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Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria
Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization
1. | Department of Mathematics, National Institute of Technology Raipur, Chhattisgarh-492010, India |
2. | Department of Business Administration, Sangji University, Wonju, Kangwon-26339, Republic of Korea |
This article studies an infinite buffer single server queueing system under $ N $-policy in which customers arrive according to a discrete-time batch Markovian arrival process. The service times of customers are independent and obey a common general discrete distribution. The idle server begins to serve the customers as soon as the queue size becomes at least $ N $ and serves the customers until the system becomes empty. We determine the queue length distribution at post-departure epoch with the help of roots of the associated characteristic equation of the vector probability generating function. Using the supplementary variable technique, we develop the system of vector difference equations to derive the queue length distribution at random epoch. An analytically simple and computationally efficient approach is also presented to compute the waiting time distribution in the queue of a randomly selected customer of an arrival batch. We also construct an expected linear cost function to determine the optimal value of $ N $ at minimum cost. Some numerical results are demonstrated for different service time distributions through the optimal control parameter to show the key performance measures.
References:
[1] |
H. K. Aksoy and S. M. Gupta,
Near optimal buffer allocation in remanufacturing systems with $N$-policy, Computers & Industrial Engineering, 59 (2010), 496-508.
doi: 10.1016/j.cie.2010.06.004. |
[2] |
A. S. Alfa, Applied Discrete-Time Queues, 2$^nd$ edition, Springer, New York, 2016.
doi: 10.1007/978-1-4939-3420-1. |
[3] |
C. Blondia, A discrete-time batch Markovian arrival process as B-ISDN traffic model, Belgian Journal of Operations Research, Statistics and Computer Science, 32 (1993), 3-23. Google Scholar |
[4] |
J. A. Buzacott and J. G. Shanthikumar, Stochastic models for production control, In Optimization Models and Concepts in Production Management, Gordon and Breach Publishers, 1995,213–255. Google Scholar |
[5] |
S. R. Chakravarthy, The batch Markovian arrival process: a review and future work, In Advances in Probability Theory and Stochastic Processes, Notable Publications, Inc., New Jersey, USA, 2001, 21–49. Google Scholar |
[6] |
K. H. Choi and B. K. Yoon,
A roots method in $GI/PH/1$ queueing model and its application, Industrial Engineering & Management Systems, 12 (2013), 281-287.
doi: 10.7232/IEMS.2013.12.3.281. |
[7] |
G. L. Curry and R. M. Feldman, Manufacturing Systems Modeling and Analysis, Springer, Berlin, Heidelberg, 2009. Google Scholar |
[8] |
H. R. Gail, S. L. Hantler and B. A. Taylor,
Spectral analysis of $M/G/1$ and $G/M/1$ type Markov chains, Advances in Applied Probability, 28 (1996), 114-165.
doi: 10.2307/1427915. |
[9] |
T. Hofkens, K. Spaey and C. Blondia, Transient analysis of the $D$-$BMAP/G/1$ queue with an application to the dimensioning of a playout buffer for VBR video, in Networking 2004, Springer, Berlin, Heidelberg, 2004, 1338–1343.
doi: 10.1007/978-3-540-24693-0_116. |
[10] |
A. J. E. M. Janssen and J. S. H. van Leeuwaarden,
Analytic computation schemes for the discrete-time bulk service queue, Queueing Systems, 50 (2005), 141-163.
doi: 10.1007/s11134-005-0402-z. |
[11] |
F. C. Jiang, D. C. Huang, C. T. Yang and F. Y. Leu,
Lifetime elongation for wireless sensor network using queue-based approaches, The Journal of Supercomputing, 59 (2012), 1312-1335.
doi: 10.1007/s11227-010-0537-5. |
[12] |
S. Kasahara, T. Takine, Y. Takahashi and T. Hasegawa,
$MAP/G/1$ queues under $N$-policy with and without vacations, Journal of the Operations Research Society of Japan, 39 (1996), 188-212.
doi: 10.15807/jorsj.39.188. |
[13] |
A. Kavusturucu and S. M. Gupta,
Expansion method for the throughput analysis of open finite manufacturing/queueing networks with $N$-Policy, Computers & Operations Research, 26 (1999), 1267-1292.
doi: 10.1016/S0305-0548(98)00107-5. |
[14] |
R. G. V. Krishna Reddy, R. Nadarajan and R. Arumuganathan,
Analysis of a bulk queue with $N$-policy multiple vacations and setup times, Computers & Operations Research, 25 (1998), 957-967.
doi: 10.1016/S0305-0548(97)00098-1. |
[15] |
A. Krishnamoorthy and T. G. Deepak,
Modified $N$-policy for $M/G/1$ queues, Computers & Operations Research, 29 (2002), 1611-1620.
doi: 10.1016/S0305-0548(00)00108-8. |
[16] |
S. Lan and Y. Tang,
An $N$-policy discrete-time $Geo/G/1$ queue with modified multiple server vacations and Bernoulli feedback, RAIRO-Operations Research, 53 (2019), 367-387.
doi: 10.1051/ro/2017027. |
[17] |
H. W. Lee, B. Y. Ahn and N. I. Park,
Decompositions of the queue length distributions in the $MAP/G/1$ queue under multiple and single vacations with $N$-Policy, Stochastic Models, 17 (2001), 157-190.
doi: 10.1081/STM-100002062. |
[18] |
H. W. Lee, S. W. Lee and J. Jongwoo,
Using factorization in analyzing $D$-$BMAP/G/1$ queues, Journal of Applied Mathematics and Stochastic Analysis, 2005 (2005), 119-132.
doi: 10.1155/JAMSA.2005.119. |
[19] |
S. S. Lee, H. W. Lee, S. H. Yoon and K. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189. Google Scholar |
[20] |
H. W. Lee and W. J. Seo, The performance of the $M/G/1$ queue under the dyadic Min($N$, $D$)-policy and its cost optimization, Performance Evaluation, 65 (2008), 742-758. Google Scholar |
[21] |
D. H. Lee and W. S. Yang,
The $N$-policy of a discrete time $Geo/G/1$ queue with disasters and its application to wireless sensor networks, Applied Mathematical Modelling, 37 (2013), 9722-9731.
doi: 10.1016/j.apm.2013.05.012. |
[22] |
D. M. Lucantoni,
New results on the single server queue with a batch Markovian arrival process, Communications in Statistics Stochastic Models, 7 (1991), 1-46.
doi: 10.1080/15326349108807174. |
[23] |
M. F. Neuts,
A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.
doi: 10.2307/3213143. |
[24] |
M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach,, John Hopkins University Press, Baltimore, MD, 1981.
![]() |
[25] |
A. Oblakova, A. Al Hanbali, R. J. Boucherie, J. C. W. V. Ommeren and W. H. M. Zijm,
An exact root-free method for the expected queue length for a class of discrete-time queueing systems, Queueing Systems, 92 (2019), 257-292.
doi: 10.1007/s11134-019-09614-1. |
[26] |
S. K. Samanta,
Waiting-time analysis of $D$-$BMAP/G/1$ queueing system, Annals of Operations Research, 284 (2020), 401-413.
doi: 10.1007/s10479-015-1974-6. |
[27] |
C. Sreenivasan, S. R. Chakravarthy and A. Krishnamoorthy,
$MAP/PH/1$ queue with working vacations, vacation interruptions and $N$ policy, Applied Mathematical Modelling, 37 (2013), 3879-3893.
doi: 10.1016/j.apm.2012.07.054. |
[28] |
K. D. Turck, S. D. Vuyst, D. Fiems, H. Bruneel and S. Wittevrongel, Efficient performance analysis of newly proposed sleep-mode mechanisms for IEEE 802.16m in case of correlated downlink traffic, Wireless Networks, 19 (2013), 831-842. Google Scholar |
[29] |
T. Y. Wang, T. H. Liu and F. M. Chang, Analysis of a random $N$-policy $Geo/G/1$ queue with the server subject to repairable breakdowns, Journal of Industrial and Production Engineering, 34 (2017), 19-29. Google Scholar |
[30] |
M. Yadin and P. Naor, Queueing systems with a removable service station, Operational Research Quarterly, 14 (1963), 393-405. Google Scholar |
[31] |
Y. Q. Zhao and L. L. Campbell,
Equilibrium probability calculations for a discrete-time bulk queue model, Queueing Systems, 22 (1996), 189-198.
doi: 10.1007/BF01159401. |
[32] |
J. A. Zhao, B. Li, C. W. Kok and I. Ahmad, MPEG-4 video transmission over wireless networks: A link level performance study, Wireless Networks, 10 (2004), 133-146. Google Scholar |
show all references
References:
[1] |
H. K. Aksoy and S. M. Gupta,
Near optimal buffer allocation in remanufacturing systems with $N$-policy, Computers & Industrial Engineering, 59 (2010), 496-508.
doi: 10.1016/j.cie.2010.06.004. |
[2] |
A. S. Alfa, Applied Discrete-Time Queues, 2$^nd$ edition, Springer, New York, 2016.
doi: 10.1007/978-1-4939-3420-1. |
[3] |
C. Blondia, A discrete-time batch Markovian arrival process as B-ISDN traffic model, Belgian Journal of Operations Research, Statistics and Computer Science, 32 (1993), 3-23. Google Scholar |
[4] |
J. A. Buzacott and J. G. Shanthikumar, Stochastic models for production control, In Optimization Models and Concepts in Production Management, Gordon and Breach Publishers, 1995,213–255. Google Scholar |
[5] |
S. R. Chakravarthy, The batch Markovian arrival process: a review and future work, In Advances in Probability Theory and Stochastic Processes, Notable Publications, Inc., New Jersey, USA, 2001, 21–49. Google Scholar |
[6] |
K. H. Choi and B. K. Yoon,
A roots method in $GI/PH/1$ queueing model and its application, Industrial Engineering & Management Systems, 12 (2013), 281-287.
