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May  2021, 17(3): 1057-1068. doi: 10.3934/jimo.2020011

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

 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  May 2021 Early access  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.

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 and Management Optimization, 2021, 17 (3) : 1057-1068. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] J. B. Keller, Time-dependent queues, SIAM Review, 24 (1982), 410-412.  doi: 10.1137/1024098. [19] D. A. Levin and Y. Peres, Markov Chains and Mixing Times, Second edition, American Mathematical Society, Providence, RI, 2017. [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. [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. [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. [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. [24] S. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Second edition. Cambridge University Press, Cambridge, 2009. [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. [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. [27] J. A. Schwarz, G. Selinka and R. Stolletz, Performance analysis of time-dependent queueing systems: Survey and classification, Omega, 63 (2016), 170-189. [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. [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. [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. [31] A. I. Zeǐfman, Quasi-ergodicity for non-homogeneous continuous-time Markov chains, J. Appl. Probab., 26 (1989), 643-648.  doi: 10.2307/3214422. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] J. B. Keller, Time-dependent queues, SIAM Review, 24 (1982), 410-412.  doi: 10.1137/1024098. [19] D. A. Levin and Y. Peres, Markov Chains and Mixing Times, Second edition, American Mathematical Society, Providence, RI, 2017. [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. [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. [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. [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. [24] S. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Second edition. Cambridge University Press, Cambridge, 2009. [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. [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. [27] J. A. Schwarz, G. Selinka and R. Stolletz, Performance analysis of time-dependent queueing systems: Survey and classification, Omega, 63 (2016), 170-189. [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. [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. [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. [31] A. I. Zeǐfman, Quasi-ergodicity for non-homogeneous continuous-time Markov chains, J. Appl. Probab., 26 (1989), 643-648.  doi: 10.2307/3214422. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
Probability of repair $Q(t)$
Probabilities $P_{00}(t)$, $P_{10}(t)$, $P_{01}(t)$ blue, green, red respectively
Probabilities $P_{11}(t)$, $P_{21}(t)$ blue and green respectively
Probabilities $P_{31}(t)$, $P_{41}(t)$, $P_{51}(t)$ blue, green, red respectively
The mean value $E(t)$
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