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A two-priority single server retrial queue with additional items
1. | Department of Mathematics, S. N. College, Chempazhanthy, Trivandrum, Kerala-695587, India |
2. | Belarusian State University, 4, Nezavisimosti Ave., Minsk, 220030, Belarus |
3. | Peoples Friendship University of Russia, 6 Miklukho-Maklaya St, Moscow, 117198, Russia |
4. | Department of Mathematics, CMS College, Kottayam-686001, India |
In this paper, we study a priority queueing-inventory problem with two types of customers. Arrival of customers follows Marked Markovian arrival process and service times have phase-type distribution with parameters depending on the type of customer in service. For service of each type of customer, a certain number of additional items are needed. High priority customers do not have waiting space and so leave the system when on their arrival a priority 1 customer is in service or the number of available additional items is less than the required threshold. Preemptive priority is assumed. Type 2 customers, encountering a busy server or idle with the number of available additional items less than a threshold, go to an orbit of infinite capacity to retry for service. The customers in orbit are non-persistent: if on retrial the server is found to be busy/idle with the number of additional items less than the threshold, this customer abandons the system with certain probability. Such a system represents an accurate enough model of many real-world systems, including wireless sensor networks and system of cognitive radio with energy harvesting and healthcare systems. The probability distribution of the system states is computed, using which several of the characteristics are derived. A detailed numerical study of the system, including the analysis of the influence of the threshold, is performed.
References:
[1] |
A. Arafa, T. Tong, M. Fu, S. Ulukus and W. Chen, Delay minimal policies in energy harvesting communication systems, IEEE Transactions on Communications, 66 (2018), 2918-2930. Google Scholar |
[2] |
J. R. Artalejo and A. Gomez Corral, Retrial Queueing Systems: A Computational Approach, Springer, Berlin, 2008.
doi: 10.1007/978-3-540-78725-9. |
[3] |
J. Baek, O. Dudina and C. Kim,
Queueing system with heterogeneous impatient customers and consumable additional items, International Journal of Applied Mathematics and Computer Science, 27 (2017), 367-384.
doi: 10.1515/amcs-2017-0026. |
[4] |
L. Boutarfa and N. Djellab,
On the performance of the $M_1, M_2/G_1, G_2/1$ retrial queue with pre-emptive resume policy, Yugoslav Journal of Operations Research, 25 (2015), 153-164.
doi: 10.2298/YJOR130217001B. |
[5] |
B. Cardoen, E. Demeulemeester and J. Belien,
Operating room planning and scheduling: A literature review, European Journal of Operational Research, 201 (2010), 921-932.
doi: 10.1016/j.ejor.2012.11.029. |
[6] |
S. R. Chakravarthy, The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Process: Notable Publications (eds. A. Krishnamoorthy, et al.), New Jersey, 2001, 21–49. Google Scholar |
[7] |
B. D. Choi and Y. Chang,
Single server retrial queues with priority customers, Mathematical and Computer Modelling, 30 (1999), 7-32.
doi: 10.1016/S0895-7177(99)00129-6. |
[8] |
E. De Cuypere, K. De Turck and D. Fiems, A queueing model of an energy harvesting sensor node with data buffering, Telecommunication Systems, 67 (2018), 281-295. Google Scholar |
[9] |
I. Dimitrious, Analysis of a priority retrial queue with dependent vacation scheme and application to power saving in wireless communication systems, The Computer Journal, 56 (2015), 1363-1380. Google Scholar |
[10] |
A. N. Dudin and V. I. Klimenok,
Queueing systems with passive servers, International Journal of Stochastic Analysis, 9 (1996), 185-204.
doi: 10.1155/S1048953396000184. |
[11] |
A. N. Dudin, M. H. Lee, O. 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 Transactions on Communications, 65 (2017), 186-199. Google Scholar |
[12] |
G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997. Google Scholar |
[13] |
G. I. Falin, J. R. Artalejo and M. Martin,
On the single server retrial queue with priority customers, Queueing Systems, 14 (1993), 439-455.
doi: 10.1007/BF01158878. |
[14] |
S. Gao, A preemptive priority retrial queue with two classes of customers and general retrial times, Operational Research, 15 (2015), 233-251. Google Scholar |
[15] |
Q. M. He, Queues with marked customers, Advances in Applied Probability, (1996), 567–587.
doi: 10.2307/1428072. |
[16] |
C. Kim, S. Dudin and V. Klimenok, The $MAP/PH/1/N$ queue with flows of customers as model for traffic control in telecommunication networks, Performance Evaluation, 66 (2009), 564-579. Google Scholar |
[17] |
C. Kim, S. Dudin and V. Klimenok, Queueing system with batch arrival of customers in sessions, Computers and Industrial Engineering, 62 (2012), 890-897. Google Scholar |
[18] |
J. Kim and B. Kim,
A survey of retrial queueing systems, Annals of Operations Research, 247 (2016), 3-36.