doi: 10.7232/IEMS.2013.12.3.281. |
[7] |
G. L. Curry and R. M. Feldman, Manufacturing Systems Modeling and Analysis, Springer, Berlin, Heidelberg, 2009. Google Scholar |
[8] |
H. R. Gail, S. L. Hantler and B. A. Taylor,
Spectral analysis of $M/G/1$ and $G/M/1$ type Markov chains, Advances in Applied Probability, 28 (1996), 114-165.
doi: 10.2307/1427915. |
[9] |
T. Hofkens, K. Spaey and C. Blondia, Transient analysis of the $D$-$BMAP/G/1$ queue with an application to the dimensioning of a playout buffer for VBR video, in Networking 2004, Springer, Berlin, Heidelberg, 2004, 1338–1343.
doi: 10.1007/978-3-540-24693-0_116. |
[10] |
A. J. E. M. Janssen and J. S. H. van Leeuwaarden,
Analytic computation schemes for the discrete-time bulk service queue, Queueing Systems, 50 (2005), 141-163.
doi: 10.1007/s11134-005-0402-z. |
[11] |
F. C. Jiang, D. C. Huang, C. T. Yang and F. Y. Leu,
Lifetime elongation for wireless sensor network using queue-based approaches, The Journal of Supercomputing, 59 (2012), 1312-1335.
doi: 10.1007/s11227-010-0537-5. |
[12] |
S. Kasahara, T. Takine, Y. Takahashi and T. Hasegawa,
$MAP/G/1$ queues under $N$-policy with and without vacations, Journal of the Operations Research Society of Japan, 39 (1996), 188-212.
doi: 10.15807/jorsj.39.188. |
[13] |
A. Kavusturucu and S. M. Gupta,
Expansion method for the throughput analysis of open finite manufacturing/queueing networks with $N$-Policy, Computers & Operations Research, 26 (1999), 1267-1292.
doi: 10.1016/S0305-0548(98)00107-5. |
[14] |
R. G. V. Krishna Reddy, R. Nadarajan and R. Arumuganathan,
Analysis of a bulk queue with $N$-policy multiple vacations and setup times, Computers & Operations Research, 25 (1998), 957-967.
doi: 10.1016/S0305-0548(97)00098-1. |
[15] |
A. Krishnamoorthy and T. G. Deepak,
Modified $N$-policy for $M/G/1$ queues, Computers & Operations Research, 29 (2002), 1611-1620.
doi: 10.1016/S0305-0548(00)00108-8. |
[16] |
S. Lan and Y. Tang,
An $N$-policy discrete-time $Geo/G/1$ queue with modified multiple server vacations and Bernoulli feedback, RAIRO-Operations Research, 53 (2019), 367-387.
doi: 10.1051/ro/2017027. |
[17] |
H. W. Lee, B. Y. Ahn and N. I. Park,
Decompositions of the queue length distributions in the $MAP/G/1$ queue under multiple and single vacations with $N$-Policy, Stochastic Models, 17 (2001), 157-190.
doi: 10.1081/STM-100002062. |
[18] |
H. W. Lee, S. W. Lee and J. Jongwoo,
Using factorization in analyzing $D$-$BMAP/G/1$ queues, Journal of Applied Mathematics and Stochastic Analysis, 2005 (2005), 119-132.
doi: 10.1155/JAMSA.2005.119. |
[19] |
S. S. Lee, H. W. Lee, S. H. Yoon and K. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189. Google Scholar |
[20] |
H. W. Lee and W. J. Seo, The performance of the $M/G/1$ queue under the dyadic Min($N$, $D$)-policy and its cost optimization, Performance Evaluation, 65 (2008), 742-758. Google Scholar |
[21] |
D. H. Lee and W. S. Yang,
The $N$-policy of a discrete time $Geo/G/1$ queue with disasters and its application to wireless sensor networks, Applied Mathematical Modelling, 37 (2013), 9722-9731.
doi: 10.1016/j.apm.2013.05.012. |
[22] |
D. M. Lucantoni,
New results on the single server queue with a batch Markovian arrival process, Communications in Statistics Stochastic Models, 7 (1991), 1-46.
doi: 10.1080/15326349108807174. |
[23] |
M. F. Neuts,
A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.
doi: 10.2307/3213143. |
[24] |
M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach,, John Hopkins University Press, Baltimore, MD, 1981.
