Abstract
The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1−p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.
Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.
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References
F. Baskett, K.M. Chandy, R.R. Muntz and F.G. Palacios, Open, closed, and mixed networks of queues with different classes of customers, J. ACM 22 (1975) 248–260.
J.L. van den Berg, Simple approximations for second moment characteristics of the sojourn time in theM/G/1 processor sharing queue, in:Messung, Modellierung und Bewertung von Rechensystemen und Netzen, eds. G. Stiege and J.S. Lie (Springer, Berlin, 1989) pp. 105–120.
J.L. van den Berg, Sojourn times in feedback and processor sharing queues, Ph.D. Thesis, Centre for Mathematics and Computer Science, Amsterdam (1990).
J.L. van den Berg and O.J. Boxma, Sojourn times in feedback queues, in:Operations Research Proceedings 1988, eds. D. Pressmar et al. (Springer, Berlin, 1989) pp. 247–257.
J.L. van den Berg and O.J. Boxma, Sojourn times in feedback and processor sharing queues, in:Teletraffic Science for New Cost-Effective Systems, Networks and Services, ed. M. Bonatti (North-Holland, Amsterdam, 1989) pp. 1467–1475.
J.L. van den Berg, O.J. Boxma and W.P. Groenendijk, Sojourn times in theM/G/1 queue with deterministic feedback, Stochastic Models 5 (1989) 115–129.
O.J. Boxma and J.W. Cohen, TheM/G/1 queue with permanent customers, IEEE J. Sel. Areas Commun. SAC-9 (1991) 179–184.
O.J. Boxma and H. Daduna, Sojourn times in queueing networks, in:Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (North-Holland, Amsterdam, 1990) pp. 401–450.
E. Çinlar,Introduction to Stochastic Processes (Prentice-Hall, Englewood Cliffs, NJ, 1975).
E.G. Coffman, R.R. Muntz and H. Trotter, Waiting time distributions for processor-sharing systems, J. ACM 17 (1970) 123–130.
J.W. Cohen, The multiple phase service network with generalized processor sharing, Acta Informatica 12 (1979) 245–284.
R.L. Disney, A note on sojourn times inM/G/1 queues with instantaneous, Bernoulli feedback, Naval Res. Log. Quarterly 28 (1981) 679–684.
R.L. Disney and P.C. Kiessler,Traffic Processes in Queueing Networks — A Markov Renewal Approach (The Johns Hopkins University Press, Baltimore, 1987).
R.L. Disney and D. König, Queueing networks: a survey of their random processes, SIAM Rev. 27 (1985) 335–403.
B.T. Doshi and J.S. Kaufman, Sojourn time in anM/G/1 queue with Bernoulli feedback, in:Queueing Theory and its Applications — Liber Amicorum for J. W. Cohen, eds. O.J. Boxma and R. Syski (North-Holland, Amsterdam, 1988) pp. 207–233.
W. Feller,An Introduction to Probability Theory and Its Applications, vol. 1 (Wiley, New York, 1950).
R.D. Foley and G.A. Klutke, Stationary increments in the accumulated work process in processor-sharing queues, J. Appl. Prob. 26 (1989) 671–677.
J.J. Hunter, Sojourn time problems in feedback queues, Queueing Systems 5 (1989) 55–76.
L. Kleinrock,Queueing Systems, Vol. 1 (Wiley, New York, 1975).
L. Kleinrock,Queueing Systems, Vol. 2 (Wiley, New York, 1976).
G.A. Klutke, P.C. Kiessler and R.L. Disney, Interoutput times in processor sharing queues with feedback, Queueing Systems 3 (1988) 363–376.
S.S. Lam and A.U. Shankar, A derivation of response time distributions for a multi-class feedback queueing system, Perform. Eval. 1 (1981) 48–61.
T.J. Ott, The sojourn-time distribution in theM/G/1 queue with processor sharing, J. Appl. Prob. 21 (1984) 360–378.
J.A.C. Resing, G. Hooghiemstra and M.S. Keane, TheM/G/1 processor sharing queue as the almost sure limit of feedback queues, J. Appl. Prob. 27 (1990) 913–918.
R. Schassberger, A new approach to theM/G/1 processor-sharing queue, Adv. Appl. Prob. 16 (1984) 202–213.
L. Takács, A single-server queue with feedback, Bell Syst. Tech. J. 42 (1963) 505–519.
H.C. Tijms,Stochastic Modeling and Analysis (Wiley, New York, 1986).
D. Voelker and G. Doetsch,Die Zweidimensionale Laplace-Transformation (Birkhäuser, Basel, 1950).
P.R. de Waal, An approximation method for a processor sharing queue with controlled arrivals and a waitbuffer, in:Proc. 28th IEEE Conf. on Decision and Control, Tampa, FL (1989).
J. Walrand,An Introduction to Queueing Networks (Prentice-Hall, Englewood Cliffs, NJ, 1988).
S.F. Yashkov, A derivation of response time distribution for aM/G/1 processor sharing queue, Problems Contr. Info. Theory 12 (1983) 133–148.
S.F. Yashkov, Processor-sharing queues: some progress in analysis, Queueing Systems 2 (1987) 1–17.
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van den Berg, J.L., Boxma, O.J. TheM/G/1 queue with processor sharing and its relation to a feedback queue. Queueing Syst 9, 365–401 (1991). https://doi.org/10.1007/BF01159223
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DOI: https://doi.org/10.1007/BF01159223