doi: 10.1007/s10479-015-2038-7. |
[19] |
V. I. Klimenok and A. N. Dudin,
Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory, Queueing Systems, 54 (2006), 245-259.
doi: 10.1007/s11134-006-0300-z. |
[20] |
A. Krishnamoorthy, B. Binitha and D. Shajin, A revisit to queueing-inventory system with reservation, cancellation and common life time, OPSEARCH, 54 (2017), 336-350. Google Scholar |
[21] |
A. Krishnamoorthy, D. Shajin and B. Lakshmy, On a queueing-inventory with reservation, cancellation, common life time and retrial, Annals of Operations Research, (2016), 1–25.
doi: 10.1007/s10479-015-1849-x. |
[22] |
A. Krishnamoorthy, D. Shajin and B. Lakshmy,
Product form solution for some queueing-inventory supply chain problem, OPSEARCH, 53 (2016), 85-102.
doi: 10.1007/s12597-015-0215-8. |
[23] |
C. Langaris and E. Moutzoukis,
A retrial queue with structured batch arrivals, priorities and server vacations, Queueing Systems, 20 (1995), 341-368.
doi: 10.1007/BF01245324. |
[24] |
M. Liu, M. Feng and C. Y. Wong, Flexible service policies for a Markov inventory system with two demand classes, International Journal of Production Economics, 151 (2014), 180-185. Google Scholar |
[25] |
K. Patil, K. De Turck and D. Fiems,
A two-queue model for optimising the value of information in energy-harvesting sensor networks, Performance Evaluation, 119 (2018), 27-42.
doi: 10.1016/j.orl.2018.04.002. |
[26] |
T. Phung-Duc, Retrial queueing models: A survey on theory and applications, (to appear) in Stochastic Operations Research in Business and Industry (eds. T. Dohi, K. Ano and S. Kasahara), WSP, 2017. Google Scholar |
[27] |
J. Walraevens, D. Claeys and T. Phung-Duc, Asymptotics of queue length distributions in priority retrial queues, Performance Evaluation, 127 (2018), 235-252. Google Scholar |
[28] |
Y. Wang, X. Tang and T. Wang,
A unified QoS and security provisioning framework for wiretap cognitive radio networks: A statistical queueing analysis approach, IEEE Transactions on Wireless Communications, 18 (2019), 1548-1565.
doi: 10.1109/TWC.2019.2893381. |
[29] |
N. Zhao and Z. Lian, A queueing-inventory system with two classes of customers, International Journal of Production Economics, 129 (2011), 225-231. Google Scholar |
show all references
References:
[1] |
A. Arafa, T. Tong, M. Fu, S. Ulukus and W. Chen, Delay minimal policies in energy harvesting communication systems, IEEE Transactions on Communications, 66 (2018), 2918-2930. Google Scholar |
[2] |
J. R. Artalejo and A. Gomez Corral, Retrial Queueing Systems: A Computational Approach, Springer, Berlin, 2008.
doi: 10.1007/978-3-540-78725-9. |
[3] |
J. Baek, O. Dudina and C. Kim,
Queueing system with heterogeneous impatient customers and consumable additional items, International Journal of Applied Mathematics and Computer Science, 27 (2017), 367-384.
doi: 10.1515/amcs-2017-0026. |
[4] |
L. Boutarfa and N. Djellab,
On the performance of the $M_1, M_2/G_1, G_2/1$ retrial queue with pre-emptive resume policy, Yugoslav Journal of Operations Research, 25 (2015), 153-164.
doi: 10.2298/YJOR130217001B. |
[5] |
B. Cardoen, E. Demeulemeester and J. Belien,
Operating room planning and scheduling: A literature review, European Journal of Operational Research, 201 (2010), 921-932.
doi: 10.1016/j.ejor.2012.11.029. |
[6] |
S. R. Chakravarthy, The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Process: Notable Publications (eds. A. Krishnamoorthy, et al.), New Jersey, 2001, 21–49. Google Scholar |
[7] |
B. D. Choi and Y. Chang,
Single server retrial queues with priority customers, Mathematical and Computer Modelling, 30 (1999), 7-32.
doi: 10.1016/S0895-7177(99)00129-6. |
[8] |
E. De Cuypere, K. De Turck and D. Fiems, A queueing model of an energy harvesting sensor node with data buffering, Telecommunication Systems, 67 (2018), 281-295. Google Scholar |
[9] |
I. Dimitrious, Analysis of a priority retrial queue with dependent vacation scheme and application to power saving in wireless communication systems, The Computer Journal, 56 (2015), 1363-1380. Google Scholar |
[10] |
A. N. Dudin and V. I. Klimenok,
Queueing systems with passive servers, International Journal of Stochastic Analysis, 9 (1996), 185-204.
doi: 10.1155/S1048953396000184. |
[11] |
A. N. Dudin, M. H. Lee, O. 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 Transactions on Communications, 65 (2017), 186-199. Google Scholar |
[12] |
G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997. Google Scholar |
[13] |
G. I. Falin, J. R. Artalejo and M. Martin,
On the single server retrial queue with priority customers, Queueing Systems, 14 (1993), 439-455.
doi: 10.1007/BF01158878. |
[14] |
S. Gao, A preemptive priority retrial queue with two classes of customers and general retrial times, Operational Research, 15 (2015), 233-251. Google Scholar |
[15] |
Q. M. He, Queues with marked customers, Advances in Applied Probability, (1996), 567–587.
doi: 10.2307/1428072. |
[16] |
C. Kim, S. Dudin and V. Klimenok, The $MAP/PH/1/N$ queue with flows of customers as model for traffic control in telecommunication networks, Performance Evaluation, 66 (2009), 564-579. Google Scholar |
[17] |
C. Kim, S. Dudin and V. Klimenok, Queueing system with batch arrival of customers in sessions, Computers and Industrial Engineering, 62 (2012), 890-897. Google Scholar |
[18] |
J. Kim and B. Kim,
A survey of retrial queueing systems, Annals of Operations Research, 247 (2016), 3-36.
doi: 10.1007/s10479-015-2038-7. |
[19] |
V. I. Klimenok and A. N. Dudin,
Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory, Queueing Systems, 54 (2006), 245-259.
doi: 10.1007/s11134-006-0300-z. |
[20] |
A. Krishnamoorthy, B. Binitha and D. Shajin, A revisit to queueing-inventory system with reservation, cancellation and common life time, OPSEARCH, 54 (2017), 336-350. Google Scholar |
[21] |
A. Krishnamoorthy, D. Shajin and B. Lakshmy, On a queueing-inventory with reservation, cancellation, common life time and retrial, Annals of Operations Research, (2016), 1–25.
doi: 10.1007/s10479-015-1849-x. |
[22] |
A. Krishnamoorthy, D. Shajin and B. Lakshmy,
Product form solution for some queueing-inventory supply chain problem, OPSEARCH, 53 (2016), 85-102.
doi: 10.1007/s12597-015-0215-8. |
[23] |
C. Langaris and E. Moutzoukis,
A retrial queue with structured batch arrivals, priorities and server vacations, Queueing Systems, 20 (1995), 341-368.
doi: 10.1007/BF01245324. |
[24] |
M. Liu, M. Feng and C. Y. Wong, Flexible service policies for a Markov inventory system with two demand classes, International Journal of Production Economics, 151 (2014), 180-185. Google Scholar |
[25] |
K. Patil, K. De Turck and D. Fiems,
A two-queue model for optimising the value of information in energy-harvesting sensor networks, Performance Evaluation, 119 (2018), 27-42.
doi: 10.1016/j.orl.2018.04.002. |
[26] |
T. Phung-Duc, Retrial queueing models: A survey on theory and applications, (to appear) in Stochastic Operations Research in Business and Industry (eds. T. Dohi, K. Ano and S. Kasahara), WSP, 2017. Google Scholar |
[27] |
J. Walraevens, D. Claeys and T. Phung-Duc, Asymptotics of queue length distributions in priority retrial queues, Performance Evaluation, 127 (2018), 235-252. Google Scholar |
[28] |
Y. Wang, X. Tang and T. Wang,
A unified QoS and security provisioning framework for wiretap cognitive radio networks: A statistical queueing analysis approach, IEEE Transactions on Wireless Communications, 18 (2019), 1548-1565.
