Queueing systems with vacations - a survey. (English) Zbl 0655.60089

In this excellent review article single-server queueing models with vacations are discussed. The author aims at (and succeeds in) providing a methodological overview with the objective to illustrate how the seemingly diverse mix of problems where vacation models arise in some form is closely related in structure and can be understood in a common framework.
After describing a large number of problems which can be addressed by vacation-type models, the author selects two models for detailed treatment. For these models, which are generalizations of the standard GI/G/1 queue, a variety of techniques used (to obtain decomposition results, in particular) and the scope and limitations of each is discussed. It is then shown how understanding the behaviour of these two models helps to analyse other vacation models using similar techniques. Some applications and a discussion of some unsolved problems conclude the paper.
Reviewer: E.A.van Doorn


60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
Full Text: DOI


[1] O.M.E. Ali and M.F. Neuts, A service system with two stages of waiting and feedback of customers, J. Appl. Prob. 21(1984)414. · Zbl 0544.60088
[2] B. Avi-Itzhak and P. Naor, Some queueing problems with the service station subject to server breakdown, Oper. Res. 10(1962)303. · Zbl 0114.34202
[3] P.H. Brill and M.J.M. Posner, Level crossing in point processes applied to queues: Single server case, Oper. Res. 25(1977)662. · Zbl 0373.60114
[4] R.B. Cooper, Queues served in cyclic order: Waiting times, BSTJ 49(1970)399. · Zbl 0208.22502
[5] R.B. Cooper and G. Murray, Queues served in cyclic order, BSTJ 48(1969)675. · Zbl 0169.20804
[6] P.J. Courtois, The M/G/1 finite capacity queue with delays, IEEE Trans. Commun. COM-28 (1980)165. · Zbl 0422.68004
[7] B.T. Doshi, An M/G/1 queue with a hybrid discipline, BSTJ 62(1983)1251.
[8] B.T. Doshi, A note on stochastic decomposition in a GI/G/1 queue with vacation or set-up times, J. Appl. Prob. 22(1985)419. · Zbl 0566.60090
[9] B.T. Doshi, An M/G/1 queue with variable vacations,Proc. Int. Conf, on Performance Modeling, Sophia Antipolis, France (1985).
[10] M. Eisenberg, Queues with periodic service and changeover time, Oper. Res. 20(1972)440. · Zbl 0245.60073
[11] A. Federgruen and L. Green, Queueing systems with service interruptions, Research Working Paper No. 84-5, Columbia University (1984).
[12] P. Franken, D. Konig, U. Arndt and V. Schmidt, Queues and Point Processes (Academic, New York, 1981).
[13] S. Fuhrmann, A note on the M/G/1 queue with server vacations, Oper. Res. 31(1981)1368.
[14] S.W. Fuhrmann and R.B. Cooper, Stochastic decomposition in an M/G/1 queue with generalized vacations, Oper. Res. 33(1985)1117. · Zbl 0585.90033
[15] S.W. Fuhrmann and R.B. Cooper, Application of decomposition principle in M/G/1 vacation model to two continuum cyclic queueing models (Especially Token Ring LANs) At&T TJ 64(1985)1091.
[16] S.W. Fuhrmann, Symmetric queues served in cyclic order, Oper. Res. Lett. (1986), to appear. · Zbl 0572.60092
[17] D.P. Gaver, Jr., A waiting line with interrupted service, including priorities, J. Roy. Stat. Soc. B24(1962)73. · Zbl 0108.31403
[18] E. Gelenbe and R. Iasnogorodski, A queue with server of walking type (autonomous service), Ann. Inst. Henry Poincare, Vol. XVI (1980) p. 63. · Zbl 0433.60086
[19] D.P. Heyman, Optimal operating policies for M/G/1 queueing systems, Oper. Res. 16(1968) 362. · Zbl 0164.47704
[20] D.P. Heyman, A priority queueing system with server interference, SIAM J. Appl. Math. 17(1969)74. · Zbl 0172.21701
[21] D.P. Heyman, The T-policy for the M/G/1 queue, Management Science 23(1977)775. · Zbl 0357.60022
[22] J. Keilson, Queues subject to service interruptions, Ann. Math. Statist. 33(1962)1314. · Zbl 0104.36805
[23] J. Keilson and L. Servi, Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules, J. Appl. Prob. (1986), to appear. · Zbl 0612.60087
[24] J. Keilson and L. Servi, Blocking probability for M/G/1 vacation systems with occupancy level dependent schedules, Oper. Res. (1986), submitted. · Zbl 0666.60097
[25] O. Kela and U. Yechiali (1985), unpublished work.
[26] T.T. Lee, M/G/1/N queue with vacation time and exhaustive service discipline, Oper. Res. 32(1984)774. · Zbl 0559.90032
[27] T.T. Lee (1984), unpublished work.
[28] T.T. Lee (1984), unpublished work.
[29] A. Lemoine, Limit theorems for generalized single server queues: The exceptional system, SIAM J. Appl. Math. 29(1975)596. · Zbl 0296.60067
[30] Y. Levy and U. Yechiali, Utilization of idle time in an M/G/1 queueing, Management Science 22(1975)202. · Zbl 0313.60067
[31] Y. Levy and U. Yechiali, M/M/S queues with server vacations, INFO 14(1976)153.
[32] H. Levy and L.O. Kleinrock, A queue with starter and a queue with vacations: Delay analysis by decomposition, Oper. Res. (1986), to appear. · Zbl 0611.60092
[33] H. Levy, Ph.D. Thesis, UCLA (1985).
[34] D. Minh, Analysis of the exceptional queueing system by the use of regenerative processes and analytical methods, Math. Oper. Res. 5(1980)147. · Zbl 0428.60097
[35] I.L. Mitrany and B. Avi-Itzhak, A many server queue with service interruptions, Oper. Res. 16(1967)628. · Zbl 0239.60094
[36] R. Morris and Y. Wang, Some results for multi-queue systems with multiple cyclic servers, Performance of Computer-Communication Systems, Zurich (1984) 245.
[37] M. Neuts and D. Lucantoni, A Markovian queue withN servers subject to breakdowns and repairs, Management Science 25(1979)849.
[38] M. Neuts and M.F. Ramalhoto, A service model in which the server is required to search for customers, J. Appl. Prob. 21(1984)157. · Zbl 0531.60089
[39] T.J. Ott, On the M/G/1 queue with additional inputs, J. Appl. Prob. 21(1984)129. · Zbl 0529.60095
[40] A.G. Pakes, A GI/M/1 queue with a modified service mechanism, Ann. Inst. Stat. Math. 24(1972)589. · Zbl 0311.60054
[41] N.H. Prabhu, Stochastic Storage Processes (Springer-Verlag, New York, 1980). · Zbl 0453.60094
[42] M. Scholl and L. Kleinrock, On the M/G/1 queue with rest periods and certain service independent queueing disciplines, Oper. Res. 31(1983)705. · Zbl 0523.60088
[43] J.G. Shanthikumar, Analysis of the control of queues with shorest processing time service discipline, J. Oper. Res. Soc. of Japan (1980) 341. · Zbl 0448.60065
[44] J.G. Shanthikumar, On the buffer behavior with Poisson arrivals, priority service and random service interruptions, IEEE Trans. on Computers C30(1981)781.
[45] J.G. Shanthikumar, Analysis of a single server queue with time and operation depenedent server failures, Adv. in Management Science 1(1982)339.
[46] J.G. Shanthikumar, Analyses of priority queues with server control, Working Paper No. 83-018, University of Arizona (1983).
[47] F.A. Van Der Duyn, M/G/1 queueing model with vacation times, ZOR 22(1978)95. · Zbl 0378.60089
[48] P.D. Welch, On a generalized M/G/1 queueing process in which the first customer of each busy period receives exceptional service, Oper. Res. 12(1964)736. · Zbl 0132.38404
[49] R. Wolff, Poisson arrivals see time averages, Oper. Res. 30(1982)223. · Zbl 0489.60096
[50] S.S. Lavenberg, The steady-state queueing time distribution for the M/G/1 finite capacity queue, Management Science 21(1975)501. · Zbl 0302.60058
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