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doi: 10.3934/jimo.2022044
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Optimization of traffic control in $ MMAP\mathit{[2]}/PH\mathit{[2]}/S$ priority queueing model with $ PH $ retrial times and the preemptive repeat policy

Department of Mathematics, Central University of Rajasthan, Ajmer, India

*Corresponding author: Vidyottama Jain

Received  June 2021 Revised  December 2021 Early access March 2022

Fund Project: The first author, Raina Raj is supported by a senior research fellowship (SRF) grant No.- 09/1131(0024)/2018-EMR-I from Council of Scientific and Industrial Research (CSIR), India

The presented study elaborates a multi-server priority queueing model considering the preemptive repeat policy and phase-type distribution ($ P\!H $) for retrial process. The incoming heterogeneous calls are categorized as handoff calls and new calls. The arrival and service processes of both types of calls follow marked Markovian arrival process ($ M\!M\!A\!P $) and $ P\!H $ distribution with distinct parameters, respectively. An arriving new call will be blocked when all the channels are occupied, and consequently will join the orbit (virtual space) to retry following $ P\!H $ distribution. When all the channels are occupied and a handoff call arrives at the system, out of the following two scenarios, one might take place. In the first scenario, if all the channels are occupied with handoff calls, the arriving handoff call will be lost from the system. While in the second one, if all the channels are occupied and at least one of them is serving a new call, the arriving handoff call will be provided service by using preemptive priority over that new call and the preempted new call will join the orbit. Behaviour of the proposed system is modelled by the level dependent quasi-birth-death $ (L\!D\!Q\!B\!D) $ process. The expressions of various performance measures have been derived for the numerical illustration. An optimization problem for optimal channel allocation and traffic control has been formulated and dealt by employing appropriate heuristic approaches.

Citation: Raina Raj, Vidyottama Jain. Optimization of traffic control in $ MMAP\mathit{[2]}/PH\mathit{[2]}/S$ priority queueing model with $ PH $ retrial times and the preemptive repeat policy. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022044
References:
[1]

J. R. ArtalejoA. Dudin and V. I. Klimenok, Stationary analysis of a retrial queue with preemptive repeated attempts, Oper. Res. Let., 28 (2001), 173-180.  doi: 10.1016/S0167-6377(01)00059-1.

[2]

J. R. Artalejo and A Gomez-Corral, Modelling communication systems with phase type service and retrial times, IEEE Commun. Lett., 11 (2007), 955-957.  doi: 10.1109/LCOMM.2007.070742.

[3]

A. Brandwajn and T. Begin, Multi-server preemptive priority queue with general arrivals and service times, Perform. Eval., 115 (2017), 150-164.  doi: 10.1016/j.peva.2017.08.003.

[4]

L. Bright and P. G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Comm. Statist. Stochastic Models, 11 (1995), 497-525.  doi: 10.1080/15326349508807357.

[5]

S. R. Chakravarthy, A retrial queueing model with thresholds and phase type retrial times, J. Appl. Math. Inf., 38 (2020), 351-373.  doi: 10.14317/jami.2020.351.

[6]

W. Chang, Preemptive priority queues, Oper. Res., 13 (1965), 820-827.  doi: 10.1287/opre.13.5.820.

[7]

T. Dayar, Analyzing Markov Chains Using Kronecker Products: Theory and Applications, SpringerBriefs in Mathematics. Springer, New York, 2012. doi: 10.1007/978-1-4614-4190-8.

[8]

S. DharmarajaV. Jindal and A. S. Alfa, Phase-type models for cellular networks supporting voice, video and data traffic, Math. Comput. Model., 47 (2008), 1167-1180.  doi: 10.1016/j.mcm.2007.07.006.

[9]

S. Drekic and D. A. Stanford, Reducing delay in preemptive repeat priority queues, J. Amer. Math. Soc., 5 (1992), 33-74.  doi: 10.1287/opre.49.1.145.11186.

[10]

A. DudinC. KimS. Dudin and O. Dudina, Priority retrial queueing model operating in random environment with varying number and reservation of servers, Appl. Math. Comput., 269 (2015), 674-690.  doi: 10.1016/j.amc.2015.08.005.

[11]

A. DudinC. KimS. Dudin and O. Dudina, Analysis and optimization of Guard Channel Policy with buffering in cellular mobile networks, Comput. Netw., 107 (2016), 258-269.  doi: 10.1016/j.comnet.2016.04.003.

[12]

A. DudinM. H. LeeO. Dudina and S. K. Lee, Analysis of priority retrial queue with many types of customers and servers reservation as a model of cognitive radio system, IEEE Trans. Commun., 107 (2016), 186-199.  doi: 10.1109/TCOMM.2016.2606379.

[13]

R. Eberhart and J. Kennedy, Particle swarm optimization, IEEE Int. Conf. Neural Net., 4 (1995), 1942-1948.  doi: 10.1109/ICNN.1995.488968.

[14]

D. Fiems and S. De Vuyst, From exhaustive vacation queues to preemptive priority queues with general interarrival times, Int. J. Appl. Math. Comput. Sci., 28 (2018), 695-704.  doi: 10.2478/amcs-2018-0053.

[15]

Q.-M. He, Queues with marked calls, Adv. Appl. Probab., 28 (1996), 567-587.  doi: 10.2307/1428072.

[16]

Q.-M. He, Fundamentals of Matrix-analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5.

[17]

Q.-M. He and A. S. Alfa, The MMAP[K]/PH[K]/1 queues with a last-come-first-served preemptive service discipline, Queueing Syst., 29 (1998), 269-291.  doi: 10.1023/A:1019140332008.

[18]

Q.-M. He and A. S. Alfa, Space reduction for a class of multi-dimensional Markov chains: A summary and some applications, INFORMS J. Comput., 30 (2018), 1-10.  doi: 10.1287/ijoc.2017.0759.

[19]

Q.-M. He and H. Li, Stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue, Queueing Syst., 44 (2003), 137-160.  doi: 10.1023/A:1024420505098.

