### Abstract

Original language | Undefined |
---|---|

Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 23 |

Publication status | Published - Apr 2013 |

### Publication series

Name | Memorandum |
---|---|

Publisher | University of Twente, Department of Applied Mathematics |

No. | 2006 |

ISSN (Print) | 1874-4850 |

ISSN (Electronic) | 1874-4850 |

### Keywords

- MSC-60J10
- EWI-23266
- METIS-296454
- IR-85560

### Cite this

*Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms*. (Memorandum; No. 2006). Enschede: University of Twente, Department of Applied Mathematics.

}

*Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms*. Memorandum, no. 2006, University of Twente, Department of Applied Mathematics, Enschede.

**Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms.** / Chen, Y.; Boucherie, Richardus J.; Goseling, Jasper.

Research output: Book/Report › Report › Professional

TY - BOOK

T1 - Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms

AU - Chen, Y.

AU - Boucherie, Richardus J.

AU - Goseling, Jasper

PY - 2013/4

Y1 - 2013/4

N2 - We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as an infinite sum of geometric terms. We present necessary conditions for the invariant measure of a random walk to be a sum of geometric terms. We demonstrate that each geometric term must individually satisfy the balance equations in the interior of the state space. We show that the geometric terms in an invariant measure must have a pairwise-coupled structure. We further show that the random walk cannot have transitions to the North, Northeast or East. Finally, we show that for an infinite sum of geometric terms to be an invariant measure at least one coefficient must be negative. This paper extends our previous work for the case of finitely many terms to that of countably many terms.

AB - We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as an infinite sum of geometric terms. We present necessary conditions for the invariant measure of a random walk to be a sum of geometric terms. We demonstrate that each geometric term must individually satisfy the balance equations in the interior of the state space. We show that the geometric terms in an invariant measure must have a pairwise-coupled structure. We further show that the random walk cannot have transitions to the North, Northeast or East. Finally, we show that for an infinite sum of geometric terms to be an invariant measure at least one coefficient must be negative. This paper extends our previous work for the case of finitely many terms to that of countably many terms.

KW - MSC-60J10

KW - EWI-23266

KW - METIS-296454

KW - IR-85560

M3 - Report

T3 - Memorandum

BT - Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -