American Institute of Mathematical Sciences

Advanced
September  2020, 16(5): 2407-2424. doi: 10.3934/jimo.2019060

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

Shunfu Jin 1, , Haixing Wu 1, , Wuyi Yue 2, and  Yutaka Takahashi 3,

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  October 2018 Revised  January 2019 Published  September 2020 Early access  May 2019

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: Cloud computing, energy conservation level, mean delay, asynchronous vacation, selectable extra service, Nash equilibrium, intelligent searching algorithm, pricing policy.
Mathematics Subject Classification: Primary: 60K25, 35Q91; Secondary: 90B18.
Citation: Shunfu Jin, Haixing Wu, Wuyi Yue, Yutaka Takahashi. Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2407-2424. doi: 10.3934/jimo.2019060
References:
[1]

S. Ahn, J. Lee, S. Park, S. Newaz and J. Choi, Competitive partial computation offloading for maximizing energy efficiency in mobile cloud computing, IEEE Access, 6 (2018), 899-912.  doi: 10.1109/ACCESS.2017.2776323.  Google Scholar

[2]

R. Buyya, A. Beloglazov and J. Abawajy, Energy-efficient management of data center resources for cloud computing: A vision, architectural elements, and open challenges, Eprint arXiv, 12 (2010), 6-17.   Google Scholar

[3]

R. Dhanwate and V. Bhagat, Improving energy efficiency on android using cloud based services, International Journal of Advance Research in Computer Science and Management Studies, 3 (2015), 75-79.   Google Scholar

[4]

B. Doshi, Queueing systems with vacations–-A survey, Queueing Systems, 1 (1986), 29-66.  doi: 10.1007/BF01149327.  Google Scholar

[5]

W. Gary, P. Wang and M. Scott, A vacation queueing model with service breakdowns, Applied Mathematical Modelling, 24 (2000), 391-400.  doi: 10.1016/S0307-904X(99)00048-7.  Google Scholar

[6]

M. Ghorbani-Mandolakani and M. Rad, ML and Bayes estimation in a Two-Phase tandem queue with a second optional service and random feedback, Communications in Statistics-Theory and Methods, 45 (2016), 2576-2591.  doi: 10.1080/03610926.2014.887107.  Google Scholar

[7] Z. Gui, J. Xia, N. Zhou and Q. Huang, How to Choose Cloud Services: Toward a Cloud Computing Cost Model, CRC Press, 2013.   Google Scholar
[8]

Z. Guo, M. Song and Q. Wang, Policy-based market-oriented cloud service management architecture, Proc. of the International Conference on Information and Management Engineering, Wuhan, China, (2011), 284–291. doi: 10.1007/978-3-642-24010-2_39.  Google Scholar

[9]

J. Hu, J. Deng and J. Wu, A green private cloud architecture with global collaboration, Telecommunication Systems, 52 (2013), 1269-1279.  doi: 10.1007/s11235-011-9639-5.  Google Scholar

[10]

S. Hussein, Y. Alkabani and H. Mohamed, Green cloud computing: Datacenters power management policies and algorithms, Proc. of the 9th IEEE International Conference on Computer Engineering and Systems, Cairo, Egypt, (2015), 421–426. doi: 10.1109/ICCES.2014.7030998.  Google Scholar

[11]

A. Jain and M. Jain, Multi-server machine repair problem with unreliable server and two types of spares under asynchronous vacation policy, International Journal of Mathematics in Operational Research, 10 (2017), 286-315.  doi: 10.1504/IJMOR.2017.083187.  Google Scholar

[12]

S. Jin, H. Wu and W. Yue, Pricing policy for a cloud registration service with a novel cloud architecture, Cluster Computing, 22 (2019), 271-283.  doi: 10.1007/s10586-018-2854-z.  Google Scholar

[13]

S. Jin, X. Ma and W. Yue, Energy-saving strategy for green cognitive radio networks with an LTE-advanced structure, Journal of Communications and Networks, 18 (2016), 610-618. Google Scholar

[14]

Z. Ma, P. Wang and W. Yue, Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with $N$-policy, setup time and multiple working vacations, Journal of Industrial and Management Optimization, 13 (2017), 1467-1481.  doi: 10.3934/jimo.2017002.  Google Scholar

[15]

K. Madan, An M/G/1 queue with second optional service, Queueing Systems, 34 (2000), 37-46.  doi: 10.1023/A:1019144716929.  Google Scholar

[16] M. Neuts, Matrix-Geometric Solutions in Stochastic Models, Johns Hopkins University Press, 1981.   Google Scholar
[17]

P. Shi, H. Wang, X. Yue, S. Yang, X. Fu and Y. Peng, Corporation architecture for multiple cloud service providers in jointcloud computing, Proc. of the 37th International Conference on Distributed Computing Systems Workshops, Atlanta, USA, (2017), 294–298. doi: 10.1109/ICDCSW.2017.9.  Google Scholar

[18]

C. Singh, M. Jain and B. Kumar, Queueing model with state-dependent bulk arrival and second optional service, International Journal of Mathematics in Operational Research, 3 (2011), 322-340.  doi: 10.1504/IJMOR.2011.040029.  Google Scholar

[19]

A. Tarabia, Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs, Journal of Communications and Networks, 7 (2017), 811-823.  doi: 10.3934/jimo.2011.7.811.  Google Scholar

[20]

C. Wei, L. Cai and J. Wang, A discrete-time Geom/G/1 retrial queue with balking customers and second optional service, Opsearch, 53 (2016), 344-357.  doi: 10.1007/s12597-015-0232-7.  Google Scholar

[21]

H. Wu, S. Jin, W. Yue and Y. Takahashi, Performance evaluation for a registration service with an energy efficient cloud architecture, Proc. of the International Conference on Queueing Theory and Network Applications, Tsukuba City, Japan, (2018), 133–141. doi: 10.1007/978-3-319-93736-6_10.  Google Scholar

[22]

K. Ye, D. Huang, X. Jiang, H. Chen and S. Wu, Virtual machine based energy-efficient data center architecture for cloud computing: A performance perspective, Proc. of the IEEE/ACM International Conference on Green Computing and Communications, Hangzhou, China, (2010), 171–178. doi: 10.1109/GreenCom-CPSCom.2010.108.  Google Scholar

show all references

References:
[1]

S. Ahn, J. Lee, S. Park, S. Newaz and J. Choi, Competitive partial computation offloading for maximizing energy efficiency in mobile cloud computing, IEEE Access, 6 (2018), 899-912.  doi: 10.1109/ACCESS.2017.2776323.  Google Scholar

[2]

R. Buyya, A. Beloglazov and J. Abawajy, Energy-efficient management of data center resources for cloud computing: A vision, architectural elements, and open challenges, Eprint arXiv, 12 (2010), 6-17.   Google Scholar

[3]

R. Dhanwate and V. Bhagat, Improving energy efficiency on android using cloud based services, International Journal of Advance Research in Computer Science and Management Studies, 3 (2015), 75-79.   Google Scholar

[4]

B. Doshi, Queueing systems with vacations–-A survey, Queueing Systems, 1 (1986), 29-66.  doi: 10.1007/BF01149327.  Google Scholar

[5]

W. Gary, P. Wang and M. Scott, A vacation queueing model with service breakdowns, Applied Mathematical Modelling, 24 (2000), 391-400.  doi: 10.1016/S0307-904X(99)00048-7.  Google Scholar

[6]

M. Ghorbani-Mandolakani and M. Rad, ML and Bayes estimation in a Two-Phase tandem queue with a second optional service and random feedback, Communications in Statistics-Theory and Methods, 45 (2016), 2576-2591.  doi: 10.1080/03610926.2014.887107.  Google Scholar

[7] Z. Gui, J. Xia, N. Zhou and Q. Huang, How to Choose Cloud Services: Toward a Cloud Computing Cost Model, CRC Press, 2013.   Google Scholar
[8]

Z. Guo, M. Song and Q. Wang, Policy-based market-oriented cloud service management architecture, Proc. of the International Conference on Information and Management Engineering, Wuhan, China, (2011), 284–291. doi: 10.1007/978-3-642-24010-2_39.  Google Scholar

[9]

J. Hu, J. Deng and J. Wu, A green private cloud architecture with global collaboration, Telecommunication Systems, 52 (2013), 1269-1279.  doi: 10.1007/s11235-011-9639-5.  Google Scholar

[10]

S. Hussein, Y. Alkabani and H. Mohamed, Green cloud computing: Datacenters power management policies and algorithms, Proc. of the 9th IEEE International Conference on Computer Engineering and Systems, Cairo, Egypt, (2015), 421–426. doi: 10.1109/ICCES.2014.7030998.  Google Scholar

[11]

A. Jain and M. Jain, Multi-server machine repair problem with unreliable server and two types of spares under asynchronous vacation policy, International Journal of Mathematics in Operational Research, 10 (2017), 286-315.  doi: 10.1504/IJMOR.2017.083187.  Google Scholar

[12]

S. Jin, H. Wu and W. Yue, Pricing policy for a cloud registration service with a novel cloud architecture, Cluster Computing, 22 (2019), 271-283.  doi: 10.1007/s10586-018-2854-z.  Google Scholar

[13]

S. Jin, X. Ma and W. Yue, Energy-saving strategy for green cognitive radio networks with an LTE-advanced structure, Journal of Communications and Networks, 18 (2016), 610-618. Google Scholar

[14]

Z. Ma, P. Wang and W. Yue, Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with $N$-policy, setup time and multiple working vacations, Journal of Industrial and Management Optimization, 13 (2017), 1467-1481.  doi: 10.3934/jimo.2017002.  Google Scholar

[15]

K. Madan, An M/G/1 queue with second optional service, Queueing Systems, 34 (2000), 37-46.  doi: 10.1023/A:1019144716929.  Google Scholar

[16] M. Neuts, Matrix-Geometric Solutions in Stochastic Models, Johns Hopkins University Press, 1981.   Google Scholar
[17]

P. Shi, H. Wang, X. Yue, S. Yang, X. Fu and Y. Peng, Corporation architecture for multiple cloud service providers in jointcloud computing, Proc. of the 37th International Conference on Distributed Computing Systems Workshops, Atlanta, USA, (2017), 294–298. doi: 10.1109/ICDCSW.2017.9.  Google Scholar

[18]

C. Singh, M. Jain and B. Kumar, Queueing model with state-dependent bulk arrival and second optional service, International Journal of Mathematics in Operational Research, 3 (2011), 322-340.  doi: 10.1504/IJMOR.2011.040029.  Google Scholar

[19]

A. Tarabia, Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs, Journal of Communications and Networks, 7 (2017), 811-823.  doi: 10.3934/jimo.2011.7.811.  Google Scholar

[20]

C. Wei, L. Cai and J. Wang, A discrete-time Geom/G/1 retrial queue with balking customers and second optional service, Opsearch, 53 (2016), 344-357.  doi: 10.1007/s12597-015-0232-7.  Google Scholar

[21]

H. Wu, S. Jin, W. Yue and Y. Takahashi, Performance evaluation for a registration service with an energy efficient cloud architecture, Proc. of the International Conference on Queueing Theory and Network Applications, Tsukuba City, Japan, (2018), 133–141. doi: 10.1007/978-3-319-93736-6_10.  Google Scholar

[22]

K. Ye, D. Huang, X. Jiang, H. Chen and S. Wu, Virtual machine based energy-efficient data center architecture for cloud computing: A performance perspective, Proc. of the IEEE/ACM International Conference on Green Computing and Communications, Hangzhou, China, (2010), 171–178. doi: 10.1109/GreenCom-CPSCom.2010.108.  Google Scholar

Figure 1.  Sleeping mechanism-based cloud architecture
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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}}. $
Table Options

Download as excel

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}}. $
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^{*}) $.
Table Options

Download as excel

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^{*}) $.
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
Table Options

Download as excel

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
[1]

Gábor Horváth, Zsolt Saffer, Miklós Telek. Queue length analysis of a Markov-modulated vacation queue with dependent arrival and service processes and exhaustive service policy. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1365-1381. doi: 10.3934/jimo.2016077
[2]

Jinsong Xu. Reversible hidden data access algorithm in cloud computing environment. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1219-1232. doi: 10.3934/dcdss.2019084
[3]

Xuena Yan, Shunfu Jin, Wuyi Yue, Yutaka Takahashi. Performance analysis and system optimization of an energy-saving mechanism in cloud computing with correlated traffic. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021106
[4]

Min Zhang, Gang Li. Multi-objective optimization algorithm based on improved particle swarm in cloud computing environment. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1413-1426. doi: 10.3934/dcdss.2019097
[5]

Yang Chen, Xiaoguang Xu, Yong Wang. Wireless sensor network energy efficient coverage method based on intelligent optimization algorithm. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 887-900. doi: 10.3934/dcdss.2019059
[6]

Chandan Mahato, Gour Chandra Mahata. Optimal replenishment, pricing and preservation technology investment policies for non-instantaneous deteriorating items under two-level trade credit policy. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021123
[7]

Wei Fu, Jun Liu, Yirong Lai. Collaborative filtering recommendation algorithm towards intelligent community. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 811-822. doi: 10.3934/dcdss.2019054
[8]

Liyuan Wang, Zhiping Chen, Peng Yang. Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1203-1233. doi: 10.3934/jimo.2020018
[9]

Bing-Bing Cao, Zai-Jing Gong, Tian-Hui You. Stackelberg pricing policy in dyadic capital-constrained supply chain considering bank's deposit and loan based on delay payment scheme. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2855-2887. doi: 10.3934/jimo.2020098
[10]

Jun Wu, Shouyang Wang, Wuyi Yue. Supply contract model with service level constraint. Journal of Industrial & Management Optimization, 2005, 1 (3) : 275-287. doi: 10.3934/jimo.2005.1.275
[11]

Kyosuke Hashimoto, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. Performance analysis of backup-task scheduling with deadline time in cloud computing. Journal of Industrial & Management Optimization, 2015, 11 (3) : 867-886. doi: 10.3934/jimo.2015.11.867
[12]

Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1927-1941. doi: 10.3934/jimo.2019036
[13]

Weidong Bao, Haoran Ji, Xiaomin Zhu, Ji Wang, Wenhua Xiao, Jianhong Wu. ACO-based solution for computation offloading in mobile cloud computing. Big Data & Information Analytics, 2016, 1 (1) : 1-13. doi: 10.3934/bdia.2016.1.1
[14]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[15]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[16]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51
[17]

Gopinath Panda, Veena Goswami, Abhijit Datta Banik, Dibyajyoti Guha. Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption. Journal of Industrial & Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851
[18]

Dong-Sheng Ma, Hua-Ming Song. Behavior-based pricing in service differentiated industries. Journal of Dynamics & Games, 2020, 7 (4) : 351-364. doi: 10.3934/jdg.2020027
[19]

Weiping Li, Haiyan Wu, Jie Yang. Intelligent recognition algorithm for social network sensitive information based on classification technology. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1385-1398. doi: 10.3934/dcdss.2019095
[20]

P. Daniele, S. Giuffrè, S. Pia. Competitive financial equilibrium problems with policy interventions. Journal of Industrial & Management Optimization, 2005, 1 (1) : 39-52. doi: 10.3934/jimo.2005.1.39

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (191)
  • HTML views (706)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences