Routing in Multi-hop Wireless Mesh Networks with Bandwidth Guarantees by bestt571

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Mesh network is "wireless mesh network ", which is a wireless multi-hop networks, ad hoc network is evolved to solve the "last mile"problem of the key technologies. In the process of evolution as the next-generation networks, wireless is an essential technology. Wireless mesh network with other cooperative communication. Is a dynamic network architecture that can scale, any two devices can be to keep the wireless Internet.

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									           Routing in Multi-hop Wireless Mesh Networks with
                        Bandwidth Guarantees

                          Ronghui Hou, King-Shan Lui, Hon Sun Chiu, Kwan L. Yeung
                     Department of Electrical and Electronic Engineering, The University of Hong Kong
                                      {rhhou, kslui,hschiu,kyeung}
                                                           Fred Baker
                                                     Cisco Research Center

Abstract                                                          b is transmitting to c, f must remain idle to avoid interfer-
   Wireless Mesh Networks (WMNs) has become an im-                ence. In other words, only one link on p can be active at
portant edge network to provide Internet access to remote         the same time. Suppose the number on a link reflects the
areas and wireless connections in a metropolitan scale. In        bandwidth (Mbits/sec) of that link when only that link
this paper, we describe our hop-by-hop bandwidth guar-            is active. Bandwidth is a concave metric in wired net-
anteed routing protocol in IEEE 802.11-based wireless             works since transmissions on all links can be concurrent.
mesh networks. Due to interference among links, band-             If the links along p were wires, the bandwidth would be
width, a well-known bottleneck metric in wired networks,          min{4, 2, 10} = 2. In the wireless situation, based on the
is neither concave nor additive in wireless networks. To          conflict graph model, for transmitting 1 Mbits along p, it
facilitate hop-by-hop routing, we develop a mechanism             takes 1 + 1 + 10 = 17 second [10]. The available band-
                                                                          4   2
for computing the available bandwidth of a path in a                                         20
                                                                  width on path p is only 17 Mbit/second. The bandwidth
distributed manner. Unfortunately, available bandwidth            metric is neither concave nor additive in WMNs.
is not isotonic, the necessary and sufficient property for             The wireless transmission interference also compli-
consistent hop-by-hop routing. To solve the problem, we           cates the problem of computing the maximum bandwidth
introduce an isotonic parameter that captures the avail-          paths in a hop-by-hop manner. The problem is that
able bandwidth metric so that packets can traverse the            the best path from Node s to Node d passes through v
maximum bandwidth path consistently according to the              does not imply that the best path from v to d is a sub-
routing tables constructed in the nodes along the path. To        path of the best path from s to d. In Fig. 1, there are
the best of our knowledge, our protocol is the first WMN           two paths from b to g. Let p1 = b → c → f → g and
hop-by-hop routing scheme that can identify bandwidth             p2 = b → d → e → g. The actual available bandwidth of
guaranteed paths.                                                                  5
                                                                  p1 and p2 are 4 and 1 Mbit/second, respectively. That
1   INTRODUCTION                                                  is, p1 is a better path from b to g. However, the band-
   A wireless mesh network (WMN) consists a large                 width of the path a → b → d → e → g is higher than
number of wireless nodes. The nodes form a wireless               the path a → b → c → f → g according to the conflict
overlay to cover the service area while a few nodes sit-          graph model which we will discuss later. In other words,
ting on the edge are wired to the Internet. As part of the        each node cannot simply select the maximum bandwidth
Internet, WMN has to support diversified multimedia ap-            path as the best path and only advertises this path to its
plications for its users. Quality-of-Service (QoS) routing,       neighbors. Besides, when an intermediate node receives
which focus on finding a path satisfying the application           a packets from the source, it should not determine the
requirements, is one of the building blocks for supporting        next hop based on the destination only. It has been for-
QoS. In this paper, we study how to find a path satisfying         mally proved that the necessary and sufficient condition
the bandwidth requirement of an incoming connection re-           that an efficient routing protocol exists is the metric con-
quest. We adopt the proactive approach, which computes            sidered has to be isotonic[27]. Unfortunately, available
the maximum bandwidth supported by the paths from a               bandwidth based on the conflict graph model does not
source to a destination prior to the requests arrive. By pre-     satisfy this property. Therefore, the design of the packet
computing the supported QoS, we can determine whether             forward mechanism based on the hop-by-hop routing is
a request is feasible upon its arrival. If this request is fea-   not trivial. To develop a hop-by-hop QoS routing proto-
sible, a resource reservation will be initiated; otherwise,       col with bandwidth guarantees, there are three problems
this request is blocked.                                          we need to solve.
   QoS routing in wireless networks is very challenging
due to the wireless transmission interference. Consider             1. How to compute the available bandwidth of a given
the simple network in Fig. 1 where all the links use the               path?
same channel. Along the path p = b → c → f → g, when                2. How to select the best paths to be advertised?
                           4   c       f   10                 provide correct information for identifying high through-
               a       b                        g             put paths [10]. The authors in [10] develop a centralized
                               d       e    3                 mechanism to compute the available bandwidth of a path
                                                              and this metric performs better than other loss-based met-
              Figure 1. A simple network.                     rics compared.
                                                                  Much attention has been paid to the problem of find-
                                                              ing routes with bandwidth concerned in wireless ad hoc
 3. How to determine the next hop when receiving a            networks [15]-[21]. The works in [15, 16, 17] con-
    packet?                                                   sider the effect of the interference of wireless commu-
  In this work, we study these problems in the 802.11         nications, and their works are all based on the TDMA
WMNs and make the following contributions.                    channel model. The choice of MAC would affect the
  • We develop a hop-by-hop mechanism to compute              overall bandwidth utilization of the network, and differ-
    the available bandwidth of a given path and formally      ent QoS routing schemes should be adopted for different
    proof the correctness. The mechanism facilitates us       MAC protocols. [18]-[21] study QoS routing in 802.11
    to establish a distributed routing algorithm.             wireless networks. The mechanisms in [18, 19, 22] are
    • We propose a new left-isotonic QoS parameter that       based on the AODV protocol, which is a reactive ap-
      captures the concept of available bandwidth so that     proach. However, a feasible path may not be identified
      hop-by-hop routing based on routing tables can be       even if it does exist in the network. The work in [21] pro-
      consistent.                                             poses a polynomial-time routing algorithm that considers
                                                              bandwidth requirements. This algorithm requires a FIFO
    • Finally, we describe how routing tables and distance    scheduling policy, in which all the packets contending
      tables are constructed in nodes and how packets are     the common channel are prioritized based on their arrival
      forwarded on the optimal paths. We formally prove       time where the highest priority packet seizes the channel
      that our protocol is correct and consistent.            first. Ref. [20] also considers how to find the path that
2    RELATED WORKS                                            provides the maximum bandwidth. We mentioned earlier
    To find a path with higher bandwidth, many re-             that if a node just advertises one path to its neighbors, its
searchers develop new link metrics to quantify the in-        neighbors may not be able to identify the maximum avail-
terference level of a link and these new link metrics are     able bandwidth path. In [20], each node keeps multiple
additive. The weight of a path becomes the sum of in-         paths to a destination so as to increase the probability of
dividual link weights of the links along the path and any     finding the best path from each node to a destination. All
routing protocol that works for an additive metric can be     of these mechanisms are not distributed in nature and can-
applied. Simulation studies show that paths found based       not be used directly in a hop-by-hop manner. A heuristic
on this kind of metrics are better than the least hop count   method for computing the maximum bandwidth path was
paths [1, 2].                                                 proposed in [23]. In this work, the bandwidth of a link
    In [3], the expected transmission count (ETX) met-        is defined as the minimum of the bandwidth for all links
ric is proposed, which computes the average number            which interfere with this link, and the bandwidth of a path
of transmission attempts required to send successfully a      is defined as the minimum of the bandwidth for all links
packet over the link. It is the earliest link metric devel-   on this path. We can easily develop a hop-by-hop rout-
oped and many other metrics are extended from it, such        ing protocol by using this method. However, this method
as mETX, ETN [4], and ETT [9]. The link metric ETT is         cannot guarantee that the found path can satisfy a certain
used for designing the path metrics WCETT (weighted           bandwidth requirement. To the best of our knowledge,
cumulative expected transmission time), iAWARE [5],           we are the first to develop a hop-by-hop routing protocol
and MIC [6]. Instead of developing an additive met-           with bandwidth guarantees.
ric, the Bottleneck Link Capacity (BLC) is based on the
expected busy time of a link for transmitting a packet        3 WIRELESS TRANSMISSION INTER-
successfully [7]. The path BLC is the minimum link              FERENCE
BLC along the path. The link metric CATT proposed                We adopt the Transmitter-Receiver Conflict Avoidance
in [8] is similar to BLC. The work in [11] proposes the       (TRCA) model [25] since it reflects the popular IEEE
interference-aware routing metric, which determines the       802.11 technologies more closely and is also used in
percentage of time each transmission wastes due to inter-     many other prior works [12, 13, 22, 23, 24]. Under this
ference from other nodes. The routing algorithm based         model, the neighbors of the transmitter and the receiver
on the proposed metric selects the path with the minimum      cannot send or receive data. A conflict graph can be used
interference.                                                 to represent the interference among wireless links. We
    However, all these works do not consider the prob-        assume links are symmetric. In the network G (V , E )
lem of providing bandwidth guarantees. Given a request        where V is the set of nodes, if u and v where u, v ∈ V
with the bandwidth requirement, we cannot determine           can communicate with each other, the link (u, v) ∈ E . De-
whether the best path selected by using these proposed        note Gc (Vc , Ec ) as the conflict graph corresponding to the
metrics can support the request. In fact, packet loss based   original network G (V , E ). Each link e in E corresponds
metrics such as ETX and its extensions do not always          to a node v(e) in Vc . Therefore, we have |E | = |Vc |.
                                               a         b
                                                                                v1     v2         v3    v4              vh


