In this section, we develop an isotonic routing metric based on uniform description of interference and load. Equivalent bandwidth is utilized to describe logical interflow interference, intraflow interference, and physical interference uniformly. MIL also takes load information into consideration to help avoid routing packets into heavy load areas. A byproduct of MIL is channel diversity expression (CDE), which quantifies the channel distribution along a path.
Interference model
Consider MRMC WMNs, where each node is equipped with multiple radio interfaces. Each radio interface is preconfigured to a certain channel; there is no channel switching. Radios configured to different channels do not interfere with each other; they can be active simultaneously. Radios belonging to the same node are configured to different channels.
In this paper, we take both physical interference and logical interference into consideration. We consider a 802.11based MAC layer, a successful transmission from node v to node u needs or will result in the silence of wireless link (s,t) satisfying the conditions given by Equations 9 and 10 [21]:

1.
During the transmission of the data packet from v to u
d\left(v,s\right)\le {R}_{h}\left(v\right)\phantom{\rule{0.3em}{0ex}}\text{or}\phantom{\rule{0.3em}{0ex}}d\left(v,t\right)\le {R}_{h}\left(v\right)\text{or}\phantom{\rule{0.3em}{0ex}}d\left(s,u\right)\le {R}_{h}\left(s\right)
(9)

2.
During the transmission of the ACK frame from u to v
\phantom{\rule{12.0pt}{0ex}}d\left(u,s\right)\le {R}_{h}\left(u\right)\phantom{\rule{0.3em}{0ex}}\text{or}\phantom{\rule{0.3em}{0ex}}d\left(u,t\right)\le {R}_{h}\left(u\right)\phantom{\rule{0.3em}{0ex}}\text{or}\phantom{\rule{0.3em}{0ex}}d\left(s,v\right)\le {R}_{h}\left(s\right)
(10)
where R_{
h
}(v) denotes carrier sensing range of node v. d(s,u) denotes the distance between nodes s and u. Note that in Equations 9 and 10, we have not considered the signal capture property. So, we utilize the physical interference model to describe the interference among different hops from the signal strength point of view. This interference model indicates that a transmission from node v to node u is successful if the SINR at receiver u is not less than a predetermined threshold γ, i.e.,
\frac{{P}_{u}\left(v\right)}{N+\sum _{q\ne v}{P}_{u}\left(q\right)}\ge \gamma
(11)
where N denotes the received background noise power, P_{
u
}(v) denotes the received signal power at node u from node v, and P_{
u
}(q) denotes the interference power from a different transmitting node q.
MIL metric definition
Two neighboring links that belong to the different flows cannot be active simultaneously when operating on the same channel; we call this interflow interference [26]. Whether a transmission is successful or not is also influenced by the physical signal power, so the equivalent bandwidth of link i under logical interflow interference and physical interference can be calculated as follows:
{B}_{\text{Inter},i}=(1{\text{CBT}}_{i})\times {B}_{\text{bas}}\times {\text{IR}}_{i}
(12)
where B_{bas} is the nominal link data rate and CBT_{
i
} is the channel busy time, which denotes the utilization of channel used by link i. CBT_{
i
} can be obtained from Equation 13:
{\text{CBT}}_{i}=\frac{\text{TotalTime}\text{IdleTime}}{\text{TotalTime}}
(13)
where TotalTime is the entire monitoring time and IdleTime is the time when no data keeps the channel busy. Analysis in [27] shows that CBT is the most precise means of measuring the utilization of channels in wireless networks, which can be acquired by passive monitoring, without introducing overhead into the networks. CBT can measure logical interference more accurately than other measures.
IR_{
i
} is interference ratio, which is given in Equation 14:
{\text{IR}}_{i}=\frac{{\text{SINR}}_{i}}{{\text{SNR}}_{i}}
(14)
where SINR_{
i
} is the signaltointerferenceplusnoise ratio and SNR_{
i
} is the signaltonoise ratio.
