where qi(Topt) = 1−FRi(Topt). Since the average number of hops in a route scales as the square-root of the number of nodes, even in a small-scale mobile ad hoc network with just 50 nodes, route lengths above 7 hops occur frequently. For a large-scale ad hoc network, the number of hops can be much larger, significantly increasing the computation time for Topt when deriving the optimal root of the h-order equation in (4.3). Therefore, in practice, it is a challenge to implement the optimal timeout caching scheme proposed in [54], particularly if the link up-times are not identically distributed.
4.3
Use Path Residual Time for Route Cache
Timeout
The most intuitive choice for the timeout of a given cached route should be set to coincide with the exact moment the communication between the source and destination nodes breaks. That is, the ideal timeout is equal to the path residual time1 [102, 103], R(h), of an h-hop route. In the ideal case, the cached route is
removed from the cache exactly when it ceases to physically exist.
However, due to the mobility of nodes in MANETs and the uncertainty of the radio propagation channel, it is often not possible to determine the exact residual time of a given link or a given path between two roaming mobile nodes. Therefore, it is pertinent to utilise statistical information of the link and path availabilities to estimate the optimal timeout. This approach has lead us to choosing the expected path residual time as an estimation of the life expectancy of a given route. Given a route with h hops, the expected path residual time, denoted by E{R(h)}, with corresponding link residual times {R1,R2,· · · ,Rh} [9, 80, 103], is given by
E{R(h)}=E{min(R1,R2,· · ·,Rh)}. (4.4)
The expected path residual time can be determined from the PDF ofR(h). Let
fRi(t) be the PDF, and FRi(t) be the CDF of the link residual time of the ith link of a given path. If all links are independent, then the distribution of min(R1,R2)
1Defined in Chapter 2: given an active path withhhops between two nodes at timet
0(which
may also have been active immediately prior to time t0), the path residual time, R(h), is the
4.3 Use Path Residual Time for Route Cache Timeout 61 is given by [70] fR(1)(t) =fR1(t), (4.5) and fR(2)(t) = fmin(R(1),R2)(t) = fR(1)(t) +fR2(t)−fR(1)(t)FR2(t)−fR2(t)FR(1)(t). (4.6) And, in general, fR(i)(t) = fmin(R(i−1),Ri)(t) = fR(i−1)(t) +fRi(t)−fR(i−1)(t)FRi(t)−fRi(t)FR(i−1)(t). (4.7)
This can be applied iteratively to derive the PDF,fR(h)(t), of the path residual
time of an h-hop path. Subsequently, we can obtain the expected value E{R(h)}. The determination of E{R(h)} depends on the information available for the constituent links. The simplest case occurs when all links can be accurately pre- dicted, such as when the MNs move in a deterministic manner. For instance, given a 5-hop route with known link residual times [7 6 8 2 5], respectively, the expected path residual time is given by E{R(5)}= min([7 6 8 2 5]) = 2.
However, if the link information is based on cache history, or if the MNs move randomly, the links are no longer deterministic. For example, let the link residual times be exponentially distributed with mean values, [7 6 8 2 5], respectively. In this case, the expected path residual time is given by E{R(5)}= 1/(1/7 + 1/6 + 1/8 + 1/2 + 1/5) = 1.0084. We explore this scenario in more detail in Section 4.3.1.
Remark 4.3.1
Using E{R(h)}) as the route cache timeout is intuitive and much more practical to implement than Topt in [54]. Calculation of E{R(h)} from (4.4) is less com-
putationally complex than the calculation of Topt by finding the roots of (4.3), particularly if the link residual times in the route are not identically distributed.
We note that, for the fourth link in [7 6 8 2 5], with the smallest mean link residual time, E{R4} = 2, the hosts of this link either move faster or have a
larger initial separation distance than others. Consequently, the path is much more affected by this link than the other links. Our strategy, based on (4.4), can readily deal with any combination of link distributions. For example, the first link of a route could be Rayleigh distributed, and the second link could be exponentially distributed.
In the following, we show how to quickly estimate the value of E{R(h)} from the mean values of the link residual times. Three example probability distributions are considered: the exponential distribution, the Rayleigh distribution and a de-
62 Choice of Timeout for Route Caching
terministic model. In all three cases, we assume that the route request times are from a Poisson process, but this assumption does not affect the choice of timeout.
4.3.1
Exponential Distributed Link Residual Time
The distributions of path and link duration in MANETs have been extensively studied. In [9, 38, 80], link and path duration PDFs are shown to vary with factors such as mobility model, relative speed, number of hops, and transmission range. However, it is suggested that, at moderate and high velocities, the exponential dis- tribution with appropriate parameterizations is a good approximation of the path duration distribution for a range of mobility models such as the Random Way- point, Reference Point Group Mobility, Freeway and Manhattan Mobility models. Our theoretical work in Chapter 3 and [103] verify that the link and path duration are exponentially distributed in the Random Walk Mobility model. Therefore, we suggest that exponential distribution is generally a good approximation of the link duration PDF.
If an arbitrary link of an h-hop route has exponentially distributed duration with parameter λi, from (4.7) the distribution of min(R1,R2) can be shown to be
given by
fR(2)(t) = (λ1+λ2)e−(λ1+λ2)tU(t), (4.8)
where U(t) is the unit step function. Iteratively, the distribution of R(h) can be shown to be given by fR(h)(t) = e− Ph i=1λit h X i=1 λi U(t). (4.9)
Thus, it can be shown that the expected path residual time is given by
E{R(h)}= Ph1
i=1λi
= Ph 1
i=11/µi
, (4.10)
where µi is the mean value of the ith link residual time.
Now, let us assume that the time between route requests is exponentially dis- tributed with parameter λa. Following [54], the expected routing delay for non-
identically exponentially distributed link residual times can be shown to be:
C(T) = 2Lh+ 2λaL h−1 X i=0 1−e−(λa+Pij=0λj)T λa+ Pi j=0λj −4λaLh 1−e−(λa+Phi=1λi)T λa+ Ph i=1λi , (4.11)
4.3 Use Path Residual Time for Route Cache Timeout 63
time from (4.10), the corresponding expected routing delay is:
C(E{R(h)}) = 2Lh+ 2λaL h−1 X i=0 1−e− λa+Pij=0λj Ph j=1λj λa+ Pi j=0λj −4λaLh 1−e−(1+λa/Phi=1λi) λa+ Ph i=1λi . (4.12) If all link residual times of an h-hop route are i.i.d. with fR(t) = λe−λt, then
(4.9) can be rewritten as:
fR(h)(t) =hλe−hλtU(t). (4.13)
It can be shown that E{R(h)} in this case is given by
E{R(h)}= 1
hλ = µ
h. (4.14)
Given the expected path residual time from (4.14), the corresponding expected routing delay is:
C(E{R(h)}) = 2Lh+ 2λaL h−1 X i=0 1−e−(i h+λahλ) λa+iλ −4λaLh 1−e−(1+λa hλ) λa+hλ . (4.15)
Comparing (4.14) with (4.2), we readily observe that the E{R(h)} is much easier to calculate than the Topt in [54], for identically exponentially distributed
links. It is the same with the non-identically distributed links, by comparing (4.10) with (4.3).
4.3.2
Rayleigh Link Residual Time
If the link residual times in an h-hop route are now Rayleigh distributed with the
ith link having PDF fRi(t) = t σ2 ie −t2/2σ2 i, from (4.7) we obtain fR(2)(t) = fmin(R(1),R2)(t) = t σ2 12 e−t2/2σ122 U(t), (4.16) where σ12 = √σ1σ2 σ2 1+σ22
. Iteratively, the distribution and the expected value of R(h) can be obtained: fR(i)(t) =fmin(R(i−1),Ri)(t) = t σ2 1i e−t2/2σ12iU(t), (4.17) whereσ1i = √σσ12,i−1σi 1,i−1+σ2i
. Once we have derived the PDFfR(i)(t), which is of Rayleigh
distribution, the expected value of E{R(h)}is:
E{R(h)}=σ1h
r π
64 Choice of Timeout for Route Caching
where σ1h given by (4.17).