[20]

V. Jain, R. Raj and S. Dharmaraja, Numerical optimization of loss system with retrial phenomenon in cellular networks, Int. J. Oper. Res., In Press, (2021). doi: 10.1504/IJOR.2020.10032709.

[21]

J. Kim and B. Kim, A survey of retrial queueing systems, Ann. Oper. Res., 247 (2016), 3-36.  doi: 10.1007/s10479-015-2038-7.

[22]

S. KirkpatrickC. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680.  doi: 10.1126/science.220.4598.671.

[23]

V. KlimenokA. Dudin and V. Vishnevsky, Priority multi-server queueing system with heterogeneous customers, Mathematics, 8 (2020), 1501-1516.  doi: 10.3390/math8091501.

[24]

A. KrishnamoorthyS. Babu and V. C. Narayanan, MAP/(PH/PH)/c queue with self-generation of priorities and non-preemptive service, Stoch. Anal. Appl., 26 (2008), 1250-1266.  doi: 10.1080/07362990802405802.

[25]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, 1999. doi: 10.1137/1.9780898719734.

[26]

F. Machihara, A bridge between preemptive and non-preemptive queueing models, Perform. Eval., 23 (1995), 93-106.  doi: 10.1016/0166-5316(94)00045-LGet.

[27]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Corrected reprint of the 1981 original. Dover Publications, Inc., New York, 1994. doi: doi.org/10.1002/net.3230130219.

[28]

T. Phung-Duc, Asymptotic analysis for Markovian queues with two types of nonpersistent retrial customers, Appl. Math. Comput., 265 (2015), 768-784.  doi: 10.1016/j.amc.2015.05.133.

[29]

T. Phung-Duc, Retrial queueing models: A survey on theory and applications, In Stochastic Operations Research in Business and Industry, T. Dohi, K. Ano, and S. Kasahara, Eds. Singapore: World Scientific, 2017. doi: arXiv:1906.09560.

[30]

T. Phung-DucK. AkutsuK. KawanishiO. Salameh and S. Wittevrongel, Queueing models for cognitive wireless networks with sensing time of secondary users, Ann. Oper. Res., 310 (2022), 641-660.  doi: 10.1007/s10479-021-04118-9.

[31]

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.  doi: 10.3934/jimo.2018030.

[32]

T. Phung-DucH. MasuyamaS. Kasahara and Y. Takahashi, A matrix continued fraction approach to multiserver retrial queues, Ann. Oper. Res., 202 (2013), 161-183.  doi: 10.1007/s10479-011-0840-4.

[33]

Y. W. Shin and D. H. Moon, Approximation of M/M/c retrial queue with PH-retrial times, Eur. J. Oper. Res., 213 (2011), 205-209.  doi: 10.1016/j.ejor.2011.03.024.

[34]

B. Sun, M. H. Lee, A. Dudin and S. Dudin, MAP+ MAP/$M_2$/N/$\infty$ Queueing System with Absolute Priority and Reservation of Servers, Math. Probl. Eng., 2014 (2014), Art. ID 813150, 15 pp. doi: 10.1155/2014/813150.

[35]

V. Torczon and M. W. Trosset, From evolutionary operation to parallel direct search: Pattern search algorithms for numerical optimization, Comput. Sci. Stat., 29 (1998), 396–401. doi: doi=10.1.1.75.5210.

show all references

References:
[1]

J. R. ArtalejoA. Dudin and V. I. Klimenok, Stationary analysis of a retrial queue with preemptive repeated attempts, Oper. Res. Let., 28 (2001), 173-180.  doi: 10.1016/S0167-6377(01)00059-1.

[2]

J. R. Artalejo and A Gomez-Corral, Modelling communication systems with phase type service and retrial times, IEEE Commun. Lett., 11 (2007), 955-957.  doi: 10.1109/LCOMM.2007.070742.

[3]

A. Brandwajn and T. Begin, Multi-server preemptive priority queue with general arrivals and service times, Perform. Eval., 115 (2017), 150-164.  doi: 10.1016/j.peva.2017.08.003.

[4]

L. Bright and P. G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Comm. Statist. Stochastic Models, 11 (1995), 497-525.  doi: 10.1080/15326349508807357.

[5]

S. R. Chakravarthy, A retrial queueing model with thresholds and phase type retrial times, J. Appl. Math. Inf., 38 (2020), 351-373.  doi: 10.14317/jami.2020.351.

[6]

W. Chang, Preemptive priority queues, Oper. Res., 13 (1965), 820-827.  doi: 10.1287/opre.13.5.820.

[7]

T. Dayar, Analyzing Markov Chains Using Kronecker Products: Theory and Applications, SpringerBriefs in Mathematics. Springer, New York, 2012. doi: 10.1007/978-1-4614-4190-8.

[8]

S. DharmarajaV. Jindal and A. S. Alfa, Phase-type models for cellular networks supporting voice, video and data traffic, Math. Comput. Model., 47 (2008), 1167-1180.  doi: 10.1016/j.mcm.2007.07.006.

[9]

S. Drekic and D. A. Stanford, Reducing delay in preemptive repeat priority queues, J. Amer. Math. Soc., 5 (1992), 33-74.  doi: 10.1287/opre.49.1.145.11186.

[10]

A. DudinC. KimS. Dudin and O. Dudina, Priority retrial queueing model operating in random environment with varying number and reservation of servers, Appl. Math. Comput., 269 (2015), 674-690.  doi: 10.1016/j.amc.2015.08.005.

[11]

A. DudinC. KimS. Dudin and O. Dudina, Analysis and optimization of Guard Channel Policy with buffering in cellular mobile networks, Comput. Netw., 107 (2016), 258-269.  doi: 10.1016/j.comnet.2016.04.003.

[12]

A. DudinM. H. LeeO. Dudina and S. K. Lee, Analysis of priority retrial queue with many types of customers and servers reservation as a model of cognitive radio system, IEEE Trans. Commun., 107 (2016), 186-199.  doi: 10.1109/TCOMM.2016.2606379.

