Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T09:19:51.719Z Has data issue: false hasContentIssue false
  • English
  • Français

The Compensation Approach for Walks With Small Steps in the Quarter Plane

Published online by Cambridge University Press:  11 January 2013

IVO J. B. F. ADAN ,
JOHAN S. H. van LEEUWAARDEN  and
KILIAN RASCHEL
IVO J. B. F. ADAN
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: i.j.b.f.adan@tue.nl, j.s.h.v.leeuwaarden@tue.nl)
JOHAN S. H. van LEEUWAARDEN
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: i.j.b.f.adan@tue.nl, j.s.h.v.leeuwaarden@tue.nl)
KILIAN RASCHEL
Affiliation:
CNRS and Université de Tours, Faculté des Sciences et Techniques, Parc de Grandmont, 37200 Tours, France (e-mail: kilian.raschel@lmpt.univ-tours.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is the first application of the compensation approach (a well-established theory in probability theory) to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane +2 with a step set that is a subset of

\[ \{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}\]
in the interior of +2. We derive an explicit expression for the generating function which turns out to be non-holonomic, and which can be used to obtain exact and asymptotic expressions for the counting numbers.

Keywords

Primary 05A15Secondary 05A16
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 161 - 183
Copyright
Copyright © Cambridge University Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adan, I. J. B. F. (1991) A compensation approach for queueing problems. PhD dissertation, Technische Universiteit Eindhoven.Google Scholar
[2]Adan, I. J. B. F., Boxma, O. J. and Resing, J. (2001) Queueing models with multiple waiting lines. Queueing Syst. 37 65–98.Google Scholar
[3]Adan, I. J. B. F., Wessels, J. and Zijm, W. H. M. (1990) Analysis of the symmetric shortest queue problem. Comm. Statist. Stochastic Models 6 691–713.Google Scholar
[4]Adan, I. J. B. F., Wessels, J. and Zijm, W. H. M. (1993) A compensation approach for two-dimensional Markov processes. Adv. Appl. Probab. 25 783–817.Google Scholar
[5]Bostan, A. and Kauers, M. (2009) Automatic classification of restricted lattice walks. In Proc. 21st International Conference on Formal Power Series and Algebraic Combinatorics, pp. 203–217.Google Scholar
[6]Bostan, A. and Kauers, M. (2010) The complete generating function for Gessel's walk is algebraic. Proc. Amer. Math. Soc. 432 3063–3078.Google Scholar
[7]Bousquet-Mélou, M. and Mishna, M. (2010) Walks with small steps in the quarter plane. Contemp. Math. 520 1–40.Google Scholar
[8]Fayolle, G., Iasnogorodski, R. and Malyshev, V. (1999) Random Walks in the Quarter-Plane, Springer.Google Scholar
[9]Fayolle, G. and Raschel, K. (2010) On the holonomy or algebraicity of generating functions counting lattice walks in the quarter-plane. Markov Process. Rel. Fields 16 485–496.Google Scholar
[10]Fayolle, G. and Raschel, K. (2012) Some exact asymptotics in the counting of walks in the quarter plane. In DMTCS Proc. AofA'12, pp. 109–124.Google Scholar
[11]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics. Cambridge University Press.Google Scholar
[12]Kurkova, I. and Malyshev, V. (1998) Martin boundary and elliptic curves. Markov Process. Rel. Fields 4 203–272.Google Scholar
[13]Kurkova, I. and Raschel, K. (2011) Explicit expression for the generating function counting Gessel's walk. Adv. Appl. Math. 47 414–433.Google Scholar
[14]Kurkova, I. and Raschel, K. (2012) On the functions counting walks with small steps in the quarter plane. Publ. Math. Inst. Hautes Études Sci. 116 69–114.Google Scholar
[15]Malyshev, V. (1971) Positive random walks and Galois theory. Uspehi Mat. Nauk 26 227–228.Google Scholar
[16]Melczer, S. and Mishna, M. (2012) Singularity analysis via the iterated kernel method. In preparation.Google Scholar
[17]Mishna, M. and Rechnitzer, A. (2009) Two non-holonomic lattice walks in the quarter plane. Theor. Comput. Sci. 410 3616–3630.Google Scholar
[18]Raschel, K. (2012) Counting walks in a quadrant: a unified approach via boundary value problems. J. Eur. Math. Soc. 14 749–777.Google Scholar
[19]Resing, J. and Rietman, R. (2004) The M/M/1 queue with gated random order of service. Statist. Neerlandica 58 97–110.Google Scholar
[20]Stanley, R. (1999) Enumerative Combinatorics, Vol. 2, Cambridge University Press.Google Scholar