Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2014 | 24 | 3 | 503-518

Tytuł artykułu

On truncations for weakly ergodic inhomogeneous birth and death processes

Treść / Zawartość

Warianty tytułu

Języki publikacji



We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt /Mt /S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.








Opis fizyczny




  • Institute of Socio-Economic Development of Territories, Russian Academy of Sciences, Gorkogo Str., 56A, Vologda, Russia
  • Department of Applied Mathematics, Vologda State University, Vologda, S. Orlova, 6, Russia
  • Department of Applied Mathematics, Vologda State University, Vologda, S. Orlova, 6, Russia
  • Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory, Moscow, Russia
  • Institute of Problems of Informatics, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, Russia
  • Institute of Problems of Informatics, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, Russia


  • Daleckij, J.L. and Krein, M.G. (1974). Stability of Solutions of Differential Equations in Banach Space, American Mathematical Society, Providence, RI.
  • Di Crescenzo, A., Giorno, V., Kumar, B.K. and Nobile, A.G. (2012). A double-ended queue with catastrophes and repairs, and a jump-diffusion approximation, Methodology and Computing in Applied Probability 14(4): 937-954, DOI: 10.1007/s11009-011-9214-2.
  • Di Crescenzo, A., Giorno, V., Nobile, A.G. and Ricciardi, L.M. (2003). On the M/M/1 queue with catastrophes and its continuous approximation, Queueing Systems 43(4): 329-347, DOI: 10.1023/A:1023261830362.
  • Di Crescenzo, A. and Nobile, A.G. (1995). Diffusion approximation to a queueing system with time dependent arrival and service rates, Queueing systems 19(1): 41-62, DOI: 10.1007/BF01148939.
  • Granovsky, B. and Zeifman, A. (2004). Nonstationary queues: Estimation of the rate of convergence, Queueing Systems 46(3): 369-388, DOI: 10.1023/B:QUES.0000027991.19758.b4.
  • Knessl, C. (2000). Exact and asymptotic solutions to a PDE that arises in time-dependent queues, Advances in Applied Probability 32(1): 256-283.
  • Knessl, C. and Yang, Y.P. (2002). An exact solution for an M (t)/M (t)/1 queue with time-dependent arrivals and service, Advances in Applied Probability 40(3): 233-248, DOI: 10.1023/A:1014786928831.
  • Mandelbaum, A. and Massey, W. (1995). Strong approximations for time-dependent queues, Mathematics of Operations Research 20(1): 33-64, DOI: 10.1287/moor.20.1.33.
  • Margolius, B. (2007a). Periodic solution to the time-inhomogeneous multi-server Poisson queue, Operations Research Letters 35(1): 125-138, DOI: 10.1016/j.orl.2005.12.008.
  • Margolius, B. (2007b). Transient and periodic solution to the time-inhomogeneous quasi-birth death process, Queueing Systems 56(3): 183-194, DOI: 10.1007/s11134-007-9027-8.
  • Massey, W. (2002). The analysis of queues with time-varying rates for telecommunication models, Telecommunication Systems 21(2): 173-204, DOI: 10.1023/A:1020990313587.
  • Massey, W. and Pender, J. (2013). Gaussian skewness approximation for dynamic rate multi-server queues with abandonment, Queueing Systems 75(2): 243-277.
  • Massey, W. and Whitt, W. (1994). On analysis of the modified offered-load approximation for the nonstationary Erlang loss model, Annals of Applied Probability 4(4): 1145-1160, DOI: 10.1214/aoap/1177004908.
  • Olwal, T.O., Djouani, K., Kogeda, O.P. and van Wyk, B.J.V (2012). Joint queue-perturbed and weakly coupled power control for wireless backbone networks, International Journal of Applied Mathematics and Computer Science 22(3): 749-764, DOI: 10.2478/v10006-012-0056-z.
  • Tan, X., Knessl, C. and Yang, Y.P. (2013). On finite capacity queues with time dependent arrival rates, Stochastic Processes and their Applications 123(6): 2175-2227, DOI: 10.1016/
  • Van Doorn, E.A., Zeifman, A.I. and Panfilova, T.L. (2010). Bounds and asymptotics for the rate of convergence of birth-death processes, Theory of Probability and Its Applications 54(1): 97-113, DOI: 10.1137/S0040585X97984097.
  • Zeifman, A. (1995a). Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes, Stochastic Processes and Their Applications 59(1): 157-173, DOI: 10.1016/0304-4149(95)00028-6.
  • Zeifman, A.I. (1985). Stability for continuous-time nonhomogeneous Markov chains, in V.V. Kalashnikov and V.M. Zolotarev (Eds.), Stability Problems for Stochastic Models, Lecture Notes in Mathematics, Vol. 1155, Springer, Berlin/Heidelberg, pp. 401-414, DOI: 10.1007/BFb0074830.
  • Zeifman, A.I. (1988). Truncation error in a birth and death system, USSR Computational Mathematics and Mathematical Physics 28(6): 210-211, DOI: 10.1016/0041-5553(88)90068-7.
  • Zeifman, A.I. (1995b). On the estimation of probabilities for birth and death processes, Journal of Applied Probability 32(3): 623-634.
  • Zeifman, A., Korolev, V., Satin, Y., Korotysheva, A. and Bening, V. (2014). Perturbation bounds and truncations for a class of Markovian queues, Queueing Systems 76(2): 205-221, DOI: 10.1007/s11134-013-9388-0.
  • Zeifman, A. and Korotysheva, A. (2012). Perturbation bounds for Mₜ/Mₜ/N queue with catastrophes, Stochastic Models 281(1): 49-62, DOI: 10.1080/15326349.2011.614900.
  • Zeifman, A., Leorato, S., Orsingher, E., Satin, Y. and Shilova, G. (2006). Some universal limits for nonhomogeneous birth and death processes, Queueing Systems 52(2): 139-151, DOI: 10.1007/s11134-006-4353-9.
  • Zeifman, A., Satin, Y. and Panfilova, T. (2013a). Limiting characteristics for finite birth-death-catastrophe processes, Mathematical Biosciences 245(1): 96-102, DOI: 10.1016/j.mbs.2013.02.009.
  • Zeifman, A., Satin, Y., Shilova, G., Korolev, V., Bening, V. and Shorgin, S. (2013b). On Mₜ/Mₜ/S type queue with group services, Proceedings of the 27th European Conference on Modeling and Simulation, Aalesund, Norway, pp. 604-609.
  • Zhang, J. and Coyle, E.J.J. (1991). The transient solution of time-dependent M/M/L queues, IEEE Transactions on Information Theory 37(6): 1690-1696, DOI: 10.1109/18.104335.

Typ dokumentu



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.