[13]

R. Eberhart and J. Kennedy, Particle swarm optimization, IEEE Int. Conf. Neural Net., 4 (1995), 1942-1948.  doi: 10.1109/ICNN.1995.488968.

[14]

D. Fiems and S. De Vuyst, From exhaustive vacation queues to preemptive priority queues with general interarrival times, Int. J. Appl. Math. Comput. Sci., 28 (2018), 695-704.  doi: 10.2478/amcs-2018-0053.

[15]

Q.-M. He, Queues with marked calls, Adv. Appl. Probab., 28 (1996), 567-587.  doi: 10.2307/1428072.

[16]

Q.-M. He, Fundamentals of Matrix-analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5.

[17]

Q.-M. He and A. S. Alfa, The MMAP[K]/PH[K]/1 queues with a last-come-first-served preemptive service discipline, Queueing Syst., 29 (1998), 269-291.  doi: 10.1023/A:1019140332008.

[18]

Q.-M. He and A. S. Alfa, Space reduction for a class of multi-dimensional Markov chains: A summary and some applications, INFORMS J. Comput., 30 (2018), 1-10.  doi: 10.1287/ijoc.2017.0759.

[19]

Q.-M. He and H. Li, Stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue, Queueing Syst., 44 (2003), 137-160.  doi: 10.1023/A:1024420505098.

[20]

V. Jain, R. Raj and S. Dharmaraja, Numerical optimization of loss system with retrial phenomenon in cellular networks, Int. J. Oper. Res., In Press, (2021). doi: 10.1504/IJOR.2020.10032709.

[21]

J. Kim and B. Kim, A survey of retrial queueing systems, Ann. Oper. Res., 247 (2016), 3-36.  doi: 10.1007/s10479-015-2038-7.

[22]

S. KirkpatrickC. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680.  doi: 10.1126/science.220.4598.671.

[23]

V. KlimenokA. Dudin and V. Vishnevsky, Priority multi-server queueing system with heterogeneous customers, Mathematics, 8 (2020), 1501-1516.  doi: 10.3390/math8091501.

[24]

A. KrishnamoorthyS. Babu and V. C. Narayanan, MAP/(PH/PH)/c queue with self-generation of priorities and non-preemptive service, Stoch. Anal. Appl., 26 (2008), 1250-1266.  doi: 10.1080/07362990802405802.

[25]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, 1999. doi: 10.1137/1.9780898719734.

[26]

F. Machihara, A bridge between preemptive and non-preemptive queueing models, Perform. Eval., 23 (1995), 93-106.  doi: 10.1016/0166-5316(94)00045-LGet.

[27]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Corrected reprint of the 1981 original. Dover Publications, Inc., New York, 1994. doi: doi.org/10.1002/net.3230130219.

[28]

T. Phung-Duc, Asymptotic analysis for Markovian queues with two types of nonpersistent retrial customers, Appl. Math. Comput., 265 (2015), 768-784.  doi: 10.1016/j.amc.2015.05.133.

[29]

T. Phung-Duc, Retrial queueing models: A survey on theory and applications, In Stochastic Operations Research in Business and Industry, T. Dohi, K. Ano, and S. Kasahara, Eds. Singapore: World Scientific, 2017. doi: arXiv:1906.09560.

[30]

T. Phung-DucK. AkutsuK. KawanishiO. Salameh and S. Wittevrongel, Queueing models for cognitive wireless networks with sensing time of secondary users, Ann. Oper. Res., 310 (2022), 641-660.  doi: 10.1007/s10479-021-04118-9.

[31]

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.  doi: 10.3934/jimo.2018030.

[32]

T. Phung-DucH. MasuyamaS. Kasahara and Y. Takahashi, A matrix continued fraction approach to multiserver retrial queues, Ann. Oper. Res., 202 (2013), 161-183.  doi: 10.1007/s10479-011-0840-4.

[33]

Y. W. Shin and D. H. Moon, Approximation of M/M/c retrial queue with PH-retrial times, Eur. J. Oper. Res., 213 (2011), 205-209.  doi: 10.1016/j.ejor.2011.03.024.

[34]

B. Sun, M. H. Lee, A. Dudin and S. Dudin, MAP+ MAP/$M_2$/N/$\infty$ Queueing System with Absolute Priority and Reservation of Servers, Math. Probl. Eng., 2014 (2014), Art. ID 813150, 15 pp. doi: 10.1155/2014/813150.

[35]

V. Torczon and M. W. Trosset, From evolutionary operation to parallel direct search: Pattern search algorithms for numerical optimization, Comput. Sci. Stat., 29 (1998), 396–401. doi: doi=10.1.1.75.5210.