         a       b       c       d
                                               c         d         Figure 3. An illustration for computing the bandwidth
    1        2       3       4       5
                                                                   of a path.
          (a) The original graph.        (b) The conflict graph.
        Figure 2. Illustration for interference model.
                                                                   is a neighbor of its previous hop and its next hop but not
                                                                   other nodes on the path. That is, in p, v2 is within the
There is a link from v(e1 ) to v(e2 ) in Gc if e1 and e2 in        communication range of v1 and v3 but not v4 , v5 , ..., vh .
E interfere with each other. For example, Fig. 2(b) is the         This limits the interference to be within 2 hops. As a
conflict graph of the linear network in Fig. 2(a). Link             result, each clique in Gc (p) contains three nodes only.
a interferes with Link b because they share a common               Since mesh routers are carefully deployed, paths in a
node, Node 2. Link a also interferes with Link c because           WMN should follow this assumption readily.
Node 3 is a neighbor of Node 2 and has to remain idle
when a is active according to the TRCA model.
    A clique is a complete graph among a set of nodes in a         4.1 Path Bandwidth Computation
graph. A maximal clique is a clique that is not contained             The following lemma gives the mechanism for com-
in any other clique. A clique in Gc contains the links             puting WB(p) in a hop-by-hop manner that no node on p
in G of which transmission cannot be concurrent. For               needs to know the information of each link on the path.
example, there are two maximal cliques, a − b − c and              L EMMA 1. Let path p =<v1 , v2 , v3 , ..., vh >, p1 =
b − c − d, in Fig. 2(b), reflecting that only one link in           <v1 , v2 , v3 , v4 >, and p = <v2 , v3 , ..., vh >.           Then,
{a, b, c} and one link in {b, c, d} can be active at the same      WB(p) = min{WB(p1 ), WB(p )}.
time to avoid interference.                                        P ROOF. Fig. 3 illustrates paths p, p , and p1 . The max-
    Ref. [10] describes how to estimate the available              imal cliques in Gc (p) are Q1 , Q2 ,. . . , Qh−3 where Qi =
bandwidth of a path based on the conflict graph model.              {(vi , vi+1 ), (vi+1 , vi+2 ), (vi+2 , vi+3 )}. Similarly, Q1 is the
The available bandwidth refers to the maximum through-             only clique in Gc (p1 ) and the maximal cliques in Gc (p )
put a path can accept where the traffic intensity on each           are Q2 , Q3 , ..., Qh−3 . By Eq. (1),
link is not larger than 1. The authors also provide a mech-
anism to find the maximum additional input rate, B (l), of
                                                                    WB(p) = min{B (Q1 ), B (Q2 ), . . . , B (Qh−3 )}
a link l based on the fraction of busy time and packet loss
                                                                          = min{min{B (Q1 )}, min{B (Q2 ), . . . , B (Qh−3 )}}
probability of that link. Let Gc (p) be the conflict graph
                                                                          = min{WB(p1 ), WB(p )}
of a path p. Let {Q1 , Q2 , ..., Q j } where j > 0 be the set
of maximal cliques in Gc (p). The available bandwidth of
p is
                                                      1 −1            As illustrated in Fig. 3, Lemma 1 suggests that when
 min1≤i≤ j {B (Qi )} where B (Qi ) = (       ∑       B (l)
                                                           ) (1)   v1 wants to find WB(p), it can compute based on the
                                           v(l)∈Qi                 WB(p ) information advertised by v2 and the available
                                                                   bandwidth of the links (v1 , v2 ), (v2 , v3 ), and (v3 , v4 ). v1
To illustrate, let B (a), B (b), B (c), and B (d) of the net-
                                                                   does not have to know the whole path information about
work in Fig. 2(a) be 50, 100, 25, and 20 pkt/sec, respec-                                                 20
tively, as in the example provided in [10]. The available          p. WB(<b, c, f , g>) in Fig. 1 is 17 . b advertises this in-
                                    1     1     1       1
bandwidth of the path is min{( 50 + 100 + 25 )−1 , ( 100 +         formation together with the bandwidth of links (b, c) and
 1     1 −1                                                        (c, f ) to a. Then, a can find WB(<a, b, c, d>), which
25 + 20 ) } = 10 pkt/sec. Due to space limitation, we              is 12 , and it can determine WB(<a, b, c, f , g>) to be
refer readers to [10] for further explanation.                         13
    Eq. (1) allows us to find the available bandwidth of a          min{ 20 , 12 } = 12 .
                                                                          17 13         13
path when the whole path is known. However, in a hop-                 Let path p = <v1 , v2 , v3 , . . . , vh > and pi be
by-hop routing protocol, we need a mechanism that de-              <vi , vi+1 , vi+2 , vi+3 >, where 1 ≤ i ≤ h − 3. By Lemma 1,
termines the available bandwidth based on local informa-           we have
tion only. We describe how to enhance the work in [10] to
compute the available bandwidth in a distributed manner                      WB(p) = min{WB(pi )|1 ≤ i ≤ h − 3}                    (2)
in the next section.
4       DISTRIBUTED PATH BANDWIDTH                                    Given two paths p and q,                                where p
        COMPUTATION AND SELECTION                                  =<v1 , . . . , vh , vh+1 > and q=<vh+1 , vh+2 , . . . , vh+n >,
   Given a path p=<v1 , v2 , ..., vh >, denote WB(p) as the        respectively, let p ⊕ q denote the concatenated
available bandwidth of the whole path p. The conflict               path <v1 , v2 , . . . , vh , vh+1 , vh+2 , . . . , vh+n > and pi be
graph of p is Gc (p). We only consider paths where a node          <vi , vi+1 , vi+2 , vi+3 >, as illustrated in Fig. 4. By Lemma
                         ph-1                      ph
                                                                      lustrated in Fig. 5. By Eq. (3),
            V1    Vh-1          Vh   Vh+1   Vh+2        Vh+3   Vh+n
                    p                                    q             WB(p ⊕ p1 ) = min{WB(p), WB(<vh−1 , vh , v, u1 >),
Figure 4. An illustration for computing the bandwidth                                       WB(<vh , v, u1 , u2 >), WB(p1 )}
of a concatenated path.                                                WB(p ⊕ p2 ) = min{WB(p), WB(<vh−1 , vh , v, g1 >),
                                                                                            WB(<vh , v, g1 , g2 >), WB(p2 )}
                                       u1      u2                         TB(p1 ) ≥ TB(p2 ) implies the bandwidth of
        s          vh           v                                d    <v, u1 , u2 > is larger than <v, g1 , g2 >.      Therefore,
                                       g1      g2                     WB(<vh , v, u1 , u2 >) ≥ WB(<vh , v, g1 , g2 >). Similarly,