![]() |
[25] |
A. Oblakova, A. Al Hanbali, R. J. Boucherie, J. C. W. V. Ommeren and W. H. M. Zijm,
An exact root-free method for the expected queue length for a class of discrete-time queueing systems, Queueing Systems, 92 (2019), 257-292.
doi: 10.1007/s11134-019-09614-1. |
[26] |
S. K. Samanta,
Waiting-time analysis of $D$-$BMAP/G/1$ queueing system, Annals of Operations Research, 284 (2020), 401-413.
doi: 10.1007/s10479-015-1974-6. |
[27] |
C. Sreenivasan, S. R. Chakravarthy and A. Krishnamoorthy,
$MAP/PH/1$ queue with working vacations, vacation interruptions and $N$ policy, Applied Mathematical Modelling, 37 (2013), 3879-3893.
doi: 10.1016/j.apm.2012.07.054. |
[28] |
K. D. Turck, S. D. Vuyst, D. Fiems, H. Bruneel and S. Wittevrongel, Efficient performance analysis of newly proposed sleep-mode mechanisms for IEEE 802.16m in case of correlated downlink traffic, Wireless Networks, 19 (2013), 831-842. Google Scholar |
[29] |
T. Y. Wang, T. H. Liu and F. M. Chang, Analysis of a random $N$-policy $Geo/G/1$ queue with the server subject to repairable breakdowns, Journal of Industrial and Production Engineering, 34 (2017), 19-29. Google Scholar |
[30] |
M. Yadin and P. Naor, Queueing systems with a removable service station, Operational Research Quarterly, 14 (1963), 393-405. Google Scholar |
[31] |
Y. Q. Zhao and L. L. Campbell,
Equilibrium probability calculations for a discrete-time bulk queue model, Queueing Systems, 22 (1996), 189-198.
doi: 10.1007/BF01159401. |
[32] |
J. A. Zhao, B. Li, C. W. Kok and I. Ahmad, MPEG-4 video transmission over wireless networks: A link level performance study, Wireless Networks, 10 (2004), 133-146. Google Scholar |






0 | 0.00114837 | 0.00255731 | 0.00122101 | 0.00057458 | 0.00030733 | 0.00580861 |
1 | 0.00196732 | 0.00413389 | 0.00211277 | 0.00132765 | 0.00067006 | 0.01021170 |
2 | 0.00268813 | 0.00542042 | 0.00287623 | 0.00214745 | 0.00100214 | 0.01413437 |
3 | 0.00336532 | 0.00676685 | 0.00362099 | 0.00302599 | 0.00135476 | 0.01813391 |
4 | 0.00407581 | 0.00806720 | 0.00442181 | 0.00388853 | 0.00172228 | 0.02217563 |
5 | 0.00480560 | 0.00931208 | 0.00519387 | 0.00478410 | 0.00209565 | 0.02619131 |
6 | 0.00545231 | 0.01038279 | 0.00590375 | 0.00577743 | 0.00247998 | 0.02999627 |
7 | 0.00550115 | 0.01037107 | 0.00599819 | 0.00616987 | 0.00258839 | 0.03062867 |
10 | 0.00518961 | 0.00969826 | 0.00573383 | 0.00601163 | 0.00253004 | 0.02916336 |
20 | 0.00355502 | 0.00663985 | 0.00392615 | 0.00413473 | 0.00174204 | 0.01999780 |
50 | 0.00113199 | 0.00211426 | 0.00125016 | 0.00131658 | 0.00055470 | 0.00636770 |
100 | 0.00016808 | 0.00031392 | 0.00018562 | 0.00019549 | 0.00008236 | 0.00094547 |
200 | 0.00000371 | 0.00000692 | 0.00000409 | 0.00000431 | 0.00000182 | 0.00002084 |
300 | 0.00000008 | 0.00000015 | 0.00000009 | 0.00000010 | 0.00000004 | 0.00000046 |
500 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.17884285 | 0.33699076 | 0.19680812 | 0.20180982 | 0.08554844 | 1.00000000 |
0 | 0.00114837 | 0.00255731 | 0.00122101 | 0.00057458 | 0.00030733 | 0.00580861 |
1 | 0.00196732 | 0.00413389 | 0.00211277 | 0.00132765 | 0.00067006 | 0.01021170 |
2 | 0.00268813 | 0.00542042 | 0.00287623 | 0.00214745 | 0.00100214 | 0.01413437 |
3 | 0.00336532 | 0.00676685 | 0.00362099 | 0.00302599 | 0.00135476 | 0.01813391 |
4 | 0.00407581 | 0.00806720 | 0.00442181 | 0.00388853 | 0.00172228 | 0.02217563 |
