A retrial queue with redundancy and unreliable server. (English) Zbl 0817.60093

This paper introduces a new version of the unreliable \(M/G/1/1\) retrial queue and obtains, amongst other things, using the theory of piecewise Markovian processes, the analogue of the Pollaczek-Khinchin formula for such a queue.


60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
90B25 Reliability, availability, maintenance, inspection in operations research
90C40 Markov and semi-Markov decision processes
Full Text: DOI


[1] A. Aissani, Unreliable queueing with repeated orders, Microelectron. Reliab. 33 (1993) 2093-2106. · Zbl 0789.60071
[2] A. Aissani,Stochastic Models of Reliability Theory (Office des Publications Universitaires, Algiers, 1992), in French.
[3] A. Aissani, Heavy loading approximation of the unreliable queue with repeated orders, in:Colloque ?Méthodes et outils d’aide á la décision?, Bejaia (1992) pp. 97-102.
[4] A. Aissani, Influence de la fiabilité des serveurs sur la distribution du nombre de clients en ?Orbite? dans un systémeM/G/1/0 avec rappels, Technologies Avancées 2 (1991) 23-28.
[5] A. Aissani, Queueing systems with a single unreliable server, Paper 756 Be-D83, Bielorussian Institute for Scientific and Technical Information, Minsk (1983).
[6] A.M. Alexandrov, A queueing system with repeated orders, Izv. Akad. Nauk SSSR, Tekhn. Kibern. 2 (1974) 86-89.
[7] R. Barlow and F. Proshan,Mathematical Theory of Reliability (Wiley, New York, 1965).
[8] F. Beichelt and. P. Franken,Reliability and Maintenance. Mathematical Methods (Nauka, Moscow, 1988), in Russian (German edition: Verlag Technik, Berlin, 1983).
[9] B.S. Dhillon and S.M. Rayapati, Human error modelling of parallel and standby redundant systems, Int. J. Syst. Sci. 19 (1988) 589-611. · Zbl 0641.60098
[10] M.V. Burlakov, Situational control in non-Markov queueing systems, Auto. Remote Contr. 50 (1989) 637-646. · Zbl 0707.60094
[11] Q.H. Choo and B.W. Conolly, New results in the theory of repeated orders queueing systems, J. Appl. Prob. 16 (1979) 631-639. · Zbl 0418.60088
[12] J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecomm. Rev. 18 (1957) 49-100.
[13] G.I. Falin, Multiphase servicing in a single-channel system for automation of experiments with repeated calls, in:Problems of Automation of Scientific Investigation in Radio Engineering and Electronics (USSR Academy of Science, Moscow, 1975).
[14] G.I. Falin, Aggregate arrival of customers in one-line system with repeated calls, Ukrainian Math. J. 28 (1976) 437-440. · Zbl 0361.60086
[15] G.I. Falin, A. single-line system with repeated orders, Izv. Akad. Nauk SSSR, Tekhn. Kibern. 2 (1980) 76-83. · Zbl 0437.60073
[16] G.I. Falin, A survey of retrial queues, Queueing Syst. 7 (1990) 127-168. · Zbl 0709.60097
[17] M.S. Finkelstein,Reliability of Systems with Imperfect Repair (Znanie, Moscow, 1991), in Russian.
[18] M. Finkelstein, Some notes on wear-dependent systems, Microelectron. Reliab. 33 (1993) 115-117.
[19] P. Franken, B. Kirstein and A. Sreller, Reliability analysis of complex systems with repair, Elektron. Informationsverarb. Kybernet. 20 (1984) 407-422. · Zbl 0583.60090
[20] B.V. Gnedenko and I.N. Kovalenko,Introduction to Queueing Theory (Israel Program for Scientific Translations, 1969).
[21] B. Gnedenko and M. Schneps-Schneppe, Statistical problems in teletraffic theory,7th Int. Teletraffic Congr., Stockholm (1973).
[22] B.V. Gnedenko, Y.K. Beliaev and A.D. Soloviev,Mathematical Methods of Reliability (Wiley, 1969).
[23] L.R. Goel, R.K. Agnihotri and R. Gupta, Stochastic analysis of a two-unit warm standby system with fault detection and inspection, Microelectron. Reliab. 30 (1990) 61-64.
[24] B.S. Greenberg,M/G/1 queueing systems with returning customers, J. Appl. Prob. 26 (1989) 152-163. · Zbl 0672.60094
[25] C. Co Henry and A. Araar, Maintenance ofM/G/1 production system, Int. J. Prod. Res. 29 (1991) 2135-2149. · Zbl 0744.90032
[26] L.E. Hsu and C.S. Tapiero, Maintenance of an unreliableM/G/1 queue-like job shop, Queueing Syst. 2 (1987). · Zbl 0652.90053
[27] J. Keilson, J. Cozzolino and H. Young, A service system with unfilled requests repeated, Oper. Res. 16(1968) 1126-1137. · Zbl 0165.52703
[28] F.P. Kelly, Auto-repeat facilities and telephone network performance, J. Roy. Statist. Soc. B 48 (1985) 123-132. · Zbl 0595.60088
[29] L. Kosten, On the influence of repeated calls in the theory of probabilities of blocking, De Ingenieur 59 (1947), in Dutch.
[30] A. Kumar and M. Agarwal, A review of standby redundant systems, IEEE Trans. Reliab. R-20 (1980) 290-294. · Zbl 0451.90071
[31] A. Lesanovsky, Stochastic ordering in a two-unit redundant system with three states of units, Math. Operationsforsch. Statist. Optim. 15 (1984) 623-636. · Zbl 0553.90052
[32] K.C. Madan, A single channel queue with bulk service subject to interruption, Microelectron. Reliab. 29 (1989) 813-818.
[33] I.S. Mikadze and R.V. Kakubav, A redundant queueing system with delay, Kibernetika (Kiev) 4 (1984) 97-101.
[34] I.S. Mikadze, A periodically controllable service system with an unreliable device, Cybernetics 24 (1988) 68-75. · Zbl 0679.60091
[35] G.S. Mokaddis and Kh.M. El-Said, Two models for two-unit warm standby redundant system with two different kinds of repair and tolerable time, Microelectron. Reliab. 30 (1990) 327-374.
[36] K. Murari and V. Goyal, Cost analysis of a two-unit warm standby reliability system with two-types of repair facilities, Microelectron. Reliab. 25 (1985) 681-689.
[37] T.E. Phipps Jr, Machine repair as a priority waiting line problem, Oper. Res. 76 (1966).
[38] S.K. Srinivasan and R. Subramanian,Probabilistic Analysis of Redundant Systems, Lecture Notes in Economic and Mathematical Systems (Springer, Berlin/Heidelberg/New York, 1980). · Zbl 0512.90034
[39] S.H. Srinivasan, A standby redundant model with non-instantaneous switchover, IEEE Trans. Reliab. R-17 (1968) 175-178.
[40] R. Subramanian and R. Natarasar, A two-unit system with imperfect repair, Math. Oper. Statist., Ser. Optimization 11 (1980) 617-622. · Zbl 0515.90039
[41] J.G.C. Templeton, Editorial introduction, Queueing Syst. 7 (1990) 125-126.
[42] S.G. Tochenaya, Priority queue with unreliable server and redundant unit, in:Teletraffic theory and Information Networks (Nauka, Moscow, 1977), in Russian.
[43] S.G. Tochenaya, Virtual waiting time in a queueing system with unreliable server and redundant unit, in:Teletraffic Theory and Networks with Controllable Elements (Nauka, Moscow, 1980), in Russian.
[44] T. Yang and J.G.C. Templeton, A survey on retrial queues, Queueing Syst. 2 (1987) 201-233. · Zbl 0658.60124
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.