Skip to main content

Computable Bounds on the Spectral Gap for Unreliable Jackson Networks

  • Paweł Lorek (a1) and Ryszard Szekli (a1)

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.

Corresponding author
Postal address: Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
∗∗ Email address:
∗∗∗ Email address:
Hide All

The work of both authors was supported by NCN Research Grant DEC-2011/01/B/ST1/01305.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 20 *
Loading metrics...

Abstract views

Total abstract views: 150 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd July 2018. This data will be updated every 24 hours.