
-
Previous Article
Analysis of Markov-modulated fluid polling systems with gated discipline
- JIMO Home
- This Issue
-
Next Article
Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem
Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization
Department of Mathematics, National Institute of Technology Raipur, Raipur-492010, India |
This paper analyzes an infinite-buffer single-server queueing system wherein customers arrive in batches of random size according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process. Based on the censoring technique, the UL-type $ RG $-factorization for the Toeplitz type block-structured Markov chain is used to obtain the prearrival epoch probabilities. The random epoch probabilities are obtained with the help of classical principle based on Markov renewal theory. The system-length distributions at outside observer's, intermediate and post-departure epochs are obtained by making relations among various time epochs. The analysis of waiting-time distribution measured in slots of an arbitrary customer in an arrival batch has also been investigated. In order to unify the results of both discrete-time and its continuous-time counterpart, we give a brief demonstration to get the continuous-time results from those of the discrete-time ones. A variety of numerical results are provided to illustrate the effect of model parameters on the performance measures.
References:
[1] |
J. Abate, G. L. Choudhury and W. Whitt,
Asymptotics for steady-state tail probabilities in structured Markov queueing models, Comm. Statist. Stochastic Models, 10 (1994), 99-143.
doi: 10.1080/15326349408807290. |
[2] |
A. S. Alfa, Applied Discrete-Time Queues, 2$^{nd}$ edition, Springer-Verlag, New York, 2016.
doi: 10.1007/978-1-4939-3420-1. |
[3] |
A. S. Alfa, J. Xue and Q. Ye,
Perturbation theory for the asymptotic decay rates in the queues with Markovian arrival process and/or Markovian service process, Queueing Syst., 36 (2000), 287-301.
doi: 10.1023/A:1011032718715. |
[4] |
J. R. Artalejo, I. Atencia and P. Moreno,
A discrete-time $Geo^{[X]}/G/1$ retrial queue with control of admission, Applied Mathematical Modelling, 29 (2005), 1100-1120.
doi: 10.1016/j.apm.2005.02.005. |
[5] |
J. R. Artalejo and Q. L. Li,
Performance analysis of a block-structured discrete-time retrial queue with state-dependent arrivals, Discrete Event Dyn. Syst., 20 (2010), 325-347.
doi: 10.1007/s10626-009-0075-6. |
[6] |
F. Avram and D. F. Chedom,
On symbolic $RG$-factorization of quasi-birth-and-death processes, TOP, 19 (2011), 317-335.
doi: 10.1007/s11750-011-0195-7. |
[7] |
A. D. Banik and U. C. Gupta,
Analyzing the finite buffer batch arrival queue under Markovian service process: $GI^{X}/MSP/1/N$, TOP, 15 (2007), 146-160.
doi: 10.1007/s11750-007-0007-2. |
[8] |
P. P. Bocharov, C. D'Apice and S. Salerno,
The stationary characteristics of the $G/MSP/1/r$ queueing system, Autom. Remote Control, 64 (2003), 288-301.
doi: 10.1023/A:1022219232282. |
[9] |
H. Bruneel and B. G. Kim, Discrete-time Models for Communication Systems including ATM, The Springer International Series in Engineering and Computer Science, 205, Kluwer Academic Publishers, Boston, 1993.
doi: 10.1007/978-1-4615-3130-2. |
[10] |
M. L. Chaudhry, A. D. Banik and A. Pacheco,
A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $GI^{[X]}/C$-$MSP/1/\infty $, Ann. Oper. Res., 252 (2017), 135-173.
doi: 10.1007/s10479-015-2026-y. |
[11] |
M. L. Chaudhry, S. K. Samanta and A. Pacheco,
Analytically explicit results for the $GI/C$-$MSP/1/\infty$ queueing system using roots, Probab. Engrg. Inform. Sci., 26 (2012), 221-244.
doi: 10.1017/S0269964811000349. |
[12] |
E. Çinlar, Introduction to Stochastic Process, Prentice Hall, New Jersey, 1975. |
[13] |
D. Freedman, Approximating Countable Markov Chains, 2$^{nd}$ edition, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8230-0. |
[14] |
Y. Gao and W. Liu,
Analysis of the $GI/Geo/c$ queue with working vacations, Applied Mechanics and Materials, 197 (2012), 534-541.
doi: 10.4028/www.scientific.net/AMM.197.534. |
[15] |
V. Goswami, U. C. Gupta and S. K. Samanta,
Analyzing discrete-time bulk-service $Geo/Geo^b/m$ queue, RAIRO Operations Research, 40 (2006), 267-284.
doi: 10.1051/ro:2006021. |
[16] |
W. K. Grassmann and D. P. Heyman,
Equilibrium distribution of block-structured Markov chains with repeating rows, J. Appl. Probab., 27 (1990), 557-576.
doi: 10.2307/3214541. |
[17] |
U. C. Gupta and A. D. Banik,
Complete analysis of finite and infinite buffer $GI/MSP/1$ queue — A computational approach, Oper. Res. Lett., 35 (2007), 273-280.
doi: 10.1016/j.orl.2006.02.003. |
[18] |
A. Horváth, G. Horváth and M. Telek,
A joint moments based analysis of networks of $MAP/MAP/1$ queues, Performance Evaluation, 67 (2010), 759-778.
doi: 10.1016/j.peva.2009.12.006. |
[19] |
J. J. Hunter, Mathematical techniques of applied probability, in Discrete-Time Models: Techniques and Applications, Operations Research and Industrial Engineering, Academic Press,
New York, 1983. |
[20] |
T. Jiang and L. Liu,
Analysis of a batch service multi-server polling system with dynamic service control, J. Ind. Manag. Optim., 14 (2018), 743-757.
