Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms

被引：5

作者：

Chen, Yanting ^{[1
,2
]}

Boucherie, Richard J. ^{[2
]}

Goseling, Jasper ^{[2
]}

机构：

[1] Hunan Univ, Coll Math & Econometr, Changsha 410082, Hunan, Peoples R China

[2] Univ Twente, Stochast Operat Res, POB 217, NL-7500 AE Enschede, Netherlands

来源：

QUEUEING SYSTEMS | 2016年 / 84卷 / 1-2期

关键词：

Random walk; Quarter-plane; Geometric terms; Error bounds; Performance measure; Tandem queue; TANDEM QUEUE; BACKGROUND STATES; STATIONARY DISTRIBUTION; COUPLED PROCESSORS; BLOCKING; SERVER; VIEWS; DECAY; MODEL;

D O I：

10.1007/s11134-016-9483-0

中图分类号：

TP39 [计算机的应用];

学科分类号：

081203 ; 0835 ;

摘要：

We consider homogeneous random walks in the quarter-plane. The necessary conditions which characterize random walks of which the invariant measure is a sum of geometric terms are provided in Chen et al. (arXiv:1304.3316, 2013, Probab Eng Informational Sci 29(02):233-251, 2015). Based on these results, we first develop an algorithm to check whether the invariant measure of a given random walk is a sum of geometric terms. We also provide the explicit form of the invariant measure if it is a sum of geometric terms. Second, for random walks of which the invariant measure is not a sum of geometric terms, we provide an approximation scheme to obtain error bounds for the performance measures. Our results can be applied to the analysis of two-node queueing systems. We demonstrate this by applying our results to a tandem queue with server slow-down.

引用

页码：21 / 48

页数：28

共 33 条

[1] A COMPENSATION APPROACH FOR 2-DIMENSIONAL MARKOV-PROCESSES [J].

ADAN, IJBF ;

WESSELS, J ;

ZIJM, WHM .

ADVANCES IN APPLIED PROBABILITY, 1993, 25 (04) :783-817

[2] Erlang arrivals joining the shorter queue [J].

Adan, Ivo J. B. F. ;

Kapodistria, Stella ;

van Leeuwaarden, Johan S. H. .

QUEUEING SYSTEMS, 2013, 74 (2-3) :273-302

[3]

[Anonymous], 1989, STOCHASTIC MODELING

[4]

[Anonymous], 1999, RANDOM WALKS QUARTER

[5]

BLANC JPC, 1985, COMM STAT STOCHASTIC, V1, P1

[6]

Boucherie RJ, 2011, INT SER OPER RES MAN, V154, P1, DOI 10.1007/978-1-4419-6472-4

[7]

Chen Y., 2013, ARXIV13043316

[8] THE INVARIANT MEASURE OF RANDOM WALKS IN THE QUARTER-PLANE: REPRESENTATION IN GEOMETRIC TERMS [J].

Chen, Yanting ;

Boucherie, Richard J. ;

Goseling, Jasper .

PROBABILITY IN THE ENGINEERING AND INFORMATIONAL SCIENCES, 2015, 29 (02) :233-251

[9]

Cohen J., 1983, Boundary Value Problems in Queueing System Analysis

[10] A multi-dimensional SRBM: geometric views of its product form stationary distribution [J].

Dai, J. G. ;

Miyazawa, Masakiyo ;

Wu, Jian .

QUEUEING SYSTEMS, 2014, 78 (04) :313-335

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