                                                                      FB(p1 ) ≥ FB(p2 ) implies WB(<vh−1 , vh , v, u1 >) ≥
      Figure 5. An illustration for path selection.                   WB(<vh−1 , vh , v, g1 >).     Together with WB(p1 ) ≥
                                                                      WB(p2 ), we can conclude that WB(p ⊕ p1 ) ≥
                                                                      WB(p ⊕ p2 ).
1, we also have                                                           To show that WB(p1 ) ≥ WB(p2 ) and FB(p1 ) ≥
 WB(p ⊕ q) = min{WB(pi )|1 ≤ i ≤ h + n − 3}                           FB(p2 ) and TB(p1 ) ≥ TB(p2 ) is also a necessary con-
                = min{min{WB(pi )|1 ≤ i ≤ h − 2},                     dition for v to determine whether p ⊕ p1 is better than
                      WB(ph−1 ), WB(ph ),                             p ⊕ p2 , we use examples to illustrate that if either one
                      min{WB(pi )|h + 1 ≤ i ≤ h + n − 3}}             of WB(p1 ) ≥ WB(p2 ), FB(p1 ) ≥ FB(p2 ), or TB(p1 ) ≥
                = min{WB(p), WB(ph−1 ), WB(ph ),                      TB(p2 ) does not hold, WB(p ⊕ p1 ) ≥ WB(p ⊕ p2 ) may
                          WB(q)}                                      not hold even the other two are satisfied. In the following
                                                       (3)            examples, let <b, c, f , g> be p1 and <b, d, e, g> be p2 .
   Eq. (3) suggests that to compute the bandwidth of the                  Case I: TB(p1 ) ≥ TB(p2 ) and FB(p1 ) ≥ FB(p2 ) but
path p ⊕ q, we just need to know the bandwidths of paths              WB(p1 ) ≥ WB(p2 ), as illustrated in Fig. 6(a), where p
p, q, ph−1 , and ph .                                                 = <a, b>. We thus have WB(p ⊕ p1 ) ≥ WB(p ⊕ p2 ),
                                                                      where WB(p ⊕ p1 ) = WB(p1 ) = 2.5 and WB(p ⊕ p2 ) =
4.2    Path Selection                                                 WB(p2 ) = 3.
    We now know how to compute the available band-                        Case II: WB(p1 ) ≥ WB(p2 ) and TB(p1 ) ≥ TB(p2 )
width of a path advertised by a neighbor. In the dis-                 but FB(p1 ) ≥ FB(p2 ), as shown in Fig. 6(b), where p =
tance vector protocol, after computing the bandwidth of               <s, a, b>. We obtain that WB(p ⊕ p1 ) ≥ WB(p ⊕ p2 ),
the paths advertised by the neighbors, a node advertises              since WB(p ⊕ p1 ) = 12 and WB(p ⊕ p2 ) = 8 .
the “best” one among them. However, this may not facili-                                      5                      3
                                                                          Case III: WB(p1 ) ≥ WB(p2 ) and FB(p1 ) ≥ FB(p2 )
tate a neighbor to identify its own best path. For example,
                                                                      but TB(p1 ) ≥ TB(p2 ), as illustrated in Fig. 1 where p =
in Fig. 1, WB(<b, c, f , g>) = 20 and WB(<a, b, c, f , g>)
                                   17                                 <a, b>. It turns out that WB(p ⊕ p1 ) ≥ WB(p ⊕ p2 ).
= 12 , while WB(<b, d, e, g>) = WB(<a, b, d, e, g>) = 1.
   13                                                                     Therefore, for v to determine for sure that it is not
From b’s perspective, the best path is <b, c, f , g>. Un-             necessary to advertise p2 , we have to check whether
fortunately, if b only advertises this path to a, a can-              WB(p1 ) ≥ WB(p2 ), FB(p1 ) ≥ FB(p2 ), and TB(p1 ) ≥
not identify its own best path, which is <a, b, d, e, g>.             TB(p2 ).
Since bandwidth is neither concave nor additive in wire-
less networks, each node cannot just advertise the maxi-                                                          10
                                                                                                10        c                f       5
mum bandwidth path to all its neighbors. A trivial way                                 10
                                                                                   a        b                                           g
for assuring all the maximum bandwidth paths can be
found is to advertise all the possible paths to a desti-                                        9         d                e       9
nation. This is definitely too expensive. To reduce the
overhead, we should not advertise those paths that would                                              (a)
not be a subpath of any maximum bandwidth path. In                                                                     60
                                                                                                     12       c                f       12
this section, we study the sufficient and necessary con-                            6        6                                               g
                                                                               s       a         b
ditions for a node to determine whether a path would be                                              24       d                e       12
the subpath of any maximum bandwidth path. Given a                                                                    12
path p=<v1 , v2 , . . . , vh >, denote FB(p) as the available                                         (b)
bandwidth of the link (v1 , v2 ), and TB(p) as the available
                                                                            Figure 6. Examples of network topologies.
bandwidth of the subpath <v1 , v2 , v3 >. Lemma 2 gives
the sufficient condition to determine a path is not worth-
while to be advertised.                                               5 QOS ROUTING PROTOCOL
L EMMA 2. Suppose that p1 and p2 are two paths from                      In this section, we describe our bandwidth-guaranteed
v to d. If WB(p1 ) ≥ WB(p2 ), TB(p1 ) ≥ TB(p2 ), and                  hop-by-hop routing protocol. The isotonicity property of
FB(p1 ) ≥ FB(p2 ), then WB(p ⊕ p1 ) ≥ WB(p ⊕ p2 ) for                 the path weight is the sufficient and necessary condition
any p that ends at v.                                                 for a routing protocol satisfying the optimality require-
P ROOF. Let p1 = <v, u1 , . . . , un , d>,                p2 =        ment [27]. In the following, we first introduce a new
<v, g1 , . . . , gm , d>, and p = <s, v1 , . . . , vh , v> as il-     QoS path metric which satisfies the isotonicity property.
We then present our path calculation mechanism and our                                          QoS to Destination
packet forwarding mechanism. We show that our rout-
ing protocol satisfies the optimality and consistency re-           Destination        ω(p)          NF(p)   NS(p)        NT(p)
quirements. Finally, we analyze the complexities of our                             ( 15 , 4, 6)      b         c          f
protocol.                                                               g
                                                                                   ( 24 , 24 , 6)
                                                                                               b        d         e
5.1    Isotonicity                                                                    7 5
                                                                   Table 2. The routing table of Node a in Fig. 6(b).
    We first give the definition of isotonicity introduced
in [27, 28].
Definition 1. Left-isotonicity The quadruplet (S, ⊕, w, )
                                                                    Node s puts all the non-dominated paths received
is left-isotonic if w(a) w(b) implies w(c ⊕ a) w(c ⊕
                                                                 from its neighbors in its distance table. Besides, s
b), for all a, b, c ∈ S, where S is a set of paths, ⊕ is the
                                                                 also keeps all the non-dominated paths found by it-
path concatenation operation, w is a function which maps
                                                                 self in its routing table. When s receives an advertise-
a path to a weight, and is the order relation.
                                                                 ment (u, d, NF(p), NS(p), ω(p)) from u which represents
    Now, we present the proposed QoS path metrics,
                                                                 a non-dominated path p from u to d, s first removes all the
called composite bandwidth as follows.
                                                                 locally recorded paths from u to d which are dominated
Definition 2. Given a path p, the composite bandwidth
                                                                 by p by comparing ω(p) with the composite bandwidths
of p, denoted by ω(p), is (ω1 (p), ω2 (p), ω3 (p)) where
                                                                 of the paths from u to d in the distance table. Denote p
ω1 (p) = WB(p), ω2 (p) = TB(p), and ω3 (p) = FB(p).
                                                                 as the path from s to d which is one-hop extended from
ω(p1 ) ω(p2 ) iff ω1 (p1 ) ≥ ω1 (p2 ), ω2 (p1 ) ≥ ω2 (p2 ),
                                                                 p. s computes the composite bandwidth of p as follows:
and ω3 (p1 ) ≥ ω3 (p2 ).
Definition 3. Given two paths p1 and p2 , if ω(p1 )                             1               1       1
ω(p2 ), we call p1 dominates p2 . If we cannot find a path                   ω1 (p ) = min{ B (s,d) + ω2 (p) , ω1 (p)}
dominating p1 , we call p1 a non-dominated path.                                1         1        1                           (4)
                                                                            ω2 (p ) = B (s,d) + ω3 (p)
T HEOREM 1. The composite bandwidth of a path is left-                       ω3 (p ) = B (s, d)
P ROOF. Let p1 = <v, u1 , u2 , . . . , un , d> and p2 =              By comparing ω(p ) with the composite bandwidths
<v, g1 , g2 , . . . , gm , d>, such that ω(p1 ) ω(p2 ). Let p3   of the paths from s to d in the routing table, s can deter-
= <s, v1 , . . . , vh , v> from s to v. Denote p = p3 ⊕ p1 and   mine whether p is a non-dominated path and remove the
p = p3 ⊕ p2 . We are going to show that ω j (p) ≥ ω j (p )       paths that are dominated by p . Node s also knows that
for all j = 1, 2, 3.                                             path p goes through the subpath <s, u, NF(p), NS(p)>.
    By Lemma 2, we have ω1 (p) ≥ ω1 (p ). Since p and            If p is a non-dominated path, s will generate an advertise-
p share the same first link, it holds that ω3 (p) = ω3 (p ).      ment (s, d, u, NF(p), ω(p )). The implementation of the
If p3 consists of more than two nodes, both p and p              above discussion is illustrated in Procedure QoS Update.
share the same next second link, so that we have ω2 (p) =            The distance table of each node keeps all the non-
ω2 (p ). Now, we consider the case that p3 is <s, v>. We         dominated paths received from its neighbors to each
have                                                             destination, and the routing table keeps all the non-
                                                                 dominated paths from itself to each destination. Each
                     1          1          1
                   ω2 (p) = B (v,u1 ) + B (s,v)                  entry of the distance table contains the same informa-
                      1          1          1                    tion as an advertisement received from a neighbor. Ta-
                   ω2 (p ) = B (v,g1 ) + B (s,v) .
                                                                 ble 1 illustrates the distance table of node a in Fig. 6(b).
   Note that B (v, u1 ) = FB(p1 ) and B (v, g1 ) = FB(p2 ).      Based on its distance table, a knows that there are two
Since B (v, u1 ) ≥ B (v, g1 ), it holds that ω2 (p) ≥ ω2 (p ).   paths from b to destination g. One path going through
   Therefore, we have ω(p3 ⊕ p1 ) ω(p3 ⊕ p2 ).                   <b, c, f > has the composite bandwidth ( 60 , 10, 12) and
5.2    Table construction                                        another one going through <b, d, e> has the composite
    The isotonicity property of the proposed QoS metric          bandwidth ( 24 , 8, 24). By (4), a finds two non-dominated
of a path allows us to develop a routing protocol that can       paths from itself to g. Table 2 illustrates the routing
identify the maximum bandwidth path from each node               table of a. NT(p) denotes the third next hop on p.
to each destination. In our routing protocol, if a node          For each path p, the source has its three-hop neighbors
finds a new non-dominated path, it will advertise this            on p. For instance, a obtains one path going through
path information to its neighbors. We call the packet car-       the subpath <a, b, c, f > with the composite bandwidth
rying the path information the route packet. For each            ( 15 , 4, 6). To ease our discussion, we use the tuple
non-dominated path p from s to d, s advertises the tu-           (s, d, NF(p), NS(p), NT(p), ω(p)) to identify an entry in
ple (s, d, NF(p), NS(p), ω(p)) to its neighbors in a route       the routing table of node s, where p is a path from s to d.
packet. NF(p) and NS(p) are the next hop and the second              We would like to use the network topology in Fig. 6(b)
next hop on p from s. Based on the information contained         as an example to illustrate the process of our path calcu-
in a route packet, each node knows the first three hops of        lation. Suppose that each node computes the maximum
a path identified. This information is necessary for con-         bandwidth path from itself to g. Initially, the distance
sistent routing, as discussed in the following subsection.       table and the routing table of each node is set to be ∅.
                                                                              Non-dominated path
                                    Destination      Neighbor
                                                                           ω(p)             NF(p)         NS(p)
                                                          b             ( 60 , 10, 12)
                                                                          11                    c             f
                                                     b                   ( 24 , 8, 24)
                                                                           5   d         e
                                       Table 1. The distance table of Node a in Fig. 6(b).