doi: 10.3934/jimo.2017073. |
[21] |
N. K. Kim, S. H. Chang and K. C. Chae,
On the relationships among queue lengths at arrival, departure, and random epochs in the discrete-time queue with D-BMAP arrivals, Oper. Res. Lett., 30 (2002), 25-32.
doi: 10.1016/S0167-6377(01)00110-9. |
[22] |
Q. Li, Y. Ying and Y. Q. Zhao,
A $BMAP/G/1$ retrial queue with a server subject to breakdowns and repairs, Ann. Oper. Res., 141 (2006), 233-270.
doi: 10.1007/s10479-006-5301-0. |
[23] |
Q. L. Li, Constructive Computation in Stochastic Models with Applications: The RGfactorization, Springer, Berlin and Tsinghua University Press, Beijing, 2010.
doi: 10.1007/978-3-642-11492-2. |
[24] |
Q. L. Li and Y. Q. Zhao,
Light-tailed asymptotics of stationary probability vectors of Markov chains of $GI/G/1$ type, Adv. in Appl. Probab., 37 (2005), 1075-1093.
doi: 10.1017/S0001867800000677. |
[25] |
Q. L. Li and Y. Q. Zhao, A $MAP/G/1$ queue with negative customers, Queueing Syst., 47(1) (2004), 5–43.
doi: 10.1023/B:QUES.0000032798.65858.19. |
[26] |
D. M. Lucantoni and M. F. Neuts,
Some steady-state distributions for the $MAP/SM/1$ queue, Comm. Statist. Stochastic Models, 10 (1994), 575-598.
doi: 10.1080/15326349408807311. |
[27] |
C. D. Meyer,
Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems, SIAM Review, 31 (1989), 240-272.
doi: 10.1137/1031050. |
[28] |
M. S. Mushtaq, S. Fowler and A. Mellouk, QoE in 5G cloud networks using multimedia services, in Proceeding of IEEE international Wireless Communication and Networking Conference (WCNC'16), Doha, Qatar, 2016.
doi: 10.1109/WCNC.2016.7565173. |
[29] |
T. Ozawa,
Analysis of queues with Markovian service processes, Stochastic Models, 20 (2004), 391-413.
doi: 10.1081/STM-200033073. |
[30] |
A. Pacheco, S. K. Samanta and M. L. Chaudhry,
A short note on the $GI/Geo/1$ queueing system, Statist. Probab. Lett., 82 (2012), 268-273.
doi: 10.1016/j.spl.2011.09.022. |
[31] |
S. K. Samanta,
Sojourn-time distribution of the $GI/MSP/1$ queueing system, OPSEARCH, 52 (2015), 756-770.
doi: 10.1007/s12597-015-0202-0. |
[32] |
S. K. Samanta, M. L. Chaudhry and A. Pacheco,
Analysis of $BMAP/MSP/1$ queue, Methodol. Comput. Appl. Probab., 18 (2016), 419-440.
doi: 10.1007/s11009-014-9429-0. |
[33] |
S. K. Samanta, M. L. Chaudhry, A. Pacheco and U. C. Gupta,
Analytic and computational analysis of the discrete-time $GI/D$-$MSP/1$ queue using roots, Comput. Oper. Res., 56 (2015), 33-40.
doi: 10.1016/j.cor.2014.10.017. |
[34] |
S. K. Samanta, U. C. Gupta and M. L. Chaudhry,
Analysis of stationary discrete-time $GI/D$-$MSP/1$ queue with finite and infinite buffers, 4OR, 7 (2009), 337-361.
doi: 10.1007/s10288-008-0088-2. |
[35] |
S. K. Samanta and Z. G. Zhang,
Stationary analysis of a discrete-time $GI/D$-$MSP/1$ queue with multiple vacations, Appl. Math. Model., 36 (2012), 5964-5975.
doi: 10.1016/j.apm.2012.01.049. |
[36] |
K. D. Turck, S. D. Vuyst, D. Fiems, H. Bruneel and 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.
doi: 10.1007/s11276-012-0504-6. |
[37] |
Y. C. Wang, J. H. Chou and S. Y. Wang,
Loss pattern of $DBMAP/DMSP/1/K$ queue and its application in wireless local communications, Appl. Math. Model., 35 (2011), 1782-1797.
doi: 10.1016/j.apm.2010.10.009. |
[38] |
Y. Wang, C. Linb and Q. L. Li,
Performance analysis of email systems under three types of attacks, Performance Evaluation, 67 (2010), 485-499.
doi: 10.1016/j.peva.2010.01.003. |
[39] |
M. Yu and A. S. Alfa,
Algorithm for computing the queue length distribution at various time epochs in $DMAP/G^{(1, a, b)}/1/N$ queue with batch-size-dependent service time, European J. Oper. Res., 244 (2015), 227-239.
doi: 10.1016/j.ejor.2015.01.056. |
[40] |
M. Zhang and Z. Hou,
Performance analysis of $MAP/G/1$ queue with working vacations and vacation interruption, Applied Mathematical Modelling, 35 (2011), 1551-1560.
doi: 10.1016/j.apm.2010.09.031. |
[41] |
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.
