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November  2020, 16(6): 2625-2649. doi: 10.3934/jimo.2019073

Admission control for finite capacity queueing model with general retrial times and state-dependent rates

Madhu Jain and  Sudeep Singh Sanga

Indian Institute of Technology Roorkee - 247 667, India

Received  March 2018 Revised  February 2019 Published  July 2019

The finite state dependent queueing model with $ F $-policy is investigated by considering the general retrial attempts. On arrival in the system, if the job finds the server engaged, it is forced to enter into the retrial orbit. After a random period of time, the job from the retrial orbit re-attempts for the service. According to $ F $-policy, as the system attains its full capacity, the arrivals are restricted to join the system until the number of jobs comes down to the prefixed threshold value '$ F $'. The supplementary variable corresponding to the remaining retrial time is used to frame the governing equations which are solved by using Laplace-Stieltjes transform and then applying the recursive method. Special models for machine repair and time-sharing queue are deduced by setting the state dependent rates. Several system indices are obtained explicitly which are further used to facilitate the sensitivity analysis by considering a numerical illustration. A cost function is constructed and minimized for evaluating the optimal threshold parameter and optimal service rate.

Keywords: Finite queue, general retrial, F-policy, supplementary variable, sensitivity analysis, cost optimization.
Mathematics Subject Classification: 60K25, 68M20, 90B22.
Citation: Madhu Jain, Sudeep Singh Sanga. Admission control for finite capacity queueing model with general retrial times and state-dependent rates. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2625-2649. doi: 10.3934/jimo.2019073
References:
[1]

I. J. B. F. Adan and V. G. Kulkarni, Single-server queue with Markov-dependent inter-arrival and service times, Queueing Syst., 45 (2003), 113-134.  doi: 10.1023/A:1026093622185.  Google Scholar

[2]

I. Adiri and B. Avi-Itzhak, A time-sharing queue, Manage. Sci., 15 (1969), 639-657.  doi: 10.1287/mnsc.15.11.639.  Google Scholar

[3]

A. Banerjee and U. C. Gupta, Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service, Perform. Eval., 69 (2012), 53-70.  doi: 10.1016/j.peva.2011.09.002.  Google Scholar

[4]

M. Boualem, N. Djellab and D. Aïssani, Stochastic bounds for a single server queue with general retrial times, Bull. Iran. Math. Soc., 40 (2014), 183-198.   Google Scholar

[5]

M. Chandrasekaran, M. Muralidhar and U. S. Dixit, Online optimization of multipass machining based on cloud computing, Int. J. Adv. Manuf. Technol., 65 (2013), 239-250.   Google Scholar

[6]

C.-J. Chang, F.-M. Chang and J.-C. Ke, Economic application in a Bernoulli $F$-policy queueing system with server breakdown, Int. J. Prod. Res., 52 (2014), 743-756.   Google Scholar

[7]

C.-J. Chang and J.-C. Ke, Randomized controlling arrival for a queueing system with subject to server breakdowns, Optimization., 64 (2015), 941-955.  doi: 10.1080/02331934.2013.804076.  Google Scholar

[8]

J. Chang and J. Wang, Unreliable M/M/1/1 retrial queues with set-up time, Qual. Technol. Quant. Manag., 3703 (2017), 1-13.  doi: 10.1080/16843703.2017.1320459.  Google Scholar

[9]

G. Choudhury and J.-C. Ke, An unreliable retrial queue with delaying repair and general retrial times under Bernoulli vacation schedule, Appl. Math. Comput., 230 (2014), 436-450.  doi: 10.1016/j.amc.2013.12.108.  Google Scholar

[10]

D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Math. Proc. Cambridge Philos. Soc., 51 (1955), 433. doi: 10.1017/S0305004100030437.  Google Scholar

[11]

S. D. Flapper, J.-P. Gayon and L. L. Lim, On the optimal control of manufacturing and remanufacturing activities with a single shared server, Eur. J. Oper. Res., 234 (2014), 86-98.  doi: 10.1016/j.ejor.2013.10.049.  Google Scholar

[12]

S. Gao, J. Wang and W. W. Li, An M/G/1 retrial queue with general retrial times, working vacations and vacation interruption, Asia-Pacific J. Oper. Res., 31 (2014), 1440006. doi: 10.1142/S0217595914400065.  Google Scholar

[13]

S. M. Gupta, Interrelationship between controlling arrival and service in queueing systems, Comput. Oper. Res., 22 (1995), 1005-1014.  doi: 10.1016/0305-0548(94)00088-P.  Google Scholar

[14]

M. Jain, An $(m, M)$ machine repair problem with spares and state dependent rates: A diffusion process approach, Microelectron. Reliab., 37 (1997), 929-933.  doi: 10.1016/S0026-2714(96)00146-1.  Google Scholar

[15]

