In the absence of capacity constraints, the cost can represent energy consumption. In a general multi-hop ad hoc network, the hop distance can be optimized so as to minimize the energy consumption. Even within a single cell of 802.11 IEEE wireless LAN one can improve the energy consumption by using multiple hops, as it has been shown not to be efficient in terms of energy consumption to use a single hop [50].
Alternatively, the cost can take into account the scaling of the nodes (as we have done in Section 1.2.2) that is obtained when there are energy constraints. As an example, assuming random deployment of nodes, where each node has data to send to another randomly selected node, the capacity (in bits per Joule) has the form f (λ) = Ω (λ/ log λ)(q−1)/2 where q is
Chapter 2
Electrostatics Approach
In the work of Toumpis et al. ([25, 48, 24, 23, 26, 52]), the authors addressed the problem of the optimal deployment of massively dense wireless ad hoc networks by analogy with Electrostatics. We shall recall below the representation of the flow conservation constraint, which is well known in Electrostatics. This derivation appears both in physics-inspired works, as well as in the road traffic literature [29].
We first consider the one dimensional case in order to explain the main concepts involved in our model and how these concepts can be extended to the two dimensional case in order to obtain the optimal deployment of the relay nodes in a wireless ad hoc network.
2.1
Fluid approximations: one-dimensional case
As a first approach we consider the line segment [0, L] as the geographical reference of the network. We consider the continuous node density function η(x), measured in nodes/m, such that the total number of nodes on a segment [ℓ0, ℓ1], denoted by N(ℓ0, ℓ1), is
N(ℓ0, ℓ1) = ℓ1
Z
ℓ0
η(x) dx.
We consider as well the continuous information density function ρ(x), measured in bps/m, generated by the nodes such that
• At location x where ρ(x) > 0 there is a fraction of data created by the sensor sources, such that the rate with which information is created in an infinitesimal area of size dε, centered at position x, is equal to ρ(x) dε.
• Similarly, at location x where ρ(x) < 0 there is a fraction of data received at the sensor destinations such that the rate with which information is received by an infinitesimal area of size dε, centered at position x, is equal to −ρ(x) dε.
We assume that the total rate at which sensor destinations have to receive data is the same as the total rate which the data is created at the sensor sources, i.e.,
L
Z
0
ρ(x) dx = 0. (2.1)
Notice that if we have an estimation of the packet loss through the network, we can put different weights to the evaluations of the function ρ in order to adequate the function to satisfy equation (2.1).
Consider the continuously differentiable traffic flow function T (x), measured in bps/m, such that its direction (positive or negative) coincides with the direction of the flow of information at point x and kT (x)k is the rate at which information propagates at position x, i.e., kT (x)k gives the total amount of traffic that is passing through the position x.
Next we present the flow conservation condition. In order to conserve the information transmitted over a line segment [ℓ0, ℓ1], it is necessary that the rate with which information is
created over the segment is equal to the rate with which information is leaving the segment, i.e.,
T (ℓ1)− T (ℓ0) =
Z ℓ1
ℓ0
ρ(x) dx.
The integral on the right hand side is equal to the quantity of information generated (if it’s positive) or demanded (if it’s negative) by the fraction of nodes over the line segment [ℓ0, ℓ1].
The expression T (ℓ1)−T (ℓ0), measured in bps/m, is equal at the rate with which information
is leaving (if it’s positive) or entering (if it’s negative) the segment [ℓ0, ℓ1]. This holding for
any line segment, it follows that necessarily, dT (x)
dx = ρ(x) for all x ∈ (0, L). (2.2)
The problem considered is to find the number of nodes N(0, L) in the line segment [0, L], needed to support the information created by the sources and received at the destinations subject to the flow conservation condition given by equation (2.2) and imposing that there is no flow of information leaving the network, i.e., T (0) = 0 and T (L) = 0. Thus the system of equations that model our problem in the one-dimensional case is given by:
Min N(0, L) = Z L
0
η(x) dx, (2.3)
subject to the following constraints dT (x)
dx = ρ(x) for all x ∈ (0, L), (2.4)
T (0) = 0 and T (L) = 0. (2.5)
Notice that in the one-dimensional case, there is no minimization problem since by using the constraints (2.4) and (2.5), we obtain just one solution. As we will see, this will not
23 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 Line Segment In fo rm at io n d en si ty ρ
Figure 2.1: The information density function over the line segment [0 m, 10 m] in Exam- ple 2.1.1, given in the first half of the line segment [0 m, 5 m] by a uniform information den- sity function generated by the sources of ρ(x) = 1 bps/m and in the second half [5 m, 10 m] by a uniform information density function received at the sensor destinations given by ρ(x) =−1 bps/m.
be the case for the two-dimensional case. Within the one-dimensional case context, we further assume that the proportion of sensor nodes η(x) in a line segment of infinitesimal size dε, centered at location x, needed as relay nodes, is proportional to the traffic flow of information that is passing through that region, i.e., η(x) dε =kT (x)kαdε where α > 0 is a
fixed number called the relay-traffic constant. Then the optimal placement of the relay nodes in the network will be given by η∗(x) = kT∗(x)kα, where the traffic flow function T∗(x) is
the optimal traffic flow function, given by the solution of the previous system of equations. Furthermore, the optimal total number of relay nodes N∗(0, L) needed to support the optimal
traffic flow function T∗(x) in the network will be
N∗(ℓ0, ℓ1) = Z ℓ1 ℓ0 η(x) dx = Z ℓ1 ℓ0 kT (x)k αdx.
Let us see an example to illustrate the previous framework.
Example 2.1.1 Suppose that we can divide the line segment [0, L] in two parts:
• In the first half [0, L/2] there will be a uniform information density function generated by the sensor sources, given by ρ(x) = 1 bps/m.
• In the second half [L/2, L] there will be a uniform information density function received at the sensor destinations given by ρ(x) =−1 bps/m (see Figure 2.1).
From the equations (2.4) and (2.5) we obtain that the optimal traffic flow function will be given by
T∗(x) =
x bps/m for all x∈ [0, L/2] L− x bps/m for all x∈ [L/2, L]
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 Line Segment T ra ffi c F lo w |T |
Figure 2.2: Optimal magnitude of the traffic flow T∗ with positive direction in Example 2.1.1
where in the first half of the line segment [0 m, 5 m] there is a uniform information density function generated by the sensor sources of ρ(x) = 1 bps/m and in the second half [5 m, 10 m] there is a uniform information density function received at the sensor destinations given by ρ(x) =−1 bps/m.
with positive direction (see Figure 2.2). If we further assume that the relay-traffic con- stant α = 2, then the optimal placement of the relay nodes needed to relay the information from the sources to the destinations on the network will be given by (see Figure 2.3)
η∗(x) =
x2 nodes for all x∈ [0, L/2]
(L− x)2 nodes for all x∈ [L/2, L]
The optimal total number of relay nodes N∗(0, L) needed to support the optimal traffic
flow T∗(x) will be given by
N∗(L) = Z L/2 0 x2dx + Z L L/2 (L− x)2dx = L3/12.
From this example we obtained a closed-form expression for the total number of nodes needed to maintain the optimal traffic flow as a function of the length of the line segment for the one-dimensional case. Within this context the problem didn’t required any minimization. This will not be the case for the two-dimensional case.