Advances in Applied Probability

Computable bounds on the spectral gap for unreliable Jackson networks

Paweł Lorek and Ryszard Szekli

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First, we use the bounds of Lawler and Sokal (1988) in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one-dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from 0. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one-dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.

Article information

Source
Adv. in Appl. Probab. Volume 47, Number 2 (2015), 402-424.

Dates
First available in Project Euclid: 25 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1435236981

Digital Object Identifier
doi:10.1239/aap/1435236981

Mathematical Reviews number (MathSciNet)
MR3360383

Zentralblatt MATH identifier
1329.60319

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
Unreliable Jackson network spectral gap exponential ergodicity Cheeger's constant

Citation

Lorek, Paweł; Szekli, Ryszard. Computable bounds on the spectral gap for unreliable Jackson networks. Adv. in Appl. Probab. 47 (2015), no. 2, 402--424. doi:10.1239/aap/1435236981. https://projecteuclid.org/euclid.aap/1435236981


Export citation

References

  • Anantharam, V. (1989). Threshold phenomena in the transient behaviour of Markovian models of communication networks and databases. Queueing Systems Theory Appl. 5, 77–98.
  • Baxendale, P. H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Prob. 15, 700–738.
  • Berenhaut, K. S. and Lund, R. (2001). Geometric renewal convergence rates from hazard rates. J. Appl. Prob. 38, 180–194.
  • Berenhaut, K. S. and Lund, R (2002). Renewal convergence rates for DHR and NWU lifetimes. Prob. Eng. Inf. Sci. 16, 67–84.
  • Blanc, J. P. C. (1985). The relaxation time of two queueing systems in series. Commun. Statist. Stoch. Models 1, 1–16.
  • Callaert, H. and Keilson, J. (1973). On exponential ergodicity and spectral structure for birth–death processes. I. Stoch. Process. Appl. 1, 187–216.
  • Chafai, D. and Joulin, A. (2013). Intertwining and commutation relations for birth–death processes. Bernoulli 19, 1855–1879.
  • Chen, M.-F. (1991). Exponential $L^2$-convergence and $L^2$-spectral gap for Markov processes. Acta Math. Sinica (N.S.) 7, 19–37.
  • Chen, M.-F. (1996). Estimation of spectral gap for Markov chains. Acta Math. Sinica (N.S.) 12, 337–360.
  • Chen, M.-F. (2005). Eigenvalues, Inequalities, and Ergodic Theory. Springer, London.
  • Chen, M.-F. (2010). Speed of stability for birth–death processes. Front. Math. China 5, 379–515.
  • Daduna, H. and Szekli, R. (1995). Dependencies in Markovian networks. Adv. Appl. Prob. 27, 226–254.
  • Daduna, H. and Szekli, R. (1996). A queueing theoretical proof of increasing property of Pólya frequency functions. Statist. Prob. Lett. 26, 233–242.
  • Daduna, H. and Szekli, R. (2008). Impact of routeing on correlation strength in stationary queueing network processes. J. Appl. Prob. 45, 846–878.
  • Daduna, H., Kulik, R., Sauer, C. and Szekli, R. (2006). Dependence ordering for queueing networks with breakdown and repair. Prob. Eng. Inf. Sci. 20, 575–594.
  • Diaconis, P. and Fill, J. A. (1990). Examples for the theory of strong stationary duality with countable state spaces. Prob. Eng. Inf. Sci. 4, 157–180.
  • Diaconis, P. and Fill, J. A. (1990). Strong stationary times via a new form of duality. Ann. Prob. 18, 1483–1522.
  • Diaconis, P. and Miclo, L. (2009). On times to quasi-stationarity for birth and death processes. J. Theoret. Prob. 22, 558–586.
  • Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1, 36–61.
  • Dieker, A. B. and Warren, J. (2010). Series Jackson networks and noncrossing probabilities. Math. Operat. Res. 35, 257–266.
  • Fayolle, G., Malyshev, V. A., Men'shikov, M. V. and Sidorenko, A. F. (1993). Lyapounov functions for Jackson networks. Math. Operat. Res. 18, 916–927.
  • Fill, J. A. (1991). Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Prob. 1, 62–87.
  • Fill, J. A. (1991). Time to stationarity for a continuous-time Markov chain. Prob. Eng. Inf. Sci. 5, 61–76.
  • Fill, J. A. (1992). Strong stationary duality for continuous-time Markov chains. I. Theory. J. Theoret. Prob. 5, 45–70.
  • Fill, J. A. (2009). On hitting times and fastest strong stationary times for skip-free and more general chains. J. Theoret. Prob. 22, 587–600.
  • Hart, A. G., Martínez S. and San Martin J. (2003). The $\lambda$-classification of continuous-time birth-and-death processes. Adv. Appl. Prob. 35, 1111–1130.
  • Hordijk, A. and Spieksma, F. (1992). On ergodicity and recurrence properties of a Markov chain with an application to an open Jackson network. Adv. Appl. Prob. 24, 343–376.
  • Ignatiouk-Robert, I. (2006). On the spectrum of Markov semigroups via sample path large deviations. Prob. Theory Relat. Fields 134, 44–80.
  • Ignatiouk-Robert, I. and Tibi, D. (2012). Explicit Lyapunov functions and estimates of the essential spectral radius for Jackson networks. Preprint. Available at http://arxiv.org/abs/1206.3066v1.
  • Iscoe, I. and McDonald, D. (1994). Asymptotics of exit times for Markov jump processes. II. Applications to Jackson networks. Ann. Prob. 22, 2168–2182.
  • Kartashov, N. V. (2000). Determination of the spectral index of ergodicity of a birth-and-death process. Ukrainian Math. J. 52, 1018–1028.
  • Kijima M. (1992). Evaluation of the decay parameter for some specialized birth–death processes. J. Appl. Prob. 29, 781–791.
  • Kroese, D. P., Scheinhardt, W. R. W. and Taylor, P. G. (2004). Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. Ann. Appl. Prob. 14, 2057–2089.
  • Lawler, G. F. and Sokal, A. D. (1988). Bounds on the $L^2$ spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309, 557–580.
  • Liggett, T. M. (1989). Exponential $L_2$ convergence of attractive reversible nearest particle systems. Ann. Prob. 17, 403–432.
  • Liu, W. and Ma, Y. (2009). Spectral gap and convex concentration inequalities for birth–death processes. Ann. Inst. H. Poincaré Prob. Statist. 45, 58–69.
  • Lorek, P. and Szekli, R. (2012). Strong stationary duality for Möbius monotone Markov chains. Queueing Systems 71, 79–95.
  • Lund, R. B., Meyn, S. P. and Tweedie, R. L. (1996). Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Prob. 6, 218–237.
  • Malyshev, V. A. and Spieksma, F. M. (1995). Intrinsic convergence rate of countable Markov chains. Markov Processes Relat. Fields 1, 203–266.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Roberts, G. O. and Tweedie, R. L. (2000). Rates of convergence of stochastically monotone and continuous time Markov models. J. Appl. Prob. 37, 359–373.
  • Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.
  • Sauer, C. (2006). Stochastic product form networks with unreliable nodes: analysis of performance and availability. Doctoral Thesis, Hamburg University.
  • Sauer, C. and Daduna, H. (2003). Availability formulas and performance measures for separable degradable networks. Econom. Quality Control 18, 165–194.
  • Sirl, D., Zhang, H. and Pollett, P. (2007). Computable bounds for the decay parameter of a birth–death process. J. Appl. Prob. 44, 476–491.
  • Van Doorn, E. A. (1981). Stochastic Monotonicity and Queueing Applications of Birth–Death Processes (Lecture Notes Statis. 4). Springer, New York.
  • Van Doorn, E. A. (1985). Conditions for exponential ergodicity and bounds for the decay parameter of a birth–death process. Adv. Appl. Prob. 17, 514–530.
  • Van Doorn, E. A. (2002). Representations for the rate of convergence of birth–death processes. Theory Prob. Math. Statist. 65, 37–43.
  • Van Doorn, E. A., Zeifman, A. I. and Panfilova, T. L. (2010). Bounds and asymptotics for the rate of convergence of birth–death processes. Theory Prob. Appl. 54, 97–113.
  • Vere-Jones, D. (1963). On the spectra of some linear operators associated with queueing systems. Z. Wahrscheinlichkeitsth. 2, 12–21.
  • Wu, L. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Prob. Theory Relat. Fields 128, 255–321.