Figure 1.  A multi-server $ \rm{\it MMAP\mathit{[2]}/PH\mathit{[2]}/S} $ model with $ \rm{\it P\!H} $ retrial times and the preemptive repeat priority policy
Figure 2.  Dependence of the dropping probability $ P_d $ and preemption probability $ P_{preempt} $ over arrival rate of a handoff call $ \lambda_{\mathcal{H}} $
Figure 3.  Dependence of the dropping probability $ P_d $ and preemption probability $ P_{preempt} $ over arrival rate of a new call $ \lambda_{\mathcal{N}} $
Figure 4.  Dependence of the dropping probability $ P_d $ and preemption probability $ P_{preempt} $ over service rate of a handoff call $ \mu_{\mathcal{H}} $
Figure 5.  Dependence of the intensity by which a retrial call is successfully connected to an available channel $ \theta_r^{succ} $ and the probability that a retrial call will exit the system without obtaining the service $ P_{leave}^{no-service} $ over retrial rate $ \theta $
Table 1.  Optimal values of $ S^* $ and $ \lambda_{\mathcal{H}}^* $ for different values of $ \mu_{\mathcal{H}} $ and $ \lambda_{\mathcal{N}} $ by applying DS Method
$ \lambda_{\mathcal{N}} = 0.1 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ 4 4 4 4 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.0575 0.15 0.22 0.2750 0.6250 0.7 1.05
$ P_d^* $ $ 1.04 \times 10^{-8} $ $ 1.71 \times 10^{-5} $ $ 4.472\times 10^{-5} $ $ 6.13 \times 10^{-5} $ $ 9.64 \times 10^{-5} $ $ 9.46 \times 10^{-5} $ $ 9.33 \times 10^{-5} $
$ P_{preempt}^* $ $ 2.04 \times 10^{-5} $ $ 3.569 \times 10^{-5} $ $ 7.413 \times 10^{-5} $ $ 9.79 \times 10^{-5} $ $ 8.69 \times 10^{-5} $ $ 8.97 \times 10^{-5} $ $ 9.27 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.2 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ 5 6 6 6 7 7 7
$ \lambda_{\mathcal{H}}^* $ 0.1750 0.525 0.65 0.75 1.25 1.4 1.55
$ P_d^* $ $ 8.72 \times 10^{-5} $ $ 4.752 \times 10^{-5} $ $ 5.67\times 10^{-5} $ $ 5.22\times 10^{-5} $ $ 6.68\times 10^{-5} $ $ 6.65 \times 10^{-5} $ $ 6.64 \times 10^{-5} $
$ P_{preempt}^* $ $ 8.78 \times 10^{-5} $ $ 8.23 \times 10^{-5} $ $ 9.88 \times 10^{-5} $ $ 9.78\times 10^{-5} $ $ 9.23\times 10^{-5} $ $ 9.58 \times 10^{-5} $ $ 9.93 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.3 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ $ - $ 6 6 6 6 6 7
$ \lambda_{\mathcal{H}}^* $ $ - $ 0.45 0.5 0.625 0.705 0.8 1.35
$ P_d^* $ $ - $ $ 2.09 \times 10^{-5} $ $ 1.109\times 10^{-5} $ $ 1.78\times 10^{-5} $ $ 1.59\times 10^{-5} $ $ 1.67 \times 10^{-5} $ $ 2.43 \times 10^{-5} $
$ P_{preempt}^* $ $ - $ $ 9.19 \times 10^{-5} $ $ 6.23 \times 10^{-5} $ $ 9.2\times 10^{-5} $ $ 9.02\times 10^{-5} $ $ 9.84 \times 10^{-5} $ $ 9.00 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.4 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ $ - $ 7 7 7 7 7 7
$ \lambda_{\mathcal{H}}^* $ $ - $ 0.625 0.76 0.75 0.875 1.1 1.2
$ P_d^* $ $ - $ $ 1.61 \times 10^{-5} $ $ 1.71\times 10^{-5} $ $ 1.53 \times 10^{-5} $ $ 1.23\times 10^{-5} $ $ 1.26 \times 10^{-5} $ $ 1.08 \times 10^{-5} $
$ P_{preempt}^* $ $ - $ $ 9.08 \times 10^{-5} $ $ 9.99 \times 10^{-5} $ $ 9.73 \times 10^{-5} $ $ 8.87 \times 10^{-5} $ $ 9.48 \times 10^{-5} $ $ 8.95 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.5 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ $ - $ 8 8 8 8 8 8
$ \lambda_{\mathcal{H}}^* $ $ - $ 0.1 0.35 0.375 0.425 0.45 0.5
$ P_d^* $ $ - $ $ 2.37\times 10^{-5} $ $ 2.88\times 10^{-5} $ $ 1.05 \times 10^{-5} $ $ 8.06\times 10^{-5} $ $ 3.39 \times 10^{-5} $ $ 3.10 \times 10^{-5} $
$ P_{preempt}^* $ $ - $ $ 1.07\times 10^{-5} $ $ 9.9 \times 10^{-5} $ $ 8.80\times 10^{-5} $ $ 9.49\times 10^{-5} $ $ 8.86 \times 10^{-5} $ $ 9.69 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.6 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ 8 $ - $ 8 8 8 $ - $ 8
$ \lambda_{\mathcal{H}}^* $ 0.25 $ - $ 0.75 0.85 1 $ - $ 1.2
$ P_d^* $ $ 1.36 \times 10^{-5} $ $ - $ $ 8.70\times 10^{-5} $ $ 3.11 \times 10^{-5} $ $ 1.03 \times 10^{-5} $ $ - $ $ 4.01 \times 10^{-5} $
$ P_{preempt}^* $ $ 1.17 \times 10^{-5} $ $ - $ $ 9.91 \times 10^{-5} $ $ 9.50\times 10^{-5} $ $ 9.84\times 10^{-5} $ $ - $ $ 9.48 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.1 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ 4 4 4 4 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.0575 0.15 0.22 0.2750 0.6250 0.7 1.05
$ P_d^* $ $ 1.04 \times 10^{-8} $ $ 1.71 \times 10^{-5} $ $ 4.472\times 10^{-5} $ $ 6.13 \times 10^{-5} $ $ 9.64 \times 10^{-5} $ $ 9.46 \times 10^{-5} $ $ 9.33 \times 10^{-5} $
$ P_{preempt}^* $ $ 2.04 \times 10^{-5} $ $ 3.569 \times 10^{-5} $ $ 7.413 \times 10^{-5} $ $ 9.79 \times 10^{-5} $ $ 8.69 \times 10^{-5} $ $ 8.97 \times 10^{-5} $ $ 9.27 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.2 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ 5 6 6 6 7 7 7
$ \lambda_{\mathcal{H}}^* $ 0.1750 0.525 0.65 0.75 1.25 1.4 1.55
$ P_d^* $ $ 8.72 \times 10^{-5} $ $ 4.752 \times 10^{-5} $ $ 5.67\times 10^{-5} $ $ 5.22\times 10^{-5} $ $ 6.68\times 10^{-5} $ $ 6.65 \times 10^{-5} $ $ 6.64 \times 10^{-5} $
$ P_{preempt}^* $ $ 8.78 \times 10^{-5} $ $ 8.23 \times 10^{-5} $ $ 9.88 \times 10^{-5} $ $ 9.78\times 10^{-5} $ $ 9.23\times 10^{-5} $ $ 9.58 \times 10^{-5} $ $ 9.93 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.3 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ $ - $ 6 6 6 6 6 7
$ \lambda_{\mathcal{H}}^* $ $ - $ 0.45 0.5 0.625 0.705 0.8 1.35
$ P_d^* $ $ - $ $ 2.09 \times 10^{-5} $ $ 1.109\times 10^{-5} $ $ 1.78\times 10^{-5} $ $ 1.59\times 10^{-5} $ $ 1.67 \times 10^{-5} $ $ 2.43 \times 10^{-5} $
$ P_{preempt}^* $ $ - $ $ 9.19 \times 10^{-5} $ $ 6.23 \times 10^{-5} $ $ 9.2\times 10^{-5} $ $ 9.02\times 10^{-5} $ $ 9.84 \times 10^{-5} $ $ 9.00 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.4 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ $ - $ 7 7 7 7 7 7
$ \lambda_{\mathcal{H}}^* $ $ - $ 0.625 0.76 0.75 0.875 1.1 1.2
$ P_d^* $ $ - $ $ 1.61 \times 10^{-5} $ $ 1.71\times 10^{-5} $ $ 1.53 \times 10^{-5} $ $ 1.23\times 10^{-5} $ $ 1.26 \times 10^{-5} $ $ 1.08 \times 10^{-5} $
$ P_{preempt}^* $ $ - $ $ 9.08 \times 10^{-5} $ $ 9.99 \times 10^{-5} $ $ 9.73 \times 10^{-5} $ $ 8.87 \times 10^{-5} $ $ 9.48 \times 10^{-5} $ $ 8.95 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.5 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ $ - $ 8 8 8 8 8 8
$ \lambda_{\mathcal{H}}^* $ $ - $ 0.1 0.35 0.375 0.425 0.45 0.5
$ P_d^* $ $ - $ $ 2.37\times 10^{-5} $ $ 2.88\times 10^{-5} $ $ 1.05 \times 10^{-5} $ $ 8.06\times 10^{-5} $ $ 3.39 \times 10^{-5} $ $ 3.10 \times 10^{-5} $
$ P_{preempt}^* $ $ - $ $ 1.07\times 10^{-5} $ $ 9.9 \times 10^{-5} $ $ 8.80\times 10^{-5} $ $ 9.49\times 10^{-5} $ $ 8.86 \times 10^{-5} $ $ 9.69 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.6 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1.0 1.2
$ S^* $ 8 $ - $ 8 8 8 $ - $ 8
$ \lambda_{\mathcal{H}}^* $ 0.25 $ - $ 0.75 0.85 1 $ - $ 1.2
$ P_d^* $ $ 1.36 \times 10^{-5} $ $ - $ $ 8.70\times 10^{-5} $ $ 3.11 \times 10^{-5} $ $ 1.03 \times 10^{-5} $ $ - $ $ 4.01 \times 10^{-5} $
$ P_{preempt}^* $ $ 1.17 \times 10^{-5} $ $ - $ $ 9.91 \times 10^{-5} $ $ 9.50\times 10^{-5} $ $ 9.84\times 10^{-5} $ $ - $ $ 9.48 \times 10^{-5} $
Table 2.  Optimal values of $ S^* $ and $ \lambda_{\mathcal{H}}^* $ for different values of $ \mu_{\mathcal{H}} $ and $ \lambda_{\mathcal{N}} $ by applying PSO method
$ \lambda_{\mathcal{N}} = 0.1 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 4 3 3 3 3 3 3
$ \lambda_{\mathcal{H}}^* $ 0.1 0.1 0.1 0.1 0.1 0.1 0.125
$ P_d^* $ $ 6.8735 \times 10^{-8} $ $ 2.0619 \times 10^{-6} $ $ 7.326 \times 10^{-7} $ $ 3.043 \times 10^{-5} $ $ 1.404 \times 10^{-7} $ $ 7.027 \times 10^{-8} $ $ 1.19 \times 10^{-7} $
$ P_{preempt}^* $ $ 6.4362 \times 10^{-5} $ $ 8.5911\times 10^{-6} $ $ 4.856 \times 10^{-6} $ $ 3.129 \times 10^{-5} $ $ 2.205 \times 10^{-6} $ $ 1.663 \times 10^{-5} $ $ 2.436 \times 10^{-6} $
$ \lambda_{\mathcal{N}} = 0.2 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 5 5 4 4 4 4 4
$ \lambda_{\mathcal{H}}^* $ 0.1250 0.1250 0.31 0.23 0.23 0.23 0.5
$ P_d^* $ $ 2.145 \times 10^{-8} $ $ 8.714 \times 10^{-12} $ $ 4.16 \times 10^{-5} $ $ 3.9 \times 10^{-4} $ $ 1.9023 \times 10^{-5} $ $ 9.9959\times 10^{-5} $ $ 2.766 \times 10^{-5} $