doi: 10.1109/TWC.2019.2893381. |
[29] |
N. Zhao and Z. Lian, A queueing-inventory system with two classes of customers, International Journal of Production Economics, 129 (2011), 225-231. Google Scholar |







2 | 0.541687 | 1.367441 | 1.684218 | 1.271213 |
4 | 0.549554 | 1.412635 | 1.703399 | 1.305689 |
6 | 0.5646816 | 1.518695 | 1.755786 | 1.403333 |
8 | 0.581552 | 1.588845 | 1.784279 | 1.466440 |
10 | 0.60139 | 1.626319 | 1.805897 | 1.509774 |
12 | 0.628908 | 1.656662 | 1.827878 | 1.551554 |
14 | 0.669292 | 1.682243 | 1.849624 | 1.590796 |
16 | 0.732318 | 1.709136 | 1.874669 | 1.631378 |
18 | 0.831686 | 1.742311 | 1.904638 | 1.674188 |
20 | 1.044837 | 1.802246 | 1.949755 | 1.728375 |
(A) Dependence of NO | ||||
2 | 15.265487 | 15.01827 | 12.685517 | 7.343982 |
4 | 15.360819 | 15.39104 | 13.30253 | 8.478718 |
6 | 15.489897 | 15.661518 | 13.504709 | 8.945458 |
8 | 15.707897 | 16.001991 | 13.742329 | 9.415827 |
10 | 16.008232 | 16.420983 | 13.98464 | 10.00213 |
12 | 16.376142 | 16.857811 | 14.170774 | 10.51655 |
14 | 16.821438 | 17.324321 | 14.475216 | 11.21957 |
16 | 17.34789 | 17.797588 | 14.7440311 | 11.92004 |
18 | 17.91796 | 18.255182 | 15.1044063 | 12.71894 |
20 | 18.51922 | 18.665273 | 15.5122786 | 13.58646 |
(B) Dependence of Nitem |
2 | 0.541687 | 1.367441 | 1.684218 | 1.271213 |
4 | 0.549554 | 1.412635 | 1.703399 | 1.305689 |
6 | 0.5646816 | 1.518695 | 1.755786 | 1.403333 |
8 | 0.581552 | 1.588845 | 1.784279 | 1.466440 |
10 | 0.60139 | 1.626319 | 1.805897 | 1.509774 |
12 | 0.628908 | 1.656662 | 1.827878 | 1.551554 |
14 | 0.669292 | 1.682243 | 1.849624 | 1.590796 |
16 | 0.732318 | 1.709136 | 1.874669 | 1.631378 |
18 | 0.831686 | 1.742311 | 1.904638 | 1.674188 |
20 | 1.044837 | 1.802246 | 1.949755 | 1.728375 |
(A) Dependence of NO | ||||
2 | 15.265487 | 15.01827 | 12.685517 | 7.343982 |
4 | 15.360819 | 15.39104 | 13.30253 | 8.478718 |
6 | 15.489897 | 15.661518 | 13.504709 | 8.945458 |
8 | 15.707897 | 16.001991 | 13.742329 | 9.415827 |
10 | 16.008232 | 16.420983 | 13.98464 | 10.00213 |
12 | 16.376142 | 16.857811 | 14.170774 | 10.51655 |
14 | 16.821438 | 17.324321 | 14.475216 | 11.21957 |
16 | 17.34789 | 17.797588 | 14.7440311 | 11.92004 |
18 | 17.91796 | 18.255182 | 15.1044063 | 12.71894 |
20 | 18.51922 | 18.665273 | 15.5122786 | 13.58646 |
(B) Dependence of Nitem |
2 | 0.075438 | 0.594164 | 0.621250 | 0.584869 |
4 | 0.069154 | 0.528047 | 0.612127 | 0.574752 |
6 | 0.053754 | 0.343713 | 0.526146 | 0.420849 |
8 | 0.044331 | 0.230872 | 0.488863 | 0.334311 |
10 | 0.041304 | 0.184474 | 0.466582 | 0.291888 |
12 | 0.039596 | 0.154385 | 0.442873 | 0.251298 |
14 | 0.039017 | 0.138246 | 0.426295 | 0.224901 |
16 | 0.038708 | 0.127744 | 0.407572 | 0.199159 |
18 | 0.038589 | 0.12157 | 0.392962 | 0.179236 |
20 | 0.038535 | 0.117919 | 0.37687 | 0.16082 |
(A) Dependence of ploss of p1loss | ||||
2 | 0.218984 | 0.596466 | 0.720285 | 0.570702 |
4 | 0.22019 | 0.599877 | 0.725534 | 0.58051 |
6 | 0.222009 | 0.60253 | 0.727949 | 0.584911 |
8 | 0.225624 | 0.60626 | 0.731018 | 0.590137 |
10 | 0.23144 | 0.611019 | 0.734444 | 0.596847 |
12 | 0.23998 | 0.616327 | 0.737665 | 0.6033 |
14 | 0.25284 | 0.62257 | 0.742163 | 0.61164 |
16 | 0.272944 | 0.630337 | 0.747311 | 0.620533 |
18 | 0.304933 | 0.641239 | 0.754921 | 0.631276 |
20 | 0.373814 | 0.662226 | 0.767272 | 0.645899 |
(B) Dependence of ploss of pitemloss |
2 | 0.075438 | 0.594164 | 0.621250 | 0.584869 |
4 | 0.069154 | 0.528047 | 0.612127 | 0.574752 |
6 | 0.053754 | 0.343713 | 0.526146 | 0.420849 |
8 | 0.044331 | 0.230872 | 0.488863 | 0.334311 |
10 | 0.041304 | 0.184474 | 0.466582 | 0.291888 |
12 | 0.039596 | 0.154385 | 0.442873 | 0.251298 |
14 | 0.039017 | 0.138246 | 0.426295 | 0.224901 |
16 | 0.038708 | 0.127744 | 0.407572 | 0.199159 |
18 | 0.038589 | 0.12157 | 0.392962 | 0.179236 |
20 | 0.038535 | 0.117919 | 0.37687 | 0.16082 |
(A) Dependence of ploss of p1loss | ||||
2 | 0.218984 | 0.596466 | 0.720285 | 0.570702 |
4 | 0.22019 | 0.599877 | 0.725534 | 0.58051 |
6 | 0.222009 | 0.60253 | 0.727949 | 0.584911 |
8 | 0.225624 | 0.60626 | 0.731018 | 0.590137 |
10 | 0.23144 | 0.611019 | 0.734444 | 0.596847 |
12 | 0.23998 | 0.616327 | 0.737665 | 0.6033 |
14 | 0.25284 | 0.62257 | 0.742163 | 0.61164 |
16 | 0.272944 | 0.630337 | 0.747311 | 0.620533 |
18 | 0.304933 | 0.641239 | 0.754921 | 0.631276 |
20 | 0.373814 | 0.662226 | 0.767272 | 0.645899 |
(B) Dependence of ploss of pitemloss |
2 | 0.036982 | 0.045292 | 0.042889 | 0.016605 |
4 | 0.037234 | 0.054023 | 0.043846 | 0.01701 |
6 | 0.03785 | 0.07846 | 0.054843 | 0.023166 |
8 | 0.038227 | 0.093433 | 0.059606 | 0.026628 |
10 | 0.038347 | 0.099601 | 0.062507 | 0.028324 |
12 | 0.038416 | 0.103602 | 0.06565 | 0.029948 |
14 | 0.038439 | 0.105748 | 0.06781 | 0.031004 |
16 | 0.038452 | 0.107144 | 0.07029 | 0.032034 |
18 | 0.038456 | 0.107965 | 0.07221 | 0.032831 |
20 | 0.038459 | 0.10845 | 0.07433 | 0.033567 |
(A) Dependence of p1busy loss | ||||
2 | 0.038456 | 0.548872 | 0.578361 | 0.568264 |
4 | 0.03192 | 0.474025 | 0.568281 | 0.557742 |
6 | 0.015904 | 0.265254 | 0.471303 | 0.397683 |
8 | 0.006105 | 0.13744 | 0.429257 | 0.307683 |
10 | 0.002956 | 0.084873 | 0.404075 | 0.263563 |
12 | 0.00118 | 0.050783 | 0.377223 | 0.22135 |
14 | 5.77E-4 | 0.032498 | 0.358485 | 0.193897 |
16 | 2.56E-4 | 0.020599 | 0.337281 | 0.167125 |
18 | 1.33E-4 | 0.013605 | 0.320755 | 0.146405 |
20 | 7.63E-5 | 0.009469 | 0.302539 | 0.127253 |
(B) Dependence of p1lack loss |
2 | 0.036982 | 0.045292 | 0.042889 | 0.016605 |
4 | 0.037234 | 0.054023 | 0.043846 | 0.01701 |
6 | 0.03785 | 0.07846 | 0.054843 | 0.023166 |
8 | 0.038227 | 0.093433 | 0.059606 | 0.026628 |
