Abstract
We study the stationary solution of a (max, plus)-linear recursion. Under subexponentiality assumptions on the input to the recursion, we obtain the tail asymptotics of certain (max, plus)-linear functionals of this solution.
(Max, plus)-linear recursions arise from FIFO queueing networks; more specifically, from stochastic event graphs. In the event graph setting, two special cases of our results are of particular interest and have already been investigated in the literature. First, the functional may correspond to the end-to-end sojourn time of a customer. Second, for two queues in tandem, the functional may correspond to the sojourn time in the second queue. Our results allow for more general networks, which we illustrate by studying the tail asymptotics of the resequencing delay due to multi-path routing.
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References
R. Agrawal, A. Makowski, and P. Nain, On a reduced load equivalence for fluid queues under subexponentiality, Queueing Syst., 33 (1999) 5–41.
S. Agrawal and R. Ramaswamy, Analysis of the resequencing delay for M/M/m systems, in Proceedings of the 1987 ACM SIGMETRICS conference on measurement and modeling of computer systems, Banff, Alberta, Canada (1987) 27–35.
S. Asmussen, C. Klüppelberg, and K. Sigman, Sampling at subexponential times, with queueing applications, Stochastic Process. Appl. 79 (1999) 265–286.
S. Asmussen, H. Schmidli, and V. Schmidt, Tail probabilities for non-standard risk and queueing processes with subexponential jumps, Adv. in Appl. Probab. 31 (1999) 422–447.
H. Ayhan, Z. Palmowski, and S. Schlegel, Cyclic queueing networks with subexponential service times, J. Appl. Probab. 41 (2004) 791–801.
F. Baccelli and P. Brémaud, Elements of queueing theory, Springer, Berlin, (2003).
F. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat, Synchronization and linearity, John Wiley & Sons Ltd., Chichester, (1992). Available from http://www.maxplus.org.
F. Baccelli and S. Foss, On the saturation rule for the stability of queues, J. Appl. Probab. 32 (1995) 494–507.
F. Baccelli and S. Foss, Moments and tails in monotone-separable stochastic networks, Ann. Appl. Probab. 14 (2004) 612–650.
F. Baccelli, S. Foss, and M. Lelarge, Asymptotics of subexponential max plus networks: the stochastic event graph case, Queueing Syst. 46 (2004) 75–96.
F. Baccelli, S. Foss, and M. Lelarge, Tails in generalized Jackson networks with subexponential service-time distributions, J. Appl. Probab. 42 (2005) 513–530.
F. Baccelli, E. Gelenbe, and B. Plateau, An end-to-end approach to the resequencing problem, Journal of the ACM, 31 (1984) 474–485.
F. Baccelli, S. Schlegel, and V. Schmidt, Asymptotics of stochastic networks with subexponential service times, Queueing Syst. 33 (1999) 205–232.
O. J. Boxma and Q. Deng, Asymptotic behaviour of the tandem queueing system with identical service times at both queues, Math. Methods Oper. Res. 52 (2000) 307–323.
A. B. Dieker, Reduced-load equivalence for queues with Gaussian input, Queueing Syst. 49 (2005) 405–414.
S. Foss and D. Korshunov, Sampling at a random time with a heavy-tailed distribution, Markov Process. Related Fields, 6 (2000) 543–568.
C. M. Goldie and C. Klüppelberg, Subexponential distributions, in A practical guide to heavy tails (Santa Barbara, CA, 1995), Birkhäuser Boston, Boston, MA, (1998) 435–459.
Y. Han and A. Makowski, Resequencing delays under multipath routing—Asymptotics in a simple queueing model, in Proc. of IEEE INFOCOM, (2006).
T. Huang and K. Sigman, Steady-state asymptotics for tandem, split-match and other feedforward queues with heavy tailed service, Queueing Syst. 33 (1999) 233–259.
A. Jean-Marie and L. Gün, Parallel queues with resequencing, Journal of the ACM, 40 (1993) 1188–1208.
P. Jelenković, P. Momvilović, and B. Zwart, Reduced load equivalence under subexponentiality, Queueing Syst. 46 (2004) 97–112.
M. Lelarge, Fluid limit of generalized Jackson queueing networks with stationary and ergodic arrivals and service times, J. Appl. Probab. 42 (2005) 491–512.
M. Lelarge, Packet reordering in networks with heavy-tailed delays, Tech. Rep. RR-5783, INRIA, (2005).
M. Lelarge, Rare events in networks, PhD thesis, Ecole polytechnique, (2005).
A. G. Pakes, On the tails of waiting-time distributions, J. Appl. Probab. 12 (1975) 555–564.
A. Scheller-Wolf and K. Sigman, Delay moments for FIFO GI/GI/s queues, Queueing Syst. 25 (1997) 77–95.
F. Toomey, Large deviations of products of random topical operators, Ann. Appl. Probab. 12 (2002) 317–333.
N. Veraverbeke, Asymptotic behaviour of Wiener-Hopf factors of a random walk, Stochastic Process. Appl. 5 (1977) 27–37.
Y. Xia and D. Tse, Analysis on packet resequencing for reliable network protocols, in Proc. of IEEE INFOCOM, (2003).
Y. Xia and D. Tse, On the large deviation of resequencing queue size: 2-M/M/1 case, in Proc. of IEEE INFOCOM, (2004).
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Dieker, A.B., Lelarge, M. Tails for (max, plus) recursions under subexponentiality. Queueing Syst 53, 213–230 (2006). https://doi.org/10.1007/s11134-006-7730-5
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DOI: https://doi.org/10.1007/s11134-006-7730-5