Outage Analysis of Asynchronous Interference Limited Wireless
3.2 System Model
3.2.2 Proposed Model for Representing Interference
3.2.2.2 Expected Numbers of Type-1 and Type-2 Interferers
Unlike [31], where only first tier of nodes are considered as effective interferers, this work does not take any assumption while calculating effective number of interferers. A generic solution is presented here to calculate the effective number of interferers, outside the guard zone,rz. However, assumption as in [31] may be useful in case of less dense network with high path loss exponent, where only few nodes outside the rz act as effective interferers.
At any time instant, the effective number of interferers in the network are all transmitters except those barred by MAC protocol (lie within rz), i.e.
effective interferes =Ks−desired transmitter−transmitters barred by MAC protocol (3.18) In a randomly deployed network, a transmitter node can be categorized as a potential interferer if the received power (Pr) from that node at receiver of interest,Rxo, is considerably high (assuming links are reciprocal).Therefore, probability for ith transmitter node to be within the guard zone
3.2 System Model
around a receiver (Pri
zRx), conditioned onr is given by Pri
zRx|r = P[i∈ rzRx|r] = P[Pr > ΨPth|r]
=
∫ PT
ΨPth
fPr(p) dp
(3.19)
where rzRx, Pth, PT, r and fPr(p) are receiver’s guard zone, detection threshold, transmit- ted power, distance between receiver node and interfering node and the pdf of received power, respectively†. The radius of guard zone can be adjusted with Ψ∈ (0,1]. Here, a trade-off exists between network capacity and interfering users. Large value of Ψ results in high interference energy, but the sum capacity also increases. A small value of Ψ restricts more users from transmitting, thereby keeping interference energy low but sum capacity is also reduced. To solve (3.19) further, it is assumed that the received power is lognormally distributed in shadowing environment. Using the pdf of lognormal RV in (3.19), we find
P[i∈rzRx|r] =
∫ PT
ΨPth
4.343 p√
2πσ2 exp [
−((p)dB− m)2 2σ2
] dp
= 1 2
[ erf
(m−(ΨPth)dB
√2σ )
−erf
(m−(PT)dB
√2σ
)] (3.20)
wherem and σ are mean and standard deviation of corresponding Gaussian distribution. Cal- culations for estimated number of effective interferers are based on average signal strength which is lognormally distributed. Considering r as a uniformly distributed RV which takes values in the range [λo,√
2Ar], whereλoand√
2Arare signal wavelength and the diagonal distance of deployment area, respectively, removing the condition on r, we get
PrizRx = P[i∈ rzRx]
=
∫ √2Ar
λo
P[i∈ rzRx|r] fr(r)dr
(3.21)
In the absence of closed form expression for the general case of (3.21), a closed form expres- sion is given for the special case of path loss exponents, η1 = η2 = η in Appendix A.1. It is enough to say, the event that a node falls in the guard zone rz, follows Bernoulli distribution with PrizRx = P[i ∈ rzRx]. Also, if N and Nr represent the total number of nodes in the network
†PT is used as the upper limit as maximum received power cannot exceedPT, even if the transmitter is in close proximity of the receiver.
3.2 System Model
and in the guard zone, respectively, then Nr follows binomial distribution with P[Nr = No] = CNN
o
(Pri
zRx
)No( 1−Pri
zRx
)N−No
.
Now the number of (interfering) transmitters inside therz of Rxo can be written as T xrzRxo = (Ks−1)PrizRxo
= (Ks−1)
(√ ℜ 2Ar−λo
) (3.22)
where ℜ is a part of the solution of (3.21) and a function of deployment area width as well as wavelength†.
As class-2 MAC protocol (CSMA/CA with reservation)‡ of [30] is used, transmitters having as- sociated receivers inrz ofRxodo not contribute to effective interference, hence, they are considered as barred. In case of randomly distributed nodes, a node is assumed to have equal probability to become either transmitter or receiver. So, the number of receivers in therz(of intended transmitter / receiver) can be assumed to be proportional to the number of transmitters in the same rz. Here, we assume the number of such receivers in the rz of Rxo can be given by
RxrzRxo = ψ1 (Ks−1)PrizRxo (3.23) whereRxrzRxo is also equal to the number of those transmitters, whose corresponding receivers are inrzofRxo. Suitable practical value of proportionality constantψ1is obtained through simulation.
Therefore, the effective number of interferers from receiver’s perspective (Ief fRxo) becomes
Ief fRxo = (Ks−1) −[
T xrzRxo +RxrzRxo
] (3.24)
Again due to MAC protocol under consideration, transmitter nodes having good link withT xo (intended transmitter) are also not able to transmit because of exposed node phenomenon. The number of such transmitters can be calculated as
T xrzT xo = Ief fRxoPri
zT xo (3.25)
Receiver nodes having good link probability withT xoare also not been able to receive the data from their corresponding transmitters. Hence, they are effectively barred and do not contribute in
†For comparison purpose in Figure 3.4, standard numerical techniques can be used to obtain ℜin (3.22) in the absence of closed form.
‡Other examples of class-2 MAC protocol are MARCH [94] and S-MAC [95].
3.2 System Model
effective interference. Same as (3.23), such receivers can be calculated as
RxrzT xo = ψ2Ief fRxo PrizT xo (3.26)
whereRxrzT xo also gives the number of those transmitters, whose corresponding receivers are inrz
ofT xo. Suitable practical value of proportionality constantψ2 also is obtained through simulation.
Now, the number of effective interferers becomes Ief f∗ = Ief fRxo −[
T xrzT xo +RxrzT xo
]
= (Ks−1) [
1−(1 +ψ1) PrizRxo ] [
1−(1 +ψ2) PrizT xo
] (3.27)
Under the assumption of homogeneity and reciprocal links, to make situation symmetric from transmitter and receiver point of view, we take ψ1 =ψ2 and PrizRxo =PrizT xo. Under the assump- tions, (3.27) simplifies to
Ief f∗ = (Ks−1) [
1−(1 +ψ1) PrizRxo ]2
(3.28) In case of other MAC protocols (class-1, class-3) of [30], a slight modification will be required accordingly.
Considering the worst case scenario of random transmission slots, where all transmitters outside the combined guard zone rzRxo+T xo transmit (activity probability = 1) and create interference, it may be considered as a valid assumption that Ief f∗ /2 will be of type-1 andIef f∗ /2 will be of type-2 interferers. So, E[IT1] = E[IT2] = Ief f = ⌊Ief f∗ /2⌋. In practical scenario, the actual number of interferers will be even less than that of calculated in (3.28), as some of them will be further barred by the other transmitters as well. In an asynchronous environment, it is extremely difficult to account for such interferers and are not included in our analysis as well as simulation.
Figure 3.4 shows the effective number of type-1 (type-2) interferers w.r.t. total number of inter- ferers (transmitters). Analytical results are in good agreement with simulation results. Simulation model is discussed in Section 5.4 in detail.