On a system with retrial queue and unreliable server

TitleOn a system with retrial queue and unreliable server
Publication TypeJournal Article
Year of Publication2020
AuthorsLebedev, EA, Sharapov, MM, Livinska, HV
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
SectionInformation Science and Cybernetics
Date Published9/2020

We consider a model of retrial queue with one unreliable server whose lifetime is an exponentially distributed random variable with the known failure rate. A two-dimensional Markov chain defines the service process in the system. Its first component indicates the number of sources of repeated calls, and the second one fixes the status of the server at the current time: the server is busy, free, and ready for maintenance or out of order. The main feature of the considered system is that the input flow rate depends on the size of the queue of repeated calls. Each of the sources of repeated calls can generate a call with the same rate. If a primary or repeated call arrives into the system and finds the server idle, its service begins immediately. If the server is busy, the call is directed to the orbit and becomes a source of retrial calls. For the service process, a condition for the existence of a stationary regime and vector-matrix formulas are found. These formulas express stationary probabilities through the model parameters in the explicit form. To control the accuracy of calculations using these formulas, an estimate of the remainder of the series is obtained, which sets the normalizing constant. The rate of the remainder decreasing to zero has an exponential upper estimation. In the case where the input flow is the Poisson one, the exact expression is obtained for a normalizing constant. The application of the obtained results is demonstrated by numerical examples, which show the dependence of the blocking probability in the stationary regime on the parameters of the system. The obtained results can be used to solve optimization problems in the class of threshold strategies.

Keywordsergodicity condition, matrix-vector representation, normalizing constant, retrial queue, stationary regime, unreliable server

1. Falin, G. I., Templeton, J. G. C. (1997). Retrial queues. London: Chapman & Hall.
2. Yang, T. & Templeton, J.G.C. (1987). A survey on retrial queues. Queueing Systems, No. 2, pp. 201-233.
3. Artalejo, J. R. & Gomez-Corral, A. (2008). Retrial queueing systems. A computational approach. Berlin: Sprin ger
4. Nazarov, A., Sztrik, J., Kvach, A. (2018, september). A survey of recent results in finite-source retrial queues with collisions. Information Technologies and Mathematical Modelling. Queueing Theory and Applications. 17th International conference and 12th workshop on Retrial Queues and Related Topics (pp. 10-15). Cham: Springer.
5. Lebedev, E. A. & Ponomarev, V. D. (2011). Retrial queues with variable service rate. Cybernetics and Systems Analysis, 47, No. 2, pp. 434-441.
6. Gomez-Corral, A. & Ramalhoto, M. F. (1999). The stationary distribution of a Markovian process arising in the theory of multiserver retrial queueing systems. Mathematical and Computer Modelling. 30, pp. 141-158.
7. Walrand, J. (1988). Introduction to Quening Networks. New York: Prentice-Hall.
8. Thiruvengadam, K. (1963). Queueing with breakdowns. Operations Research, 11, pp. 62-71.
9. Li, W., Shi, D. and Chao, X. (1997). Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations. J. of Applied Probability, 34, pp. 546-555.
10. Artalejo, J. R. (1994). New results in retrial queueing systems with breakdowns of the servers. Statistica Neerlandica, 48, pp. 23-36.
11. Wartenhorst, P. (1995). N parallel queueing systems with server breakdown and repair. European J. Operational Research, Elsevier, 82(2), pp. 302-322.
12. Vinod, B. (1985). Unreliable queueing systems. Computers and Operations Research, 12, pp. 322-340.
13. Wang, J., Cao, J. & Li, Q. (2001). Reliability analysis of the retrial queue with server breakdowns and re pairs. Queueing Systems, No. 4, pp. 363-380.
14. Artalejo, J. & Falin, G. (2002). Standard and retrial queueing systems: a comparative analysis. Revista Matemática Complutense, 15, No. 1, pp. 101-129.