American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A stochastic model and social optimization of a blockchain system based on a general limited batch service queue
  • JIMO Home
  • This Issue
  • Next Article
    A mathematical formulation and heuristic approach for the heterogeneous fixed fleet vehicle routing problem with simultaneous pickup and delivery
doi: 10.3934/jimo.2020011

On limiting characteristics for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs

Sherif I. Ammar 1,2, , Alexander Zeifman 3,, , Yacov Satin 4, , Ksenia Kiseleva 4, and  Victor Korolev 5,6,

1. 

Department of Mathematics, Faculty of Science, Menofia University, Shebin El Kom, Egypt
2. 

Department of Mathematics, College of Science, Taibah University, Medinah, Saudi Arabia
3. 

Vologda State University, Institute of Informatics Problems of the FRC CSC RAS, Vologda Research Center RAS, Russia
4. 

Vologda State University, Russia
5. 

Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Russia
6. 

Institute of Informatics Problems of FRC CSC RAS, Hangzhou Dianzi University, China

* Corresponding author: Alexander Zeifman

Received  November 2018 Revised  May 2019 Published  January 2020

In this paper, we display a method for the computation of convergence bounds for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs when all parameters varying with time. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence and the main limiting characteristics of the queue-length process are obtained. Finally a numerical example is presented to show the effect of parameters.

Keywords: Two-processor heterogeneous system, catastrophes, repairs, bounds, rate of convergence.
Mathematics Subject Classification: Primary: 60J27, 60J28.
Citation: Sherif I. Ammar, Alexander Zeifman, Yacov Satin, Ksenia Kiseleva, Victor Korolev. On limiting characteristics for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020011
References:
[1]

S. I. Ammar, Transient behavior of a two-processor heterogeneous system with catastrophes, server failures and repairs, Applied Mathematical Modelling, 38 (2014), 2224-2234.  doi: 10.1016/j.apm.2013.10.033.  Google Scholar

[2]

S. I. Ammar and Y. F. Alharbi, Time-dependent analysis for a two-processor heterogeneous system with time-varying arrival and service rates, Applied Mathematical Modelling, 54 (2018), 743-751.  doi: 10.1016/j.apm.2017.10.021.  Google Scholar

[3]

M. Armony and A. R. Ward, Fair dynamic routing in large-scale heterogeneous-server systems, Oper. Res., 58 (2010), 624-637.  doi: 10.1287/opre.1090.0777.  Google Scholar

[4]

S. R. Chakravarthy, A catastrophic queueing model with delayed action, Applied Mathematical Modelling, 46 (2017), 631-649.  doi: 10.1016/j.apm.2017.01.089.  Google Scholar

[5]

A. Y. Chen and E. Renshaw, The $M|M|1$ queue with mass exodus and mass arrives when empty, J. Appl. Prob., 34 (1997), 192-207.  doi: 10.2307/3215186.  Google Scholar

[6]

A. Chen and E. Renshaw, Markov bulk-arriving queues with state-dependent control at idle time, Adv. Appl. Prob., 36 (2004), 499-524.  doi: 10.1017/S0001867800013586.  Google Scholar

[7]

A. Chen, P. Pollett, J. P. Li and H. J. Zhang, Markovian bulk-arrival and bulk-service queues with state-dependent control, Queueing Syst., 64 (2010), 267-304.  doi: 10.1007/s11134-009-9162-5.  Google Scholar

[8]

Ju. L. Daleckij and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, Vol. 43. American Mathematical Society, Providence, R.I., 1974.  Google Scholar

[9]

P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, The Annals of Applied Probability, 6 (1996), 695-750.  doi: 10.1214/aoap/1034968224.  Google Scholar

[10]

P. Diaconis and L. Saloff-Coste, Separation cut-offs for birth and death chains, The Annals of Applied Probability, 16 (2006), 2098-2122.  doi: 10.1214/105051606000000501.  Google Scholar

[11]

A. Di Crescenzo, V. Giorno and A. G. Nobile, Constructing transient birth-death processes by means of suitable transformations, Applied Mathematics and Computation, 281 (2016), 152-171.  doi: 10.1016/j.amc.2016.01.058.  Google Scholar

[12]

A. Di Crescenzo, V. Giorno, B. K. Kumar and A. G. Nobile, A time-non-homogeneous double-ended queue with failures and repairs and its continuous approximation, Mathematics, 6 (2018), 1-23.  doi: 10.3390/math6050081.  Google Scholar

[13]

V. Giorno, A. G. Nobile and S. Spina, On some time non-homogeneous queueing systems with catastrophes, Applied Mathematics and Computation, 245 (2014), 220-234.  doi: 10.1016/j.amc.2014.07.076.  Google Scholar

[14]

B. L. Granovsky and A. I. Zeifman, The $N$-limit of spectral gap of a class of birth-death Markov chains, Appl. Stoch. Models in Business and Industry, 16 (2000), 235-248.  doi: 10.1002/1526-4025(200010/12)16:4<235::AID-ASMB415>3.0.CO;2-S.  Google Scholar

[15]

B. L. Granovsky and A. Zeifman, Nonstationary queues: Estimation of the rate of convergence, Queueing Systems, 46 (2004), 363-388.  doi: 10.1023/B:QUES.0000027991.19758.b4.  Google Scholar

[16]

L. Green and P. Kolesar, The pointwise stationary approximation for queues with nonstationary arrivals, Manag. Sci., 37 (1991), 84-97.  doi: 10.1287/mnsc.37.1.84.  Google Scholar

[17]

N. V. Kartashov, Criteria for uniform ergodicity and strong stability of Markov chains with a common phase space, Teor. Veroyatnost. i Mat. Statist., (1984), 65–81,151.  Google Scholar

[18]

J. B. Keller, Time-dependent queues, SIAM Review, 24 (1982), 410-412.  doi: 10.1137/1024098.  Google Scholar

[19]

D. A. Levin and Y. Peres, Markov Chains and Mixing Times, Second edition, American Mathematical Society, Providence, RI, 2017.  Google Scholar

[20]

J. P. Li and A. Y. Chen, The decay parameter and invariant measures for Markovian bulk-arrival queues with control at idle time, Methodology and Computing in Applied Probability, 15 (2013), 467-484.  doi: 10.1007/s11009-011-9252-9.  Google Scholar

[21]

A. Mandelbaum and W. A. Massey, Strong approximations for time-dependent queues, Math. Oper. Res., 20 (1995), 33-64.  doi: 10.1287/moor.20.1.33.  Google Scholar

[22]

W. A. Massey and W. Whitt, Uniform acceleration expansions for Markov chains with time-varying rates, Ann. Appl. Probab., 8 (1998), 1130-1155.  doi: 10.1214/aoap/1028903375.  Google Scholar

[23]

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Ann. Appl. Probab., 25 (1993), 518-548.  doi: 10.2307/1427522.  Google Scholar

[24] S. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Second edition. Cambridge University Press, Cambridge, 2009.   Google Scholar
[25]

A. Yu. Mitrophanov, Stability and exponential convergence of continuous-time Markov chains, J. Appl. Probab., 40 (2003), 970-979.  doi: 10.1239/jap/1067436094.  Google Scholar

[26]

A. Yu. Mitrophanov, The spectral gap and perturbation bounds for reversible continuous-time Markov chains, J. Appl. Probab., 41 (2004), 1219-1222.  doi: 10.1239/jap/1101840568.  Google Scholar

[27]

J. A. Schwarz, G. Selinka and R. Stolletz, Performance analysis of time-dependent queueing systems: Survey and classification, Omega, 63 (2016), 170-189.   Google Scholar

[28]

È. A. van Doorn, A. I. Zeǐfman and T. L. Panfilova, Estimates and asymptotics for the rate of convergence of birth-death processes, Th. Prob. Appl., 54 (2010), 97-113.   Google Scholar

[29]

È. A. van Doorn, Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem, Journal of applied probability, 52 (2015), 278-289.  doi: 10.1239/jap/1429282622.  Google Scholar