If all link residual times of an h-hop route are identically Rayleigh distributed with fR(t) = σt2e−t 2/2σ2 , we obtain E{R(h)}= r π 2hσ= µ √ h. (4.19)
It can be seen that E{R(h)}for Rayleigh distributed link residual times is similar to that for the exponential distribution in (4.2). Consequently, a similar analysis can be made, as in section 4.3.1.
4.3.3
Deterministic Link Residual Time
In many mobile networks, nodes exhibit some degree of regularity in their mobility pattern [88]. For example, a car travelling on a highway, or a tank travelling across a battlefield, is likely to maintain its direction and speed for some period of time. By exploiting the non-random travelling patterns (and GPS location information) of mobile users, we can calculate the exact link residual time between a pair of mobile nodes. That is, we can assume the link residual time is deterministic instead of random. If mobile nodes move with constant velocity, the path residual time is given by
R(h) = min(R1,R2,· · ·,Rh). (4.20)
The residual time of the wireless link between two nodes with transmission range r, which are moving with speed, Vi and Vj at angles θi and θj, respectively,
can be estimated as [88]:
R= −(ab+cd) +
p
(a2+c2)r2−(ad−bc)2
a2+c2 (4.21)
where (xi, yi) is the coordinate of mobile node i, a = Vicos(θi)−Vjcos(θj), b =
xi−xj, c=Visin(θi)−Vjsin(θj), and d=yi−yj.
Let the route request time again be exponentially distributed with fa(t) =
λae−λat. Then, the corresponding expected routing delay can be shown to be:
C(R(h) ) = 2Lh e−λaR(h). (4.22)
In Section 4.3.2 and Section 4.3.3, we demonstrate that, apart form exponential distribution, E{R(h)} can be readily derived for many link distributions, such as Rayleigh distributed and deterministic links. Though theoretically the method in [54] can also deal with arbitrary link distributions, we found it is hard to obtain the corresponding expressions of Topt.
4.3 Use Path Residual Time for Route Cache Timeout 65
4.3.4
Implementation Issues
Generally there are two different ways to implement the proposed timeout scheme in on-demand routing, depending on whether or not the MNs in the network are equipped with location-aided system. The performance of the solution is also slightly different with different on-demand routing protocols.
Figure 4.2: Comparison of the actual link residual time and the recorded duration of the link in the cache.
The first method is based on recorded link information. Without a location-aid information system, such as GPS, MNs can only use the recorded link lifetimes to estimate the expected path residual time. This method does not require any extra hardware or control overhead packets, but it does require more CPU time. Every MN keeps a link cache table, which is separate from the route cache table. For the link between a given host node and a specific neighbouring node, the host node records a series of link residual times, approximated by the actual duration of the link in the cache, that is, the time between an active link entering the cache and the noted time of failure to transmit a packet over this link, as shown in Fig. 4.2. In the on-demand routing protocol, AODV, the delay before the link is removed from cache is negligible because of the periodic broadcasting of link information between neighbouring nodes. But in other on-demand routing protocols without link broadcasting, such as DSR, the delay is small only when the network traffic
is reasonably high, which means small µa. However, if the traffic is so low that
µa Àµ, the cached route is likely to expire before the next route request arrives.
In this case, the cache timeout is not as important, because the source node needs to discover a new route, anyway.
Once the link residual times have been recorded, the mean link residual time is
input to (4.10) as parameter, µi. The source node sets the timeout to E{R(h)},
based on the parameter,µi, which is included in the routing request packet (RREQ)
66 Choice of Timeout for Route Caching
RREQ and RREP packets, rather than being periodically broadcast in packets dedicated to this task, the number of routing control packets is not increased by link information dissemination.
Alternatively, estimation ofE{R(h)} can be achieved by prediction from GPS information. This is especially useful when MNs are travelling in a non-random pat- tern (e.g., in Random Waypoint Model MNs move long distances without changing direction), where the link residual time is given by (4.21). In other random mobil- ity models, such as the Random Walk Mobility model, the expected link residual time can be estimated using velocity and initial separation distance information as discussed in Chapter 3 and [101, 102]. The location information is included in the routing control packets when the source node initiates a route discovery process, and it can be overheard by any intermediate nodes.
In this section, we have presented a practical method for choosing the route cache timeout for the minimum routing delay. In the next section, we will compare our proposed route cache scheme to other cache schemes.