Systems and Means of Informatics
2020, Volume 30, Issue 3, pp 14-31
STATIONARY CHARACTERISTICS OF THE TWO-NODE MARKOVIAN TANDEM QUEUEING SYSTEM WITH GENERAL RENOVATION
- L.A. Meykhanadzhyan
- I. S. Zaryadov
- T.A. Milovanova
Abstract
Consideration is given to the Markovian tandem queueing system with two finite-capacity heterogeneous nodes, say node 1 and node 2. The output of node 1 is the input into node 2. Each node is a single-server queue with a Poisson incoming flow of customers and service times having Erlang distribution. The service discipline is FIFO (first in, first out). General renovation is implemented in each node which implies that upon a service completion, a customer may remove a random number of customers from the queue (if any is available), with a given probability distribution; removed customers leave the system. Using the matrix-geometric technique, one derives the joint stationary distribution of the nodes' states. A recursive algorithm for computation of the stationary loss probabilities under the head-of-the-queue renovation is also proposed.
[+] References (16)
- Chydzinski, A., and L. Chrost. 2011. Analysis of AQM queues with queue size based packet dropping. Int. J. Appl. Math. Comp. 21(3):567-577. doi: 10.2478/v10006- 011-0045-7.
- Zaryadov, I. S., R. V. Razumchik, and T.A. Milovanova. 2016. Stationary waiting time distribution in G/M/n/r with random renovation policy. Comm. Com. Inf. Sc. 678:418-429. doi: 10.1007/978-3-319-51917-3^1.
- Chydzinski, A., and P. Mrozowski. 2016. Queues with dropping functions and general arrival processes. PLoS ONE 11 (3):e0150702. doi: 10.1371/journal.pone.0150702.
- Konovalov, M., and R. Razumchik. 2018. Comparison of two active queue management schemes through the M/D/l/N queue. Informatika i ee Primeneniya - Inform. Appl. 12(4): 9-15. doi: 10.14357/19922264180402.
- Zaryadov, I., E. Bogdanova, T. Milovanova, S. Matushenko, and D. Pyatkina. 2018. Stationary characteristics of the GI/M/l queue with general renovation and feedback. 10th Congress (International) on Ultra Modern Telecommunications and Control Systems and Workshops. IEEE. 1-6. doi: 10.1109/icumt.2018.8631244.
- Konovalov, M. G., and R. V. Razumchik. 2018. Numerical analysis of improved access restriction algorithms in a Gl/G/l/N system. J. Commun. Technol. El. 63(6):616- 625. doi: 10.1134/S1064226918060141.
- Zaryadov, I. S., L. A. Meykhanadzhyan, and T. A. Milovanova. 2019. Statsionarnye kharakteristiki obsluzhivaniya v sisteme GI/MSP/n/го s obobshchennym obnovleniem [Stationary characteristics of the GI/MSP/n/то queue with general renovation]. Sistemy i Sredstva Informatiki - Systems and Means of Informatics 29(4):50-64. doi: 10.14357/08696527190405.
- Hilquias, V. C. C., I. S. Zaryadov, V. V. Tsurlukov, T. A. Milovanova, E. V. Bogdanova, A. V. Korolkova, and D. S. Kulyabov. 2019. The general renovation as the active queue management mechanism. Some aspects and results. Comm. Com. Inf. Sc. 1141:488-502. doi: 10.1007/978-3-030-36625-4^9.
- Zaryadov, I. S. 2010. The Gl/M/n/infinity queuing system with generalized renovation. Automat. Rem. Contr. 71(4):663-671. doi: 10.1134/S0005117910040077.
- Zhu, Y. 1994. Markovian queueing networks in a random environment. Oper. Res. Lett. 15(1): 11-17. doi: 10.1016/0167-6377(94)90009-4.
- Economou, A. 2005. Generalized product-form stationary distributions for Markov chains in random environment with queueing application. Adv. Appl. Probab. 37(1): 185-211. doi: 10.1239/aap/1113402405.
- Vishnevskii, V. M., A. N. Dudin, and V. I. Klimenok. 2018. Stokhasticheskie sistemy s korrelirovannymi potokami. Teoriya i primenenie v telekommunikatsionnykh setyakh [Stochastic systems with correlated streams. Theory and applications in telecommunication networks]. Moscow: Tekhnosfera. 564 p.
- Bocharov, P. P., and A. V. Pechinkin. 1995. Teoriya massovogo obsluzhivaniya [Queueing theory]. Moscow: RUDN. 529 p.
- Bocharov, P.P. 1985. A queueing system of limited capacity with distributions of phase type depending on the queue state. Automat. Rem. Contr. 46:1229-1236.
- Bocharov, P. P., C. D'Apice, A. V. Pechinkin, and S. Salerno. 2003. The stationary characteristics of the G/MSP/1/r queueing system. Automat. Rem. Contr. 64(2):288- 301. doi: 10.1023/A:1022219232282.
- Pechinkin, A. V., and R. V. Razumchik. 2018. Sistemy massovogo obsluzhivaniya v diskretnom vremeni [Discrete time queuing systems]. Moscow: Fizmatlit. 432 p.
[+] About this article
Title
STATIONARY CHARACTERISTICS OF THE TWO-NODE MARKOVIAN TANDEM QUEUEING SYSTEM WITH GENERAL RENOVATION
Journal
Systems and Means of Informatics
Volume 30, Issue 3, pp 14-31
Cover Date
2020-11-10
DOI
10.14357/08696527200302
Print ISSN
0869-6527
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
queueing system; tandem; general renovation; queue management
Authors
L.A. Meykhanadzhyan  , I. S. Zaryadov  ,  , and T.A. Milovanova
Author Affiliations
Department of Data Analysis and Machine Learning, Financial University under the Government of the Russian Federation, 49 Leningradsky Prosp., Moscow 125993, Russian Federation
Peoples' Friendship University of Russia (RUDN University), 6 Miklukho- Maklaya Str., Moscow 117198, 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
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