In the first step, f finds an one-hop path from itself to                     not advertise path p1 = <vk−1 , vk , . . . , vn > to vk−2 , there
g with the composite bandwidth (12, 12, 12). It will add                    must exist a path p2 = <vk−1 , g1 , g2 , . . . , gm , vn > which
an entry ( f , g, g, g, g, (12, 12, 12)) in its routing table and           dominates p1 . We thus have ω(p2 ) ω(p1 ). Denote
generate an advertisement R f = ( f , g, g, g, (12, 12, 12)).               p = <v1 , . . . , vk−1 >. By Theorem 1, we have p ⊕ p2
When node c receives R f , it includes this path in-                        p ⊕ p1 . This means that p ⊕ p2 has larger available
formation contained in R f in its distance table. By                        bandwidth than that of path <v1 , . . . , vn >, implying that
using Eq. (4), it obtains a new path with the com-                          <v1 , . . . , vn > is not the maximum bandwidth path, which
posite bandwidth (10, 10, 60). It then adds an entry                        leads to contradiction.
(c, g, f , g, g, (10, 10, 60)) in its routing table and gener-
ates an advertisement Rc = (c, g, f , g, (10, 10, 60)). When                Procedure QoS Update of Node s
b receives Rc , it will find a new path with the com-                        /*
                                                                            s receives advertisement (u, d, NF(p), NS(p), ω(p))
posite bandwidth ( 60 , 10, 12). Similarly, b will also re-
                           11                                               */
ceive an advertisement Rd = (c, g, e, g, (6, 6, 12)) from                    1: for each path p1 from u to d in the distance table of s do
d, so that it will obtain another path with the com-                         2:     if ω(p) ω(p1 ) then
                                                                             3:         Remove p1 from the distance table
posite bandwidth ( 24 , 8, 24). Both paths from b to g
                             5                                               4: p ←< s, u > ⊕p
are non-dominated paths and there are two entries in                         5: Find ω(p ) using Eq. (4)
                                                                             6: for each path p2 from s to d in the routing table of s do
its routing table represented by (b, g, c, f , g, ( 60 , 10, 12))
                                                        11                   7:     if ω(p ) ω(p2 ) then
and (b, g, d, e, g, ( 24 , 8, 24)).        Node b thus gen-                  8:         Remove p2 from the routing table
                           5                                                 9:     else
erates two advertisements (b, g, c, f , ( 60 , 10, 12)) and
                                                                            10:          if ω(p2 ) ω(p ) then
                                                                            11:              return
(b, g, d, e, ( 24 , 8, 24)). When a receives two advertise-
                  5                                                         12: Advertise (s, d, u, NF(p), ω(p ))
ments from b, it will obtain the distance table illustrated
in Table 1. From the routing table of a illustrated by
Table 2, node a will generate two advertisements Ra1 =                          When a request carrying a bandwidth requirement
(a, g, b, c, ( 15 , 4, 6)) and Ra2 = (a, g, b, d, ( 24 , 24 , 6)). If       comes in, the node can determine whether there is a
                 4                                   7 5
s receives Ra1 first, it will obtain a path with the                         path to support this request by looking up the routing ta-
                                                                            ble. A resource reservation mechanism should be used
composite bandwidth ( 12 , 3, 6), and adds an entry
                                  5                                         to reserve the bandwidth and nodes should advertise new
(s, g, a, b, c, ( 12 , 3, 6)) in its routing table. When s re-
                    5                                                       available bandwidth information as the existing Internet
ceives the advertisement Ra2 later, it will obtain another                  routing mechanisms.
path with the composite bandwidth ( 3 , 3, 6). This path                    5.3 Packet forwarding
dominates the path contained in the routing table of s,                         Suppose that node s wants to transmit traffic to d along
node s thus removes the existing entry from its routing ta-                 the widest path p = <s, v1 , . . . , vn , d>. Then, each node vi
ble and adds a new entry (s, g, a, b, d, ( 8 , 3, 6)) in its rout-
                                              3                             on this path should make the consistent decision so that
ing table. Therefore, the routing table of s just records                   the traffic does travel along p. However, as mentioned
one path from itself to g.                                                  earlier, the widest path from vi to d may not be a subpath
T HEOREM 2. Our routing protocol satisfies the opti-                         on p. If vi selects the next hop according to its widest
mality requirement.                                                         path to d, the traffic may not be sent along the best path
P ROOF. We now prove that each node v1 must find the                         from s to d. In this subsection, we present the consistent
maximum bandwidth path to destination vn , denoted by                       hop-by-hop packet forwarding mechanism.
<v1 , v2 , . . . , vn >. Suppose that the widest (maximum                       In a traditional hop-by-hop routing protocol, a packet
bandwidth) path between v1 and vn is unique, we now                         carries the destination of the packet and when a node re-
prove that each on-path node vi must advertise the in-                      ceives a packet, it looks up the next hop by the destina-
formation of the subpath <vi , . . . , vn > to vi−1 , where                 tion only. In our mechanism, apart from the destination,
i = 2, . . . , n − 1, by induction.                                         a packet also carries a Routing Field which specifies the
    As the basic step, since vn−1 is a direct neighbor of vn ,              next three hops the packet should traverse. For example,
it must advertise the information of path <vn−1 , vn > to                   when node s in Fig. 6(b) wants to send a packet to g, by
vn−2 .                                                                      looking up the routing table, the Routing Field <a, b, d>
    For the inductive step, assume that vk advertises the                   will be put in the packet. The packet is sent to a, which
information of path <vk , . . . , vn > to vk−1 . If vk−1 does               is also the first node in the Routing Field.
   When a receives the packet from s, it should for-                   p is a non-dominated path by comparing p with other
ward the packet along subpath <a, b, d> as specified in                 paths contained in its routing table, as illustrated in Lines
the Routing Field. It should locate the path p where                   6-11 of Procedure QoS Update. There are at most O (A 2 )
NF(p) = b and NS(p) = d in its routing table. Table 2                  paths in its routing table, and so it takes O (A 2 ) time to de-
shows that NT(p) = e. Then, it updates the Routing Field               termine whether p is a non-dominated path. Therefore,
to <b, d, e> and send it to e.                                         the spacial and time complexities are both O (A 3 ).
L EMMA 3. If two paths share the same first two links
and have different available bandwidth, they cannot be
                                                                       6 PERFORMANCE EVALUATION
non-dominated paths at the same time.                                      In this section, we investigate the performance of our
P ROOF. Let paths p1 = <s, a, b, u1 , . . . , un , d> and p2 =         routing algorithm, called the composite bandwidth rout-
<s, a, b, g1 , . . . , gm , d>. p1 and p2 share the same first two      ing algorithm (CBRA), and compare it with the exist-
links. We have TB(p1 ) = TB(p2 ) and FB(p1 ) = FB(p2 ).                ing hop-by-hop routing algorithms. The widest-shortest
Since either WB(p1 ) > WB(p2 ) or WB(p2 ) > WB(p1 ),                   path algorithm, called WSP, is a well known hop-by-
p1 and p2 would not be non-dominated paths at the same                 hop routing algorithm. In WSP, each node computes the
time.                                                                  minimum-hop path from itself to the destination. Given
                                                                       several paths with the same hop count, the source selects
T HEOREM 3. Our routing protocol satisfies the consis-                  the one with the maximum bandwidth as the best path.
tency requirement.                                                     The available bandwidth of each path is computed based
P ROOF. Assume that node v1 wants to transmit a data                   on the conflict graph model. If the bandwidth of each link
packet over path <v1 , v2 , . . . , vn >. If v2 transmits the data     is the same, the minimum-hop path is also the widest. We
packet received from v1 over path <v2 , . . . , vn >, our rout-        also investigate the performance of the heuristic maxi-
ing protocol satisfies the consistency requirement.                     mum capacity path algorithm (HCP) proposed in [23]. In
   In our protocol, a node v1 identifies a path                         HCP, the bandwidth of each link is defined as the min-
<v1 , v2 , . . . , vn > because v2 advertises <v2 , . . . , vn >. In   imum of the bandwidth for all the links which interfere
other words, <v2 , . . . , vn > must be a non-dominated path           with this link. Moreover, the bandwidth of a path is de-
of v2 and is kept in the routing table of v2 . By Lemma 3,             fined as the minimum of the bandwidth for all the links on
there should be one path in v2 ’s routing table that goes              this path. We consider static wireless mesh networks with
through the subpath <v2 , v3 , v4 >.                                   n nodes uniformly deployed in a 2500m × 2500m region.
   Since <v2 , v3 , v4 , . . . , vn > is the only path that tra-       Each node has a fixed transmission range of 500m. The
verses the nodes specified in the routing field of the                   available bandwidth of each link is uniformly distributed
packet, v2 should forward the packet over the path se-                 in the range [50, 500]. We compute the maximum band-
lected by v1 .                                                         width path between any two nodes. The available band-
5.4     The complexities                                               width of the best path found by the algorithm is used for
    In the first step of the path computation, all the one-             evaluating the accuracy performance of the algorithm.
hop non-dominated paths have been found. In step k,                        We investigate how much improvement our algorithm
all the k-hop non-dominated paths have been found. The                 can achieve when compared with WSP and HCP. De-
process of path computation will terminate after |V | − 1              note Bcb , Bws , and Bhc as the maximum bandwidth of the
steps. The maximum available bandwidth path from each                  paths between two given nodes found by CBRA, WSP,
node to a destination must go through the non-dominated                and HCP, respectively. We compute the ratio of Bcb to
path from each on-path node to the destination.                        Bws , denoted by rcb,ws , and the ratio of Bcb to Bhc , de-
    Given a destination, we are going to analyze the spa-              noted by rcb,hc . The higher the ratio, the larger improve-
cial and time complexities for each node to find all the                ment produced by our algorithm. Given a pair of nodes,
non-dominated paths from itself to a destination. Denote               the distance between them is the minimum number of
A as the average number of the neighbors of each node.                 hops between them. The smaller the distance, the smaller
We first analyze the spacial complexity, which actually                 the interference, so that the higher probability that the
denotes the storage requirement of the distance table and              minimum-hop path is the widest. For instance, an one-
routing table in each node. By Lemma 3, there is only                  hop path is highly likely to be the widest, which will be
one non-dominated path going through the given first two                demonstrated by our simulation results. Therefore, we in-
links on this path. This means that the maximum number                 vestigate the impact of the pair distance on the improve-
of non-dominated paths from each node to a destination                 ment of our algorithm. We compute rcb,ws and rcb,hc under
is less than A 2 . The storage requirement for the routing             different node pair distances.
table and distance table of each node to each destination                  We have implemented an event-driven simulator. Our
are O (A 2 ) and O (A 3 ), respectively.                               simulator is single-threaded, and only one event with the
    When node s receives an advertisement, it should first              earliest execution time is performed at any give time,
update its distance table, as illustrated in Lines 1-3 of Pro-         which is the same as NS2. Our simulator considers the ef-
cedure QoS Update. Since there are at most O (A 3 ) paths              fect of the wireless transmission interference but not the
from the neighbors of s to d, it takes O (A 3 ) to remove              802.11 protocol overhead (such as DCF) on the perfor-
the paths dominated by p from the distance table of s.                 mance of the routing protocols. In the path computation
Afterwards, s finds a new path p and determines whether                 process, all nodes in the network broadcast the route in-
formation from itself to the destination to all its neigh-               The network   The number of messages
bors. According to the TRCA model, when a node v is                       topology     CBRA WSP        HCP
advertising the route information to the network, all the                 (50, 300)      71     69       67
nodes within three-hop neighborhood must not be adver-