doi: 10.1023/B:WINE.0000013078.74259.13. |
[42] |
Y. Q. Zhao, Censoring technique in studying block-structured Markov chains, in Advance in Algorithmic Methods for Stochastic Models, Notable Publications, 2000, 417–433. Google Scholar |
[43] |
Y. Q. Zhao, W. Li and W. J. Braun,
Infinite block-structured transition matrices and their properties, Adv. in Appl. Probab., 30 (1998), 365-384.
doi: 10.1239/aap/1035228074. |
[44] |
Y. Q. Zhao and D. Liu,
The censored Markov chain and the best augmentation, Journal of Applied Probability, 33 (1996), 623-629.
doi: 10.1017/S0021900200100063. |
show all references
References:
[1] |
J. Abate, G. L. Choudhury and W. Whitt,
Asymptotics for steady-state tail probabilities in structured Markov queueing models, Comm. Statist. Stochastic Models, 10 (1994), 99-143.
doi: 10.1080/15326349408807290. |
[2] |
A. S. Alfa, Applied Discrete-Time Queues, 2$^{nd}$ edition, Springer-Verlag, New York, 2016.
doi: 10.1007/978-1-4939-3420-1. |
[3] |
A. S. Alfa, J. Xue and Q. Ye,
Perturbation theory for the asymptotic decay rates in the queues with Markovian arrival process and/or Markovian service process, Queueing Syst., 36 (2000), 287-301.
doi: 10.1023/A:1011032718715. |
[4] |
J. R. Artalejo, I. Atencia and P. Moreno,
A discrete-time $Geo^{[X]}/G/1$ retrial queue with control of admission, Applied Mathematical Modelling, 29 (2005), 1100-1120.
doi: 10.1016/j.apm.2005.02.005. |
[5] |
J. R. Artalejo and Q. L. Li,
Performance analysis of a block-structured discrete-time retrial queue with state-dependent arrivals, Discrete Event Dyn. Syst., 20 (2010), 325-347.
doi: 10.1007/s10626-009-0075-6. |
[6] |
F. Avram and D. F. Chedom,
On symbolic $RG$-factorization of quasi-birth-and-death processes, TOP, 19 (2011), 317-335.
doi: 10.1007/s11750-011-0195-7. |
[7] |
A. D. Banik and U. C. Gupta,
Analyzing the finite buffer batch arrival queue under Markovian service process: $GI^{X}/MSP/1/N$, TOP, 15 (2007), 146-160.
doi: 10.1007/s11750-007-0007-2. |
[8] |
P. P. Bocharov, C. D'Apice and S. Salerno,
The stationary characteristics of the $G/MSP/1/r$ queueing system, Autom. Remote Control, 64 (2003), 288-301.
doi: 10.1023/A:1022219232282. |
[9] |
H. Bruneel and B. G. Kim, Discrete-time Models for Communication Systems including ATM, The Springer International Series in Engineering and Computer Science, 205, Kluwer Academic Publishers, Boston, 1993.
doi: 10.1007/978-1-4615-3130-2. |
[10] |
M. L. Chaudhry, A. D. Banik and A. Pacheco,
A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $GI^{[X]}/C$-$MSP/1/\infty $, Ann. Oper. Res., 252 (2017), 135-173.
doi: 10.1007/s10479-015-2026-y. |
[11] |
M. L. Chaudhry, S. K. Samanta and A. Pacheco,
Analytically explicit results for the $GI/C$-$MSP/1/\infty$ queueing system using roots, Probab. Engrg. Inform. Sci., 26 (2012), 221-244.
doi: 10.1017/S0269964811000349. |
[12] |
E. Çinlar, Introduction to Stochastic Process, Prentice Hall, New Jersey, 1975. |
[13] |
D. Freedman, Approximating Countable Markov Chains, 2$^{nd}$ edition, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8230-0. |
[14] |
Y. Gao and W. Liu,
Analysis of the $GI/Geo/c$ queue with working vacations, Applied Mechanics and Materials, 197 (2012), 534-541.
doi: 10.4028/www.scientific.net/AMM.197.534. |
[15] |
V. Goswami, U. C. Gupta and S. K. Samanta,
Analyzing discrete-time bulk-service $Geo/Geo^b/m$ queue, RAIRO Operations Research, 40 (2006), 267-284.
doi: 10.1051/ro:2006021. |
[16] |
W. K. Grassmann and D. P. Heyman,
Equilibrium distribution of block-structured Markov chains with repeating rows, J. Appl. Probab., 27 (1990), 557-576.
doi: 10.2307/3214541. |
[17] |
U. C. Gupta and A. D. Banik,
Complete analysis of finite and infinite buffer $GI/MSP/1$ queue — A computational approach, Oper. Res. Lett., 35 (2007), 273-280.
doi: 10.1016/j.orl.2006.02.003. |
[18] |
A. Horváth, G. Horváth and M. Telek,
A joint moments based analysis of networks of $MAP/MAP/1$ queues, Performance Evaluation, 67 (2010), 759-778.
doi: 10.1016/j.peva.2009.12.006. |
[19] |
J. J. Hunter, Mathematical techniques of applied probability, in Discrete-Time Models: Techniques and Applications, Operations Research and Industrial Engineering, Academic Press,
New York, 1983. |
[20] |
T. Jiang and L. Liu,
Analysis of a batch service multi-server polling system with dynamic service control, J. Ind. Manag. Optim., 14 (2018), 743-757.
doi: 10.3934/jimo.2017073. |
[21] |
N. K. Kim, S. H. Chang and K. C. Chae,
On the relationships among queue lengths at arrival, departure, and random epochs in the discrete-time queue with D-BMAP arrivals, Oper. Res. Lett., 30 (2002), 25-32.
