The Annals of Applied Probability

Moments and tails in monotone-separable stochastic networks

François Baccelli and Serguei Foss

Full-text: Open access

Abstract

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework, which was first introduced to derive the stability region for stochastic networks with stationary and ergodic driving sequences, is revisited. It contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. Our purpose is the analysis of the tails of the stationary state variables in the particular case of i.i.d. driving sequences. For this, we establish general comparison relationships between networks of this class and the GI/GI/1/∞ queue. We first use this to show that two classical results of the asymptotic theory for GI/GI/1/∞ queues can be directly extended to this framework. The first one concerns the existence of moments for the stationary state variables. We establish that for all α≥1, the (α+1)-moment condition for service times is necessary and sufficient for the existence of the α-moment for the stationary maximal dater (typically the time to empty the network when stopping further arrivals) in any network of this class. The second one is a direct extension of Veraverbeke”s tail asymptotic for the stationary waiting times in the GI/GI/1/∞ queue. We show that under subexponential assumptions for service times, the stationary maximal dater in any such network has tail asymptotics which can be bounded from below and from above by a multiple of the integrated tails of service times. In general, the upper and the lower bounds do not coincide. Nevertheless, exact asymptotics can be obtained along the same lines for various special cases of networks, providing direct extensions of Veraverbeke”s tail asymptotic for the stationary waiting times in the GI/GI/1/∞ queue. We exemplify this on tandem queues (maximal daters and delays in stations) as well as on multiserver queues.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 2 (2004), 612-650.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1082737105

Digital Object Identifier
doi:10.1214/105051604000000044

Mathematical Reviews number (MathSciNet)
MR2052896

Zentralblatt MATH identifier
1048.60067

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 90B15: Network models, stochastic 60F10: Large deviations

Keywords
Queueing network generalized Jackson network ergodicity subexponential random variable tail asymptotics Veraverbeke”s theorem

Citation

Baccelli, François; Foss, Serguei. Moments and tails in monotone-separable stochastic networks. Ann. Appl. Probab. 14 (2004), no. 2, 612--650. doi:10.1214/105051604000000044. https://projecteuclid.org/euclid.aoap/1082737105


Export citation

References

  • Anantharam, V. (1989). How large delays build up in a $\mathit{GI}/G/1$ queue. QUESTA 5 345–368.
  • Asmussen, S. and Klüppelberg, C. (1996). Large deviations results in the presence of heavy tails, with applications to insurance risk. Stochastic Process. Appl. 64 103–125.
  • Asmussen, S., Klüppelberg, C. and Sigman, K. (1998). Sampling at subexponential times, with queueing applications. Stochastic Process. Appl. 79 265–286.
  • Asmussen, S., Schmidli, H. P. and Schmidt, V. (1999). Tail probabilities for nonstandard risk and queueing processes. Stochastic Process. Appl. 79 265–286.
  • Baccelli, F. and Foss, S. (1994). Ergodicity of Jackson-type queueing networks. QUESTA 17 5–72.
  • Baccelli, F. and Foss, S. (1995). On the saturation rule for the stability of queues. J. Appl. Probab. 32 494–507.
  • Baccelli, F., Foss, S. and Lelarge, M. (2004). Tails in generalized Jackson networks with subexponential distributions. J. Appl. Probab. To appear.
  • Baccelli, F., Foss, S. and Lelarge, M. (2004). Tails in $(\max , +)$ systems with subexponential service distributions. QUESTA. To appear.
  • Baccelli, F., Makowski, A. and Shwartz, A. (1989). The fork join queue and related systems with synchronization constraints: Stochastic dominance, approximations and computable bounds. Adv. in Appl. Prob. 21 629–660.
  • Baccelli, F., Schlegel, S. and Schmidt, V. (1999). Asymptotics of stochastic networks with subexponential service times. QUESTA 33 205–232.
  • Boxma, O. J. and Deng, Q. (2000). Asymptotics behaviour of the tandem queueing system with identical service times at both queues. Math. Methods Oper. Res. 52 307–323.
  • Boxma, O. J., Deng, Q. and Zwart, A. P. (1999). Waiting-time asymptotics for the $M/G/2$ queue with heterogeneous servers. Memorandum COSOR 99-20, Eindhoven Univ. Technology.
  • Embrechts, P., Klüppelberg, C. and Mikosch, Th. (1997). Modeling Extremal Events. Springer, New York.
  • Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55–72.
  • Foss, S. (1980). Approximation of multichannel queueing systems. Siberian Math. J. 21 132–140.
  • Foss, S. and Korshunov, D. (2000). Sampling at a random time with a heavy-tailed distribution. Markov Process. Related Fields 6 534–568.
  • Foss, S. and Korshunov, D. (2003). Moments and tails in multi-server queues. Unpublished manuscript.
  • Gunawardena, J., ed. (1998). Idempotency. Cambridge Univ. Press.
  • Huang, T. and Sigman, K. (1999). Steady state asymptotics for tandem, split-match and other feedforward queues with heavy tailed service. QUESTA 33 233–259.
  • Sigman, K. (1999). A primer on heavy-tailed
  • distributions. Queueing Systems Theory Appl.
  • Scheller-Wolf, A. and Sigman, K. (1997). Delay moments for FIFO $\mathit{GI}/\mathit{GI}/c$ queues. QUESTA 25 77–95.
  • Veraverbeke, N. (1977). Asymptotic behavior of Wiener–Hopf factors of a random walk. Stochastic Process. Appl. 5 27–37.