Entropy solution of fractional dynamic cloud computing system associated with finite boundary condition
- Rabha W Ibrahim^{1}Email author,
- Hamid A Jalab^{1} and
- Abdullah Gani^{1}
Received: 21 February 2016
Accepted: 28 April 2016
Published: 4 May 2016
Abstract
Cloud computing is relevant for the applications transported as services over the hardware and for the Internet and systems software in the datacenters that deliver those services. The major problem for this state is computing the capacity and the amplitude of the dynamic system of these services. In this effort, we process an algorithm based on fractional differential stochastic equation (fractional Fokker-Planck equation (FFPE)) to find the fractional entropy solutions. Our tool is based on Mellin-Laplace transforms. Also, we suggest a fractional functional entropy formula by using the Tsallis entropy. Approximate outcomes are illustrated and discussed. The convergence of the method is investigated.
Keywords
fractional calculus fractional dynamical system fractional entropy cloud computing1 Introduction
Fractional calculus has many applications, not only in mathematics, but in other sciences, engineering, economics, and social studies. It covenants with differential and integral operators involving arbitrary powers; real and complex. It is associated with many well-known names such as Abel, Caputo, Euler, Grunwald, Hadamard, Hardy, Heaviside, Jumarie, Laplace, Leibniz, Letnikov, Liouville, Riemann, Riesz, and Weyl. The central purpose, or, at least, one of the chief purposes in considering and studying fractional calculus, is the circumstance that fractional calculus appears to be fairly significant in the investigation of some problems which arise in fractal space-time physics. We have physical schemes at three unalike stages of thought: microscopic, megascopic, and macroscopic. The fractional calculus approximates the classical calculus, and it includes non-commutative derivatives, which appears to be fairly reliable on using non-commutative geometry. This development leads one to generalize the information theory of fractional order. The books of Oldham and Spanier [1], Srivastava and Owa [2], Oustaloup [3], Miller and Ross [4], Samko et al. [5], Kiryakova [6], Mainardi [7], Podlubny [8], Hilfer [9], Zaslavsky [10], Kilbas et al. [11], Magin [12], Sabatier et al. [13], Hilfer [14], Mainardi [15], Monje et al. [16], Klafter et al. [17], Tarasov [18], Baleanu et al. [19], Yang [20], Jumarie [21, 22], etc. have enriched all areas of applied sciences. However, certain mathematical problems remain and baffle us. The complications and most of the recognized key mathematical problems in the field have been determined up to a point. There were practically no applied formulations of requirements in different areas. The developments in these areas continue [1–22]. The central substantial advantage of fabricating a procedure of fractional differential equations (ordinary and partial) in scientific modeling is their nonlocal property. It is recognized that the normal derivative is a local, linear operator, while the fractional derivative is nonlocal and non-linear. As a result the subsequent formulation of a system is predisposed by not only its current formal feature, but consistently by all of its preceding ones.
The theory of entropy was introduced in the area of thermodynamics in the 19th century and was utilized by Shannon to improve the information theory. Entropy is a conventional statistic computing concept, showing uniformity and complexity, which achieves promising applications to a widespread variety of reasonable and noisy time series data. The development was motivated by data length restraints that are commonly challenging. Investigators stressed its employment and amplification, and its utility to differentiate associated stochastic processes and models. They deliberated its impact and are stimulated to apply it in a statistically usable manner, such as marginal probability distributions and other methods. The major outcome is that the density of information so convoluted is formulated by the derivative of the function or its fractional derivative, depending upon whether it is differentiable or not. As regards information theory, one may compare the perspectives between the probability density and the derivative of a function. Fractional entropy appeared due to Tsallis (1988) (see [23]). Many investigators published different studies to improve this concept (see [24–32]).
Recently, cloud computing (CC) has developed as one of the newest and most general network computing models in various areas, such as academic circles, governments, information industry, etc. It is now essential for the new compeers information technology modification, and it expresses the progress of great scales, increasing focus on the relevance in IT studies. In an environment of CC, data is stockpiled on the cloud and manipulators can attain the influential computing capability from the cloud, devoid of getting those costly substructures. A manipulator would purchase the service of CC and attain demand as extended as suggested to the cloud service supplier and paying the lowest price. The experimental results show that the entropy is the best method to select the service of CC (see [32, 33]). This study leads to the probability capacity and the amplitude of the dynamic system of these services.
In this work, we develop an algorithm based on the fractional differential stochastic equation (FFPE) to find the fractional entropy solutions. These solutions are employed to compute the capacity and the amplitude of fractional dynamic systems. Our tool is based on the Mellin-Laplace transforms. Moreover, we propose the fractional functional entropy formulated by the Tsallis concept of entropy. Numerical results are presented for illustration. The convergence of the method is investigated.
2 Processing
Our approach deals with the following concepts.
2.1 The fractional Fokker-Planck equation (FFPE)
2.2 The fractional entropy
2.3 The fractional transform
2.4 The entropy system
The cloud computing entropy system can be constructed as a multi-agent system. Therefore, it must deal with the world’s natural affinity to ailment. Various applications involve a set of agents that are independently autonomous. In this case, each agent concludes its activities established by its own formula, capacity, and the environment. Typically, agents deliberately openly recognize one another additionally, aim approximately at this sensitivity, and then perform a sensible action. In the FFPE model assessment, coordination as an organism is referred to by the environment; agents undergo modification by the capacity of the system. Procedures in the environment produce assemblies that the agents recognize, thus authorizing ordered performance at the agent level. Naturally, these procedures increase ailment and chaos at the agent level, so that the system converts to being less ordered over time. Entropy introduces a good concept describing such a system. The minimization method for circumventing discreteness agents has the consequence of choosing at each time step the capacity that best centers on the agents. Thus, the state of the asymptotic system completely depends on the entropy solution of the FFPE model. We aim to study and determine the global behavior (self-organization) of the systems by using the capacity.
2.5 Approximate solutions
3 Results and discussion
4 Application
The job scheduling scheme is interesting and one of the essential research fields in cloud computing. It plays a similar role itself in cloud computing. The job scheduling system is accountable to choose the best appropriate resources in a cloud computing users’ jobs, by compelling various static and dynamic parameter constraints of the cloud into the deliberation. In this section, we deal with a model that describes the job scheduling system based on the queuing property and cost function considering the users, providers, and the quality of the system (QoS). By utilizing a cloud computing environment, we may assume it as a very influential server. This server holds the user’s jobs. For each job one may have a different QoS obligation; typically, the user’s jobs have various agencies to be treated. Therefore, we can classify the jobs’ urgencies into several classes. Customarily, since the cloud computes resources, customers continuously deliberate which cloud computing resource can encounter their job QoS supplies for computing (such as the paid time of job ruining, the calculating capacity), and how much the cost is that they must feed for the cloud computing resources.
The cost function with \(\pmb{\tau=1}\) , \(\pmb{\lambda=2}\) , \(\pmb{\rho=2}\)
Capacity (Λ) | Time | \(\boldsymbol{\chi_{i}}\) ( 14 ) | Cost: ( ν = 1) | ν = 0.75 | ν = 0.5 | ν = 0.25 |
---|---|---|---|---|---|---|
1 | 0.1 | 30 | 70.37 | 57.444 | 39.712 | 19.412 |
2 | 0.25 | 60 | 191.25 | 156.122 | 107.928 | 52.758 |
3 | 0.55 | 90 | 203.78 | 166.351 | 115 | 56.215 |
4 | 0.75 | 120 | 420.16 | 342.687 | 237.11 | 115.9 |
5 | 1 | 150 | 550.89 | 449.7 | 310.88 | 151.969 |
Table 1 shows the initial service rate and expectant service rate for each group in the queue with changed significance. Fractional calculus is utilized to minimize the cost. The decreasing of the fractional value \(\nu\in(0,1] \) implies the minimization of the cost function.
4.1 Discrete cloud system
5 Conclusion
We introduced a technique based on a class of fractional differential stochastic equations (fractional Fokker-Planck equations) to discuss the fractional entropy solutions of fractional dynamical systems. We showed that the concept of the transforms (see [42–49]) is very useful to complete our investigation. We applied the relation between Mellin and Laplace transforms. The approximate result is utilized to perform in a cloud computing environment system. We imposed straightforwardly a differential service adapted job scheduling system in the cloud computing setting. Examination and outcomes demonstrated that our method for the job scheduling system cannot only ensure the QoS supplies of the cloud computing service jobs, but it also can create the maximum profits for the system, with minimizing the cost.
Declarations
Acknowledgements
The authors would like to thank the referees for giving useful suggestions for improving the work. This research is supported by Project UM.C/625/1/HIR/MOE/FCSIT/03.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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