5 | 0.00480560 | 0.00931208 | 0.00519387 | 0.00478410 | 0.00209565 | 0.02619131 |
6 | 0.00545231 | 0.01038279 | 0.00590375 | 0.00577743 | 0.00247998 | 0.02999627 |
7 | 0.00550115 | 0.01037107 | 0.00599819 | 0.00616987 | 0.00258839 | 0.03062867 |
10 | 0.00518961 | 0.00969826 | 0.00573383 | 0.00601163 | 0.00253004 | 0.02916336 |
20 | 0.00355502 | 0.00663985 | 0.00392615 | 0.00413473 | 0.00174204 | 0.01999780 |
50 | 0.00113199 | 0.00211426 | 0.00125016 | 0.00131658 | 0.00055470 | 0.00636770 |
100 | 0.00016808 | 0.00031392 | 0.00018562 | 0.00019549 | 0.00008236 | 0.00094547 |
200 | 0.00000371 | 0.00000692 | 0.00000409 | 0.00000431 | 0.00000182 | 0.00002084 |
300 | 0.00000008 | 0.00000015 | 0.00000009 | 0.00000010 | 0.00000004 | 0.00000046 |
500 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.17884285 | 0.33699076 | 0.19680812 | 0.20180982 | 0.08554844 | 1.00000000 |
0 | 0.00222695 | 0.00529393 | 0.00242011 | 0.00090775 | 0.00048977 | 0.01133850 |
1 | 0.00122073 | 0.00248229 | 0.00141283 | 0.00079891 | 0.00049111 | 0.00640588 |
2 | 0.00105468 | 0.00179357 | 0.00112890 | 0.00103029 | 0.00047010 | 0.00547755 |
3 | 0.00089757 | 0.00190533 | 0.00096004 | 0.00118332 | 0.00048134 | 0.00542760 |
4 | 0.00096295 | 0.00185514 | 0.00111938 | 0.00116282 | 0.00051063 | 0.00561091 |
5 | 0.00111788 | 0.00198211 | 0.00120152 | 0.00116430 | 0.00051719 | 0.00598300 |
6 | 0.00106199 | 0.00200132 | 0.00116720 | 0.00115114 | 0.00049567 | 0.00587732 |
0 | 0.00162452 | 0.00345778 | 0.00174092 | 0.00103506 | 0.00052768 | 0.00838598 |
1 | 0.00234181 | 0.00477333 | 0.00250811 | 0.00179593 | 0.00085359 | 0.01227277 |
2 | 0.00300104 | 0.00603923 | 0.00322414 | 0.00261562 | 0.00118356 | 0.01606358 |
3 | 0.00366851 | 0.00729361 | 0.00397075 | 0.00344304 | 0.00152948 | 0.01990539 |
4 | 0.00435868 | 0.00849806 | 0.00471593 | 0.00428708 | 0.00188379 | 0.02374353 |
5 | 0.00500101 | 0.00957287 | 0.00541216 | 0.00520452 | 0.00224699 | 0.02743755 |
10 | 0.00486647 | 0.00905994 | 0.00535865 | 0.00563201 | 0.00238020 | 0.02729727 |
20 | 0.00330525 | 0.00617334 | 0.00365030 | 0.00384423 | 0.00161965 | 0.01859277 |
50 | 0.00105246 | 0.00196571 | 0.00116233 | 0.00122408 | 0.00051573 | 0.00592030 |
100 | 0.00015627 | 0.00029187 | 0.00017258 | 0.00018175 | 0.00007658 | 0.00087904 |
200 | 0.00000345 | 0.00000643 | 0.00000380 | 0.00000401 | 0.00000169 | 0.00001938 |
300 | 0.00000008 | 0.00000014 | 0.00000008 | 0.00000009 | 0.00000003 | 0.00000042 |
500 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.17895473 | 0.33796971 | 0.19708476 | 0.20065459 | 0.08533621 | 1.00000000 |
0 | 0.00222695 | 0.00529393 | 0.00242011 | 0.00090775 | 0.00048977 | 0.01133850 |
1 | 0.00122073 | 0.00248229 | 0.00141283 | 0.00079891 | 0.00049111 | 0.00640588 |
2 | 0.00105468 | 0.00179357 | 0.00112890 | 0.00103029 | 0.00047010 | 0.00547755 |
3 | 0.00089757 | 0.00190533 | 0.00096004 | 0.00118332 | 0.00048134 | 0.00542760 |
4 | 0.00096295 | 0.00185514 | 0.00111938 | 0.00116282 | 0.00051063 | 0.00561091 |
5 | 0.00111788 | 0.00198211 | 0.00120152 | 0.00116430 | 0.00051719 | 0.00598300 |
6 | 0.00106199 | 0.00200132 | 0.00116720 | 0.00115114 | 0.00049567 | 0.00587732 |
0 | 0.00162452 | 0.00345778 | 0.00174092 | 0.00103506 | 0.00052768 | 0.00838598 |
1 | 0.00234181 | 0.00477333 | 0.00250811 | 0.00179593 | 0.00085359 | 0.01227277 |
2 | 0.00300104 | 0.00603923 | 0.00322414 | 0.00261562 | 0.00118356 | 0.01606358 |
3 | 0.00366851 | 0.00729361 | 0.00397075 | 0.00344304 | 0.00152948 | 0.01990539 |
4 | 0.00435868 | 0.00849806 | 0.00471593 | 0.00428708 | 0.00188379 | 0.02374353 |
5 | 0.00500101 | 0.00957287 | 0.00541216 | 0.00520452 | 0.00224699 | 0.02743755 |
10 | 0.00486647 | 0.00905994 | 0.00535865 | 0.00563201 | 0.00238020 | 0.02729727 |
20 | 0.00330525 | 0.00617334 | 0.00365030 | 0.00384423 | 0.00161965 | 0.01859277 |
50 | 0.00105246 | 0.00196571 | 0.00116233 | 0.00122408 | 0.00051573 | 0.00592030 |
100 | 0.00015627 | 0.00029187 | 0.00017258 | 0.00018175 | 0.00007658 | 0.00087904 |
200 | 0.00000345 | 0.00000643 | 0.00000380 | 0.00000401 | 0.00000169 | 0.00001938 |
300 | 0.00000008 | 0.00000014 | 0.00000008 | 0.00000009 | 0.00000003 | 0.00000042 |
500 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.17895473 | 0.33796971 | 0.19708476 | 0.20065459 | 0.08533621 | 1.00000000 |
0 | 0.00174343 | 0.00277470 | 0.00221558 | 0.00262957 | 0.00132854 | 0.01069181 |
1 | 0.00097452 | 0.00160317 | 0.00122966 | 0.00163056 | 0.00074979 | 0.00618770 |
2 | 0.00083109 | 0.00141140 | 0.00103647 | 0.00156583 | 0.00062849 | 0.00547327 |
3 | 0.00547327 | 0.00133665 | 0.00103247 | 0.00165829 | 0.00060400 | 0.00546386 |
4 | 0.00085171 | 0.00139904 | 0.00105950 | 0.00167026 | 0.00063105 | 0.00561157 |
5 | 0.00090939 | 0.00152759 | 0.00113284 | 0.00173509 | 0.00068207 | 0.00598698 |
6 | 0.00089418 | 0.00148113 | 0.00111396 | 0.00171177 | 0.00066572 | 0.00586676 |
0 | 0.00128465 | 0.00209323 | 0.00162002 | 0.00213136 | 0.00097544 | 0.00810471 |
1 | 0.00187638 | 0.00307986 | 0.00235698 | 0.00326222 | 0.00141681 | 0.01199226 |
2 | 0.00245421 | 0.00402943 | 0.00307416 | 0.00440903 | 0.00184105 | 0.01580788 |
3 | 0.00303820 | 0.00499321 | 0.00379932 | 0.00556828 | 0.00227196 | 0.01967097 |
4 | 0.00362127 | 0.00597046 | 0.00452247 | 0.00673302 | 0.00270420 | 0.02355141 |
5 | 0.00418155 | 0.00691160 | 0.00521393 | 0.00790838 | 0.00311484 | 0.02733030 |
10 | 0.00415110 | 0.00687971 | 0.00516200 | 0.00810034 | 0.00307831 | 0.02737146 |
20 | 0.00282716 | 0.00468171 | 0.00351552 | 0.00552180 | 0.00209539 | 0.01864157 |
50 | 0.00090022 | 0.00149075 | 0.00111941 | 0.00175825 | 0.00066721 | 0.00593585 |
100 | 0.00013366 | 0.00022135 | 0.00016621 | 0.00026106 | 0.00009907 | 0.00088135 |
200 | 0.00000295 | 0.00000488 | 0.00000366 | 0.00000576 | 0.00000218 | 0.00001943 |
300 | 0.00000006 | 0.00000010 | 0.00000008 | 0.00000012 | 0.00000005 | 0.00000042 |
500 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.15214569 | 0.25164805 | 0.18938631 | 0.29385469 | 0.11296526 | 1.00000000 |
0 | 0.00174343 | 0.00277470 | 0.00221558 | 0.00262957 | 0.00132854 | 0.01069181 |
1 | 0.00097452 | 0.00160317 | 0.00122966 | 0.00163056 | 0.00074979 | 0.00618770 |
2 | 0.00083109 | 0.00141140 | 0.00103647 | 0.00156583 | 0.00062849 | 0.00547327 |
3 | 0.00547327 | 0.00133665 | 0.00103247 | 0.00165829 | 0.00060400 | 0.00546386 |
4 | 0.00085171 | 0.00139904 | 0.00105950 | 0.00167026 | 0.00063105 | 0.00561157 |
5 | 0.00090939 | 0.00152759 | 0.00113284 | 0.00173509 | 0.00068207 | 0.00598698 |
6 | 0.00089418 | 0.00148113 | 0.00111396 | 0.00171177 | 0.00066572 | 0.00586676 |
0 | 0.00128465 | 0.00209323 | 0.00162002 | 0.00213136 | 0.00097544 | 0.00810471 |
1 | 0.00187638 | 0.00307986 | 0.00235698 | 0.00326222 | 0.00141681 | 0.01199226 |
2 | 0.00245421 | 0.00402943 | 0.00307416 | 0.00440903 | 0.00184105 | 0.01580788 |