[30]

W. Whitt, The pointwise stationary approximation for $Mt/Mt/s$ queues is asymptotically correct as the rates increase, Manag. Sci., 37 (1991), 251-376.  doi: 10.1287/mnsc.37.3.307.  Google Scholar

[31]

A. I. Zeǐfman, Quasi-ergodicity for non-homogeneous continuous-time Markov chains, J. Appl. Probab., 26 (1989), 643-648.  doi: 10.2307/3214422.  Google Scholar

[32]

A. I. Zeǐfman, On the estimation of probabilities for birth and death processes, J. Appl. Probab., 32 (1995), 623-634.  doi: 10.2307/3215117.  Google Scholar

[33]

A. I. Zeǐfman, A. V. Korotysheva, Y. A. Satin and S. Y. Shorgin, On stability for nonstationary queueing systemswith catastrophes, Informatika i Ee Primeneniya [Informatics and its Applications], 4 (2010), 9-15.   Google Scholar

[34]

A. I. Zeǐfman, A. V. Korotysheva, T. L. Panfilova and S. Y. Shorgin, Stability bounds for some queueing systems with catastrophes, Informatika i Ee Primeneniya [Informatics and its Applications], 5 (2011), 27-33.   Google Scholar

[35]

A. I. Zeǐfman and A. V. Korotysheva, Perturbation bounds for $M_t|M_t|N$ queue with catastrophes, Stochastic Models, 28 (2012), 49-62.  doi: 10.1080/15326349.2011.614900.  Google Scholar

[36]

A. Zeifman, Y. Satin, V. Korolev and S. Shorgin, On truncations for weakly ergodic inhomogeneous birth and death processes, Int. J. Appl. Math. Comp. Sci., 24 (2014), 503-518.  doi: 10.2478/amcs-2014-0037.  Google Scholar

[37]

A. Zeifman, Y. Satin, A. Korotysheva, V. Korolev, S. Shorgin and R. Razumchik, Ergodicity and perturbation bounds for inhomogeneous birth and death processes with additional transitions from and to origin, Int. J. Appl. Math. Comput. Sci., 25 (2015), 787-802.  doi: 10.1515/amcs-2015-0056.  Google Scholar

[38]

A. I. Zeǐfman, Ya. Satin, A. Korotysheva, V. Korolev and V. Bening, On a class of Markovian queuing systems described by inhomogeneous birth-and-death processes with additional transitions, Doklady Mathematics, 94 (2016), 502-505.   Google Scholar

[39]

A. Zeifman, A. Korotysheva, Y. Satin, R. Razumchik, V. Korolev and S. Shorgin, Ergodicity and uniform in time truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin, Stochastic Models, 33 (2017), 598-616.  doi: 10.1080/15326349.2017.1362654.  Google Scholar

[40]

A. I. Zeǐfman, A. Korotysheva, V. Korolev and Y. Satin, Truncation bounds for approximations of inhomogeneous continuous-time Markov chains, Theory of Probability & Its Applications, 61 (2017), 513-520.   Google Scholar

[41]

A. Zeifman, Y. Satin, K. Kiseleva, V. Korolev and T. Panfilova, On limiting characteristics for a non-stationary two-processor heterogeneous system, Applied Mathematics and Computation, 351 (2019), 48-65.  doi: 10.1016/j.amc.2019.01.032.  Google Scholar

[42]

L. N. Zhang and J. P. Li, The M/M/c queue with mass exodus and mass arrivals when empty, Journal of Applied Probability, 52 (2015), 990-1002.  doi: 10.1239/jap/1450802748.  Google Scholar

show all references

References:
[1]

S. I. Ammar, Transient behavior of a two-processor heterogeneous system with catastrophes, server failures and repairs, Applied Mathematical Modelling, 38 (2014), 2224-2234.  doi: 10.1016/j.apm.2013.10.033.  Google Scholar

[2]

S. I. Ammar and Y. F. Alharbi, Time-dependent analysis for a two-processor heterogeneous system with time-varying arrival and service rates, Applied Mathematical Modelling, 54 (2018), 743-751.  doi: 10.1016/j.apm.2017.10.021.  Google Scholar

[3]

M. Armony and A. R. Ward, Fair dynamic routing in large-scale heterogeneous-server systems, Oper. Res., 58 (2010), 624-637.  doi: 10.1287/opre.1090.0777.  Google Scholar

[4]

S. R. Chakravarthy, A catastrophic queueing model with delayed action, Applied Mathematical Modelling, 46 (2017), 631-649.  doi: 10.1016/j.apm.2017.01.089.  Google Scholar

[5]

A. Y. Chen and E. Renshaw, The $M|M|1$ queue with mass exodus and mass arrives when empty, J. Appl. Prob., 34 (1997), 192-207.  doi: 10.2307/3215186.  Google Scholar

[6]

A. Chen and E. Renshaw, Markov bulk-arriving queues with state-dependent control at idle time, Adv. Appl. Prob., 36 (2004), 499-524.  doi: 10.1017/S0001867800013586.  Google Scholar

[7]

A. Chen, P. Pollett, J. P. Li and H. J. Zhang, Markovian bulk-arrival and bulk-service queues with state-dependent control, Queueing Syst., 64 (2010), 267-304.  doi: 10.1007/s11134-009-9162-5.  Google Scholar

[8]

Ju. L. Daleckij and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, Vol. 43. American Mathematical Society, Providence, R.I., 1974.  Google Scholar

[9]

P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, The Annals of Applied Probability, 6 (1996), 695-750.  doi: 10.1214/aoap/1034968224.  Google Scholar

[10]

P. Diaconis and L. Saloff-Coste, Separation cut-offs for birth and death chains, The Annals of Applied Probability, 16 (2006), 2098-2122.  doi: 10.1214/105051606000000501.  Google Scholar

[11]

A. Di Crescenzo, V. Giorno and A. G. Nobile, Constructing transient birth-death processes by means of suitable transformations, Applied Mathematics and Computation, 281 (2016), 152-171.  doi: 10.1016/j.amc.2016.01.058.  Google Scholar

[12]

A. Di Crescenzo, V. Giorno, B. K. Kumar and A. G. Nobile, A time-non-homogeneous double-ended queue with failures and repairs and its continuous approximation, Mathematics, 6 (2018), 1-23.  doi: 10.3390/math6050081.  Google Scholar

[13]

V. Giorno, A. G. Nobile and S. Spina, On some time non-homogeneous queueing systems with catastrophes, Applied Mathematics and Computation, 245 (2014), 220-234.  doi: 10.1016/j.amc.2014.07.076.  Google Scholar

[14]

B. L. Granovsky and A. I. Zeifman, The $N$-limit of spectral gap of a class of birth-death Markov chains, Appl. Stoch. Models in Business and Industry, 16 (2000), 235-248.  doi: 10.1002/1526-4025(200010/12)16:4<235::AID-ASMB415>3.0.CO;2-S.  Google Scholar

[15]

B. L. Granovsky and A. Zeifman, Nonstationary queues: Estimation of the rate of convergence, Queueing Systems, 46 (2004), 363-388.  doi: 10.1023/B:QUES.0000027991.19758.b4.  Google Scholar

[16]

L. Green and P. Kolesar, The pointwise stationary approximation for queues with nonstationary arrivals, Manag. Sci., 37 (1991), 84-97.  doi: 10.1287/mnsc.37.1.84.  Google Scholar

[17]

N. V. Kartashov, Criteria for uniform ergodicity and strong stability of Markov chains with a common phase space, Teor. Veroyatnost. i Mat. Statist., (1984), 65–81,151.  Google Scholar

[18]

J. B. Keller, Time-dependent queues, SIAM Review, 24 (1982), 410-412.  doi: 10.1137/1024098.  Google Scholar

[19]

D. A. Levin and Y. Peres, Markov Chains and Mixing Times, Second edition, American Mathematical Society, Providence, RI, 2017.  Google Scholar

[20]

J. P. Li and A. Y. Chen, The decay parameter and invariant measures for Markovian bulk-arrival queues with control at idle time, Methodology and Computing in Applied Probability, 15 (2013), 467-484.  doi: 10.1007/s11009-011-9252-9.  Google Scholar

[21]

A. Mandelbaum and W. A. Massey, Strong approximations for time-dependent queues, Math. Oper. Res., 20 (1995), 33-64.  doi: 10.1287/moor.20.1.33.  Google Scholar

[22]

W. A. Massey and W. Whitt, Uniform acceleration expansions for Markov chains with time-varying rates, Ann. Appl. Probab., 8 (1998), 1130-1155.  doi: 10.1214/aoap/1028903375.  Google Scholar

[23]

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Ann. Appl. Probab., 25 (1993), 518-548.  doi: 10.2307/1427522.  Google Scholar

[24] S. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Second edition. Cambridge University Press, Cambridge, 2009.   Google Scholar
[25]

A. Yu. Mitrophanov, Stability and exponential convergence of continuous-time Markov chains, J. Appl. Probab., 40 (2003), 970-979.  doi: 10.1239/jap/1067436094.  Google Scholar

[26]

A. Yu. Mitrophanov, The spectral gap and perturbation bounds for reversible continuous-time Markov chains, J. Appl. Probab., 41 (2004), 1219-1222.  doi: 10.1239/jap/1101840568.  Google Scholar

[27]

J. A. Schwarz, G. Selinka and R. Stolletz, Performance analysis of time-dependent queueing systems: Survey and classification, Omega, 63 (2016), 170-189.   Google Scholar

[28]

È. A. van Doorn, A. I. Zeǐfman and T. L. Panfilova, Estimates and asymptotics for the rate of convergence of birth-death processes, Th. Prob. Appl., 54 (2010), 97-113.   Google Scholar

[29]

È. A. van Doorn, Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem, Journal of applied probability, 52 (2015), 278-289.  doi: 10.1239/jap/1429282622.  Google Scholar

[30]

W. Whitt, The pointwise stationary approximation for $Mt/Mt/s$ queues is asymptotically correct as the rates increase, Manag. Sci., 37 (1991), 251-376.  doi: 10.1287/mnsc.37.3.307.  Google Scholar

[31]

A. I. Zeǐfman, Quasi-ergodicity for non-homogeneous continuous-time Markov chains, J. Appl. Probab., 26 (1989), 643-648.  doi: 10.2307/3214422.  Google Scholar

[32]

A. I. Zeǐfman, On the estimation of probabilities for birth and death processes, J. Appl. Probab., 32 (1995), 623-634.  doi: 10.2307/3215117.  Google Scholar

[33]

A. I. Zeǐfman, A. V. Korotysheva, Y. A. Satin and S. Y. Shorgin, On stability for nonstationary queueing systemswith catastrophes, Informatika i Ee Primeneniya [Informatics and its Applications], 4 (2010), 9-15.   Google Scholar

[34]

A. I. Zeǐfman, A. V. Korotysheva, T. L. Panfilova and S. Y. Shorgin, Stability bounds for some queueing systems with catastrophes, Informatika i Ee Primeneniya [Informatics and its Applications], 5 (2011), 27-33.   Google Scholar

[35]

A. I. Zeǐfman and A. V. Korotysheva, Perturbation bounds for $M_t|M_t|N$ queue with catastrophes, Stochastic Models, 28 (2012), 49-62.  doi: 10.1080/15326349.2011.614900.  Google Scholar

[36]

A. Zeifman, Y. Satin, V. Korolev and S. Shorgin, On truncations for weakly ergodic inhomogeneous birth and death processes, Int. J. Appl. Math. Comp. Sci., 24 (2014), 503-518.  doi: 10.2478/amcs-2014-0037.  Google Scholar

[37]

A. Zeifman, Y. Satin, A. Korotysheva, V. Korolev, S. Shorgin and R. Razumchik, Ergodicity and perturbation bounds for inhomogeneous birth and death processes with additional transitions from and to origin, Int. J. Appl. Math. Comput. Sci., 25 (2015), 787-802.  doi: 10.1515/amcs-2015-0056.  Google Scholar