M. Jain and A. Bhagat, Transient analysis of finite F-policy retrial queues with delayed repair and threshold recovery, Natl. Acad. Sci. Lett., 38 (2015), 257-261.  doi: 10.1007/s40009-014-0337-1.  Google Scholar

[16]

M. Jain and S. S. Sanga, Performance modeling and ANFIS computing for finite buffer retrial queue under F-policy, in Proceedings of Sixth International Conference on Soft Computing for Problem Solving, Patiala, India, 2017,248–258. doi: 10.1007/978-981-10-3325-4_25.  Google Scholar

[17]

M. Jain and S. S. Sanga, Control F-policy for fault tolerance machining system with general retrial attempts, Natl. Acad. Sci. Lett., 40 (2017), 359-364.  doi: 10.1007/s40009-017-0573-2.  Google Scholar

[18]

M. Jain and S. S. Sanga, $F$-policy for M/M/1/K retrial queueing model with state-dependent rates, in Performance Prediction and Analytics of Fuzzy, Reliability and Queuing Models (eds. K. Deep, M. Jain and S. Salhi), Springer, Singapore, 2019,127–138. doi: 10.1007/978-981-13-0857-4_9.  Google Scholar

[19]

M. Jain, S. S. Sanga and R. K. Meena, Control F-policy for Markovian retrial queue with server breakdowns, in 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES), New Delhi, India, 2016, 1–5. doi: 10.1109/ICPEICES.2016.7853083.  Google Scholar

[20]

M. Jain, G. C. Sharma and V. Rani, M/M/R+r machining system with reneging, spares and interdependent controlled rates, Int. J. Math. Oper. Res., 6 (2014), 655-679.  doi: 10.1504/IJMOR.2014.065422.  Google Scholar

[21]

M. Jain, G. C. Sharma and R. Sharma, Optimal control of (N, F) policy for unreliable server queue with multi-optional phase repair and start-up, Int. J. Math. Oper. Res., 4 (2012), 152-174.  doi: 10.1504/IJMOR.2012.046375.  Google Scholar

[22]

M. Jain, G. C. Sharma and C. Shekhar, Processor-shared service systems with queue-dependent processors, Comput. Oper. Res., 32 (2005), 629-645.  doi: 10.1016/j.cor.2003.08.009.  Google Scholar

[23]

M. Jain, C. Shekhar and S. Shukla, Queueing analysis of machine repair problem with controlled rates and working vacation under F-Policy, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci., 86 (2016), 21-31.  doi: 10.1007/s40010-015-0233-1.  Google Scholar

[24]

J. C. Ke, C. H. Wu and Z. G. Zhang, Recent developments in vacation queueing models: A short survey, Int. J. Oper. Res., 7 (2010), 3-8.   Google Scholar

[25]

J. Kim and B. Kim, The processor-sharing queue with bulk arrivals and phase-type services, Perform. Eval., 64 (2007), 277-297.  doi: 10.1016/j.peva.2006.05.006.  Google Scholar

[26]

E. R. Kumara and S. Dharsana, Analysis of M/M/1 queueing system with state dependent arrival and detainment of retracted customers, Malaya J. Mat., (2015), 89–98. Google Scholar

[27]

C. Lee, On moment stability properties for a class of state-dependent stochastic networks, J. Korean Stat. Soc., 40 (2011), 325-336.  doi: 10.1016/j.jkss.2010.12.003.  Google Scholar

[28]

C.-D. Liou, Optimization analysis of the machine repair problem with multiple vacations and working breakdowns, J. Ind. Manag. Optim., 11 (2014), 83-104.  doi: 10.3934/jimo.2015.11.83.  Google Scholar

[29]

W. A. Massey, The analysis of queues with time-varying rates for telecommunication models, Telecommun. Syst., 21 (2002), 173-204.   Google Scholar

[30]

P. Moreno, An M/G/1 retrial queue with recurrent customers and general retrial times, Appl. Math. Comput., 159 (2004), 651-666.  doi: 10.1016/j.amc.2003.09.019.  Google Scholar

[31]

P. R. Parthasarathy and R. Sudhesh, Time-dependent analysis of a single-server retrial queue with state-dependent rates, Oper. Res. Lett., 35 (2007), 601-611.  doi: 10.1016/j.orl.2006.12.005.  Google Scholar

[32]

T. Phung-Duc and K. Kawanishi, Multiserver retrial queue with setup time and its application to data centers, J. Ind. Manag. Optim., 15 (2019), 15-35.   Google Scholar

[33]

J. Rodrigues, S. M. Prado, N. Balakrishnan and F. Louzada, Flexible M/G/1 queueing system with state dependent service rate, Oper. Res. Lett., 44 (2016), 383-389.  doi: 10.1016/j.orl.2016.03.011.  Google Scholar

[34]

K. H. Wang, Cost analysis of the M/M/R machine-repair problem with mixed standby spares, Microelectron. Reliab., 33 (1993), 1293-1301.  doi: 10.1016/0026-2714(93)90131-H.  Google Scholar

[35]

K. H. Wang, C. C. Kuo and W. L. Pearn, Optimal control of an M/G/1/K queueing system with combined F policy and startup time, J. Optim. Theory Appl., 135 (2007), 285-299.  doi: 10.1007/s10957-007-9253-6.  Google Scholar

[36]

K.-H. Wang, C.-C. Kuo and W. L. Pearn, A recursive method for the $F$-policy G/M/1/K queueing system with an exponential startup time, Appl. Math. Model., 32 (2008), 958-970.  doi: 10.1016/j.apm.2007.02.023.  Google Scholar

[37]

K.-H. Wang and B. D. Sivazlian, Cost analysis of the M/M/R machine repair problem with spares operating under variable service rates, Microelectron. Reliab., 32 (1992), 1171-1183.  doi: 10.1016/0026-2714(92)90035-J.  Google Scholar

[38]

D.-Y. Yang, F.-M. Chang and J.-C. Ke, On an unreliable retrial queue with general repeated attempts and J optional vacations, Appl. Math. Model., 40 (2016), 3275-3288.  doi: 10.1016/j.apm.2015.10.023.  Google Scholar

[39]

D.-Y. Yang and P.-K. Chang, A parametric programming solution to the F-policy queue with fuzzy parameters, Int. J. Syst. Sci., 46 (2015), 590-598.  doi: 10.1080/00207721.2013.792975.  Google Scholar

[40]

D.-Y. Yang and Y.-D. Chang, Sensitivity analysis of the machine repair problem with general repeated attempts, Int. J. Comput. Math., 95 (2018), 1761-1774.  doi: 10.1080/00207160.2017.1336230.  Google Scholar

[41]

C. Yeh, Y.-T. Lee, C.-J. Chang and F.-M. Chang, Analysis of a two-phase queue system with <p, F>- policy, Qual. Technol. Quant. Manag., 14 (2017), 178-194.   Google Scholar

show all references

References:
[1]

I. J. B. F. Adan and V. G. Kulkarni, Single-server queue with Markov-dependent inter-arrival and service times, Queueing Syst., 45 (2003), 113-134.  doi: 10.1023/A:1026093622185.  Google Scholar

[2]

I. Adiri and B. Avi-Itzhak, A time-sharing queue, Manage. Sci., 15 (1969), 639-657.  doi: 10.1287/mnsc.15.11.639.  Google Scholar

[3]

A. Banerjee and U. C. Gupta, Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service, Perform. Eval., 69 (2012), 53-70.  doi: 10.1016/j.peva.2011.09.002.  Google Scholar

[4]

M. Boualem, N. Djellab and D. Aïssani, Stochastic bounds for a single server queue with general retrial times, Bull. Iran. Math. Soc., 40 (2014), 183-198.   Google Scholar

[5]

M. Chandrasekaran, M. Muralidhar and U. S. Dixit, Online optimization of multipass machining based on cloud computing, Int. J. Adv. Manuf. Technol., 65 (2013), 239-250.   Google Scholar

[6]

C.-J. Chang, F.-M. Chang and J.-C. Ke, Economic application in a Bernoulli $F$-policy queueing system with server breakdown, Int. J. Prod. Res., 52 (2014), 743-756.   Google Scholar

[7]

C.-J. Chang and J.-C. Ke, Randomized controlling arrival for a queueing system with subject to server breakdowns, Optimization., 64 (2015), 941-955.  doi: 10.1080/02331934.2013.804076.  Google Scholar

[8]

J. Chang and J. Wang, Unreliable M/M/1/1 retrial queues with set-up time, Qual. Technol. Quant. Manag., 3703 (2017), 1-13.  doi: 10.1080/16843703.2017.1320459.  Google Scholar

[9]

G. Choudhury and J.-C. Ke, An unreliable retrial queue with delaying repair and general retrial times under Bernoulli vacation schedule, Appl. Math. Comput., 230 (2014), 436-450.  doi: 10.1016/j.amc.2013.12.108.  Google Scholar

[10]

D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Math. Proc. Cambridge Philos. Soc., 51 (1955), 433. doi: 10.1017/S0305004100030437.  Google Scholar

[11]

S. D. Flapper, J.-P. Gayon and L. L. Lim, On the optimal control of manufacturing and remanufacturing activities with a single shared server, Eur. J. Oper. Res., 234 (2014), 86-98.  doi: 10.1016/j.ejor.2013.10.049.  Google Scholar

[12]

S. Gao, J. Wang and W. W. Li, An M/G/1 retrial queue with general retrial times, working vacations and vacation interruption, Asia-Pacific J. Oper. Res., 31 (2014), 1440006. doi: 10.1142/S0217595914400065.  Google Scholar

[13]

S. M. Gupta, Interrelationship between controlling arrival and service in queueing systems, Comput. Oper. Res., 22 (1995), 1005-1014.  doi: 10.1016/0305-0548(94)00088-P.  Google Scholar

[14]

M. Jain, An $(m, M)$ machine repair problem with spares and state dependent rates: A diffusion process approach, Microelectron. Reliab., 37 (1997), 929-933.  doi: 10.1016/S0026-2714(96)00146-1.  Google Scholar

[15]

M. Jain and A. Bhagat, Transient analysis of finite F-policy retrial queues with delayed repair and threshold recovery, Natl. Acad. Sci. Lett., 38 (2015), 257-261.  doi: 10.1007/s40009-014-0337-1.  Google Scholar

[16]

M. Jain and S. S. Sanga, Performance modeling and ANFIS computing for finite buffer retrial queue under F-policy, in Proceedings of Sixth International Conference on Soft Computing for Problem Solving, Patiala, India, 2017,248–258. doi: 10.1007/978-981-10-3325-4_25.  Google Scholar

[17]

M. Jain and S. S. Sanga, Control F-policy for fault tolerance machining system with general retrial attempts, Natl. Acad. Sci. Lett., 40 (2017), 359-364.  doi: 10.1007/s40009-017-0573-2.  Google Scholar

[18]

M. Jain and S. S. Sanga, $F$-policy for M/M/1/K retrial queueing model with state-dependent rates, in Performance Prediction and Analytics of Fuzzy, Reliability and Queuing Models (eds. K. Deep, M. Jain and S. Salhi), Springer, Singapore, 2019,127–138. doi: 10.1007/978-981-13-0857-4_9.  Google Scholar

[19]

M. Jain, S. S. Sanga and R. K. Meena, Control F-policy for Markovian retrial queue with server breakdowns, in 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES), New Delhi, India, 2016, 1–5. doi: 10.1109/ICPEICES.2016.7853083.  Google Scholar

[20]

M. Jain, G. C. Sharma and V. Rani, M/M/R+r machining system with reneging, spares and interdependent controlled rates, Int. J. Math. Oper. Res., 6 (2014), 655-679.  doi: 10.1504/IJMOR.2014.065422.  Google Scholar

[21]

M. Jain, G. C. Sharma and R. Sharma, Optimal control of (N, F) policy for unreliable server queue with multi-optional phase repair and start-up, Int. J. Math. Oper. Res., 4 (2012), 152-174.  doi: 10.1504/IJMOR.2012.046375.  Google Scholar

[22]

M. Jain, G. C. Sharma and C. Shekhar, Processor-shared service systems with queue-dependent processors, Comput. Oper. Res., 32 (2005), 629-645.  doi: 10.1016/j.cor.2003.08.009.  Google Scholar

[23]

M. Jain, C. Shekhar and S. Shukla, Queueing analysis of machine repair problem with controlled rates and working vacation under F-Policy, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci., 86 (2016), 21-31.  doi: 10.1007/s40010-015-0233-1.  Google Scholar

[24]

J. C. Ke, C. H. Wu and Z. G. Zhang, Recent developments in vacation queueing models: A short survey, Int. J. Oper. Res., 7 (2010), 3-8.   Google Scholar

[25]

J. Kim and B. Kim, The processor-sharing queue with bulk arrivals and phase-type services, Perform. Eval., 64 (2007), 277-297.  doi: 10.1016/j.peva.2006.05.006.  Google Scholar

[26]

E. R. Kumara and S. Dharsana, Analysis of M/M/1 queueing system with state dependent arrival and detainment of retracted customers, Malaya J. Mat., (2015), 89–98. Google Scholar

[27]

C. Lee, On moment stability properties for a class of state-dependent stochastic networks, J. Korean Stat. Soc., 40 (2011), 325-336.  doi: 10.1016/j.jkss.2010.12.003.  Google Scholar

[28]

C.-D. Liou, Optimization analysis of the machine repair problem with multiple vacations and working breakdowns, J. Ind. Manag. Optim., 11 (2014), 83-104.  doi: 10.3934/jimo.2015.11.83.  Google Scholar

[29]

W. A. Massey, The analysis of queues with time-varying rates for telecommunication models, Telecommun. Syst., 21 (2002), 173-204.   Google Scholar

[30]

P. Moreno, An M/G/1 retrial queue with recurrent customers and general retrial times, Appl. Math. Comput., 159 (2004), 651-666.  doi: 10.1016/j.amc.2003.09.019.  Google Scholar

[31]

P. R. Parthasarathy and R. Sudhesh, Time-dependent analysis of a single-server retrial queue with state-dependent rates, Oper. Res. Lett., 35 (2007), 601-611.  doi: 10.1016/j.orl.2006.12.005.  Google Scholar

[32]

T. Phung-Duc and K. Kawanishi, Multiserver retrial queue with setup time and its application to data centers, J. Ind. Manag. Optim., 15 (2019), 15-35.   Google Scholar

[33]

J. Rodrigues, S. M. Prado, N. Balakrishnan and F. Louzada, Flexible M/G/1 queueing system with state dependent service rate, Oper. Res. Lett., 44 (2016), 383-389.  doi: 10.1016/j.orl.2016.03.011.  Google Scholar

[34]

K. H. Wang, Cost analysis of the M/M/R machine-repair problem with mixed standby spares, Microelectron. Reliab., 33 (1993), 1293-1301.  doi: 10.1016/0026-2714(93)90131-H.  Google Scholar

[35]

K. H. Wang, C. C. Kuo and W. L. Pearn, Optimal control of an M/G/1/K queueing system with combined F policy and startup time, J. Optim. Theory Appl., 135 (2007), 285-299.  doi: 10.1007/s10957-007-9253-6.  Google Scholar

[36]

K.-H. Wang, C.-C. Kuo and W. L. Pearn, A recursive method for the $F$-policy G/M/1/K queueing system with an exponential startup time, Appl. Math. Model., 32 (2008), 958-970.  doi: 10.1016/j.apm.2007.02.023.  Google Scholar

[37]

K.-H. Wang and B. D. Sivazlian, Cost analysis of the M/M/R machine repair problem with spares operating under variable service rates, Microelectron. Reliab., 32 (1992), 1171-1183.  doi: 10.1016/0026-2714(92)90035-J.  Google Scholar

[38]

D.-Y. Yang, F.-M. Chang and J.-C. Ke, On an unreliable retrial queue with general repeated attempts and J optional vacations, Appl. Math. Model., 40 (2016), 3275-3288.  doi: 10.1016/j.apm.2015.10.023.  Google Scholar

[39]

D.-Y. Yang and P.-K. Chang, A parametric programming solution to the F-policy queue with fuzzy parameters, Int. J. Syst. Sci., 46 (2015), 590-598.  doi: 10.1080/00207721.2013.792975.  Google Scholar

[40]

D.-Y. Yang and Y.-D. Chang, Sensitivity analysis of the machine repair problem with general repeated attempts, Int. J. Comput. Math., 95 (2018), 1761-1774.  doi: 10.1080/00207160.2017.1336230.  Google Scholar

[41]

C. Yeh, Y.-T. Lee, C.-J. Chang and F.-M. Chang, Analysis of a two-phase queue system with <p, F>- policy, Qual. Technol. Quant. Manag., 14 (2017), 178-194.   Google Scholar