3 | 0.00303820 | 0.00499321 | 0.00379932 | 0.00556828 | 0.00227196 | 0.01967097 |
4 | 0.00362127 | 0.00597046 | 0.00452247 | 0.00673302 | 0.00270420 | 0.02355141 |
5 | 0.00418155 | 0.00691160 | 0.00521393 | 0.00790838 | 0.00311484 | 0.02733030 |
10 | 0.00415110 | 0.00687971 | 0.00516200 | 0.00810034 | 0.00307831 | 0.02737146 |
20 | 0.00282716 | 0.00468171 | 0.00351552 | 0.00552180 | 0.00209539 | 0.01864157 |
50 | 0.00090022 | 0.00149075 | 0.00111941 | 0.00175825 | 0.00066721 | 0.00593585 |
100 | 0.00013366 | 0.00022135 | 0.00016621 | 0.00026106 | 0.00009907 | 0.00088135 |
200 | 0.00000295 | 0.00000488 | 0.00000366 | 0.00000576 | 0.00000218 | 0.00001943 |
300 | 0.00000006 | 0.00000010 | 0.00000008 | 0.00000012 | 0.00000005 | 0.00000042 |
500 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.15214569 | 0.25164805 | 0.18938631 | 0.29385469 | 0.11296526 | 1.00000000 |
0 | 0.00148277 | 0.00363672 | 0.00162886 | 0.00053541 | 0.00029061 | 0.00757437 |
1 | 0.00122073 | 0.00248229 | 0.00141283 | 0.00079891 | 0.00049111 | 0.00640588 |
2 | 0.00105468 | 0.00179357 | 0.00112890 | 0.00103029 | 0.00047010 | 0.00547755 |
3 | 0.00089757 | 0.00190533 | 0.00096004 | 0.00118332 | 0.00048134 | 0.00542760 |
4 | 0.00096295 | 0.00185514 | 0.00111938 | 0.00116282 | 0.00051063 | 0.00561091 |
5 | 0.00111788 | 0.00198211 | 0.00120152 | 0.00116430 | 0.00051719 | 0.00598300 |
6 | 0.00106199 | 0.00200132 | 0.00116720 | 0.00115114 | 0.00049567 | 0.00587732 |
0 | 0.00109382 | 0.00243612 | 0.00116304 | 0.00054705 | 0.00029262 | 0.00553266 |
1 | 0.00187471 | 0.00393963 | 0.00201337 | 0.00126468 | 0.00063839 | 0.00973078 |
2 | 0.00256220 | 0.00516670 | 0.00274151 | 0.00204630 | 0.00095506 | 0.01347178 |
3 | 0.00320810 | 0.00645095 | 0.00345180 | 0.00288409 | 0.00129132 | 0.01728624 |
4 | 0.00388575 | 0.00769135 | 0.00421561 | 0.00370673 | 0.00164183 | 0.02114127 |
5 | 0.00458192 | 0.00887902 | 0.00495214 | 0.00456082 | 0.00199793 | 0.02497183 |
10 | 0.00498603 | 0.00928331 | 0.00549189 | 0.00575755 | 0.00243227 | 0.02795104 |
20 | 0.00339147 | 0.00633438 | 0.00374553 | 0.00394451 | 0.00166190 | 0.01907779 |
50 | 0.00107991 | 0.00201699 | 0.00119265 | 0.00125601 | 0.00052918 | 0.00607475 |
100 | 0.00016034 | 0.00029948 | 0.00017708 | 0.00018649 | 0.00007857 | 0.00090197 |
200 | 0.00000353 | 0.00000660 | 0.00000390 | 0.00000411 | 0.00000173 | 0.00001989 |
300 | 0.00000008 | 0.00000015 | 0.00000009 | 0.00000009 | 0.00000003 | 0.00000044 |
500 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.17895473 | 0.33796971 | 0.19708476 | 0.20065459 | 0.08533621 | 1.00000000 |
0 | 0.00148277 | 0.00363672 | 0.00162886 | 0.00053541 | 0.00029061 | 0.00757437 |
1 | 0.00122073 | 0.00248229 | 0.00141283 | 0.00079891 | 0.00049111 | 0.00640588 |
2 | 0.00105468 | 0.00179357 | 0.00112890 | 0.00103029 | 0.00047010 | 0.00547755 |
3 | 0.00089757 | 0.00190533 | 0.00096004 | 0.00118332 | 0.00048134 | 0.00542760 |
4 | 0.00096295 | 0.00185514 | 0.00111938 | 0.00116282 | 0.00051063 | 0.00561091 |
5 | 0.00111788 | 0.00198211 | 0.00120152 | 0.00116430 | 0.00051719 | 0.00598300 |
6 | 0.00106199 | 0.00200132 | 0.00116720 | 0.00115114 | 0.00049567 | 0.00587732 |
0 | 0.00109382 | 0.00243612 | 0.00116304 | 0.00054705 | 0.00029262 | 0.00553266 |
1 | 0.00187471 | 0.00393963 | 0.00201337 | 0.00126468 | 0.00063839 | 0.00973078 |
2 | 0.00256220 | 0.00516670 | 0.00274151 | 0.00204630 | 0.00095506 | 0.01347178 |