For a single flow, along with its routing path, the links that are close to and interfering with each other cannot transmit simultaneously, which is termed as intraflow interference. They can be viewed as a virtual link, the equivalent bandwidth of the virtual link under logical intraflow interference is
{B}_{\text{Intra},\mathit{\text{ij}}}=\frac{{b}_{i}\times {b}_{j}}{{b}_{i}+{b}_{j}}
(15)
where b_{
i
} and b_{
j
} are the available bandwidth of links i and j, respectively.
The impact of interflow interference on link capacity can be conveniently integrated with the intraflow interference by substituting b_{
i
} and b_{
j
} in Equation 15 with the equivalent bandwidth calculated from Equation 12; the equivalent bandwidth of the virtual link under various interference can be defined as
{B}_{\mathit{\text{ij}}}=\frac{{B}_{\text{Inter},i}\times {B}_{\text{Inter},j}}{{B}_{\text{Inter},i}+{B}_{\text{Inter},j}}
(16)
If equivalent bandwidth above is directly used in the routing metric, just as the case in [22], it may result in nonisotonic property.
In order to achieve isotonicity, we regard equivalent bandwidth calculated from Equation 16 as a single link’s bandwidth. If interference exists between links that are within 2 hops, for the first link of the path, i.e., the link originates from the source, it has no previous link, so its equivalent bandwidth can be calculated from Equation 12. For the second link of the path, say link l, with previous link m, its equivalent bandwidth can be calculated as below:
{B}_{l}=\left\{\begin{array}{cc}{B}_{\text{Inter},l}& \text{CH}\left(l\right)\ne \text{CH}\left(m\right)\\ \frac{{B}_{\text{Inter},m}\times {B}_{\text{Inter},l}}{{B}_{\text{Inter},m}+{B}_{\text{Inter},l}}& \text{CH}\left(l\right)=\text{CH}\left(m\right)\end{array}\right.
(17)
From Equation 17, we can see that if link l and link m use different channels, which means that they are attached to different interfaces on the same node, they will not affect each other and can transmit simultaneously, so the equivalent bandwidth of link l is only affected by interflow interference and physical interference, and has nothing to do with link m. If link l and link m use the same channel, which means that they are attached to the same interface on the same node, link l must keep silent while link m is transmitting, so the equivalent bandwidth of link l is affected by intraflow interference from link m, interflow interference, and physical interference. For the third link of the path and links after it, say link k, link k may interfere with its previous link j and link j’s previous link i; the equivalent bandwidth of link k can be defined as
\phantom{\rule{13.0pt}{0ex}}{B}_{k}=\left\{\begin{array}{cc}{B}_{\text{Inter},k}& \text{CH}\left(k\right)\ne \text{CH}\left(j\right),\text{CH}\left(k\right)\ne \text{CH}\left(i\right)\\ \frac{{B}_{\text{Inter},i}\times {B}_{\text{Inter},k}}{{B}_{\text{Inter},i}+{B}_{\text{Inter},k}}& \text{CH}\left(k\right)\ne \text{CH}\left(j\right),\text{CH}\left(k\right)=\text{CH}\left(i\right)\\ \frac{{B}_{\text{Inter},j}\times {B}_{\text{Inter},k}}{{B}_{\text{Inter},j}+{B}_{\text{Inter},k}}& \text{CH}\left(k\right)=\text{CH}\left(j\right),\text{CH}\left(k\right)\ne \text{CH}\left(i\right)\\ \frac{{B}_{\mathit{\text{ij}}}\times {B}_{\text{Inter},k}}{{B}_{\mathit{\text{ij}}}+{B}_{\text{Inter},k}}& \text{CH}\left(k\right)=\text{CH}\left(j\right),\text{CH}\left(k\right)=\text{CH}\left(i\right)\end{array}\right.
(18)
From Equation 18, we can see that four cases may happen in the calculation of link k’s equivalent bandwidth. In the first case, B_{
k
} has nothing to do with link j or link i, because they use different channels. In the second and third cases, B_{
k
} is only related to one previous link, link i or j. In the last case, B_{
k
} is related to both links j and i. Of course, in all four cases, B_{
k
} is also affected by interflow interference and physical interference.
