Reference for citation: Песчанский А. И. Стационарные характеристики системы M/G/1/m с учетом контроля качества обслуживания // Изв. вузов. Приборостроение. 2024. Т. 67, № 2. С. 133—144. DOI: 10.17586/0021-3454-2024-67-2-133-144.

Abstract. A semi-Markov model is constructed to describe the functioning of a single-component servicing system with a storage device of finite capacity, in which the quality of service of requests is monitored. In case of an unsatisfactory result, repeated servicing of the application is carried out until satisfactory quality is achieved. The stationary distribution of the nested Markov chain is found, the stationary characteristics of the system are determined, depending on the probability of high-quality service of requests: stationary distribution of the queue over time, the average stationary sojourn times in states, the average queue length, the average request sojourn time in the queue and in the system.

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

References:

1. Bocharov P.P., Pechinkin A.V. Teoriya massovogo obsluzhivaniya (Queuing Theory), Moscow, 1995, 529 р. (in Russ.)
2. Gnedenko B.V., Kovalenko I.N. Vvedeniye v teoriyu massovogo obsluzhivaniya (Introduction to Queuing Theory), Moscow, 1987, 336 р. (in Russ.)
3. Klimov G.P. Stokhasticheskiye sistemy obsluzhivaniya (Stochastic Queuing Systems), Moscow, 1966, 244 р. (in Russ.)
4. Kovalenko I.N. Itogi Nauki. Seriya "Teoriya Veroyatnostei. Matematicheskaya Statistika. Teoreticheskaya Kibernetika. 1970", Moscow, 1971, рр. 5–109. (in Russ.)
5. Matveev V.F., Ushakov V.G. Sistemy massovogo obsluzhivaniya (Queuing Systems), Moscow, 1984, 240 р. (in Russ.)
6. Borovkov A.A. Veroyatnostnyye protsessy v teorii massovogo obsluzhivaniya (Probabilistic Processes in Queuing Theory), Moscow, 1971, 368 р. (in Russ.)
7. Ivchenko G.I., Kashtanov V.A., Kovalenko I.N. Teoriya massovogo obsluzhivaniya (Queuing theory), Moscow, 1982, 256 р. (in Russ.)
8. Peschansky A.I. Appl. Mathematics, 2011, no. 4(2), pp. 403–409, DOI: 10.4236/am.2011.24049.
9. Peschansky A.I. Vestnik. SevNTU: Ser. Informatika, elektronika, svyaz', 2011, no. 114, pp. 47–52. (in Russ.)
10. Kendall D. Ann. Math. Statistics, 1953, no. 3(24), pp. 338–354.
11. Peschansky A.I. Journal of Instrument Engineering, 2023, no. 9(66), pp. 715–730, DOI: 10.17586/0021-3454-2023-66-9-715-730. (in Russ.)
12. Korolyuk V.S., Turbin A.F. Protsessy markovskogo vosstanovleniya v zadachakh nadezhnosti sistem (Markov Recovery Processes in System Reliability Problems), Kyiv, 1982, 236 р.
13. Korlat A.N., Kuznetsov V.N., Novikov M.I., Turbin A.F. Polumarkovskiye modeli vosstanavlivayemykh sistem i sistem massovogo obsluzhivaniya (Semi-Markov Models of Recoverable Systems and Queuing Systems), Kishinev, 1991, 276 р. (in Russ.)
14. Beichelt F., Franken P. Zuverlässigkeit und Instanphaltung, Mathematische Methoden, Berlin, VEB Verlag Technik, 1983, 392 s.
15. Raynshke K., Ushakov I.A. Otsenka nadezhnosti sistem s ispol'zovaniyem grafov (Assessing the Reliability of Systems Using Graphs), Moscow, 1988, 208 р. (in Russ.)