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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) $
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Figure 2.  Probabilities $ P_{00}(t) $, $ P_{10}(t) $, $ P_{01}(t) $ blue, green, red respectively
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Figure 3.  Probabilities $ P_{11}(t) $, $ P_{21}(t) $ blue and green respectively
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Figure 4.  Probabilities $ P_{31}(t) $, $ P_{41}(t) $, $ P_{51}(t) $ blue, green, red respectively
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Figure 5.  The mean value $ E(t) $
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