Figure 1.  State transition diagram.($ n: $ number of jobs present in the system, $ i: $ status of the server)
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Figure 2.   
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Figure 3.   
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Figure 4.  TC vs. F for different distributions ($ Exp $, $ E_{3} $ and $ D $) when $ \gamma = 0.5 $
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Figure 5.  Total cost TC for varying $ \mu $ and $ F $ for three distribution by taking three cost sets-I for MRP
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Figure 6.  Total cost TC for varying $ \mu $ and $ \gamma $ for exponential distribution by taking three cost sets-Ⅱ, Ⅲ and Ⅳ
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Figure 7.  Total cost TC for varying $ \mu $ and $ \gamma $ for Erlang-3 distribution by taking three cost sets-Ⅱ, Ⅲ and Ⅳ
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Figure 8.  Total cost TC for varying $ \mu $ and $ \gamma $ for deterministic distribution by taking three cost sets-Ⅱ, Ⅲ and Ⅳ
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Figure 9.   
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Figure 10.   
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Figure 11.  TC vs. F for different distributions ($ Exp $, $ E_{3} $ and $ D $) when $ \gamma = 3 $
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Figure 12.  Total cost TC for varying $ \mu $ and F for three distribution by taking three cost sets- Ⅰ for TSM
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Figure 13.  Total cost TC for varying $ \mu $ and $ \gamma $ for exponential distribution by taking three cost sets- Ⅱ, Ⅲ and Ⅳ
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Figure 14.  Total cost TC for varying $ \mu $ and $ \gamma $ for Erlang-3 distribution by taking three cost sets- Ⅱ, Ⅲ and Ⅳ
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Figure 15.  Total cost TC for varying $ \mu $ and $ \gamma $ for deterministic distribution by taking three cost sets- Ⅱ, Ⅲ and Ⅳ
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Table 1.  Various performance measures for varying values of $ \mu $ for MRP
E[Nq] PI PSB TC
μ Exp Е3 D Exp Е3 D Exp Е3 D Exp Е3 D
1 3.164 3.110 3.060 0.303 0.335 0.354 0.697 0.665 0.646 532.25 543.43 547.66
2 2.043 2.013 1.964 0.492 0.543 0.569 0.508 0.457 0.431 473.61 506.62 518.70
3 1.482 1.477 1.436 0.580 0.643 0.G72 0.420 0.357 0.328 431.77 485.08 505.76
4 1.155 1.166 1.135 0.633 0.699 0.730 0.367 0.301 0.270 403.18 469.82 499.15
5 0.940 0.963 0.941 0.671 0.735 0.768 0.329 0.265 0.232 385.15 459.00 496.05
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E[Nq] PI PSB TC
μ Exp Е3 D Exp Е3 D Exp Е3 D Exp Е3 D
1 3.164 3.110 3.060 0.303 0.335 0.354 0.697 0.665 0.646 532.25 543.43 547.66
2 2.043 2.013 1.964 0.492 0.543 0.569 0.508 0.457 0.431 473.61 506.62 518.70
3 1.482 1.477 1.436 0.580 0.643 0.G72 0.420 0.357 0.328 431.77 485.08 505.76
4 1.155 1.166 1.135 0.633 0.699 0.730 0.367 0.301 0.270 403.18 469.82 499.15
5 0.940 0.963 0.941 0.671 0.735 0.768 0.329 0.265 0.232 385.15 459.00 496.05
Table 2.  Various performance measures for varying values of $ \lambda $ for MRP
λ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
1 1.132 1.060 0.998 0.704 0.771 0.793 0.296 0.229 0.207 562.12 599.89 618.34
2 1.742 1.604 1.542 0.644 0.687 0.702 0.356 0.313 0.298 677.66 662.77 667.59
3 2.161 2.026 1.985 0.582 0.617 0.626 0.418 0.383 0.374 711.49 688.00 689.76
4 2.494 2.372 2.347 0.530 0.559 0.564 0.470 0.441 0.436 729.36 704.04 704.74
5 2.767 2.661 2.645 0.486 0.510 0.514 0.514 0.490 0.486 741.01 715.85 716.15
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λ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
1 1.132 1.060 0.998 0.704 0.771 0.793 0.296 0.229 0.207 562.12 599.89 618.34
2 1.742 1.604 1.542 0.644 0.687 0.702 0.356 0.313 0.298 677.66 662.77 667.59
3 2.161 2.026 1.985 0.582 0.617 0.626 0.418 0.383 0.374 711.49 688.00 689.76
4 2.494 2.372 2.347 0.530 0.559 0.564 0.470 0.441 0.436 729.36 704.04 704.74
