Infinite-Server Queue Model \(MMAP_{k}(t)|G_{k}|\infty \) with Time Varying Marked Map Arrivals of Customers and Occurrence of Catastrophes

  • Ruben KerobyanEmail author
  • Khanik Kerobyan
  • Carol Shubin
  • Phu Nguyen
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1231)


In the present paper, the infinite-server queue model \(MMAP_{k}(t)|G_{k}|\infty \) in transient MMAP random environment with time varying marked MAP arrival of k types of customers subject to catastrophes is considered. The transient joint probability generating functions (PGF) of the number of different types of customers present in the model at moment t and the number of different types of customers departing from the system in the time interval (0, t] are found. The Laplace-Stieltjes transform (LST) of total volume of customers being in service at moment t is defined. The basic differential equations for joint probability generating functions of the number of busy servers and served customers for transient and stationary random environment are obtained.


Marked MAP Infinite-server queue Catastrophes MMAP random environment 



This work was supported by “Data Science Program with Career Support and Connections to Industry,” NSF Award 1842386 grant.


  1. 1.
    Paxson, V., Floyd, S.: Wide-area traffic: the failure of Poisson modeling. In: Proceedings of the ACM, pp. 257–268 (1994)Google Scholar
  2. 2.
    Neuts, M.F.: A versatile Markovian point process. J. App. Prob. 16(4), 764–779 (1979)MathSciNetCrossRefGoogle Scholar
  3. 3.
    He, Q.-M.: Queues with marked customers. Adv. App Prob. 30, 365–372 (1996)zbMATHGoogle Scholar
  4. 4.
    Cordeiro, J.D., Kharoufeh, J.P.: Batch Markovian Arrival Processes (BMAP). In: Cochran, J., Cox, T., Keskinocak, P., Kharoufeh, J.P., Smith, J.C. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2011)Google Scholar
  5. 5.
    Chakravarthy, S.R.: The batch Markovian arrival process: a review and future work. In: Krishnamoorthy, A., Raju, N., Ramaswami, V. (eds.) Advances in Probability Theory and Stochastic Processes, pp. 21–39. Notable Publications, New York City (2000) Google Scholar
  6. 6.
    Artalejo, J.R., Gomez-Corral, A., He, Q.M.: Markovian arrivals in stochastic modelling: a survey and some new results. SORT 34(2), 101–144 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Pacheco, A., Tang, C.H.L., Prabhu, N.U.: Markov-Additive Processes and Semi-regenerative Phenomena. World Scientific, Singapore (2009)Google Scholar
  8. 8.
    Breuer, L.: Introduction to stochastic processes. University of Kent (2014)Google Scholar
  9. 9.
    He, Q.: Fundamentals of Matrix-Analytic Methods. Springer, New York (2014). Scholar
  10. 10.
    Shanbhag, D.N.: On innite-server queues with batch arrivals. J. Appl. Prob. 9, 208–213 (1966)Google Scholar
  11. 11.
    Brown, M., Ross, S.: Some results for infinite server poisson queues. J. Appl. Prob 6, 604–611 (1969)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Postan, M.Ya.: Flow of serviced requests in infinite-channel queueing systems in a transient mode. Probl. Inform. Trans. 13(4), 309–313 (1977)Google Scholar
  13. 13.
    Eick, S.G., Massey, W.A., Whitt, W.: The physics of the \(M(t)|G|\infty \) queue. Oper. Res. 41, 731–742 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Massey, W.A.: The analysis of queues with time-varying rates for telecommunication models. Telecom. Syst. 21(2–4), 173–204 (2002)CrossRefGoogle Scholar
  15. 15.
    Whitt, W.: Heavy-traffic fluid limits for periodic infinite-server queues. Queueing Syst. 84, 111–143 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Economou, A., Fakinos, D.: Alternative approaches for the transient analysis of Markov chains with catastrophes. J. Stat. Theory Pract. 2(2), 183–197 (2008). Scholar
  17. 17.
    Kerobyan, K.: Infinite-server \(M|G|\infty \) queueing models with catastrophes, 12 December 2018.
  18. 18.
    Kerobyan, K., Kerobyan, R.: Transient analysis of infinite-server queue \(MMAP_k(t)|G_k|\infty \) with marked MAP arrival and disasters. In: Proceedings of 7th International Conference on “HET-NETs 2013”, November 2013, Ilkley, UK, pp. 11–13 (2013)Google Scholar
  19. 19.
