American Institute of Mathematical Sciences

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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  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:
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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

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

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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

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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

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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

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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

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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

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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

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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
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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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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^{*}) $.
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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
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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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