10 | 0.038347 | 0.099601 | 0.062507 | 0.028324 |
12 | 0.038416 | 0.103602 | 0.06565 | 0.029948 |
14 | 0.038439 | 0.105748 | 0.06781 | 0.031004 |
16 | 0.038452 | 0.107144 | 0.07029 | 0.032034 |
18 | 0.038456 | 0.107965 | 0.07221 | 0.032831 |
20 | 0.038459 | 0.10845 | 0.07433 | 0.033567 |
(A) Dependence of p1busy loss | ||||
2 | 0.038456 | 0.548872 | 0.578361 | 0.568264 |
4 | 0.03192 | 0.474025 | 0.568281 | 0.557742 |
6 | 0.015904 | 0.265254 | 0.471303 | 0.397683 |
8 | 0.006105 | 0.13744 | 0.429257 | 0.307683 |
10 | 0.002956 | 0.084873 | 0.404075 | 0.263563 |
12 | 0.00118 | 0.050783 | 0.377223 | 0.22135 |
14 | 5.77E-4 | 0.032498 | 0.358485 | 0.193897 |
16 | 2.56E-4 | 0.020599 | 0.337281 | 0.167125 |
18 | 1.33E-4 | 0.013605 | 0.320755 | 0.146405 |
20 | 7.63E-5 | 0.009469 | 0.302539 | 0.127253 |
(B) Dependence of p1lack loss |
0.1 | 0.758933 | 2.592361 | 3.261523 | 2.397430 |
0.2 | 0.549554 | 1.412635 | 1.703399 | 1.305689 |
0.3 | 0.441755 | 0.987633 | 1.162907 | 0.911481 |
0.4 | 0.372968 | 0.765297 | 0.886588 | 0.705013 |
0.5 | 0.324288 | 0.627350 | 0.718082 | 0.576937 |
0.6 | 0.287627 | 0.532879 | 0.604271 | 0.489305 |
0.7 | 0.258835 | 0.463871 | 0.522087 | 0.425367 |
0.8 | 0.235527 | 0.411117 | 0.459869 | 0.376551 |
0.9 | 0.216218 | 0.369402 | 0.411081 | 0.337998 |
1 | 0.199928 | 0.335543 | 0.371769 | 0.306743 |
(A) Dependence of NO | ||||
0.1 | 14.631235 | 15.054343 | 13.047518 | 8.294399 |
0.2 | 15.360819 | 15.391041 | 13.302530 | 8.478718 |
0.3 | 15.760671 | 15.548513 | 13.408673 | 8.583818 |
0.4 | 16.021155 | 15.649013 | 13.470799 | 8.656368 |
0.5 | 16.206952 | 15.721749 | 13.512953 | 8.710798 |
0.6 | 16.347252 | 15.778002 | 13.544014 | 8.753684 |
0.7 | 16.457467 | 15.823323 | 13.568133 | 8.788605 |
0.8 | 16.546609 | 15.860874 | 13.587549 | 8.817730 |
0.9 | 16.620347 | 15.892632 | 13.603596 | 8.842468 |
1 | 16.682445 | 15.919918 | 13.617128 | 8.863788 |
(B) Dependence of Nitem |
0.1 | 0.758933 | 2.592361 | 3.261523 | 2.397430 |
0.2 | 0.549554 | 1.412635 | 1.703399 | 1.305689 |
0.3 | 0.441755 | 0.987633 | 1.162907 | 0.911481 |
0.4 | 0.372968 | 0.765297 | 0.886588 | 0.705013 |
0.5 | 0.324288 | 0.627350 | 0.718082 | 0.576937 |
0.6 | 0.287627 | 0.532879 | 0.604271 | 0.489305 |
0.7 | 0.258835 | 0.463871 | 0.522087 | 0.425367 |
0.8 | 0.235527 | 0.411117 | 0.459869 | 0.376551 |
0.9 | 0.216218 | 0.369402 | 0.411081 | 0.337998 |
1 | 0.199928 | 0.335543 | 0.371769 | 0.306743 |
(A) Dependence of NO | ||||
0.1 | 14.631235 | 15.054343 | 13.047518 | 8.294399 |
0.2 | 15.360819 | 15.391041 | 13.302530 | 8.478718 |
0.3 | 15.760671 | 15.548513 | 13.408673 | 8.583818 |
0.4 | 16.021155 | 15.649013 | 13.470799 | 8.656368 |
0.5 | 16.206952 | 15.721749 | 13.512953 | 8.710798 |
0.6 | 16.347252 | 15.778002 | 13.544014 | 8.753684 |
0.7 | 16.457467 | 15.823323 | 13.568133 | 8.788605 |
0.8 | 16.546609 | 15.860874 | 13.587549 | 8.817730 |
0.9 | 16.620347 | 15.892632 | 13.603596 | 8.842468 |
1 | 16.682445 | 15.919918 | 13.617128 | 8.863788 |
(B) Dependence of Nitem |
0.1 | 0.086069 | 0.557286 | 0.619555 | 0.588222 |
0.2 | 0.069154 | 0.528047 | 0.612127 | 0.574752 |
0.3 | 0.061659 | 0.506329 | 0.607792 | 0.566495 |
0.4 | 0.057424 | 0.489551 | 0.604719 | 0.560590 |
0.5 | 0.054707 | 0.476222 | 0.602361 | 0.556081 |
0.6 | 0.052819 | 0.465381 | 0.600469 | 0.552495 |
0.7 | 0.051434 | 0.456390 | 0.598905 | 0.549560 |
0.8 | 0.050375 | 0.448809 | 0.597586 | 0.547105 |
0.9 | 0.049540 | 0.442329 | 0.596454 | 0.545017 |
1 | 0.048865 | 0.436726 | 0.595472 | 0.543215 |
(A) Dependence of p1loss | ||||
0.1 | 0.187902 | 0.582932 | 0.711872 | 0.560124 |
0.2 | 0.220190 | 0.599877 | 0.725534 | 0.580510 |
0.3 | 0.241291 | 0.606780 | 0.731699 | 0.590660 |
0.4 | 0.256630 | 0.610762 | 0.735528 | 0.597193 |
0.5 | 0.268448 | 0.613446 | 0.738236 | 0.601898 |
0.6 | 0.277902 | 0.615420 | 0.740290 | 0.605506 |
0.7 | 0.285673 | 0.616952 | 0.741918 | 0.608389 |
0.8 | 0.292192 | 0.618187 | 0.743249 | 0.610759 |
0.9 | 0.297748 | 0.619209 | 0.744361 | 0.612748 |
1 | 0.302546 | 0.620072 | 0.745308 | 0.614446 |
(B) Dependence of pitemloss |
0.1 | 0.086069 | 0.557286 | 0.619555 | 0.588222 |
0.2 | 0.069154 | 0.528047 | 0.612127 | 0.574752 |
0.3 | 0.061659 | 0.506329 | 0.607792 | 0.566495 |
0.4 | 0.057424 | 0.489551 | 0.604719 | 0.560590 |
0.5 | 0.054707 | 0.476222 | 0.602361 | 0.556081 |
0.6 | 0.052819 | 0.465381 | 0.600469 | 0.552495 |
0.7 | 0.051434 | 0.456390 | 0.598905 | 0.549560 |
0.8 | 0.050375 | 0.448809 | 0.597586 | 0.547105 |
0.9 | 0.049540 | 0.442329 | 0.596454 | 0.545017 |
1 | 0.048865 | 0.436726 | 0.595472 | 0.543215 |
(A) Dependence of p1loss | ||||
0.1 | 0.187902 | 0.582932 | 0.711872 | 0.560124 |
0.2 | 0.220190 | 0.599877 | 0.725534 | 0.580510 |
0.3 | 0.241291 | 0.606780 | 0.731699 | 0.590660 |
0.4 | 0.256630 | 0.610762 | 0.735528 | 0.597193 |
0.5 | 0.268448 | 0.613446 | 0.738236 | 0.601898 |
0.6 | 0.277902 | 0.615420 | 0.740290 | 0.605506 |
0.7 | 0.285673 | 0.616952 | 0.741918 | 0.608389 |
0.8 | 0.292192 | 0.618187 | 0.743249 | 0.610759 |
0.9 | 0.297748 | 0.619209 | 0.744361 | 0.612748 |
1 | 0.302546 | 0.620072 | 0.745308 | 0.614446 |
(B) Dependence of pitemloss |
0.1 | 0.036557 | 0.050253 | 0.043100 | 0.016471 |
0.2 | 0.037234 | 0.054023 | 0.043846 | 0.017010 |
0.3 | 0.037534 | 0.056868 | 0.044325 | 0.017340 |
0.4 | 0.037703 | 0.059078 | 0.044682 | 0.017576 |
0.5 | 0.037812 | 0.060838 | 0.044963 | 0.017757 |
0.6 | 0.037887 | 0.062272 | 0.045192 | 0.017900 |
0.7 | 0.037943 | 0.063462 | 0.045384 | 0.018018 |
0.8 | 0.037985 | 0.064466 | 0.045547 | 0.018116 |
0.9 | 0.038018 | 0.065324 | 0.045688 | 0.018199 |
1.0 | 0.038045 | 0.066067 | 0.045811 | 0.018271 |
(A) Dependence of p1busy loss | ||||
0.1 | 0.049511 | 0.507033 | 0.576455 | 0.571751 |
0.2 | 0.031920 | 0.474025 | 0.568281 | 0.557742 |
0.3 | 0.024125 | 0.449461 | 0.563466 | 0.549155 |
0.4 | 0.019721 | 0.430472 | 0.560037 | 0.543014 |
0.5 | 0.016895 | 0.415384 | 0.557398 | 0.538324 |
0.6 | 0.014932 | 0.403110 | 0.555277 | 0.534595 |
0.7 | 0.013491 | 0.392928 | 0.553521 | 0.531543 |
0.8 | 0.012390 | 0.384343 | 0.552038 | 0.528990 |
0.9 | 0.011521 | 0.377005 | 0.550766 | 0.526817 |
1.0 | 0.010820 | 0.370659 | 0.549661 | 0.524944 |
(B) Dependence of p1lack loss |
0.1 | 0.036557 | 0.050253 | 0.043100 | 0.016471 |
0.2 | 0.037234 | 0.054023 | 0.043846 | 0.017010 |
0.3 | 0.037534 | 0.056868 | 0.044325 | 0.017340 |
0.4 | 0.037703 | 0.059078 | 0.044682 | 0.017576 |
0.5 | 0.037812 | 0.060838 | 0.044963 | 0.017757 |
0.6 | 0.037887 | 0.062272 | 0.045192 | 0.017900 |
0.7 | 0.037943 | 0.063462 | 0.045384 | 0.018018 |
0.8 | 0.037985 | 0.064466 | 0.045547 | 0.018116 |
0.9 | 0.038018 | 0.065324 | 0.045688 | 0.018199 |
1.0 | 0.038045 | 0.066067 | 0.045811 | 0.018271 |
(A) Dependence of p1busy loss | ||||
0.1 | 0.049511 | 0.507033 | 0.576455 | 0.571751 |
0.2 | 0.031920 | 0.474025 | 0.568281 | 0.557742 |
0.3 | 0.024125 | 0.449461 | 0.563466 | 0.549155 |
0.4 | 0.019721 | 0.430472 | 0.560037 | 0.543014 |
0.5 | 0.016895 | 0.415384 | 0.557398 | 0.538324 |
0.6 | 0.014932 | 0.403110 | 0.555277 | 0.534595 |
0.7 | 0.013491 | 0.392928 | 0.553521 | 0.531543 |
0.8 | 0.012390 | 0.384343 | 0.552038 | 0.528990 |
0.9 | 0.011521 | 0.377005 | 0.550766 | 0.526817 |
1.0 | 0.010820 | 0.370659 | 0.549661 | 0.524944 |
(B) Dependence of p1lack loss |
0.1 | 1.5 | 0.959861 | 3.382764 | 4.300035 | 3.145578 |
0.2 | 0.75 | 1.328719 | 3.583343 | 4.430688 | 3.384015 |
0.3 | 0.5 | 1.628854 | 3.746986 | 4.534785 | 3.574155 |
0.4 | 0.375 | 1.881123 | 3.887829 | 4.623184 | 3.731486 |
0.5 | 0.3 | 2.098026 | 4.012896 | 4.700989 | 3.865063 |
0.6 | 0.25 | 2.287694 | 4.126279 | 4.771088 | 3.980639 |
0.7 | 0.2143 | 2.455731 | 4.230599 | 4.835292 | 4.082103 |
0.8 | 0.1875 | 2.606171 | 4.327642 | 4.89482 | 4.172209 |
0.9 | 0.1667 | 2.742017 | 4.418685 | 4.950534 | 4.252983 |
1 | 0.1500 | 2.865570 | 4.504674 | 5.003066 | 4.325962 |
0.1 | 1.5 | 0.959861 | 3.382764 | 4.300035 | 3.145578 |
0.2 | 0.75 | 1.328719 | 3.583343 | 4.430688 | 3.384015 |
0.3 | 0.5 | 1.628854 | 3.746986 | 4.534785 | 3.574155 |
0.4 | 0.375 | 1.881123 | 3.887829 | 4.623184 | 3.731486 |
0.5 | 0.3 | 2.098026 | 4.012896 | 4.700989 | 3.865063 |
0.6 | 0.25 | 2.287694 | 4.126279 | 4.771088 | 3.980639 |
0.7 | 0.2143 | 2.455731 | 4.230599 | 4.835292 | 4.082103 |
0.8 | 0.1875 | 2.606171 | 4.327642 | 4.89482 | 4.172209 |
0.9 | 0.1667 | 2.742017 | 4.418685 | 4.950534 | 4.252983 |
1 | 0.1500 | 2.865570 | 4.504674 | 5.003066 | 4.325962 |
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