doi: 10.1016/S0167-6377(01)00110-9. |
[22] |
Q. Li, Y. Ying and Y. Q. Zhao,
A $BMAP/G/1$ retrial queue with a server subject to breakdowns and repairs, Ann. Oper. Res., 141 (2006), 233-270.
doi: 10.1007/s10479-006-5301-0. |
[23] |
Q. L. Li, Constructive Computation in Stochastic Models with Applications: The RGfactorization, Springer, Berlin and Tsinghua University Press, Beijing, 2010.
doi: 10.1007/978-3-642-11492-2. |
[24] |
Q. L. Li and Y. Q. Zhao,
Light-tailed asymptotics of stationary probability vectors of Markov chains of $GI/G/1$ type, Adv. in Appl. Probab., 37 (2005), 1075-1093.
doi: 10.1017/S0001867800000677. |
[25] |
Q. L. Li and Y. Q. Zhao, A $MAP/G/1$ queue with negative customers, Queueing Syst., 47(1) (2004), 5–43.
doi: 10.1023/B:QUES.0000032798.65858.19. |
[26] |
D. M. Lucantoni and M. F. Neuts,
Some steady-state distributions for the $MAP/SM/1$ queue, Comm. Statist. Stochastic Models, 10 (1994), 575-598.
doi: 10.1080/15326349408807311. |
[27] |
C. D. Meyer,
Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems, SIAM Review, 31 (1989), 240-272.
doi: 10.1137/1031050. |
[28] |
M. S. Mushtaq, S. Fowler and A. Mellouk, QoE in 5G cloud networks using multimedia services, in Proceeding of IEEE international Wireless Communication and Networking Conference (WCNC'16), Doha, Qatar, 2016.
doi: 10.1109/WCNC.2016.7565173. |
[29] |
T. Ozawa,
Analysis of queues with Markovian service processes, Stochastic Models, 20 (2004), 391-413.
doi: 10.1081/STM-200033073. |
[30] |
A. Pacheco, S. K. Samanta and M. L. Chaudhry,
A short note on the $GI/Geo/1$ queueing system, Statist. Probab. Lett., 82 (2012), 268-273.
doi: 10.1016/j.spl.2011.09.022. |
[31] |
S. K. Samanta,
Sojourn-time distribution of the $GI/MSP/1$ queueing system, OPSEARCH, 52 (2015), 756-770.
doi: 10.1007/s12597-015-0202-0. |
[32] |
S. K. Samanta, M. L. Chaudhry and A. Pacheco,
Analysis of $BMAP/MSP/1$ queue, Methodol. Comput. Appl. Probab., 18 (2016), 419-440.
doi: 10.1007/s11009-014-9429-0. |
[33] |
S. K. Samanta, M. L. Chaudhry, A. Pacheco and U. C. Gupta,
Analytic and computational analysis of the discrete-time $GI/D$-$MSP/1$ queue using roots, Comput. Oper. Res., 56 (2015), 33-40.
doi: 10.1016/j.cor.2014.10.017. |
[34] |
S. K. Samanta, U. C. Gupta and M. L. Chaudhry,
Analysis of stationary discrete-time $GI/D$-$MSP/1$ queue with finite and infinite buffers, 4OR, 7 (2009), 337-361.
doi: 10.1007/s10288-008-0088-2. |
[35] |
S. K. Samanta and Z. G. Zhang,
Stationary analysis of a discrete-time $GI/D$-$MSP/1$ queue with multiple vacations, Appl. Math. Model., 36 (2012), 5964-5975.
doi: 10.1016/j.apm.2012.01.049. |
[36] |
K. D. Turck, S. D. Vuyst, D. Fiems, H. Bruneel and 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.
doi: 10.1007/s11276-012-0504-6. |
[37] |
Y. C. Wang, J. H. Chou and S. Y. Wang,
Loss pattern of $DBMAP/DMSP/1/K$ queue and its application in wireless local communications, Appl. Math. Model., 35 (2011), 1782-1797.
doi: 10.1016/j.apm.2010.10.009. |
[38] |
Y. Wang, C. Linb and Q. L. Li,
Performance analysis of email systems under three types of attacks, Performance Evaluation, 67 (2010), 485-499.
doi: 10.1016/j.peva.2010.01.003. |
[39] |
M. Yu and A. S. Alfa,
Algorithm for computing the queue length distribution at various time epochs in $DMAP/G^{(1, a, b)}/1/N$ queue with batch-size-dependent service time, European J. Oper. Res., 244 (2015), 227-239.
doi: 10.1016/j.ejor.2015.01.056. |
[40] |
M. Zhang and Z. Hou,
Performance analysis of $MAP/G/1$ queue with working vacations and vacation interruption, Applied Mathematical Modelling, 35 (2011), 1551-1560.
doi: 10.1016/j.apm.2010.09.031. |
[41] |
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.
doi: 10.1023/B:WINE.0000013078.74259.13. |
[42] |
Y. Q. Zhao, Censoring technique in studying block-structured Markov chains, in Advance in Algorithmic Methods for Stochastic Models, Notable Publications, 2000, 417–433. Google Scholar |
[43] |
Y. Q. Zhao, W. Li and W. J. Braun,
Infinite block-structured transition matrices and their properties, Adv. in Appl. Probab., 30 (1998), 365-384.
doi: 10.1239/aap/1035228074. |
[44] |
Y. Q. Zhao and D. Liu,
The censored Markov chain and the best augmentation, Journal of Applied Probability, 33 (1996), 623-629.
doi: 10.1017/S0021900200100063. |


0 | 0.147931 | 0.087562 | 0.141983 | 0.215337 | 0.592813 |
1 | 0.017322 | 0.008928 | 0.017209 | 0.019057 | 0.062516 |
2 | 0.008854 | 0.004021 | 0.009012 | 0.007208 | 0.029094 |
3 | 0.007634 | 0.003498 | 0.007749 | 0.006361 | 0.025243 |
4 | 0.007169 | 0.003300 | 0.007241 | 0.006043 | 0.023753 |
5 | 0.012534 | 0.006493 | 0.012352 | 0.013954 | 0.045334 |
10 | 0.005820 | 0.002639 | 0.005923 | 0.004720 | 0.019101 |
30 | 0.000272 | 0.000123 | 0.000275 | 0.000218 | 0.000887 |
60 | 0.000002 | 0.000001 | 0.000002 | 0.000002 | 0.000007 |
150 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.268062 | 0.144522 | 0.263029 | 0.324388 | 1.000000 |
0 | 0.147931 | 0.087562 | 0.141983 | 0.215337 | 0.592813 |
1 | 0.017322 | 0.008928 | 0.017209 | 0.019057 | 0.062516 |
2 | 0.008854 | 0.004021 | 0.009012 | 0.007208 | 0.029094 |
3 | 0.007634 | 0.003498 | 0.007749 | 0.006361 | 0.025243 |
4 | 0.007169 | 0.003300 | 0.007241 | 0.006043 | 0.023753 |
5 | 0.012534 | 0.006493 | 0.012352 | 0.013954 | 0.045334 |
10 | 0.005820 | 0.002639 | 0.005923 | 0.004720 | 0.019101 |
30 | 0.000272 | 0.000123 | 0.000275 | 0.000218 | 0.000887 |
60 | 0.000002 | 0.000001 | 0.000002 | 0.000002 | 0.000007 |
150 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.268062 | 0.144522 | 0.263029 | 0.324388 | 1.000000 |
0 | 0.111295 | 0.065475 | 0.106245 | 0.160319 | 0.443334 |
1 | 0.022916 | 0.010687 | 0.022938 | 0.019981 | 0.076522 |
2 | 0.013339 | 0.006155 | 0.013392 | 0.011322 | 0.044209 |
3 | 0.013437 | 0.006222 | 0.013490 | 0.011506 | 0.044656 |
4 | 0.013841 | 0.006409 | 0.013895 | 0.011853 | 0.045999 |
5 | 0.014556 | 0.006778 | 0.014600 | 0.012641 | 0.048575 |
10 | 0.006965 | 0.003204 | 0.007013 | 0.005865 | 0.023048 |
30 | 0.000402 | 0.000181 | 0.000410 | 0.000321 | 0.001314 |
60 | 0.000004 | 0.000002 | 0.000004 | 0.000003 | 0.000012 |
150 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.279037 | 0.142937 | 0.275076 | 0.302949 | 1.000000 |
0 | 0.111295 | 0.065475 | 0.106245 | 0.160319 | 0.443334 |
1 | 0.022916 | 0.010687 | 0.022938 | 0.019981 | 0.076522 |
2 | 0.013339 | 0.006155 | 0.013392 | 0.011322 | 0.044209 |
3 | 0.013437 | 0.006222 | 0.013490 | 0.011506 | 0.044656 |
4 | 0.013841 | 0.006409 | 0.013895 | 0.011853 | 0.045999 |
5 | 0.014556 | 0.006778 | 0.014600 | 0.012641 | 0.048575 |
10 | 0.006965 | 0.003204 | 0.007013 | 0.005865 | 0.023048 |
30 | 0.000402 | 0.000181 | 0.000410 | 0.000321 | 0.001314 |
60 | 0.000004 | 0.000002 | 0.000004 | 0.000003 | 0.000012 |
150 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.279037 | 0.142937 | 0.275076 | 0.302949 | 1.000000 |
0 | 0.103342 | 0.060768 | 0.098612 | 0.148741 | 0.411462 |
1 | 0.025961 | 0.012561 | 0.025829 | 0.024745 | 0.089096 |
2 | 0.013329 | 0.006179 | 0.013370 | 0.011447 | 0.044325 |
3 | 0.013265 | 0.006142 | 0.013316 | 0.011358 | 0.044081 |
4 | 0.013661 | 0.006326 | 0.013714 | 0.011699 | 0.045400 |
5 | 0.016859 | 0.008165 | 0.016802 | 0.016105 | 0.057931 |
10 | 0.007064 | 0.003265 | 0.007104 | 0.006021 | 0.023455 |
30 | 0.000442 | 0.000202 | 0.000450 | 0.000366 | 0.001460 |
60 | 0.000004 | 0.000002 | 0.000004 | 0.000003 | 0.000013 |
150 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.279037 | 0.142937 | 0.275076 | 0.302949 | 1.000000 |
0 | 0.103342 | 0.060768 | 0.098612 | 0.148741 | 0.411462 |
1 | 0.025961 | 0.012561 | 0.025829 | 0.024745 | 0.089096 |
2 | 0.013329 | 0.006179 | 0.013370 | 0.011447 | 0.044325 |
3 | 0.013265 | 0.006142 | 0.013316 | 0.011358 | 0.044081 |
4 | 0.013661 | 0.006326 | 0.013714 | 0.011699 | 0.045400 |
5 | 0.016859 | 0.008165 | 0.016802 | 0.016105 | 0.057931 |
10 | 0.007064 | 0.003265 | 0.007104 | 0.006021 | 0.023455 |
30 | 0.000442 | 0.000202 | 0.000450 | 0.000366 | 0.001460 |
60 | 0.000004 | 0.000002 | 0.000004 | 0.000003 | 0.000013 |
150 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.279037 | 0.142937 | 0.275076 | 0.302949 | 1.000000 |
0 | 0.032873 | 0.019458 | 0.031552 | 0.047853 | 0.131736 |
1 | 0.019883 | 0.011780 | 0.019101 | 0.028996 | 0.079761 |
2 | 0.019751 | 0.011709 | 0.018985 | 0.028835 | 0.079280 |
3 | 0.020343 | 0.012060 | 0.019554 | 0.029699 | 0.081656 |
4 | 0.020959 | 0.012425 | 0.020146 | 0.030599 | 0.084130 |
5 | 0.011311 | 0.006712 | 0.010884 | 0.016550 | 0.045457 |
10 | 0.010060 | 0.005977 | 0.009689 | 0.014744 | 0.040469 |
30 | 0.000495 | 0.000297 | 0.000481 | 0.000737 | 0.002009 |
60 | 0.000005 | 0.000003 | 0.000005 | 0.000007 | 0.000020 |
150 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.248718 | 0.147678 | 0.239427 | 0.364177 | 1.000000 |
0 | 0.032873 | 0.019458 | 0.031552 | 0.047853 | 0.131736 |
1 | 0.019883 | 0.011780 | 0.019101 | 0.028996 | 0.079761 |
2 | 0.019751 | 0.011709 | 0.018985 | 0.028835 | 0.079280 |
3 | 0.020343 | 0.012060 | 0.019554 | 0.029699 | 0.081656 |
4 | 0.020959 | 0.012425 | 0.020146 | 0.030599 | 0.084130 |
5 | 0.011311 | 0.006712 | 0.010884 | 0.016550 | 0.045457 |
10 | 0.010060 | 0.005977 | 0.009689 | 0.014744 | 0.040469 |
30 | 0.000495 | 0.000297 | 0.000481 | 0.000737 | 0.002009 |
60 | 0.000005 | 0.000003 | 0.000005 | 0.000007 | 0.000020 |
150 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.248718 | 0.147678 | 0.239427 | 0.364177 | 1.000000 |
0 | 0.034508 | 0.020385 | 0.033070 | 0.050094 | 0.138057 |
1 | 0.009993 | 0.005467 | 0.009024 | 0.012961 | 0.037445 |
2 | 0.008841 | 0.005188 | 0.008367 | 0.012412 | 0.034807 |
3 | 0.008718 | 0.005172 | 0.008340 | 0.012581 | 0.034811 |
4 | 0.008830 | 0.005239 | 0.008473 | 0.012846 | 0.035388 |
5 | 0.008691 | 0.005166 | 0.008366 | 0.012724 | 0.034948 |
10 | 0.006267 | 0.003757 | 0.006088 | 0.009341 | 0.025452 |
30 | 0.003144 | 0.001879 | 0.003045 | 0.004660 | 0.012727 |
60 | 0.000421 | 0.000252 | 0.000408 | 0.000625 | 0.001705 |
100 | 0.000033 | 0.000020 | 0.000032 | 0.000049 | 0.000133 |
190 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.248718 | 0.147678 | 0.239427 | 0.364177 | 1.000000 |
0 | 0.034508 | 0.020385 | 0.033070 | 0.050094 | 0.138057 |
1 | 0.009993 | 0.005467 | 0.009024 | 0.012961 | 0.037445 |
2 | 0.008841 | 0.005188 | 0.008367 | 0.012412 | 0.034807 |
3 | 0.008718 | 0.005172 | 0.008340 | 0.012581 | 0.034811 |
4 | 0.008830 | 0.005239 | 0.008473 | 0.012846 | 0.035388 |
5 | 0.008691 | 0.005166 | 0.008366 | 0.012724 | 0.034948 |
10 | 0.006267 | 0.003757 | 0.006088 | 0.009341 | 0.025452 |
30 | 0.003144 | 0.001879 | 0.003045 | 0.004660 | 0.012727 |
60 | 0.000421 | 0.000252 | 0.000408 | 0.000625 | 0.001705 |
100 | 0.000033 | 0.000020 | 0.000032 | 0.000049 | 0.000133 |
190 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.248718 | 0.147678 | 0.239427 | 0.364177 | 1.000000 |
0 | 0.177488 | 0.104369 | 0.169315 | 0.255259 | 0.706432 |
1 | 0.031834 | 0.014078 | 0.032731 | 0.024080 | 0.102724 |
2 | 0.020658 | 0.009143 | 0.021320 | 0.015664 | 0.066785 |
3 | 0.013410 | 0.005939 | 0.013880 | 0.010188 | 0.043417 |
4 | 0.008708 | 0.003859 | 0.009033 | 0.006625 | 0.028224 |
5 | 0.005655 | 0.002507 | 0.005877 | 0.004308 | 0.018347 |
10 | 0.000655 | 0.000290 | 0.000683 | 0.000500 | 0.002129 |
15 | 0.000076 | 0.000034 | 0.000079 | 0.000058 | 0.000247 |
20 | 0.000009 | 0.000004 | 0.000009 | 0.000007 | 0.000029 |
25 | 0.000001 | 0.000000 | 0.000001 | 0.000001 | 0.000003 |
50 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.268240 | 0.144548 | 0.263085 | 0.324127 | 1.000000 |
0 | 0.177488 | 0.104369 | 0.169315 | 0.255259 | 0.706432 |
1 | 0.031834 | 0.014078 | 0.032731 | 0.024080 | 0.102724 |
2 | 0.020658 | 0.009143 | 0.021320 | 0.015664 | 0.066785 |
3 | 0.013410 | 0.005939 | 0.013880 | 0.010188 | 0.043417 |
4 | 0.008708 | 0.003859 | 0.009033 | 0.006625 | 0.028224 |
5 | 0.005655 | 0.002507 | 0.005877 | 0.004308 | 0.018347 |
10 | 0.000655 | 0.000290 | 0.000683 | 0.000500 | 0.002129 |
15 | 0.000076 | 0.000034 | 0.000079 | 0.000058 | 0.000247 |
20 | 0.000009 | 0.000004 | 0.000009 | 0.000007 | 0.000029 |
25 | 0.000001 | 0.000000 | 0.000001 | 0.000001 | 0.000003 |
50 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.268240 | 0.144548 | 0.263085 | 0.324127 | 1.000000 |
0 | 0.133257 | 0.077530 | 0.125951 | 0.188221 | 0.524958 |
1 | 0.051790 | 0.022946 | 0.051871 | 0.039576 | 0.166183 |
2 | 0.033578 | 0.014882 | 0.033952 | 0.025651 | 0.108062 |
3 | 0.021782 | 0.009656 | 0.022187 | 0.016637 | 0.070262 |
4 | 0.014135 | 0.006268 | 0.014481 | 0.010796 | 0.045680 |
5 | 0.009176 | 0.004069 | 0.009443 | 0.007009 | 0.029697 |
10 | 0.001061 | 0.000471 | 0.001104 | 0.000811 | 0.003446 |
15 | 0.000123 | 0.000055 | 0.000128 | 0.000094 | 0.000400 |
20 | 0.000014 | 0.000006 | 0.000015 | 0.000011 | 0.000046 |
25 | 0.000002 | 0.000001 | 0.000002 | 0.000001 | 0.000005 |
50 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.280718 | 0.142893 | 0.275506 | 0.300882 | 1.000000 |
0 | 0.133257 | 0.077530 | 0.125951 | 0.188221 | 0.524958 |
1 | 0.051790 | 0.022946 | 0.051871 | 0.039576 | 0.166183 |
2 | 0.033578 | 0.014882 | 0.033952 | 0.025651 | 0.108062 |
3 | 0.021782 | 0.009656 | 0.022187 | 0.016637 | 0.070262 |
4 | 0.014135 | 0.006268 | 0.014481 | 0.010796 | 0.045680 |
5 | 0.009176 | 0.004069 | 0.009443 | 0.007009 | 0.029697 |
10 | 0.001061 | 0.000471 | 0.001104 | 0.000811 | 0.003446 |
15 | 0.000123 | 0.000055 | 0.000128 | 0.000094 | 0.000400 |
20 | 0.000014 | 0.000006 | 0.000015 | 0.000011 | 0.000046 |
25 | 0.000002 | 0.000001 | 0.000002 | 0.000001 | 0.000005 |
50 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.280718 | 0.142893 | 0.275506 | 0.300882 | 1.000000 |
0 | 0.111612 | 0.064802 | 0.105303 | 0.157092 | 0.438808 |
1 | 0.058730 | 0.027593 | 0.058203 | 0.052204 | 0.196731 |
2 | 0.038411 | 0.017807 | 0.038510 | 0.032991 | 0.127719 |
3 | 0.025082 | 0.011509 | 0.025373 | 0.020975 | 0.082940 |
4 | 0.016359 | 0.007448 | 0.016665 | 0.013400 | 0.053872 |
5 | 0.010660 | 0.004824 | 0.010920 | 0.008593 | 0.034997 |
10 | 0.001243 | 0.000555 | 0.001292 | 0.000964 | 0.004055 |
15 | 0.000144 | 0.000064 | 0.000151 | 0.000111 | 0.000470 |
20 | 0.000017 | 0.000007 | 0.000017 | 0.000013 | 0.000054 |
25 | 0.000002 | 0.000001 | 0.000002 | 0.000001 | 0.000006 |
50 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.280718 | 0.142893 | 0.275506 | 0.300882 | 1.000000 |
0 | 0.111612 | 0.064802 | 0.105303 | 0.157092 | 0.438808 |
1 | 0.058730 | 0.027593 | 0.058203 | 0.052204 | 0.196731 |
2 | 0.038411 | 0.017807 | 0.038510 | 0.032991 | 0.127719 |
3 | 0.025082 | 0.011509 | 0.025373 | 0.020975 | 0.082940 |
4 | 0.016359 | 0.007448 | 0.016665 | 0.013400 | 0.053872 |
5 | 0.010660 | 0.004824 | 0.010920 | 0.008593 | 0.034997 |
10 | 0.001243 | 0.000555 | 0.001292 | 0.000964 | 0.004055 |
15 | 0.000144 | 0.000064 | 0.000151 | 0.000111 | 0.000470 |
20 | 0.000017 | 0.000007 | 0.000017 | 0.000013 | 0.000054 |
25 | 0.000002 | 0.000001 | 0.000002 | 0.000001 | 0.000006 |
50 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.280718 | 0.142893 | 0.275506 | 0.300882 | 1.000000 |
0 | 0.088744 | 0.052185 | 0.084658 | 0.127630 | 0.353216 |
1 | 0.056838 | 0.033672 | 0.054596 | 0.082865 | 0.227970 |
2 | 0.036522 | 0.021762 | 0.035273 | 0.053821 | 0.147378 |
3 | 0.023528 | 0.014083 | 0.022821 | 0.034965 | 0.095397 |
4 | 0.015187 | 0.009123 | 0.014781 | 0.022720 | 0.061811 |
5 | 0.009819 | 0.005914 | 0.009581 | 0.014765 | 0.040079 |
10 | 0.001124 | 0.000682 | 0.001104 | 0.001712 | 0.004622 |
15 | 0.000130 | 0.000079 | 0.000128 | 0.000199 | 0.000535 |
20 | 0.000015 | 0.000009 | 0.000015 | 0.000023 | 0.000062 |
25 | 0.000002 | 0.000001 | 0.000002 | 0.000002 | 0.000007 |
50 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.248718 | 0.147678 | 0.239427 | 0.364177 | 1.000000 |
0 | 0.088744 | 0.052185 | 0.084658 | 0.127630 | 0.353216 |
1 | 0.056838 | 0.033672 | 0.054596 | 0.082865 | 0.227970 |
2 | 0.036522 | 0.021762 | 0.035273 | 0.053821 | 0.147378 |
3 | 0.023528 | 0.014083 | 0.022821 | 0.034965 | 0.095397 |
4 | 0.015187 | 0.009123 | 0.014781 | 0.022720 | 0.061811 |
5 | 0.009819 | 0.005914 | 0.009581 | 0.014765 | 0.040079 |
10 | 0.001124 | 0.000682 | 0.001104 | 0.001712 | 0.004622 |
15 | 0.000130 | 0.000079 | 0.000128 | 0.000199 | 0.000535 |
20 | 0.000015 | 0.000009 | 0.000015 | 0.000023 | 0.000062 |
25 | 0.000002 | 0.000001 | 0.000002 | 0.000002 | 0.000007 |
50 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.248718 | 0.147678 | 0.239427 | 0.364177 | 1.000000 |
0 | 0.088744 | 0.052185 | 0.084658 | 0.127630 | 0.353216 |
1 | 0.028093 | 0.015539 | 0.025592 | 0.037079 | 0.106303 |
2 | 0.020283 | 0.012108 | 0.019471 | 0.029323 | 0.081185 |
3 | 0.016689 | 0.010130 | 0.016294 | 0.025037 | 0.068151 |
4 | 0.014155 | 0.008612 | 0.013897 | 0.021520 | 0.058184 |
5 | 0.012056 | 0.007334 | 0.011861 | 0.018415 | 0.049667 |
10 | 0.005375 | 0.003268 | 0.005294 | 0.008231 | 0.022168 |
20 | 0.001064 | 0.000647 | 0.001048 | 0.001629 | 0.004388 |
30 | 0.000211 | 0.000128 | 0.000207 | 0.000323 | 0.000869 |
50 | 0.000008 | 0.000005 | 0.000008 | 0.000013 | 0.000034 |
80 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.248718 | 0.147678 | 0.239427 | 0.364177 | 1.000000 |
0 | 0.088744 | 0.052185 | 0.084658 | 0.127630 | 0.353216 |
1 | 0.028093 | 0.015539 | 0.025592 | 0.037079 | 0.106303 |
2 | 0.020283 | 0.012108 | 0.019471 | 0.029323 | 0.081185 |
3 | 0.016689 | 0.010130 | 0.016294 | 0.025037 | 0.068151 |
4 | 0.014155 | 0.008612 | 0.013897 | 0.021520 | 0.058184 |
5 | 0.012056 | 0.007334 | 0.011861 | 0.018415 | 0.049667 |
10 | 0.005375 | 0.003268 | 0.005294 | 0.008231 | 0.022168 |
20 | 0.001064 | 0.000647 | 0.001048 | 0.001629 | 0.004388 |
30 | 0.000211 | 0.000128 | 0.000207 | 0.000323 | 0.000869 |
50 | 0.000008 | 0.000005 | 0.000008 | 0.000013 | 0.000034 |
80 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
Sum | 0.248718 | 0.147678 | 0.239427 | 0.364177 | 1.000000 |
[1] |
Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170 |
[2] |
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 |
[3] |
Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020167 |
[4] |
Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020339 |
[5] |
Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112 |
[6] |
Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264 |
[7] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[8] |
Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117 |
[9] |
Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 |
[10] |
Zsolt Saffer, Miklós Telek, Gábor Horváth. Analysis of Markov-modulated fluid polling systems with gated discipline. Journal of Industrial & Management Optimization, 2021, 17 (2) : 575-599. doi: 10.3934/jimo.2019124 |
[11] |
Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020166 |
[12] |
Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 |
[13] |
Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021022 |
[14] |
Honglin Yang, Jiawu Peng. Coordinating a supply chain with demand information updating. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020181 |
[15] |
Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021010 |
[16] |
Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020165 |
[17] |
Xi Zhao, Teng Niu. Impacts of horizontal mergers on dual-channel supply chain. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020173 |
[18] |
Mahir Demir, Suzanne Lenhart. A spatial food chain model for the Black Sea Anchovy, and its optimal fishery. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 155-171. doi: 10.3934/dcdsb.2020373 |
[19] |
Wenyan Zhuo, Honglin Yang, Leopoldo Eduardo Cárdenas-Barrón, Hong Wan. Loss-averse supply chain decisions with a capital constrained retailer. Journal of Industrial & Management Optimization, 2021, 17 (2) : 711-732. doi: 10.3934/jimo.2019131 |
[20] |
Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 |
2019 Impact Factor: 1.366
Tools
Article outline
Figures and Tables
[Back to Top]