandwidth consumption while maintains the good performance. We take the number of service provided by cloud computing servers as resources, and find an optimal strategy mapping these resources with the demands proposed by end users. In game theory, grand coalition is usually used to maximize to payoff and Shapley theorem is used to achieve fairly allocation [19, 20]. So these tools are mainly used in our analysis.

The rest of the paper is organized as follows. In Section 2, the game model is described. Game solution and Shapley value is analyzed in Section 3. A conclusion is drawn in Section 4.

2 The Game model description

In this Section, a cooperative differential game model, which was inspired by the works in [1-4], will be built to analyze the resource allocation problem in green cloud computing. Firstly, some definitions will be proposed which are related to our analysis. Then the key parameters are analyzed. Lastly, a brief differential game model is built.

Definition 1. In the green cloud computing platform, the number of server is , , . The serving ability of these servers can be divided into two parts which are the number of basic services providing , and the number of extra services providing , at a certain time point. The accumulated services amount provided by the green cloud computing platform by time denotes , .

From definition 1, we can build the following differential equation:

, given , (1)

Where represents the probability of service providing is being failed.

As game players, each green cloud computing server seeks to minimize its utility function of discounted sum of the cost of increasing bandwidth and energy consumption. So we propose the following definition.

Definition 2. In the green cloud computing platform, the cost of service providing is related to two elements, namely bandwidth and energy consumption. For a certain server, The bandwidth consumption can be written as and energy consumption can be written as . Where, are constant factors and denotes the extra services provided by server .

Here, we need to clarify that the reason why we use quadratic model representing energy cost is explained as the precipitous discharge in battery life for lithium-ion batteries [21].

The optimization problem of green cloud computing server is

(2)

s. t. (3)

Where and are the common discount rates.

In green cloud computing platform, servers need cooperative with each other to achieve cost sharing and improve network performance. We choose this formulation for activate the servers participating in cooperative. Obviously, each player’s cost depends on its bandwidth and energy consumption.

3 Game solution and Shapley value

In [2], the authors proposed a way to deal with the problem of sharing resources. They claimed that the steps of fairly sharing resource could divide into the following three ones.

Computation of the cooperative game characteristic function values.

Based on the Shapley value, allocate the total cooperative payoff among all the players.

Allocation over time of each player’s Shapley value having the characteristic of being time-consistent.

Using their method, we solve our game solution problems. In the following Section, We first introduce two related theorems. Then solve the resource allocation problem.

3.1 Overview of related theory

Considering the model mentioned above which is of infinite horizon, we will solve a series of dynamic optimization programs using feedback Nash equilibrium. The dynamic optimization programs technique was described by Bellman [20], and the technique is given in Theorem 1 below.

Theorem 1. (Bellman’s Dynamic Programming) A set of control constitutes an optimal solution to the infinite-horizon control problem (2) and (3) if there exist continuously differentiable functions defined on which satisfies the following Bellman equation:

(4)

The Shapley value is used to distribute the total payoff to the players in the condition of they all collaborating. The computing formula is described in Theorem 2 [22].

Theorem 2. (Shapley value) Given a coalitional game , and The Shapley value of player can be written as:

(5)

3.2 Cooperative game characteristic function values

In order to get the cooperative game characteristic function values, we need to compute the optimal cost of grand coalition, the feedback Nash equilibrium and the optimal cost for intermediate coalitions.

3.2.1 The optimal cost of grand coalition

The grand coalition figures out a standard dynamic programming problem which consists of minimizing the sum of all players’ costs subject to service providing dynamic. Using the theorem 1, we obtain the Bellman equation:

(6)

where represents the Bellman function of this problem as defined in Theorem 1. For convenience, from now on we omit the time argument when no ambiguity may arise.

In order to get the optimal strategy, we differentiate the right hand side of formula (6) respecting to and equating to zero. We can get the equation (7):

(7)

Assuming, and then we can get that.

Substituting, and by their values in (6) gives

(8)

Then we can get:

(9)

Lastly, the result could be as following:

(10)

And the optimal strategy could be written as:

, (11)

The optimal trajectory of the service serving can be obtained as [2]:

(12)

3.2.2 The feedback Nash equilibrium

Using the Theorem 1, we can get the feedback Nash equilibrium can be written as:

(13)

Differentiating the right hand side of formula (13) respecting toand equating to zero.

(14)

Assuming , and then we can get that .

Substituting , and by their values in (13) gives:

(15)

Then we can get:

(16)

The result could be as following:

(17)

And the optimal strategy could be written as:

, (18)

Owning to the game is symmetric, so, ,

We can get the final outcome being:

(19)

3.2.3 The optimal cost for intermediate coalitions

According to the Theorem 1, we can write the intermediate coalitions as:

(20)

Differentiating the right hand side of formula (20) respecting to and equating to zero.

(21)

Assuming , and then we can get that .

Substituting , and by their values in (20) gives:

(22)

Then we can get:

(23)

The result could be as following:

(24)

And the optimal strategy could be written as:

, (25)

For , asumming:

(26)

We can get the final outcome being:

(27)

According to the method of calculating characteristic function values proposed in [2], we get the cooperative game characteristic function values can be:

(28)

(29)

3.3 Shapley value calculation and payoff allocation

In Section 3.1, we have given the definition of Shapley value. Here we rewrite it as:

(30)

For simple the calculation process, we set.

Since the game is symmetric. For each of the player, the Shapley value is the same, so we can get:

, (31)

3.4 Imputation distribution procedure values

In [2], the authors defined the imputation distribution procedure (IDP) being, and for time-consistent, , it can be written as:

(32)

Using the Shapley values and the formula (32), we can get the final allocation could be:

, (33)

4 Numerical simulation and analysis

In this Section, we just simulate the model that we built. Considering a scenario in which three Internet of Thing’ nodes,andwant to providing services. We plot the different theoretical results regarding different energy cost parameter and bandwidth cost parameter. According to the formulas (11), (18), (26), using different kinds of service models namely grand coalition, intermediate coalitions and feedback Nash equilibrium, we can get that the total number of service that the whole cloud computing provide can be written as:

(34)

(35)

(36)

Using the parameter values provided in table 1, we plot the different theoretical results regarding different energy cost parameter and bandwidth cost parameter.

Table 1. Parameters being used

10

50

0.5

0.05

[0-5]

[0-5]

20

30

Fig.2 shows the impact of bandwidth cost parameter on the number of services when energy cost parameter fixed. We can see that the number of services decreases with the increasing of bandwidth cost parameter. The explanation is as follows. The bigger bandwidth cost parameter is, the more attentions are paid on bring down bandwidth cost. For a constant bandwidth cost parameter, the number of services is the smallest in grand coalition and is the biggest in non-cooperative one. This represents that the bandwidth is most efficient in grand coalition. In another word, the grand coalition can inspire participators working cooperatively aiming at bringing down the bandwidth cost self-consciously.

Fig. 2 Impact of bandwidth cost parameter on the number of services when energy cost parameter fixed

Fig. 3 Impact of energy cost parameter on the number of services when bandwidth cost parameter fixed

Fig.3 shows the impact of energy cost parameter on the number of services when bandwidth cost parameter fixed. We can see that the number of services increases with the increasing of bandwidth cost parameter. The explanation is as follows. The smaller bandwidth cost parameter is, the more attentions are paid on bring down energy cost, for if there is no energy, no node can work. At a constant energy cost parameter, the number of services is the smallest in grand coalition and is the biggest in non-cooperative one. The explanation is as the bandwidth one.

Fig. 4 shows that the number of provided services varies according to different cost parameters, which helps the cloud computing operators getting the idea about the values of cost parameters that could be used to trade-off between bandwidth and energy consumption. As can be seen in Fig. 4, when, the number of provided services can obtain intersection and then the cost parameters can be used to trade-off energy and bandwidth consumption with cloud computing throughput optimization.

Fig. 4 Comparison of energy cost parameter and bandwidth cost parameter impacted on the number of services

5 Conclusion

In this paper, resource allocation in green cloud computing was analyzed. Cloud computing is a new field in which resources is stored and retrieved without any limitation, for it works all the time. Then the problem comes that this work model will waste lots of energy. So green cloud computing is proposed to solve this problem. We took the number of service provided by cloud computing servers as resources, and aimed at finding an optimal strategy mapping these resources with the demands proposed by end users. We chose cooperative differential game model as our analysis tool. Using the grand coalition, payoff is maximized and cost is minimized. Turning to Shapley theorem, fairly allocation is achieved among all the green cloud computing servers.

Acknowledgments

We gratefully acknowledge anonymous reviewers who read drafts and made many helpful suggestions. This work is supported by the Project supported by the Foundation for Key Program of Ministry of Education, P. R. China (No.311007), National Science Foundation Project of P. R. China (No. 61170014, 61202079).