3 | 0.00320810 | 0.00645095 | 0.00345180 | 0.00288409 | 0.00129132 | 0.01728624 |
4 | 0.00388575 | 0.00769135 | 0.00421561 | 0.00370673 | 0.00164183 | 0.02114127 |
5 | 0.00458192 | 0.00887902 | 0.00495214 | 0.00456082 | 0.00199793 | 0.02497183 |
10 | 0.00498603 | 0.00928331 | 0.00549189 | 0.00575755 | 0.00243227 | 0.02795104 |
20 | 0.00339147 | 0.00633438 | 0.00374553 | 0.00394451 | 0.00166190 | 0.01907779 |
50 | 0.00107991 | 0.00201699 | 0.00119265 | 0.00125601 | 0.00052918 | 0.00607475 |
100 | 0.00016034 | 0.00029948 | 0.00017708 | 0.00018649 | 0.00007857 | 0.00090197 |
200 | 0.00000353 | 0.00000660 | 0.00000390 | 0.00000411 | 0.00000173 | 0.00001989 |
300 | 0.00000008 | 0.00000015 | 0.00000009 | 0.00000009 | 0.00000003 | 0.00000044 |
500 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.17895473 | 0.33796971 | 0.19708476 | 0.20065459 | 0.08533621 | 1.00000000 |
0 | 0.00047483 | 0.00077372 | 0.00059879 | 0.00078790 | 0.00036054 | 0.00299578 |
1 | 0.00076314 | 0.00119978 | 0.00097247 | 0.00140311 | 0.00061681 | 0.00495532 |
2 | 0.00110386 | 0.00172634 | 0.00140867 | 0.00210776 | 0.00091559 | 0.00726221 |
3 | 0.00147097 | 0.00227929 | 0.00190462 | 0.00286252 | 0.00123500 | 0.00975240 |
4 | 0.00185631 | 0.00287564 | 0.00242983 | 0.00368458 | 0.00157512 | 0.01242148 |
5 | 0.00226328 | 0.00349641 | 0.00297861 | 0.00455765 | 0.00193035 | 0.01522630 |
10 | 0.00342239 | 0.00526449 | 0.00449032 | 0.00694631 | 0.00290268 | 0.02302618 |
20 | 0.00253361 | 0.00388041 | 0.00331762 | 0.00511929 | 0.00214814 | 0.01699908 |
50 | 0.00115231 | 0.00176460 | 0.00150943 | 0.00233341 | 0.00097752 | 0.00773727 |
100 | 0.00031795 | 0.00048690 | 0.00041649 | 0.00064384 | 0.00026972 | 0.00213489 |
200 | 0.00002421 | 0.00003707 | 0.00003171 | 0.00004902 | 0.00002053 | 0.00016254 |
300 | 0.00000184 | 0.00000282 | 0.00000241 | 0.00000373 | 0.00000156 | 0.00001237 |
500 | 0.00000001 | 0.00000002 | 0.00000001 | 0.00000002 | 0.00000001 | 0.00000007 |
1000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.14901642 | 0.22858763 | 0.19513341 | 0.30097685 | 0.12628569 | 1.000000 |
0 | 0.00047483 | 0.00077372 | 0.00059879 | 0.00078790 | 0.00036054 | 0.00299578 |
1 | 0.00076314 | 0.00119978 | 0.00097247 | 0.00140311 | 0.00061681 | 0.00495532 |
2 | 0.00110386 | 0.00172634 | 0.00140867 | 0.00210776 | 0.00091559 | 0.00726221 |
3 | 0.00147097 | 0.00227929 | 0.00190462 | 0.00286252 | 0.00123500 | 0.00975240 |
4 | 0.00185631 | 0.00287564 | 0.00242983 | 0.00368458 | 0.00157512 | 0.01242148 |
5 | 0.00226328 | 0.00349641 | 0.00297861 | 0.00455765 | 0.00193035 | 0.01522630 |
10 | 0.00342239 | 0.00526449 | 0.00449032 | 0.00694631 | 0.00290268 | 0.02302618 |
20 | 0.00253361 | 0.00388041 | 0.00331762 | 0.00511929 | 0.00214814 | 0.01699908 |
50 | 0.00115231 | 0.00176460 | 0.00150943 | 0.00233341 | 0.00097752 | 0.00773727 |
100 | 0.00031795 | 0.00048690 | 0.00041649 | 0.00064384 | 0.00026972 | 0.00213489 |
200 | 0.00002421 | 0.00003707 | 0.00003171 | 0.00004902 | 0.00002053 | 0.00016254 |
300 | 0.00000184 | 0.00000282 | 0.00000241 | 0.00000373 | 0.00000156 | 0.00001237 |
500 | 0.00000001 | 0.00000002 | 0.00000001 | 0.00000002 | 0.00000001 | 0.00000007 |
1000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
sum | 0.14901642 | 0.22858763 | 0.19513341 | 0.30097685 | 0.12628569 | 1.000000 |
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