Reference for citation: Peschansky А. I. Stationary characteristics of an unlimited queue single-channel system with regard to service quality control. Journal of Instrument Engineering. 2023. Vol. 66, N 9. P. 715—730 (in Russian). DOI: 10.17586/0021-3454-2023-66-9-715-730.

Abstract. A service system is studied, in which, upon completion of servicing of each application, instant control of the quality of its service is carried out. In case of unsatisfactory quality, the application is sent for repeated services, which are carried out until the quality is considered satisfactory. A semi-Markov process of system functioning is constructed, and the stationary distribution of the embedded Markov chain is found. Analogues of the Pollaczek — Khinchin formula for generating functions of the stationary probabilities are obtained. The system stationary characteristics depending on the probability of quality service are determined: the stationary distribution of the queue over time, the average stationary duration of stay in a state, the average number of requests in the queue and system, the average duration of the request stay in the queue and system.

Keywords: single-server queuing system, infinite queue, quality control, re-service, stationary characteristics: final probabilities, sojourn times in states, average number of requests and sojourn time in queue and system

References:

1. Gnedenko B.V., Kovalenko I.N. Vvedeniye v teoriyu massovogo obsluzhivaniya (Introduction to Queuing Theory), Moscow, 1987, 336 р. (in Russ.)
2. Bocharov P.P., Pechinkin A.V. Teoriya massovogo obsluzhivaniya (Queuing Theory), Moscow, 1995, 529 р. (in Russ.)
3. Klimov G.P. Stokhasticheskiye sistemy obsluzhivaniya (Stochastic Queuing Systems), Moscow, 1966, 244 р. (in Russ.)
4. Kovalenko I.N. Teoriya massovogo obsluzhivaniya. Itogi nauki. Seriya Teoriya veroyatnostey. Matematicheskaya statistika. Teoreticheskaya kibernetika (Theory of Queuing. The Results of Science. Series Theory of Probability. Math statistics. Theoretical cybernetics), Moscow, 1971, рр. 5–109. (in Russ.)
5. Butko T.K. Primeneniye analiticheskikh metodov v teorii veroyatnostey (Application of Analytical Methods in Probability Theory), Kyiv, 1983, рр. 17–27. (in Russ.)
6. Chen Y., Whitt W. Queueing Systems, 2020, vol. 94, рр. 327–356, DOI: 10.1007/s11134-020-09649-9.
7. Korolyuk V.S., Turbin A.F. Protsessy markovskogo vosstanovleniya v zadachakh nadezhnosti sistem (Markov Reconstruction Processes in System Reliability Problems), Kyiv, 1982, 236 р. (in Russ.)
8. Korolyuk V.S. Stokhasticheskiye modeli sistem (Stochastic Systems Models), Kyiv, 1989, 208 р. (in Russ.)
9. Korlat A.N., Kuznetsov V.N., Novikov M.I., Turbin A.F. Polumarkovskiye modeli vosstanavlivayemykh sistem i sistem massovogo obsluzhivaniya (Semi-Markovian Models of Recoverable Systems and Queuing Systems), Kishinev, 1991, 276 р. (in Russ.)
10. Peschansky A.I., Kovalenko A.I. Sistemnyye tekhnologii (System Technologies), Dnepropetrovsk, 2011, no. 4(75), pp. 129–139. (in Russ.)
11. Peschansky A.I. Vestnik Sevastopol'skogo Gosudarstvennogo Tekhnologicheskogo Universiteta: Avtomatizatsiya protsessov i upravleniye (Bulletin of the Sevastopol State Technological University: Process Automation and Control), 2012, no. 125, pp. 55–62. (in Russ.)
12. Peschansky A.I. Applied Mathematics, 2011, no. 4(2), pp. 403–409, DOI: 10.4236/am.2011.24049.
13. Peschansky A.I. Vestnik Sevastopol'skogo Natsional'nogo Tekhnicheskogo Universiteta: seriya Informatika, elektronika, svyaz' (Bulletin of the Sevastopol National Technical University. Series Informatics, Electronics, Communications), 2011, no. 114, pp. 47–52. (in Russ.)
14. Kendall D. Ann. Math. Statistics, 1953, no. 3(24), pp. 338–354.
15. Beichelt F., Franken P. Zuverlässigkeit und Instanphaltung, Mathematische Methoden, Berlin, VEB Verlag Technik, 1983, 392 p.
16. Raynshke K., Ushakov I.A. Otsenka nadezhnosti sistem s ispol'zovaniyem grafov (System Reliability Assessment Using Graphs), Moscow, 1988, 208 р. (in Russ.)