tising at the same time. For instance, node v has an one-                 (50, 400)     1953   1210     997
hop neighbor v1 , a two-hop neighbor v2 , and a three-hop                 (50, 500)     5082   2450    1541
neighbor v3 . If they all have a packet to be advertised at               (50, 600)     7552   2915    2376
time t and the simulator lets v to advertise at time t, the             Table 3. The communication overhead.
simulator will change the execution time of the packets
in v1 , v2 , and v3 to be t + ∆t, where ∆t is the transmission            The network         The time period
delay of each packet. Sequentially, our simulator will re-                 topology       CBRA WSP HCP
sort the event list according to the execution time of all
the events. In our simulation, each packet has a fixed                       (50, 300)       127      124    121
transmission delay ∆t = 2.                                                  (50, 400)      2287     1478 1236
    Each advertisement message contains one path infor-                     (50, 500)      4890     2669 1421
mation. We use the number of messages generated totally                     (50, 600)      7463     2895 3862
for the path computation process to denote the communi-
                                                                             Table 4. The convergence speed.
cation overhead produced by the algorithm. We also test
the convergence speed of these algorithms.
    We first fix the transmission range of each node, and          algorithm changes a little, as shown in Figure 8(c).
change the number of nodes in the network. The simula-               We discussed the performance comparison of CBRA
tion results are shown in Fig. 7. Fig. 7(a) studies the ac-      with WSP in the above. Figures 7(b), 8(b), and 8(d) illus-
curacy performance between CBRA and WSP. When the                trate the performance comparison of CBRA with HCP.
distance between two nodes is 1 hop, there is no improve-        The simulation results show that HCP performs worse
ment. When the distance of the node pair is larger than          than the WSP algorithm, since the improvement of our
3 hops, for instance, in the 100-node networks, the aver-        algorithm with HCP is higher than that with WSP. There-
age improvement over WSP is more than 25%. We can                fore, the bottleneck capacity metric of each link proposed
also observe that the improvement generally increases as         in [23] cannot work well in the wireless networks.
the distance of pairs and the network size increase. How-
                                                                     Table 3 and Table 4 illustrate the communication over-
ever, considering the 100-node and 150-node networks,
                                                                 head and the convergence speed produced by the algo-
the improvement for the 9-hop pairs is smaller than those
                                                                 rithms. In our simulation, each advertisement packet
node pairs with smaller distances, such as 8-hop pairs.
                                                                 contains the information of one path only. The first col-
It is because the number of paths between 9-hop node
                                                                 umn represents the network topology. For instance, tuple
pairs is very small. As a result, the probability that the
                                                                 (50, 300) denotes 50-node network with 300 transmission
minimum-hop path is the widest increases. Nevertheless,
                                                                 range of each node. In our algorithm, the nodes need to
the improvement in 150-node networks and that in 200-
                                                                 exchange more route information, so that the communi-
node networks are almost the same. When the network
                                                                 cation overhead of our algorithm is the highest, and the
is sparse, the number of the paths between a node pair
                                                                 convergence speed is the slowest. In practice, route ad-
is small, so that the chance that the shortest path is the
                                                                 vertisement can be piggybacked in data packets. On the
widest is higher. As the network becomes dense, there
                                                                 other hand, we can also reduce the number of advertise-
are more paths between a node pair, and so it is less likely
                                                                 ment messages by putting more path information into a
that the shortest path is also the widest path. This implies
                                                                 single message. For example, instead of advertising each
that the improvement of our algorithm increases with the
                                                                 non-dominated path individually, we can group the non-
node degree. However, if the network is too crowded, the
                                                                 dominated paths to the same destination together to re-
number of the shortest paths is also large, which affects
                                                                 duce overhead and convergence time.
the improvement of our algorithm.
    We then fix the number of nodes and change the trans-         7 CONCLUSIONS
mission range of each node to investigate the behavior              In this paper, we have studied the problem of design-
of our algorithm. Figures 8(a) and 8(c) illustrate the           ing a hop-by-hop routing protocol with bandwidth re-
simulation results in 50-node networks and in 100-node           quirement in wireless mesh networks. We presented the
networks, respectively. These results also demonstrate           mechanism for path selection and packet forwarding. We
the characteristics of our algorithm as aforementioned.          have formally proved that our routing protocol satisfies
When increasing the transmission range in the 50-node            the optimality and consistency guarantees, so as to en-
network, the node density increases, so that the improve-        sure the proper operation of our protocol. By doing sim-
ment of CBRA comparing with WSP increases, as shown              ulations, we investigated the performance of our routing
in Figure 8(a). When the transmission range in the 100-          protocol. The simulation results showed that our routing
node network is more than 500 m, the network is too              protocol works better than the existing hop-by-hop rout-
crowded (we can observe that the maximum distance be-            ing protocols for finding the maximum bandwidth path.
tween nodes is 7 hops), so that the improvement of our           The widest path may have larger hop count than the short-
                            1.4                                                                                    3.5
                                                                                                                    3        150−node
  Average improvement (%)

