On max-min fair flow optimization in wireless
mesh networks
Michal Pioro, Mateusz Zotkiewicz, Barbara Staehle, Dirk Staehle and Di Yuan
Linköping University Post Print
N.B.: When citing this work, cite the original article.
Original Publication:
Michal Pioro, Mateusz Zotkiewicz, Barbara Staehle, Dirk Staehle and Di Yuan, On max-min
fair flow optimization in wireless mesh networks, 2014, Ad hoc networks, (13), 134-152.
http://dx.doi.org/10.1016/j.adhoc.2011.05.003
Copyright: Elsevier
http://www.elsevier.com/
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-104112
1
On max-min fair flow optimization in wireless mesh networks
Micha l Pi´oro
a,b
, Mateusz
˙
Zotkiewicz
b
, Barbara Staehle
c
, Dirk Staehle
c
, Di Yuan
d
a
Lund University, Sweden
b
Warsaw University of Technology, Poland
c
Wuerzburg University, Germany
d
Link
¨
oping University, Sweden
Abstract
The paper is devoted to modeling wireless mesh networks (WMN) through mixed-integer pro-
gramming (MIP) formulations that allow to precisely characterize the link data rate capacity and
transmission scheduling using the notion of time slots. Such MIP models are formulated for sev-
eral cases of the modulation and coding schemes (MCS) assignment. We present a general way
of solving the max-min fairness (MMF) traffic objective for WMN using the formulated capacity
models. Thus the paper combines WMN radio link modeling with a non-standard way of dealing
with uncertain traffic, a combination that has not, to our knowledge, been treated so far by exact
optimization models. We discuss several ways, including a method based on the so called com-
patible or independent sets, of solving the arising MIP problems. We also present an extensive
numerical study that illustrates the running time efficiency of different solution approaches, and
the influence of the MCS selection options and the number of time slots on traffic performance of
a WMN. Exact joint optimization modeling of the WMN capacity and the MMF traffic objectives
forms the main contribution of the paper.
Keywords: wireless mesh network, max-min fairness, mixed-integer programming
Email addresses: Michal.Pioro@eit.lth.se (Micha l Pi´oro), mzotkiew@tele.pw.edu.pl (Mateusz
˙
Zotkiewicz),
bstaehle@informatik.uni-wuerzburg.de (Barbara Staehle), dstaehle@informatik.uni-wuerzburg.de (Dirk
Staehle), diyua@itn.liu.se (Di Yuan)
Preprint submitted to Elsevier 22nd June 2011
Contents
1 Introduction 4
2 Survey of the field 6
3 Notation 8
4 Modeling link capacity space 12
4.1 Single MCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Single MCS, node capacity split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Static allocation of MCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Dynamic allocation of MCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.5 A simplified interference model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 MMF optimization of demand flows 18
5.1 Application of the MMF approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 The considered MMF problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Non-compact formulation and compatible sets generation . . . . . . . . . . . . . . 21
5.4 Practicability of the models’ solutions . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 A heuristic approach to SA/SI 25
6.1 Phase 1: adaptive modulation and coding . . . . . . . . . . . . . . . . . . . . . . . 26
6.2 Phase 2: max-min fair throughput computation . . . . . . . . . . . . . . . . . . . . 27
7 Numerical study 30
7.1 Example networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.2 Computational efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.3 Traffic efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.4 Practical hints on problem solving . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.5 Summary of the numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
8 Conclusion 37
Appendix A Derivation of formulation (15) 42
Appendix B MMF algorithm for the CS formulations 43
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1. Introduction
Wireless mesh networks (WMN) offer common and affordable access to the Internet in metropoli-
tan and residential areas. The core of a WMN consists of a set of fixed mesh nodes – routers and
Internet gateways – interconnected by radio links that typically follow the Wi-Fi IEEE 802.11-
family standards. Other standards, such as Bluetooth IEEE 802.15.5, WiMAX IEEE 802.16a,
and IEEE 802.20 can also support WMNs. Mesh clients, being either fixed or mobile, connect
to mesh routers to obtain access to Internet gateways either over direct links or via multi-hop
mesh routes. WMN is a cost-efficient approach for Internet access with a bandwidth in the range
of 50-200 Mbps. The WMN solution is competitive to the wired Internet access offered by cable
network providers or by mobile operators. WMNs are decentralized, non-hierarchical networks,
typically deployed by communities of users (see [1, 4]), and based on commonly available off-the-
shelf wireless communication equipment (see [2, 3, 5]). The idea of WMN stems from the ad-hoc
networking paradigm and, as such, fits very well the decentralized philosophy of the Internet. For
comprehensive surveys of WMN, we refer to [7, 19].
Although WMNs are relatively cheap and easy to deploy, achieving efficient and fair resource
allocation is not straightforward. One particular issue is how to effectively allocate the offered
network capacity among the routes between the gateways and the routers. Without network
optimization, which ideally should be simple, fast, and distributed, a WMN can behave poorly,
delivering significantly lower throughput than it can potentially achieve. For WMN, network
optimization tasks range from transmission scheduling, through channel assignment, transmission
power adjustment and rate adaptation, to routing. In this context, traffic engineering, which
is a key aspect in operating communication networks, is of high significance to WMN. From
the mathematical optimization standpoint, traffic engineering in WMN poses challenges that are
not present in classical network flow optimization, necessitating novel modeling and optimization
concepts to account for:
• transmission scheduling on radio links realized by the MAC (medium access control) layer
using multiple access schemes such as CSMA (contention-aware carrier sense multiple access,
IEEE 802.11) or TDMA (contention-free reservation-based time division multiple access,
IEEE 802.16)
• need of (dynamic) channel assignment in multi-channel WMNs
• possibility of node power control for interference mitigation
• possibility of link rate control for adapting the transmission to the channel propagation and
interference conditions
• uncertain nature of traffic to be routed between WMN routers and gateways.
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