Queueing Systems

, Volume 74, Issue 2–3, pp 235–272 | Cite as

Wireless three-hop networks with stealing II: exact solutions through boundary value problems

  • Fabrice Guillemin
  • Charles Knessl
  • Johan S. H. van Leeuwaarden


We study the stationary distribution of a random walk in the quarter plane arising in the study of three-hop wireless networks with stealing. Our motivation is to find exact tail asymptotics (beyond logarithmic estimates) for the marginal distributions, which requires an exact solution for the bivariate generating function describing the stationary distribution. This exact solution is determined via the theory of boundary value problems. Although this is a classical approach, the present random walk exhibits some salient features. In fact, to determine the exact tail asymptotics, the random walk presents several unprecedented challenges related to conformal mappings and analytic continuation. We address these challenges by formulating a boundary value problem different from the one usually seen in the literature.


Random walk Quarter plane Riemann–Hilbert problem  Analytic continuation Asymptotic analysis 



The work of CK was partly supported by NSA Grants H 98230-08-1-0102 and H 98230-11-1-0184. JvL is supported by an ERC Starting Grant.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Fabrice Guillemin
    • 1
  • Charles Knessl
    • 2
  • Johan S. H. van Leeuwaarden
    • 3
  1. 1.Orange LabsLannionFrance
  2. 2.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  3. 3.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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