$ P_{preempt}^* $ $ 2.028 \times 10^{-6} $ $ 7.0814 \times 10^{-7} $ $ 9.468\times 10^{-5} $ $ 2.462 \times 10^{-5} $ $ 1.75 \times 10^{-6} $ $ 1.315 \times 10^{-6} $ $ 7.91 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.3 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 6 6 5 5 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.125 0.125 0.395 0.45 0.5 0.68 0.7
$ P_d^* $ $ 4.02 \times 10^{-11} $ $ 5.73 \times 10^{-11} $ $ 6.534 \times 10^{-6} $ $ 5.289 \times 10^{-5} $ $ 4.252 \times 10^{-5} $ $ 1.29 \times 10^{-5} $ $ 7.925 \times 10^{-5} $
$ P_{preempt}^* $ $ 1.0162 \times 10^{-4} $ $ 1.714 \times 10^{-4} $ $ 3.46 \times 10^{-4} $ $ 3.172 \times 10^{-4} $ $ 2.905 \times 10^{-4} $ $ 5.848 \times 10^{-4} $ $ 4.517 \times 10^{-4} $
$ \lambda_{\mathcal{N}} = 0.4 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 6 6 5 5 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.125 0.125 0.3 0.43 0.465 0.549 0.6
$ P_d^* $ $ 1.14 \times 10^{-10} $ $ 1.52 \times 10^{-9} $ $ 1.50\times 10^{-7} $ $ 5.1 \times 10^{-4} $ $ 3.5 \times 10^{-4} $ $ 4.4 \times 10^{-4} $ $ 3.8 \times 10^{-4} $
$ P_{preempt}^* $ $ 3.981 \times 10^{-5} $ $ 7.284 \times 10^{-5} $ $ 2.3 \times 10^{-5} $ $ 5.6 \times 10^{-5} $ $ 4.82 \times 10^{-5} $ $ 5.63 \times 10^{-5} $ $ 5.4 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.5 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 7 7 6 6 6 5 5
$ \lambda_{\mathcal{H}}^* $ 0.1 0.1 0.6335 0.6335 0.85 0.15 0.4992
$ P_d^* $ $ 3.58 \times 10^{-9} $ $ 4.7 \times 10^{-11} $ $ 9.2 \times 10^{-5} $ $ 3.1 \times 10^{-5} $ $ 8.8 \times 10^{-5} $ $ 3.16 \times 10^{-10} $ $ 1.56 \times 10^{-5} $
$ P_{preempt}^* $ $ 8.08 \times 10^{-5} $ $ 1.63 \times 10^{-5} $ $ 7.43 \times 10^{-5} $ $ 3.98 \times 10^{-5} $ $ 7.6 \times 10^{-5} $ $ 6.34 \times 10^{-5} $ $ 6.07 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.6 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 8 8 7 7 6 6 6
$ \lambda_{\mathcal{H}}^* $ 0.25 0.755 0.125 0.112 0.7195 0.8894 1
$ P_d^* $ $ 7.90 \times 10^{-12} $ $ 7.33 \times 10^{-6} $ $ 9.32 \times 10^{-13} $ $ 9.67 \times 10^{-14} $ $ 3.26 \times 10^{-5} $ $ 5.94 \times 10^{-5} $ $ 6.24 \times 10^{-5} $
$ P_{preempt}^* $ $ 9.48 \times 10^{-5} $ $ 9.22 \times 10^{-5} $ $ 5.78 \times 10^{-4} $ $ 4.85 \times 10^{-4} $ $ 6.62 \times 10^{-4} $ $ 9.27 \times 10^{-4} $ $ 9.72 \times 10^{-4} $
$ \lambda_{\mathcal{N}} = 0.1 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 4 3 3 3 3 3 3
$ \lambda_{\mathcal{H}}^* $ 0.1 0.1 0.1 0.1 0.1 0.1 0.125
$ P_d^* $ $ 6.8735 \times 10^{-8} $ $ 2.0619 \times 10^{-6} $ $ 7.326 \times 10^{-7} $ $ 3.043 \times 10^{-5} $ $ 1.404 \times 10^{-7} $ $ 7.027 \times 10^{-8} $ $ 1.19 \times 10^{-7} $
$ P_{preempt}^* $ $ 6.4362 \times 10^{-5} $ $ 8.5911\times 10^{-6} $ $ 4.856 \times 10^{-6} $ $ 3.129 \times 10^{-5} $ $ 2.205 \times 10^{-6} $ $ 1.663 \times 10^{-5} $ $ 2.436 \times 10^{-6} $
$ \lambda_{\mathcal{N}} = 0.2 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 5 5 4 4 4 4 4
$ \lambda_{\mathcal{H}}^* $ 0.1250 0.1250 0.31 0.23 0.23 0.23 0.5
$ P_d^* $ $ 2.145 \times 10^{-8} $ $ 8.714 \times 10^{-12} $ $ 4.16 \times 10^{-5} $ $ 3.9 \times 10^{-4} $ $ 1.9023 \times 10^{-5} $ $ 9.9959\times 10^{-5} $ $ 2.766 \times 10^{-5} $
$ P_{preempt}^* $ $ 2.028 \times 10^{-6} $ $ 7.0814 \times 10^{-7} $ $ 9.468\times 10^{-5} $ $ 2.462 \times 10^{-5} $ $ 1.75 \times 10^{-6} $ $ 1.315 \times 10^{-6} $ $ 7.91 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.3 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 6 6 5 5 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.125 0.125 0.395 0.45 0.5 0.68 0.7
$ P_d^* $ $ 4.02 \times 10^{-11} $ $ 5.73 \times 10^{-11} $ $ 6.534 \times 10^{-6} $ $ 5.289 \times 10^{-5} $ $ 4.252 \times 10^{-5} $ $ 1.29 \times 10^{-5} $ $ 7.925 \times 10^{-5} $
$ P_{preempt}^* $ $ 1.0162 \times 10^{-4} $ $ 1.714 \times 10^{-4} $ $ 3.46 \times 10^{-4} $ $ 3.172 \times 10^{-4} $ $ 2.905 \times 10^{-4} $ $ 5.848 \times 10^{-4} $ $ 4.517 \times 10^{-4} $
