Systems and Means of Informatics
2017, Volume 27, Issue 3, pp 37-51
TRUNCATION BOUNDS FOR A CLASS OF INHOMOGENEOUS BIRTH AND DEATH QUEUEING MODELS WITH ADDITIONAL TRANSITIONS
- A. I. Zeifman
- A. V. Korotysheva
- Ya.A. Satin
- K. M. Kiseleva
- R. V. Razumchik
- V. Yu. Korolev
- S. Ya. Shorgin
Abstract
The paper considers the computation of limiting characteristics for a class of inhomogeneous birth-death processes with possible transitions from and to origin. The authors study the general situation of the slower (nonexponential) decreasing of intensities of transitions from state 0 to state k as k ^ то. The authors consider the situation of weak ergodicity and obtain bounds on the rate of convergence in weighted norm and, moreover, uniform in time bounds on the rate of approximations by truncated processes. The inhomogeneous M/M/S queueing model with additional transitions is studied as an example.
[+] References (12)
- Chen, A. Y., and E. Renshaw. 1997. The M/M/1 queue with mass exodus and mass arrives when empty. J. Appl. Prob. 34:192-207.
- Chen, A. Y., and E. Renshaw. 2004. Markov bulk-arriving queues with state-dependent control at idle time. Adv. Appl. Probab. 36:499-524.
- Chen, A. Y., P. Pollett, J. P. Li, and H. J. Zhang. 2010. Markovian bulk-arrival and bulk-service queues with state-dependent control. Queueing Syst. 64:267-304.
- Li, J. P., and A. Y. Chen. 2013. The decay parameter and invariant measures for Markovian bulk-arrival queues with control at idle time. Methodol. Comput. Appl. 15:467-484.
- Zeifman, A. I., A. V. Korotysheva, Ya. A. Satin, V. Yu. Korolev, S. Ya. Shorgin, and R. V. Razumchik. 2015. Ergodicity and perturbation bounds for inhomogeneous birth and death processes with additional transitions from and to origin. Int. J. Appl. Math. Comp. 25:787-802.
- Zhang, L., and J. P. Li. 2015. The M/M/c queue with mass exodus and mass arrivals when empty. J. Appl. Probab. 52:990-1002.
- Zeifman, A. I., Y. A. Satin, V. Yu. Korolev, and S. Ya. Shorgin. 2014. On truncations for weakly ergodic inhomogeneous birth and death processes. Int. J. Appl. Math. Comp. 24:503-518.
- Zeifman, A. I., A. V. Korotysheva, V. Yu. Korolev, and Ya. A. Satin. 2016. Truncation bounds for approximations of inhomogeneous continuous-time Markov chains. Theor. Probab. Appl. 61:563-569.
- Zeifman, A.I., Y. A. Satin, A. V. Korotysheva, V. Yu. Korolev, and V.E. Bening. 2016. On a class of Markovian queuing systems described by inhomogeneous birth-and- death processes with additional transitions. Dokl. Math. 94:502-505.
- Granovsky, B. L., and A. I. Zeifman. 2004. Nonstationary queues: Estimation of the rate of convergence. Queueing Syst. 46:363-388.
- Zeifman, A.I., S. Leorato, E. Orsingher, Y. A. Satin, and G. N. Shilova. 2006. Some universal limits for nonhomogeneous birth and death processes. Queueing Syst. 52:139-151.
- Van Doorn, E. A., A. I. Zeifman, and T. L. Panfilova. 2010. Bounds and asymptotics for the rate of convergence of birth-death processes. Theor. Probab. Appl. 54:97-113.
[+] About this article
Title
TRUNCATION BOUNDS FOR A CLASS OF INHOMOGENEOUS BIRTH AND DEATH QUEUEING MODELS WITH ADDITIONAL TRANSITIONS
Journal
Systems and Means of Informatics
Volume 27, Issue 3, pp 37-51
Cover Date
2017-09-30
DOI
10.14357/08696527170304
Print ISSN
0869-6527
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
inhomogeneous process; birth-death process; approximations; truncations; ergodicity; bounds; queueing systems
Authors
A. I. Zeifman  ,  ,  ,
A. V. Korotysheva , Ya.A. Satin ,
K. M. Kiseleva ,  ,
R. V. Razumchik , ,
V. Yu. Korolev ,  ,
and S. Ya. Shorgin
Author Affiliations
Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
Institute of Informatics Problems, Federal Research Center "Computer Science
and Control", Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
ISEDT RAS, 56-A Gorky Str. Vologda 160001, Russian Federation
Peoples' Friendship University of Russia (RUDN University), 6 Miklukho- Maklaya Str., Moscow 117198, Russian Federation
Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 1-52 Leninskiye Gory, GSP-1, Moscow 119991, Russian Federation
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