[38]

A. I. Zeǐfman, Ya. Satin, A. Korotysheva, V. Korolev and V. Bening, On a class of Markovian queuing systems described by inhomogeneous birth-and-death processes with additional transitions, Doklady Mathematics, 94 (2016), 502-505.   Google Scholar

[39]

A. Zeifman, A. Korotysheva, Y. Satin, R. Razumchik, V. Korolev and S. Shorgin, Ergodicity and uniform in time truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin, Stochastic Models, 33 (2017), 598-616.  doi: 10.1080/15326349.2017.1362654.  Google Scholar

[40]

A. I. Zeǐfman, A. Korotysheva, V. Korolev and Y. Satin, Truncation bounds for approximations of inhomogeneous continuous-time Markov chains, Theory of Probability & Its Applications, 61 (2017), 513-520.   Google Scholar

[41]

A. Zeifman, Y. Satin, K. Kiseleva, V. Korolev and T. Panfilova, On limiting characteristics for a non-stationary two-processor heterogeneous system, Applied Mathematics and Computation, 351 (2019), 48-65.  doi: 10.1016/j.amc.2019.01.032.  Google Scholar

[42]

L. N. Zhang and J. P. Li, The M/M/c queue with mass exodus and mass arrivals when empty, Journal of Applied Probability, 52 (2015), 990-1002.  doi: 10.1239/jap/1450802748.  Google Scholar

Figure 1.  Probability of repair $ Q(t) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Probabilities $ P_{00}(t) $, $ P_{10}(t) $, $ P_{01}(t) $ blue, green, red respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Probabilities $ P_{11}(t) $, $ P_{21}(t) $ blue and green respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Probabilities $ P_{31}(t) $, $ P_{41}(t) $, $ P_{51}(t) $ blue, green, red respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The mean value $ E(t) $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ahmed M. K. Tarabia. Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, 2011, 7 (4) : 811-823. doi: 10.3934/jimo.2011.7.811
[2]

Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009
[3]

Dequan Yue, Wuyi Yue. A heterogeneous two-server network system with balking and a Bernoulli vacation schedule. Journal of Industrial & Management Optimization, 2010, 6 (3) : 501-516. doi: 10.3934/jimo.2010.6.501
[4]

Louis Caccetta, Syarifah Z. Nordin. Mixed integer programming model for scheduling in unrelated parallel processor system with priority consideration. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 115-132. doi: 10.3934/naco.2014.4.115
[5]

Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021
[6]

Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial & Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199
[7]

Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 227-245. doi: 10.3934/dcdsb.2017012
[8]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1
[9]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503
[10]

Dequan Yue, Jun Yu, Wuyi Yue. A Markovian queue with two heterogeneous servers and multiple vacations. Journal of Industrial & Management Optimization, 2009, 5 (3) : 453-465. doi: 10.3934/jimo.2009.5.453
[11]

Qingshan Zhang, Yuxiang Li. Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2751-2759. doi: 10.3934/dcdsb.2015.20.2751
[12]

Denis Dmitriev, Jonathan Jedwab. Bounds on the growth rate of the peak sidelobe level of binary sequences. Advances in Mathematics of Communications, 2007, 1 (4) : 461-475. doi: 10.3934/amc.2007.1.461
[13]

Jau-Chuan Ke, Hsin-I Huang, Chuen-Horng Lin. Analysis on a queue system with heterogeneous servers and uncertain patterns. Journal of Industrial & Management Optimization, 2010, 6 (1) : 57-71. doi: 10.3934/jimo.2010.6.57
[14]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[15]

Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure & Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49
[16]

Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107
[17]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[18]

Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004
[19]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[20]

Emmanuel Gobet, Mohamed Mrad. Convergence rate of strong approximations of compound random maps, application to SPDEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4455-4476. doi: 10.3934/dcdsb.2018171

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (35)
  • HTML views (120)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences