Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service

Shunfu Jin; Haixing Wu; Wuyi Yue; Yutaka Takahashi

doi:10.3934/jimo.2019060

Article Contents

2020, Volume 16, Issue 5: 2407-2424. Doi: 10.3934/jimo.2019060

This issue Previous Article A dynamic lot sizing model with production-or-outsourcing decision under minimum production quantities Next Article Inverse quadratic programming problem with

$ l_1 $

norm measure

Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service

1.
School of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, China
2.
Department of Intelligence and Informatics, Konan University, Kobe 658-8501, Japan
3.
Graduate School of Informatics, Kyoto University, Kyoto 606-8225, Japan

Received: September 30, 2018

Revised: December 31, 2018

Early access: May 2019

Published: August 2020

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Abstract

Cloud computing makes it possible for application providers to provide services seamlessly and application users to receive services adaptively. By offering services that give users an initial experience, application providers can usually attract more users. This research proposes a type of sleeping mechanism-based cloud architecture where an experience service and an enrollment service are provided on one virtual machine (VM). Accordingly, we model the cloud architecture as a queue with an asynchronous multi-vacation and a selectable extra service. We also analyze the queueing model in the steady state by constructing a three-dimensional Markov chain. Following this, we evaluate the system performance of the proposed cloud architecture based on the energy conservation level of the system and the mean delay of the visitors who select the enrollment service. Moreover, we study the Nash equilibrium strategy of visitors by building an individual welfare function, and develop an improved intelligent search algorithm to investigate the socially optimal strategy of visitors. Aiming to achieve a social optimum, we formulate a pricing policy with a reasonable enrollment fee.

Keywords:

Mathematics Subject Classification: Primary: 60K25, 35Q91; Secondary: 90B18.

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Figure 1. Sleeping mechanism-based cloud architecture

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Figure 2. Trends for the mean delay of the visitors who select the enrollment service

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Figure 3. Trends for the energy conservation level of the cloud system

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Figure 4. Trends for the individual welfare $ G_{ind}(\lambda) $

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Figure 5. Trends for the social welfare $ G_{soc}(\lambda) $

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Table 1. Iterative algorithm to calculate the rate matrix $ \mathit{\boldsymbol{R}} $.

Step 1: Setting the error precision $ \varepsilon $ (for example, $ \varepsilon=10^{-8} $). Initialize $ c, \ \lambda,\ \mu_1, $
$ \mu_2, \ \theta $ and $ q $ as needed. Initialize the rate matrix $ \mathit{\boldsymbol{R}}={\boldsymbol{0}} $ with an order of
$ m \times m $, where $ m=\left(\dfrac {1}{2}(c+1)(c+2)\right) $.
Step 2: Tackle $ \mathit{\boldsymbol{Q}} $ by using the consistency technique formula and get $ \mathit{\boldsymbol{Q}}_{c+1,c}' $, $ \mathit{\boldsymbol{Q}}_{c,c}' $
and $ \mathit{\boldsymbol{Q}}'_{c,c+1}\ \! $.
$ \mathit{\boldsymbol{Q}}'=\mathit{\boldsymbol{Q}}/U $,
$ \mathit{\boldsymbol{Q}}_{c+1,c}' =\mathit{\boldsymbol{Q}}_{c+1,c}/U $,
$ \mathit{\boldsymbol{Q}}_{c,c}' =\mathit{\boldsymbol{Q}}_{c,c}/U $,
$ \mathit{\boldsymbol{Q}}'_{c,c+1}\ \! =\mathit{\boldsymbol{Q}}_{c,c+1}/U $.
Step 3: Calculate $ \mathit{\boldsymbol{R}}^* $ by
$ \mathit{\boldsymbol{R}}^*=\mathit{\boldsymbol{R}}^2\times\mathit{\boldsymbol{Q}}_{c+1,c}'+\mathit{\boldsymbol{R}}\times(\mathit{\boldsymbol{I}}+\mathit{\boldsymbol{Q}}_{c,c}')+\mathit{\boldsymbol{Q}}'_{c,c+1}\ \! $.
%$ \mathit{\boldsymbol{I}} $ is an identity matrix. %
Step 4: While{$ \|\|\mathit{\boldsymbol{R}}-\mathit{\boldsymbol{R}}^*\|\|_\infty$}>ε
% $ \|\|{\mathit{\boldsymbol{R}}}-\mathit{\boldsymbol{R}}^\|\|_\infty = {\max} \Big \{\sum\limits^m_{i=1} \sum\limits^m_{j=1}\|r_{i,j}-r_{i,j}^\| \Big \} $, where $ r_{i,j} $ and $ r_{i,j}^{*} $ are
%elements in $ \mathit{\boldsymbol{R}} $ and $ \mathit{\boldsymbol{R}}^{*} $ respectively.
$ \mathit{\boldsymbol{R}}=\mathit{\boldsymbol{R}}^* $,
$ \mathit{\boldsymbol{R}}^*=\mathit{\boldsymbol{R}}^2\times\mathit{\boldsymbol{Q}}_{c+1,c}'+\mathit{\boldsymbol{R}}\times(\mathit{\boldsymbol{I}}+\mathit{\boldsymbol{Q}}_{c,c}')+\mathit{\boldsymbol{Q}}'_{c,c+1}\ \! $.
Step 5: $ \mathit{\boldsymbol{R}}=\mathit{\boldsymbol{R}}^* $,
Step 6: Output $ \mathit{\boldsymbol{R}}. $

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Table 2. Improved Bat algorithm to obtain $ \lambda^{*} $ and $ G_{soc}(\lambda^{*}) $.

Step 1: Set the number $ N $ of bats, loudness $ A_0 $, pulse rate $ R_0 $, the maximum
search frequency $ f_{max} $, the minimum search frequency $ f_{min} $, upper search
bound $ U_b $, lower search bound $ L_b $, the minimum moving step $ step_{min} $,
volume attenuation coefficient $ \eta $, searching frequency enhancement factor $ \phi $.
Set the initial number of iterations as $ iter=1 $, the maximum iterations
as $ iter_{max} $.
Step 2: Initialize the position, the loudness and the pulse rate for each bat.
For $ i = 1 : N $
$ \lambda_i = L_b +(U_b-L_b)*rand $
% $ rand $ returns a sample in the interval (0, 1) from the "uniform''
% distribution. %
$ A_i= A_0 $
$ r_i =R_0 $
Endfor
Step 3: Calculate the fitness for each bat.
$ G_{soc}(\lambda_i) = \lambda_i \big( R_1+ q R_2 - \varepsilon \big(qE[T_1]+(1-q)E[T_2]\big) \big)+\psi E[S], $
$ i \in \{1,2,\ldots,N\} $,
$ {\lambda^{}}=\underset {i \in \{1,2,\ldots,N\}} {\rm argmax} \{G_{soc}(\lambda_i)\} $ % $ \lambda^{} $ is present optimal position.
Step 4: Calculate the position and the fitness for each bat.
For $ i=1:N $
$ f_i = f_{min} +(f_{max}-f_{min})*rand $
$ v_i = v_i + (\lambda_i -\lambda^{*})f_i $
$ \lambda_i = \lambda_i+v_i $
If $ r_i$< rand
$ \lambda_i = \lambda^{} + (1/(2 \times iter)+ step_{min}) randn $
% $ randn $ returns a sample from the "standard normal'' distribution.
Endif
$ G'_{soc}(\lambda_i) = \lambda_i \big( R_1+ q R_2 - \varepsilon \big(qE[T_1]+(1-q)E[T_2]\big) \big)+\psi E[S] $
If $ \big(G'_{soc}(\lambda_i)>G_{soc}(\lambda_i)\big)$ and $ \big(A_i>rand )$
$ G_{soc}(\lambda_i) = G'_{soc}(\lambda_i) $
$ A_i = \eta A_i $
$ r_i = R_0(1-exp(-\phi \times iter)) $
Endif
Endfor
Step 5: Select the optimal position among all the bats.
$ \lambda^{*}=\underset {i \in \{1,2,\ldots,N\}} {\rm argmax} \{G_{soc}(\lambda_i)\} $.
Step 6: Check iterations.
If $ iter< iter_{max} $
$ iter=iter+1 $, go to Step 4
Endif
Step 7: Output the optimal position $ \lambda^{} $ and the maximum fitness $ G_{soc}(\lambda^{}) $.

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Table 3. Numerical results for the enrollment fee

Sleeping parameter	Enrollment	Socially optimal	Maximum social	Enrollment
$ (\theta) $	probability $ (q) $	arrival rate $ (\lambda^{*}) $	welfare $ (G_{soc}(\lambda^*)) $	fee $ (f) $
no sleep	0.3	2.1256	73.0759	114.5963
no sleep	0.4	1.8489	68.6578	92.8360
no sleep	0.5	1.6465	65.3641	79.3976
0.8	0.3	2.0560	65.4861	105.0306
0.8	0.4	1.7981	61.9026	85.2437
0.8	0.5	1.6020	59.2212	73.3388
0.2	0.3	1.8647	49.7374	78.4550
0.2	0.4	1.6420	47.9301	69.9957
0.2	0.5	1.4728	46.6180	60.8772

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