    Kerobyan, R., Kerobyan, K., Covington, R.: Infinite-server queue model \(MMAP_{k}(t)|G_{k}|\infty \) with time varying marked MAP arrivals and catastrophes. In: Proceedings of the 18th International Conference on Named Aafter A. F. Terpugov, Information Technologies ad Mathematical Modelling, ITMM , 26–30 June 2019, Saratov, Russia (2019)Google Scholar
  20. 20.
    Pang, G., Whitt, W.: Innite-server queue with batch arrivals and dependent service times. Prob. Eng. Inf. Sci. 26, 197–220 (2012)CrossRefGoogle Scholar
  21. 21.
    Ramaswami, V., Neuts, M.F.: Some explicit formulas and computational methods for infinite-server queues with phase-type arrival. J. Appl. Prob. 17, 498–514 (1980)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ramaswami, V.: The \(N/G/\infty \) queue. Technical report, Department of Mathematics, Drexel University, Philadelphia, PA (1978)Google Scholar
  23. 23.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  24. 24.
    Moiseev, A., Nazarov, A.: Infinite-server Queueing Systems and Networks. Publ. NTL, Tomsk (2015)Google Scholar
  25. 25.
    Lisovskaya, E., Moiseeva, S., Pagano, M.: Multiclass GI/GI/\(\infty \) queueing systems with random resource requirements. In: Dudin, A., Nazarov, A., Moiseev, A. (eds.) ITMM/WRQ -2018. CCIS, vol. 912, pp. 129–142. Springer, Cham (2018). Scholar
  26. 26.
    Masuyama, H.: Studies on algorithmic analysis of queues with batch Markovian arrival streams. Ph.D., Thesis, Kyoto University (2003)Google Scholar
  27. 27.
    Nazarov, A., Moiseeva, S.: Asymptotic Analysis Method in Queueing Theory. NTL, Tomsk (2006)Google Scholar
  28. 28.
    Moiseev, A.: Asymptotic analysis of queueing system \(MAP/GI/\infty \) with high-rate arrivals. Tomsk State Univ. J. Control Comput. Sci. 3(32), 56–65 (2015)Google Scholar
  29. 29.
    D’Auria, B.: \(M/M/\infty \) queues in semi-Markovian random environment. Queueing Syst. 58, 221–237 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Fralix, B.H., Adan, I.J.B.F.: An infinite-server queue influenced by a semi-Markovian environment. Queueing Syst. 61, 65–84 (2009)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Linton, D., Purdue, P.: An \(M|G|\infty \) queue with m customer types subject to periodic clearing. Opsearch 16, 80–88 (1979)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Purdue, P., Linton, D.: An infinite-server queue subject to an extraneous phase process and related models. J. Appl. Prob. 18, 236–244 (1981)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kerobyan, K., Covington, R., Kerobyan, R., Enakoutsa, K.: An infinite-server queueing \(MMAP_k|G_k|\infty \) model in semi-markov random environment subject to catastrophes. In: Dudin, A., Nazarov, A., Moiseev, A. (eds.) ITMM/WRQ -2018. CCIS, vol. 912, pp. 195–212. Springer, Cham (2018). Scholar
  34. 34.
    Kerobyan, K., Kerobyan, R., Enakoutsa, K.: Analysis of an infinite-server queue \(MAP_k|G_k|\infty \) in random environment with k markov arrival streams and random volume of customers. In: Dudin, A., Nazarov, A., Moiseev, A. (eds.) ITMM/WRQ -2018. CCIS, vol. 912, pp. 305–320. Springer, Cham (2018). Scholar
  35. 35.
    Tikhonenko, O.M.: Queueing models in computer systems. Universitetoe, Minsk (1990)Google Scholar
  36. 36.
    Tikhonenko, O.M.: Computer Systems Probability Analysis. Akademicka Oficyna Wydawnicza EXIT, Warsaw (2006)Google Scholar
  37. 37.
    Tikhonenko, O.M., Tikhonenko-Kȩdziak, A.: Multi-server closed queueing system with limited buffer size. J. Appl. Math. Comp. Mech. 16(1), 117–125 (2017)CrossRefGoogle Scholar
  38. 38.
    Tikhonenko, O.M.: Basics of queueing theory. Lecture notes, TSU (2013)Google Scholar
  39. 39.
    Moiseev, A., Moiseeva, S., Lisovskaya, E.: Infinite-server queueing tandem with MMPP arrival and random capacity of customers. In: Proceedings of the 31st European Conference on Modelling and Simulation Budapest, Hungary 23–26 May 2017Google Scholar
  40. 40.
    Naumov, V., Samuylov, K.: On the modeling of queue systems with multiple resources. Proc. RUDN. 3, 60–63 (2014)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Ruben Kerobyan
    • 1
    Email author
  • Khanik Kerobyan
    • 2
  • Carol Shubin
    • 2
  • Phu Nguyen
    • 2
  1. 1.University of California San DiegoSan DiegoUSA
  2. 2.California State University NorthridgeNorthridgeUSA

Personalised recommendations