Abstract
We consider two queues in tandem, each with an exponential server, and with deterministic arrivals to the first queue. We obtain an explicit solution for the steady state distribution of the process (N1(t), N2(t), Y(t)), where Nj(t) is the queue length in the jth queue and Y(t) measures the time elapsed since the last arrival. Then we obtain the marginal distributions of (N1(t), N2(t)) and of N2(t). We also evaluate the solution in various limiting cases, such as heavy traffic.
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I.J.B.F. Adan, J. Wessels and W.H.M. Zijm, Analysis of the symmetric shortest queue problem, Comm. Statist. Stochastic Models 6 (1990) 691-713.
I.J.B.F. Adan, J. Wessels and W.H.M. Zijm, Analysis of the asymmetric shortest queue problem, Queueing Systems 8 (1991) 1-58.
I. Adan and Y. Zhao, Analyzing GI/Er/1 queues, Oper. Res. Lett. 19 (1996) 183-190.
J.P.C. Blanc, The relaxation time of two queueing systems in series, Comm. Statist. Stochastic Models 1 (1985) 1-16.
O.J. Boxma and G.J. van Houtum, The compensation approach applied to a 2 × 2 switch, Probab. Engrg. Inform. Sci. 7 (1993) 471-493.
R.L. Dobrushin and E.A. Pechersky, Large deviations for tandem queueing systems, J. Appl. Math. Stochastic Anal. 7 (1994) 301-330.
G.J. Foschini, Equilibria for diffusion models for pairs of communicating computers-symmetric case, IEEE Trans. Inform. Theory 28 (1982) 273-284.
A. Ganesh and V. Anantharam, Stationary tail probabilities in exponential server tandems with renewal arrivals, Queueing Systems 22 (1996) 203-248.
C. Knessl and C. Tier, Asymptotic properties of first passage times for tandem Jackson networks I: Buildup of large queue lengths, Comm. Statist. Stochastic Models 11 (1995) 133-162.
C. Knessl and C. Tier, Asymptotic properties of first passage times for two tandem queues: Time to empty the system, Comm. Statist. Stochastic Models 12 (1996) 633-672.
C. Knessl and C. Tier, A diffusion model for two tandem queues with general renewal input, to appear in Comm. Statist. Stochastic Models.
G.F. Newell, Approximate Behavior of Tandem Queues (Springer, New York, 1979).
G.J. van Houtum, New approaches for multi-dimensional queueing systems, Ph.D. thesis, Eindhoven University of Technology (1995).
Y. Zhao and W.K. Grassmann, Queueing analysis of a jockeying model, Oper. Res. 43 (1995) 520-529.
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Knessl, C. An explicit solution to a tandem queueing model. Queueing Systems 30, 261–272 (1998). https://doi.org/10.1023/A:1019169105601
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DOI: https://doi.org/10.1023/A:1019169105601