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Time-dependent analysis of M/G/1 vacation models with exhaustive service

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Abstract

We analyze the time-dependent process in severalM/G/1 vacation models, and explicitly obtain the Laplace transform (with respect to an arbitrary point in time) of the joint distribution of server state, queue size, and elapsed time in that state. Exhaustive-serviceM/G/1 systems with multiple vacations, single vacations, an exceptional service time for the first customer in each busy period, and a combination ofN-policy and setup times are considered. The decomposition property in the steady-state joint distribution of the queue size and the remaining service time is demonstrated.

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References

  1. R.B. Cooper, Queues served in cyclic order: waiting times, Bell Syst. Tech. J. 49 (1970) 399–413.

    Google Scholar 

  2. B.T. Doshi, A note on stochastic decomposition in aGI/G/1 queue with vacations or setup times, J. Appl. Prob. 22 (1985) 419–428.

    Article  Google Scholar 

  3. B.T. Doshi, Queueing systems with vacations — a survey, Queueing Systems 1 (1986) 29–66.

    Article  Google Scholar 

  4. B.T. Doshi, Single-server queues with vacations, in:Stochastic Analysis of Computer and Communication Systems, H. Takagi (ed.) (Elsevier Science, Amsterdam, 1990) to appear.

    Google Scholar 

  5. B.T. Doshi, Conditional and unconditional distributions forM/G/1 type queues with server vacations, preprint (1989).

  6. S.W. Fuhrmann and R.B. Cooper, Stochastic decompositions in theM/G/1 queue with generalized vacations, Oper. Res. 33 (1985) 1117–1129.

    Google Scholar 

  7. N.K. Jaiswal,Priority Queues (Academic Press, New York, 1968).

    Google Scholar 

  8. J. Keilson and A. Kooharian, On time dependent queueing processes, Ann. Math. Stat. 31 (1960) 104–112.

    Google Scholar 

  9. J. Keilson and R. Ramaswamy, The backlog and depletion-time process forM/G/1 vacation models with exhaustive service discipline, J. Appl. Prob. 25 (1988) 404–412.

    Article  Google Scholar 

  10. J. Keilson and L.D. Servi, Dynamics of theM/G/1 vacation model, Oper. Res. 35 (1987) 575–582.

    Google Scholar 

  11. L. Kleinrock and M.O. Scholl, Packet switching in radio channels: new conflict-free access schemes, IEEE Trans. Commun. COM-28 (1980) 1015–1029.

    Article  Google Scholar 

  12. H. Levy and L. Kleinrock, A queue with starter and a queue with vacations: delay analysis by decomposition, Oper. Res. 34 (1986) 426–436.

    Google Scholar 

  13. Y. Levy and U. Yechiali, Utilization of idle time in anM/G/1 queueing system, Management Sci. 22 (1975) 202–211.

    Google Scholar 

  14. D.L. Minh, Transient solutions for some exhaustiveM/G/1 queues with generalized independent vacations, Europ. J. Oper. Res. 36 (1988) 197–201.

    Article  Google Scholar 

  15. M.F. Neuts, Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues, Advan. Appl. Prob. 18 (1986) 952–990.

    Article  Google Scholar 

  16. M. Scholl and L. Kleinrock, On theM/G/1 queue with rest periods and certain service-independent queueing disciplines, Oper. Res. 31 (1983) 705–719.

    Article  Google Scholar 

  17. J.G. Shanthikumar, On stochastic decomposition inM/G/1 type queues with generalized server vacations, Oper. Res. 36 (1988) 566–569.

    Google Scholar 

  18. C.E. Skinner, A priority queueing system with server-walking time, Oper. Res. 15 (1967) 278–285.

    Google Scholar 

  19. B.W. Stuck and E. Arthurs,A Computer and Communications Network Performance Analysis Primer (Prentice-Hall, Englewood Cliffs, NJ, 1985).

    Google Scholar 

  20. L. Takács,Introduction to the Theory of Queues (Oxford University Press, New York, 1962).

    Google Scholar 

  21. H. Takagi,Analysis of Polling Systems (The MIT Press, Cambridge, MA, 1986).

    Google Scholar 

  22. H. Takagi, Queueing analysis of polling models, ACM Computing Surveys 20 (1988) 5–28.

    Article  Google Scholar 

  23. J. Teghem, Jr., Control of the service process in a queueing system, Europ. J. Oper. Res. 23 (1986) 141–158.

    Article  Google Scholar 

  24. P.D. Welch, On a generalizedM/G/1 queueing process in which the first customer of each busy period receives exceptional service, Oper. Res. 12 (1964) 736–752.

    Google Scholar 

  25. D.M.G. Wishart, An application of ergodic theorems in the theory of queues,Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability, vol. 2 (University of California Press, Berkeley and Loss Angeles, 1961) pp. 581–592.

    Google Scholar 

  26. M. Yadin and P. Naor, Queueing systems with a removable service station, Oper. Res. Quarterly 14 (1963) 393–405.

    Google Scholar 

Download references

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  1. IBM Research, Tokyo Research Laboratory, Sanbancho YS Building, 5-11 Sanban-cho, Chiyoda-ku, 102, Tokyo, Japan

    Hideaki Takagi

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  1. Hideaki Takagi
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Takagi, H. Time-dependent analysis of M/G/1 vacation models with exhaustive service. Queueing Syst 6, 369–389 (1990). https://doi.org/10.1007/BF02411484

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  • DOI: https://doi.org/10.1007/BF02411484

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