Abstract
We consider a Jackson network consisting of three first-in-first-out (FIFO)M/M/1 queues. When customers leave the first queue they can be routed to either the second or third queue. Thus, a customer that traverses the network by going from the first to the second to the third queue, can be overtaken by another customer that is routed from the first queue directly to the third. We study the distribution of the sojourn time of a customer through the three node network, in the heavy traffic limit. A three term heavy traffic asymptotic approximation to the sojourn time density is derived. The leading term shows that the nodes decouple in the heavy traffic limit. The next two terms, however, do show the dependence of the sojourn times at the individual nodes and give quantitative measures of the effects of overtaking.
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References
E.G. Coffman, Jr., G. Fayolle and I. Mitrani, Sojourn times in a tandem queue with overtaking: reduction to a boundary value problem, Commun. Statist.-Stochastic Models 2 (1986) 43–65.
G. Fayolle, R. Iasnogorodski and I. Mitrani, The distribution of sojourn times in a queueing network with overtaking: reduction to a boundary value problem, in:Performance '83, eds. A.K. Agrawala and S.K. Tripathi (North-Holland, 1983) pp. 477–486.
R.D. Foley and P.C. Kiessler, Positive correlations in a three-node Jackson queueing network, Adv. Appl. Probab. 21 (1989) 241–242.
F.P. Kelly, The dependence of sojourn times in closed queueing networks, in:Mathematical Computer Performance and Reliability, eds. G. Iazeolla, P.J. Courtois, and A. Hordijk (North-Holland, 1984) pp. 111–121.
P.C. Kiessler, B. Melamed, M. Yadin and R.D. Foley, Analysis of a three node queueing network, Queueing Systems 3 (1988) 53–72.
C. Knessl and J.A. Morrison, Heavy traffic analysis of the sojourn time in tandem queues with overtaking, SIAM J. Appl. Math., to appear.
C. Knessl and C. Tier, Approximations to the moments of the sojourn time in a tandem queue with overtaking, Commun. Statist. — Stochastic Models 6 (1990) 499–524.
J. McKenna, Asymptotic expansions of the sojourn time distribution functions in closed, product-form queueing networks, J. ACM 34 (1987) 985–1003.
J.A. Morrison, Response-time distribution for a processor-sharing system, SIAM J. Appl. Math. 45 (1985) 152–167.
J.A. Morrison, Asymptotic analysis of the waiting-time distribution for a large closed processor-sharing system, SIAM J. Appl. Math. 46 (1986) 140–170.
M.I. Reiman, The heavy traffic diffusion approximation for sojourn times in Jackson networks, in:Applied Probability — Computer Science, The Interface, eds. R.L. Disney and T.J. Ott (Birkhäuser, Boston, 1982) pp. 409–421.
M.I. Reiman and B. Simon, Light traffic limits of sojourn time distributions in Markovian queueing networks, Commun. Statist.-Stochastic Models 4 (1988) 191–233.
B. Simon and R.D. Foley, Some results on sojourn times in acyclic Jackson networks, Manag. Sci. 25 (1979) 1027–1034.
W. Whitt, The amount of overtaking in a network of queues, Networks 14 (1984) 411–426.
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Knessl, C., Morrison, J.A. Heavy-traffic analysis of the sojourn time in a three node Jackson network with overtaking. Queueing Syst 8, 165–182 (1991). https://doi.org/10.1007/BF02412248
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DOI: https://doi.org/10.1007/BF02412248