Abstract
This paper considers a heterogeneous M/G/2 queue. The service times at server 1 are exponentially distributed, and at server 2 they have a general distribution B(⋅). We present an exact analysis of the queue length and waiting time distribution in case B(⋅) has a rational Laplace–Stieltjes transform. When B(⋅) is regularly varying at infinity of index −ν, we determine the tail behaviour of the waiting time distribution. This tail is shown to be semi-exponential if the arrival rate is lower than the service rate of the exponential server, and regularly varying at infinity of index 1−ν if the arrival rate is higher than that service rate.
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References
J. Abate, G.L. Choudhury and W. Whitt, Waiting time tail probabilities in queues with long-tail service-time distributions, Queueing Systems 16 (1994) 311-338.
J. Abate and W.Whitt, Asymptotics for M/G/1 low-priority waiting-time tail probabilities, Queueing Systems 25 (1997) 173-233.
R. Agrawal, A. Makowski and P. Nain, On a reduced load equivalence for fluid queues under subexponentiality, Queueing Systems 33 (1999) 5-41.
N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation (Cambridge Univ. Press, Cambridge, 1987).
S.C. Borst, O.J. Boxma and P.R. Jelenković, Generalized processor sharing with long-tailed traffic sources, in: Teletraffic Engineering in a Competitive World, Proceedings of ITC-16, Edinburgh, UK, eds. P. Key and D. Smith (North-Holland, Amsterdam, 1999) pp. 345-354.
S.C. Borst and A.P. Zwart, A reduced-peak equivalence for queues with a mixture of light-tailed and heavy-tailed input flows, SPOR-Report 2000-04, Eindhoven University of Technology (2000).
O.J. Boxma and J.W. Cohen, Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions, Queueing Systems 33 (1999) 177-204.
O.J. Boxma, Q. Deng and J.A.C. Resing, Polling systems with regularly varying service and/or switchover times, Adv. Performance Anal. 3 (2000) 71-107.
J.W. Cohen, Some results on regular variation for distributions in queueing and fluctuation theory, J. Appl. Probab. 10 (1973) 343-353.
J.W. Cohen, The Single Server Queue, 2nd ed. (North-Holland, Amsterdam, 1982).
J.W. Cohen, A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution, J. Appl. Math. Stochastic Anal. 11 (1998) 339-355.
T. Daniëls, Asymptotic behaviour of queueing systems, Ph.D. thesis, Universiteit Antwerpen (1999).
G. Doetsch, Introduction to the Theory and Applications of the Laplace Transformation (Springer, New York, 1974).
V. Dumas and A. Simonian, Asymptotic bounds for the fluid queue fed by sub-exponential on/off sources, Adv. in Appl. Probab. 32 (2000) 244-255.
S.G. Foss and D.A. Korshunov, On waiting time distribution in GI/GI/2 queue system with heavy tailed service times, Unpublished manuscript (2000).
R. Haji and G.F. Newell, A relation between stationary queue length and waiting time distributions, J. Appl. Probab. 8 (1971) 617-620.
J. Kiefer and J. Wolfowitz, On the characteristics of the general queueing process with applications to random walk, Ann. Math. Statist. 27 (1956) 147-161.
C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, An integral equation approach to the M/G/2 queue, Oper. Res. 38 (1990) 506-518.
A.G. Pakes, On the tails of waiting-time distributions, J. Appl. Probab. 12 (1975) 555-564.
S. Resnick and G. Samorodnitsky, Steady state distribution of the buffer content for M/G/∞ input fluid queues, Report No. 1242, Department of ORIE, Cornell University (1999).
A. Scheller-Wolf, Further delay moment results for FIFO multiserver queues, Queueing Systems 34 (2000) 387-400.
A. Scheller-Wolf and K. Sigman, Delay moments for FIFO GI/GI/s queues, Queueing Systems 25 (1997) 77-95.
A. Shwartz and A. Weiss, Large Deviations for Performance Analysis (Chapman and Hall, London, 1995).
W.G.L. Sutton, The asymptotic expansion of a function whose operational equivalent is known, J. London Math. Soc. 9 (1934) 131-137.
E.G. Titchmarsh, The Theory of Functions (Oxford Univ. Press, London, 1952).
W. Whitt, The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution, Queueing Systems 36 (2000) 71-87.
A.P. Zwart, S.C. Borst and M. Mandjes, Exact asymptotics for fluid queues fed by multiple heavytailed on-off flows, SPOR-Report 2000-14, Eindhoven University of Technology, to appear in Ann. Appl. Probab.; shortened version in: Proc. of Infocom 2001, Anchorage, AK, 2000, pp. 279-288.
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Boxma, O., Deng, Q. & Zwart, A. Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers. Queueing Systems 40, 5–31 (2002). https://doi.org/10.1023/A:1017913826973
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DOI: https://doi.org/10.1023/A:1017913826973