5 2.767 2.661 2.645 0.486 0.510 0.514 0.514 0.490 0.486 741.01 715.85 716.15
Table 3.  Various performance measures for varying values of $ \gamma $ for MRP
γ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
0.5 0.581 0.633 0.628 0.745 0.799 0.830 0.255 0.201 0.170 384.56 448.74 500.10
0.6 0.562 0.627 0.637 0.733 0.784 0.817 0.267 0.216 0.183 365.13 423.37 477.68
0.7 0.544 0.617 0.643 0.723 0.771 0.805 0.277 0.229 0.195 348.73 403.30 459.70
0.8 0.527 0.606 0.645 0.714 0.760 0.795 0.286 0.240 0.205 334.57 386.37 444.12
0.9 0.510 0.593 0.644 0.706 0.750 0.784 0.294 0.250 0.216 322.17 371.45 429.80
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γ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
0.5 0.581 0.633 0.628 0.745 0.799 0.830 0.255 0.201 0.170 384.56 448.74 500.10
0.6 0.562 0.627 0.637 0.733 0.784 0.817 0.267 0.216 0.183 365.13 423.37 477.68
0.7 0.544 0.617 0.643 0.723 0.771 0.805 0.277 0.229 0.195 348.73 403.30 459.70
0.8 0.527 0.606 0.645 0.714 0.760 0.795 0.286 0.240 0.205 334.57 386.37 444.12
0.9 0.510 0.593 0.644 0.706 0.750 0.784 0.294 0.250 0.216 322.17 371.45 429.80
Table 4.  Cost sets with different cost elements (in ＄) for MRP
Cost Set $ C_{I} $ $ C_{B} $ $ C_{H} $ $ C_{F} $ $ C_{O} $
Ⅰ 30 30 50 70 40
Ⅱ 10 10 120 15 90
Ⅲ 15 5 120 15 90
Ⅳ 20 20 100 15 90
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Cost Set $ C_{I} $ $ C_{B} $ $ C_{H} $ $ C_{F} $ $ C_{O} $
Ⅰ 30 30 50 70 40
Ⅱ 10 10 120 15 90
Ⅲ 15 5 120 15 90
Ⅳ 20 20 100 15 90
Table 5.  Searching the optimal F for MRP for different $ \gamma $
γ $\left( {{F^*},TC\left( {{F^*},\mu } \right)} \right)$
Exp E3 D
0.3 (12,852.01) (14,898.49) (15,937.18)
0.5 (10,821.20) (12,848.52) (13,874.16)
0.7 (8,806.92) (10,821.04) (11,836.66)
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γ $\left( {{F^*},TC\left( {{F^*},\mu } \right)} \right)$
Exp E3 D
0.3 (12,852.01) (14,898.49) (15,937.18)
0.5 (10,821.20) (12,848.52) (13,874.16)
0.7 (8,806.92) (10,821.04) (11,836.66)
Table 6.  Searching the $ \mu^* $ by quasi-Newton method for exponential distribution ($ F^* = 10 $, $ \gamma = 0.5 $)
Iterations F* μ TC(F*, μ) Max. tolerance
0 10 1 853.525 9.87E+01
1 10 2 842.70 5.69E+01
2 10 1.6344 823.783 3.5E+01
3 10 1.0491 821.311 8.28
4 10 1.5255 821.207 2.27
5 10 1.4917 821.199 6.35E-02
6 10 1.4990 821.199 4Л2Е-04
7 10 1.4988 821.199 1.02E-05
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Iterations F* μ TC(F*, μ) Max. tolerance
0 10 1 853.525 9.87E+01
1 10 2 842.70 5.69E+01
2 10 1.6344 823.783 3.5E+01
3 10 1.0491 821.311 8.28
4 10 1.5255 821.207 2.27
5 10 1.4917 821.199 6.35E-02
6 10 1.4990 821.199 4Л2Е-04
7 10 1.4988 821.199 1.02E-05
Table 7.  Minimum cost $ (F^*, \mu^*, TC(F^*, \mu^*)) $ for $ \gamma = 0.3, 0.5 $ and 0.7 for MRP
γ $(F^*, \mu^*, TC(F^*, \mu^*))$
Exp E3 D
0.3 (12, 1.601,851.05) (14, 1.668,897.73) (15, 1.323,936.78)
0.5 (10, 1.499,821.20) (12, 1.579,847.89) (13, 1.691,872.05)
0.7 (8, 1.489,806.90) (10, 1.543,820.77) (11, 1.627,834.89)
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γ $(F^*, \mu^*, TC(F^*, \mu^*))$
Exp E3 D
0.3 (12, 1.601,851.05) (14, 1.668,897.73) (15, 1.323,936.78)
0.5 (10, 1.499,821.20) (12, 1.579,847.89) (13, 1.691,872.05)
0.7 (8, 1.489,806.90) (10, 1.543,820.77) (11, 1.627,834.89)
Table 8.  Various performance measures for varying values of $ \mu $ for TSM
μ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
6 0.876 0.995 1.041 0.552 0.557 0.564 0.448 0.443 0.436 274.65 307.11 323.69
8 0.476 0.570 0.623 0.654 0.656 0.659 0.346 0.344 0.341 241.50 270.93 290.14
10 0.286 0.352 0.399 0.721 0.722 0.723 0.279 0.278 0.277 237.26 261.73 280.32
12 0.187 0.235 0.272 0.767 0.767 0.768 0.233 0.233 0.232 247.50 267.75 284.72
14 0.131 0.166 0.196 0.800 0.800 0.801 0.200 0.200 0.199 265.16 282.20 297.46
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μ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
6 0.876 0.995 1.041 0.552 0.557 0.564 0.448 0.443 0.436 274.65 307.11 323.69
8 0.476 0.570 0.623 0.654 0.656 0.659 0.346 0.344 0.341 241.50 270.93 290.14
10 0.286 0.352 0.399 0.721 0.722 0.723 0.279 0.278 0.277 237.26 261.73 280.32
12 0.187 0.235 0.272 0.767 0.767 0.768 0.233 0.233 0.232 247.50 267.75 284.72
14 0.131 0.166 0.196 0.800 0.800 0.801 0.200 0.200 0.199 265.16 282.20 297.46
Table 9.  Various performance measures for varying values of $ \lambda $ for TSM
λ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
2 0.107 0.124 0.138 0.800 0.800 0.800 0.200 0.200 0.200 193.36 201.93 208.81
4 0.783 0.948 1.014 0.613 0.619 0.628 0.387 0.381 0.372 341.33 391.91 418.07
6 1.726 1.843 1.829 0.493 0.516 0.535 0.507 0.484 0.465 504.56 547.33 559.06
8 2.340 2.357 2.303 0.426 0.456 0.478 0.574 0.544 0.522 589.23 614.24 619.62
10 2.737 2.699 2.632 0.380 0.412 0.434 0.620 0.588 0.566 634.09 649.61 652.77
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λ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
2 0.107 0.124 0.138 0.800 0.800 0.800 0.200 0.200 0.200 193.36 201.93 208.81
4 0.783 0.948 1.014 0.613 0.619 0.628 0.387 0.381 0.372 341.33 391.91 418.07
6 1.726 1.843 1.829 0.493 0.516 0.535 0.507 0.484 0.465 504.56 547.33 559.06
8 2.340 2.357 2.303 0.426 0.456 0.478 0.574 0.544 0.522 589.23 614.24 619.62
10 2.737 2.699 2.632 0.380 0.412 0.434 0.620 0.588 0.566 634.09 649.61 652.77
Table 10.  Various performance measures for varying values of $ \gamma $ for TSM
γ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
1 0.286 0.352 0.399 0.721 0.722 0.723 0.279 0.278 0.277 237.26 261.73 280.32
1.5 0.225 0.256 0.276 0.719 0.721 0.722 0.281 0.279 0.278 214.93 226.13 234.03
2 0.195 0.212 0.223 0.718 0.720 0.721 0.282 0.280 0.279 204.02 210.31 214.43
2.5 0.177 0.188 0.194 0.717 0.719 0.720 0.283 0.281 0.280 197.58 201.59 204.06
3 0.165 0.173 0.177 0.716 0.718 0.719 0.284 0.282 0.281 193.35 196.12 197.75
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γ E[Nq] PI PSB TC
Exp E3 D Exp E3 D Exp E3 D Exp E3 D
1 0.286 0.352 0.399 0.721 0.722 0.723 0.279 0.278 0.277 237.26 261.73 280.32
1.5 0.225 0.256 0.276 0.719 0.721 0.722 0.281 0.279 0.278 214.93 226.13 234.03
2 0.195 0.212 0.223 0.718 0.720 0.721 0.282 0.280 0.279 204.02 210.31 214.43
2.5 0.177 0.188 0.194 0.717 0.719 0.720 0.283 0.281 0.280 197.58 201.59 204.06
3 0.165 0.173 0.177 0.716 0.718 0.719 0.284 0.282 0.281 193.35 196.12 197.75
Table 11.  Cost sets with different cost elements (in ＄) for TSM
Cost Set $ C_{I} $ $ C_{B} $ $ C_{H} $ $ C_{F} $ $ C_{O} $
Ⅰ 30 40 120 60 90
Ⅱ 10 10 120 15 90
Ⅲ 15 5 120 15 90
Ⅳ 10 10 100 15 110
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Cost Set $ C_{I} $ $ C_{B} $ $ C_{H} $ $ C_{F} $ $ C_{O} $
Ⅰ 30 40 120 60 90
Ⅱ 10 10 120 15 90
Ⅲ 15 5 120 15 90
Ⅳ 10 10 100 15 110
Table 12.  Searching the optimal $ F $ for TSM
γ $(F^* \mu^*, TC(F^*, \mu^*))$
Exp E3 D
1 (2,822.28) (1,858.94) (1,872.35)
3 (5,665.29) (5,678.51) (5,687.12)
5 (6,627.96) (6,632.94) (6,635.92)
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γ $(F^* \mu^*, TC(F^*, \mu^*))$
Exp E3 D
1 (2,822.28) (1,858.94) (1,872.35)
3 (5,665.29) (5,678.51) (5,687.12)
5 (6,627.96) (6,632.94) (6,635.92)
Table 13.  Searching the $ \mu^* $ by quasi-Newton method for exponential distribution ($ F^* = 5 $, $ \gamma = 3 $)
Iterations F* μ TC(F*, μ) Max. tolerance
0 5 8 665.286 1.58E+01
1 5 7 660.172 7.35
2 5 7.3170 669.221 1.16
3 5 7.2739 659.194 6.86E-02
4 5 7.2712 659.194 7.04E-04
5 5 7.2712 659.194 0
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Iterations F* μ TC(F*, μ) Max. tolerance
0 5 8 665.286 1.58E+01
1 5 7 660.172 7.35
2 5 7.3170 669.221 1.16
3 5 7.2739 659.194 6.86E-02
4 5 7.2712 659.194 7.04E-04
5 5 7.2712 659.194 0
Table 14.  Minimum cost $ (F^* \mu^*, TC(F^*, \mu^*)) $ for $ \gamma = 1,3 $ and 5 for TSM
γ $(F^* \mu^*, TC(F^*, \mu^*))$
Exp E3 D
1 (2, 6.169,803.45) (1, 5.854,809.85) (1, 6.092,820.80)
3 (5, 7.271,659.19) (5, 7.341,673.70) (5, 7.378,682.92)
5 (6, 7.012,615.23) (6, 7.048,621.22) (6, 7.068,624.76)
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γ $(F^* \mu^*, TC(F^*, \mu^*))$
Exp E3 D
1 (2, 6.169,803.45) (1, 5.854,809.85) (1, 6.092,820.80)
3 (5, 7.271,659.19) (5, 7.341,673.70) (5, 7.378,682.92)
5 (6, 7.012,615.23) (6, 7.048,621.22) (6, 7.068,624.76)
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