Based on equivalent bandwidth above, MIL routing metric for path p can be defined as
\text{MIL}\left(p\right)=\sum _{k\in p}\overline{{L}_{k}}\times \frac{S}{{B}_{k}}
(19)
where S is the packet size and \overline{{L}_{k}} is the average load of link k. Route oscillation caused by loadaware routing metric may result in continuous route selection and handoff, so it has great effect on the network performances, and it may even disturb normal operation of the networks. In this paper, average load is used in the place of instantaneous load, here load means buffer queue length of the link’s end node; the node will sample its own load periodically and calculate average load from the current sample value and previous value. Say link k uses current sample load value L_{k  cur} and previous value L_{k  pre} to obtain average load \overline{{L}_{k}} through exponential weighted moving average scheme:
\overline{{L}_{k}}=(1\theta )\times {L}_{k\text{cur}}+\theta \times {L}_{k\text{pre}}
(20)
where θ is the moving exponent.
Isotonicity demonstration
MIL is an isotonic metric which takes load and various interference into consideration, and it can detect heavy load and heavy interference areas in the network and guide packets to bypass these areas. As intraflow interference exists between links that are within 2 hops, transmission on link k may interfere with that on its previous link j and link j’s previous link i, so we define equivalent bandwidth which can be calculated from Equations 12, 17, or 18. The expression of equivalent bandwidth is similar as CSC in [16]; the only difference is that CSC equals to w_{2}, w_{3}, w_{2}+w_{3} or w_{1} which are constant, and the value of our equivalent bandwidth is not constant. Thus, we use the same virtual network method in [16] to achieve isotonicity. As the combination of channel assignments for precedent links within 2 hops is finite, it is possible to introduce virtual nodes to represent all channel assignment states. By doing this, routing metric can be translated into isotonic weight assignment to the virtual links. Since the routing metric of a path is the same as the aggregated link weight of the corresponding virtual links and the weight assignments of the virtual links are isotonic, efficient algorithms can find minimum weight paths. More details can be found in ([16], Section 6).
Channel diversity expression
In this paper, we propose a new quantity CDE to describe the real channel distribution along paths, which is defined as follows:
\text{CDE}\left(p\right)=\sum _{k\in p}\frac{{B}_{k}}{{B}_{\text{bas}}}
(21)
where B_{bas} is the nominal link data rate and B_{
k
} denotes the equivalent bandwidth of link k on path p calculated from Equations 12, 17, or 18. Higher CDE means that channel distribution is more uniform, and interference is lower. Suppose the interference range is 2 hops, nominal data rate is 2 Mbps, CDE value for path c in Figure 4 is 5.5, and CDE value for path d is 6.17, so CDE can describe the actual channel diversity.
We further illustrate that CDE is a better index for the channel distribution using example in Figure 5. When selecting path from source node S to destination node D, there are parallel flows E → F → G and H → K → L, which are within the interference range of transmissions on S → A → C → D and S → B → C → D, respectively. The number on each link denotes corresponding channel assigned to the link; now, we can see that flow H → K → L will not interfere with transmission on path S → B → C → D, as they use orthogonal channels. Intraflow interference exists between links S → B and B → C. Flow E → F → G will totally interfere with transmission on path S → A → C → D, as link E → F interferes with S → A and link F → G interferes with A → C, but there is no intraflow interference on path S → A → C → D. If transmission on path E→F→G occupies not less than 50% of the total monitoring time, transmission on S → A → C → D will be largely affected, then path S → B → C → D should be selected. Suppose transmission on path E → F → G occupies 50% of the total monitoring time. If CDI which takes nothing about interflow interference is used, it will select path S → A → C → D, as this path has no intraflow interference. If CDC is used, the CDC values for paths S → A → C → D and S → B → C → D are equivalent, it cannot select which one is better. When CDE is applied, the CDE values for paths are
S\to A\to C\to D:0.5+0.5+1.0=2.0
S\to B\to C\to D:1.0+0.5+1.0=2.5
From the calculation results, we can see path S → B → C → D is better than path S → A → C → D, which matches the analysis above. Thus, CDE can describe the actual channel distribution along paths.