$ \lambda_{\mathcal{N}} = 0.4 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 6 6 5 5 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.125 0.125 0.3 0.43 0.465 0.549 0.6
$ P_d^* $ $ 1.14 \times 10^{-10} $ $ 1.52 \times 10^{-9} $ $ 1.50\times 10^{-7} $ $ 5.1 \times 10^{-4} $ $ 3.5 \times 10^{-4} $ $ 4.4 \times 10^{-4} $ $ 3.8 \times 10^{-4} $
$ P_{preempt}^* $ $ 3.981 \times 10^{-5} $ $ 7.284 \times 10^{-5} $ $ 2.3 \times 10^{-5} $ $ 5.6 \times 10^{-5} $ $ 4.82 \times 10^{-5} $ $ 5.63 \times 10^{-5} $ $ 5.4 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.5 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 7 7 6 6 6 5 5
$ \lambda_{\mathcal{H}}^* $ 0.1 0.1 0.6335 0.6335 0.85 0.15 0.4992
$ P_d^* $ $ 3.58 \times 10^{-9} $ $ 4.7 \times 10^{-11} $ $ 9.2 \times 10^{-5} $ $ 3.1 \times 10^{-5} $ $ 8.8 \times 10^{-5} $ $ 3.16 \times 10^{-10} $ $ 1.56 \times 10^{-5} $
$ P_{preempt}^* $ $ 8.08 \times 10^{-5} $ $ 1.63 \times 10^{-5} $ $ 7.43 \times 10^{-5} $ $ 3.98 \times 10^{-5} $ $ 7.6 \times 10^{-5} $ $ 6.34 \times 10^{-5} $ $ 6.07 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.6 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 8 8 7 7 6 6 6
$ \lambda_{\mathcal{H}}^* $ 0.25 0.755 0.125 0.112 0.7195 0.8894 1
$ P_d^* $ $ 7.90 \times 10^{-12} $ $ 7.33 \times 10^{-6} $ $ 9.32 \times 10^{-13} $ $ 9.67 \times 10^{-14} $ $ 3.26 \times 10^{-5} $ $ 5.94 \times 10^{-5} $ $ 6.24 \times 10^{-5} $
$ P_{preempt}^* $ $ 9.48 \times 10^{-5} $ $ 9.22 \times 10^{-5} $ $ 5.78 \times 10^{-4} $ $ 4.85 \times 10^{-4} $ $ 6.62 \times 10^{-4} $ $ 9.27 \times 10^{-4} $ $ 9.72 \times 10^{-4} $
Table 3.  Optimal values of $ S^* $ and $ \lambda_{\mathcal{H}}^* $ for different values of $ \mu_{\mathcal{H}} $ and $ \lambda_{\mathcal{N}} $ by applying SA method
$ \lambda_{\mathcal{N}} = 0.1 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 4 3 3 3 3 3 3
$ \lambda_{\mathcal{H}}^* $ 0.11 0.11 0.11 0.125 0.125 0.125 0.125
$ P_d^* $ $ 1.11 \times 10^{-7} $ $ 3.906 \times 10^{-6} $ $ 8.263 \times 10^{-7} $ $ 5.430 \times 10^{-5} $ $ 2.044 \times 10^{-7} $ $ 4.49 \times 10^{-5} $ $ 1.19 \times 10^{-7} $
$ P_{preempt}^* $ $ 7.83 \times 10^{-5} $ $ 1.5\times 10^{-7} $ $ 6.568 \times 10^{-6} $ $ 4.291 \times 10^{-5} $ $ 3.025 \times 10^{-6} $ $ 2.34 \times 10^{-5} $ $ 2.436 \times 10^{-6} $
$ \lambda_{\mathcal{N}} = 0.2 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 5 5 4 4 4 4 4
$ \lambda_{\mathcal{H}}^* $ 0.12 0.12 0.3 0.2 0.2 0.2 0.5
$ P_d^* $ $ 1.52 \times 10^{-8} $ $ 1.81 \times 10^{-12} $ $ 8.35 \times 10^{-6} $ $ 4.76 \times 10^{-5} $ $ 1.89023 \times 10^{-5} $ $ 9.69959\times 10^{-5} $ $ 2.2766 \times 10^{-5} $
$ P_{preempt}^* $ $ 6.487 \times 10^{-6} $ $ 5.57 \times 10^{-7} $ $ 4.002\times 10^{-5} $ $ 2.462 \times 10^{-5} $ $ 1.675 \times 10^{-6} $ $ 1.4315 \times 10^{-6} $ $ 7.891 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.3 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 6 6 5 5 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.12 0.12 0.4 0.45 0.5 0.6 0.7
$ P_d^* $ $ 3.9902 \times 10^{-11} $ $ 5.473 \times 10^{-11} $ $ 6.1534 \times 10^{-6} $ $ 5.289 \times 10^{-5} $ $ 4.252 \times 10^{-5} $ $ 9.29 \times 10^{-5} $ $ 7.925 \times 10^{-5} $
$ P_{preempt}^* $ $ 1.0162 \times 10^{-4} $ $ 1.714 \times 10^{-4} $ $ 3.46 \times 10^{-4} $ $ 3.172 \times 10^{-4} $ $ 2.905 \times 10^{-4} $ $ 1.848 \times 10^{-4} $ $ 4.517 \times 10^{-4} $
$ \lambda_{\mathcal{N}} = 0.4 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 6 6 5 5 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.12 0.12 0.3 0.4 0.45 0.55 0.6
$ P_d^* $ $ 1.214 \times 10^{-10} $ $ 1.652 \times 10^{-9} $ $ 1.50\times 10^{-7} $ $ 5.12 \times 10^{-4} $ $ 3.5 \times 10^{-4} $ $ 4.4 \times 10^{-4} $ $ 3.8 \times 10^{-4} $
$ P_{preempt}^* $ $ 4.39 \times 10^{-5} $ $ 7.0284 \times 10^{-5} $ $ 2.3 \times 10^{-5} $ $ 5.6 \times 10^{-5} $ $ 4.682 \times 10^{-5} $ $ 5.163 \times 10^{-5} $ $ 5.4 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.5 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 7 7 6 6 6 5 5
$ \lambda_{\mathcal{H}}^* $ 0.1 0.1 0.6 0.6 0.8 0.1 0.5
$ P_d^* $ $ 3.58 \times 10^{-9} $ $ 4.7 \times 10^{-11} $ $ 9.1 \times 10^{-5} $ $ 3.0 \times 10^{-5} $ $ 8.5 \times 10^{-5} $ $ 2.16 \times 10^{-10} $ $ 1.5 \times 10^{-5} $
$ P_{preempt}^* $ $ 8.08 \times 10^{-5} $ $ 1.63 \times 10^{-5} $ $ 7.3 \times 10^{-5} $ $ 3.8 \times 10^{-5} $ $ 7.3 \times 10^{-5} $ $ 6.34 \times 10^{-5} $ $ 6.1 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.6 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 8 8 7 7 6 6 6
$ \lambda_{\mathcal{H}}^* $ 0.2 0.8 0.1 0.1 0.7 0.8 1
$ P_d^* $ $ 7.30 \times 10^{-12} $ $ 7.03 \times 10^{-6} $ $ 9.1 \times 10^{-13} $ $ 9.6 \times 10^{-14} $ $ 3.2 \times 10^{-5} $ $ 5.9 \times 10^{-5} $ $ 6.2 \times 10^{-5} $
$ P_{preempt}^* $ $ 9.14 \times 10^{-5} $ $ 9.02 \times 10^{-5} $ $ 5.17 \times 10^{-4} $ $ 4.8 \times 10^{-4} $ $ 6.6 \times 10^{-4} $ $ 9.2 \times 10^{-4} $ $ 9.7 \times 10^{-4} $
$ \lambda_{\mathcal{N}} = 0.1 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 4 3 3 3 3 3 3
$ \lambda_{\mathcal{H}}^* $ 0.11 0.11 0.11 0.125 0.125 0.125 0.125
$ P_d^* $ $ 1.11 \times 10^{-7} $ $ 3.906 \times 10^{-6} $ $ 8.263 \times 10^{-7} $ $ 5.430 \times 10^{-5} $ $ 2.044 \times 10^{-7} $ $ 4.49 \times 10^{-5} $ $ 1.19 \times 10^{-7} $
$ P_{preempt}^* $ $ 7.83 \times 10^{-5} $ $ 1.5\times 10^{-7} $ $ 6.568 \times 10^{-6} $ $ 4.291 \times 10^{-5} $ $ 3.025 \times 10^{-6} $ $ 2.34 \times 10^{-5} $ $ 2.436 \times 10^{-6} $
$ \lambda_{\mathcal{N}} = 0.2 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 5 5 4 4 4 4 4
$ \lambda_{\mathcal{H}}^* $ 0.12 0.12 0.3 0.2 0.2 0.2 0.5
$ P_d^* $ $ 1.52 \times 10^{-8} $ $ 1.81 \times 10^{-12} $ $ 8.35 \times 10^{-6} $ $ 4.76 \times 10^{-5} $ $ 1.89023 \times 10^{-5} $ $ 9.69959\times 10^{-5} $ $ 2.2766 \times 10^{-5} $
$ P_{preempt}^* $ $ 6.487 \times 10^{-6} $ $ 5.57 \times 10^{-7} $ $ 4.002\times 10^{-5} $ $ 2.462 \times 10^{-5} $ $ 1.675 \times 10^{-6} $ $ 1.4315 \times 10^{-6} $ $ 7.891 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.3 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 6 6 5 5 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.12 0.12 0.4 0.45 0.5 0.6 0.7
$ P_d^* $ $ 3.9902 \times 10^{-11} $ $ 5.473 \times 10^{-11} $ $ 6.1534 \times 10^{-6} $ $ 5.289 \times 10^{-5} $ $ 4.252 \times 10^{-5} $ $ 9.29 \times 10^{-5} $ $ 7.925 \times 10^{-5} $
$ P_{preempt}^* $ $ 1.0162 \times 10^{-4} $ $ 1.714 \times 10^{-4} $ $ 3.46 \times 10^{-4} $ $ 3.172 \times 10^{-4} $ $ 2.905 \times 10^{-4} $ $ 1.848 \times 10^{-4} $ $ 4.517 \times 10^{-4} $
$ \lambda_{\mathcal{N}} = 0.4 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 6 6 5 5 5 5 5
$ \lambda_{\mathcal{H}}^* $ 0.12 0.12 0.3 0.4 0.45 0.55 0.6
$ P_d^* $ $ 1.214 \times 10^{-10} $ $ 1.652 \times 10^{-9} $ $ 1.50\times 10^{-7} $ $ 5.12 \times 10^{-4} $ $ 3.5 \times 10^{-4} $ $ 4.4 \times 10^{-4} $ $ 3.8 \times 10^{-4} $
$ P_{preempt}^* $ $ 4.39 \times 10^{-5} $ $ 7.0284 \times 10^{-5} $ $ 2.3 \times 10^{-5} $ $ 5.6 \times 10^{-5} $ $ 4.682 \times 10^{-5} $ $ 5.163 \times 10^{-5} $ $ 5.4 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.5 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 7 7 6 6 6 5 5
$ \lambda_{\mathcal{H}}^* $ 0.1 0.1 0.6 0.6 0.8 0.1 0.5
$ P_d^* $ $ 3.58 \times 10^{-9} $ $ 4.7 \times 10^{-11} $ $ 9.1 \times 10^{-5} $ $ 3.0 \times 10^{-5} $ $ 8.5 \times 10^{-5} $ $ 2.16 \times 10^{-10} $ $ 1.5 \times 10^{-5} $
$ P_{preempt}^* $ $ 8.08 \times 10^{-5} $ $ 1.63 \times 10^{-5} $ $ 7.3 \times 10^{-5} $ $ 3.8 \times 10^{-5} $ $ 7.3 \times 10^{-5} $ $ 6.34 \times 10^{-5} $ $ 6.1 \times 10^{-5} $
$ \lambda_{\mathcal{N}} = 0.6 $, $ \mu_{\mathcal{N}}=1 $
$ \mu_{\mathcal{H}} $ 0.5 0.6 0.7 0.8 0.9 1 1.2
$ S^* $ 8 8 7 7 6 6 6
$ \lambda_{\mathcal{H}}^* $ 0.2 0.8 0.1 0.1 0.7 0.8 1
$ P_d^* $ $ 7.30 \times 10^{-12} $ $ 7.03 \times 10^{-6} $ $ 9.1 \times 10^{-13} $ $ 9.6 \times 10^{-14} $ $ 3.2 \times 10^{-5} $ $ 5.9 \times 10^{-5} $ $ 6.2 \times 10^{-5} $
$ P_{preempt}^* $ $ 9.14 \times 10^{-5} $ $ 9.02 \times 10^{-5} $ $ 5.17 \times 10^{-4} $ $ 4.8 \times 10^{-4} $ $ 6.6 \times 10^{-4} $ $ 9.2 \times 10^{-4} $ $ 9.7 \times 10^{-4} $
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