                                                                                         Average improvement (%)
                            1.3                                                                                              200−node




                            1.1                                              50−node
                                                                             100−node                              1.5
                             1                                                                                      1
                                  1     2     3    4   5   6    7    8    9 10 11 12                                     1   2      3   4    5  6    7    8    9 10 11 12
                                            Distance between s/d pairs (Hop−count)                                               Distance between s/d pairs (Hop−count)

                                      (a) Average ratio of CBRA to WSP.                                                  (b) Average ratio of CBRA to HCP.
                                  Figure 7. The accuracy performance comparison with the fixed transmission range.

                            1.4                                                                                    1.8
                                                                              500−R                                                                                500−R
                                                                              400−R                                                                                400−R
                                                                              600−R                                                                                600−R
                                                                                         Average improvement (%)
  Average improvement (%)

                            1.3                                                                                    1.6

                            1.2                                                                                    1.4

                            1.1                                                                                    1.2

                             1                                                                                      1
                                  1     2      3   4    5  6    7    8    9 10 11 12                                     1   2      3   4    5  6    7    8    9 10 11 12
                                            Distance between s/d pairs (Hop−count)                                               Distance between s/d pairs (Hop−count)

(a) Average ratio of CBRA to WSP in 50-node networks.                                   (b) Average ratio of CBRA to HCP in 50-node networks.

                            1.8                                                                                    2.5
  Average improvement (%)

                                                                                         Average improvement (%)



                             1                                                                                      1
                                  1     2      3   4    5  6    7    8    9 10 11 12                                     1   2      3   4    5  6    7    8    9 10 11 12
                                            Distance between s/d pairs (Hop−count)                                               Distance between s/d pairs (Hop−count)

(c) Average ratio of CBRA to WSP in 100-node networks. (d) Average ratio of CBRA to HCP in 100-node networks.
                                             Figure 8. The accuracy performance comparison with the nodes number.
est path. In other words, the widest path may consume                           mobile ad hoc networks,” Computer Communication, vol. 29, pp.
more network resources and result in larger transmission                        1316-1329, 2006.
delay than the shortest path. In addition to the band-                     [18] Q. Xue and A. Ganz, “Ad hoc QoS on-demand routing (AQOR)
width requirement, a request generally also has the hop                         in mobile ad hoc networks,” J. Parallel and Distributed Comput-
count constraint. We would like to consider the hop count                       ing, vol. 63, pp. 154-165, 2003.
requirement together with the bandwidth requirement in                     [19] L. Chen and W.B. Heinzelman, “QoS-aware routing based on
our future work. Moreover, multi-channel communica-                             bandwidth estimation for mobile ad hoc networks,” IEEE J. Se-
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protocol to the multi-channel networks.                                         tion,” IEEE Symposium on Computers and Communication, pp.
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