Figure 1.
State transition diagram.($ n: $ number of jobs present in the system, $ i: $ status of the server)
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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
Indian Institute of Technology Roorkee - 247 667, India |
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 and Management Optimization,
2020, 16
(6)
: 2625-2649.
doi: 10.3934/jimo.2019073
![]()
References:
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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. |
| [2] |
I. Adiri and B. Avi-Itzhak,
A time-sharing queue, Manage. Sci., 15 (1969), 639-657.
doi: 10.1287/mnsc.15.11.639. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
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| [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. |
| [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. |
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S. M. Gupta,
Interrelationship between controlling arrival and service in queueing systems, Comput. Oper. Res., 22 (1995), 1005-1014.
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| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [29] |
W. A. Massey,
The analysis of queues with time-varying rates for telecommunication models, Telecommun. Syst., 21 (2002), 173-204.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
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. |
| [2] |
I. Adiri and B. Avi-Itzhak,
A time-sharing queue, Manage. Sci., 15 (1969), 639-657.
doi: 10.1287/mnsc.15.11.639. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [29] |
W. A. Massey,
The analysis of queues with time-varying rates for telecommunication models, Telecommun. Syst., 21 (2002), 173-204.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
Figure 5.
Total cost TC for varying $ \mu $ and $ F $ for three distribution by taking three cost sets-I for MRP
Figure 6.
Total cost TC for varying $ \mu $ and $ \gamma $ for exponential distribution by taking three cost sets-Ⅱ, Ⅲ and Ⅳ
Figure 7.
Total cost TC for varying $ \mu $ and $ \gamma $ for Erlang-3 distribution by taking three cost sets-Ⅱ, Ⅲ and Ⅳ
Figure 8.
Total cost TC for varying $ \mu $ and $ \gamma $ for deterministic distribution by taking three cost sets-Ⅱ, Ⅲ and Ⅳ
Figure 12.
Total cost TC for varying $ \mu $ and F for three distribution by taking three cost sets- Ⅰ for TSM
Figure 13.
Total cost TC for varying $ \mu $ and $ \gamma $ for exponential distribution by taking three cost sets- Ⅱ, Ⅲ and Ⅳ
Figure 14.
Total cost TC for varying $ \mu $ and $ \gamma $ for Erlang-3 distribution by taking three cost sets- Ⅱ, Ⅲ and Ⅳ
Figure 15.
Total cost TC for varying $ \mu $ and $ \gamma $ for deterministic distribution by taking three cost sets- Ⅱ, Ⅲ and Ⅳ
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 |
| 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 |
| 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 |
| 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 | |||||
| Ⅰ | 30 | 30 | 50 | 70 | 40 |
| Ⅱ | 10 | 10 | 120 | 15 | 90 |
| Ⅲ | 15 | 5 | 120 | 15 | 90 |
| Ⅳ | 20 | 20 | 100 | 15 | 90 |
| Cost Set | |||||
| 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 | |||
| 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) |
| γ | |||
| 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 |
| 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 |
| γ | |||
| 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) |
| γ | |||
| 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 |
| 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 |
| 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 |
| 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 | |||||
| Ⅰ | 30 | 40 | 120 | 60 | 90 |
| Ⅱ | 10 | 10 | 120 | 15 | 90 |
| Ⅲ | 15 | 5 | 120 | 15 | 90 |
| Ⅳ | 10 | 10 | 100 | 15 | 110 |
| Cost Set | |||||
| 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 | |||
| 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) |
| γ | |||
| 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 |
| 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 |
| γ | |||
| 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) |
| γ | |||
| 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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