Performance Analysis

Outage Analysis of Asynchronous Interference Limited Wireless

3.3 Performance Analysis

to use of MAC protocols [29].

3.2.6 Channel Model

In our analysis, we are considering an asynchronized system with frequency-flat fading. Only receiver node is assumed to have complete CSI. Regarding the signals xi transmitted by a trans- mitter node to the corresponding receiver, we assume that E(xi) = 0 and E(|xi|²) = 1, for i = 0, 1, 2,· · · K_s-1.

For this direct transmission setup, we define the system model by following set of equations :

y(t) = √

E_sda_sdx_o(t) + Z_Id (3.32)

Here ZId includes the effect of interference from all other sources and noise term. ZId is given by

Z_Id =

I_{ef f}^∗

∑

j=1

√E_jda_jdxj

(t^′)

+ n(t) (3.33)

where y(t) is the received signal at the destination terminal, x_o(t) is the desired transmitted signal from corresponding source and xj(t^′) is the signal received from jth interferer, where t ≤ t^′≤ (t+T_b), a_sd (a_jd) denotes the channel coefficient between source (interferer) and destination, modeled as zero mean circularly symmetric complex Gaussian RV with unit variance, |ajd| is Rayleigh distributed and |ajd|² is exponentially distributed. Esd (Ejd) is the average signal energy received at the destination terminal over one symbol duration including path loss and shadowing.

In (3.33), n(t) denotes system noise modeled as AWGN with power spectral density N_o.

3.3 Performance Analysis

The interference models are classified into two groups in [1]. Our model falls in the second group, which deals with the network performance affected by the interference. Based on the proposed metric EINR (3.1), correct reception of desired frame in terms of type-1 and type-2 interferers is possible only if

E_id N_o+

( E_IT

1 +E_IT

2

) > β ;∀i = 1,2,3...N_b (3.34)

Using, (3.3), (3.29) andEid=Pir Tb, desired as well as interference signal energy can be written

3.3 Performance Analysis

as

E_id = P_ir T_b = K_o r^η1

( 1 +^r_g

)_η2 P_Te^λεT_b|a_id|² (3.35)

EI_Tt =

I_Tt

∑

j=1

PjτjTb =

I_Tt

∑

j=1

Ko

r_j^η1 (

1 +^r_g^j

)η2 PT e^λετjTb|ajd|² (3.36) where P_ir, P_j and r_j are the desired signal power, power received from jth interferer at the des- tination node and the distance between that interferer and destination node, respectively and t ∈ {1, 2}. The constant λis given by ^ln(10)₁₀ .

Using (3.35) and (3.36) in (3.34), we get

EIN Ri =

Ko

r^η1(

1+^r_g)η2P_Te^λεT_b|a_id|² No+

I∑T1

j=1

Ko

r_j^η1 (

1+^rj_g

)η2PTe^λεj (

τ_ij^T1Tb

)|aijd|²+

I∑T2

m=1

Ko

rm^η1

( 1+^rm_g

)η2PTe^λεm (

τ_im^T2Tb

)|aimd|²

(3.37) which gives the accurate expression ofEINRparameter in an asynchronized interference limited environment and for the channel model considered here.

3.3.1 Analysis Under Approximation

It is difficult to calculate pdf ofEINR, having (3.37) in the current form. To modify (3.37), we replace I_T1 and I_T2 by their expected values E[I_T1] and E[I_T2] respectively^†, as derived in Section 3.2.2.2 where this is denoted as I_{ef f}. With the approximation and some further modifications, (3.37) becomes

EIN R_i =

Ko

r^η1(

1+^r_g)η2P_Te^λεT_b|a_id|² N_o+T_b

I∑_{ef f}

j=1

Ko

rjη1

(

1+^rj_g)η2P_Te^λεj|a_ijd|²(

τ_ij^T1 +τ_ij^T2

) (3.38)

Even in this modified form, use of (3.38) for calculation ofEINRpdf requires the distribution of sum of lognormally distributed RVs (interference power element) with one sided random weights.

To the best of our knowledge, no closed form expression for such kind of distribution is known.

†Justification concerning this replacement can be found in Appendix A.2.

3.3 Performance Analysis

However, in [4, 9, 100, 101] another lognormal distribution is used as an approximation for sum of lognormal RVs. To compute the mean and variance (or even higher moments) of resulting distribution, several approaches are proposed in literature [102,103]. As the present case deals with lognormal RVs with one sided random weights, so, in order to solve (3.38), we use GMM method for linear combination of lognormal RVs [31, 32]. To apply GMM method, we modify (3.38) as

EIN R_i= ( |a_id|²

r^η1( 1+^r_g)η2

No

KoPTTb

) e^−λε+

I∑ef f

j=1

(r rj

)^η1(

g+r g+rj

)^η2

|a_ijd|²(

τ_ij^T1 +τ_ij^T2 )

e^λ(εj−ε)

(3.39)

which can further be written in compact form as

EIN Ri= |a_id|² Υoe^−λε+

I∑ef f

j=1

Υij e^λ(εj−ε)

(3.40)

where

Υo=



r^η1 (

1 +_g^r )η2

No

KoPT Tb





Υ_ij = (r

r_j )^η1(

g+r g+r_j

)^η2

|a_ijd|²[

τ_ij^T1 +τ_ij^T2 ]

Denominator of (3.40) is a combination of lognormal RVs weighted by one-sided RVs. Here, we take a lognormal approximation for the denominator as

Υ_oe^−λε+

Ief f

∑

j=1

Υ_ij e^λ(εj−ε) = e^λ~i =Ne (3.41) where ~ is a Gaussian RV, whose mean and variance can be calculated through GMM method (details are given in Appendix A.3) andNe is the combination of (I_{ef f} + 1) RVs with random and independent weights. Now, the EIN R_i can be re-written as

EIN Ri = |aid|²

e^λ~i (3.42)

i.e., in form of a ratio between two RVs, the numerator is a unit mean exponential RV and denom- inator is a lognormally distributed RV.

3.3 Performance Analysis

3.3.2 Outage Probability Calculation

This section calculates the outage probability to observe the effects of interference on network performance using EINR parameter. The outage probability is defined as the probability that received SINR (in our case EINR) falls below some specified thresholdβ. Outage probability, Pout

can be written as

P_out =P(EIN R < β) (3.43)

If the received EINR for a bit falls below the threshold β, bit is considered lost. Threshold β is a dimensionless quantity, depends on the specific application’s QoS. It may be noted that while dealing withEINRparameter, we essentially talk of outage at the bit level. Using (3.42) and (3.43), Pout forith bit can be written as

P_out^biti = P

[|a_id|²

e^λ~i < β ]

(3.44) Using corresponding pdf expressions, The pdf ofEINR can be written as [104]

f_{EIN Ri}(w) =

∫ _∞

0

√ 1

2πσ_λ~i exp



−

(

lnNe −m_λ~i )2

2σ_λ²_~

i



 exp

(−wNe )

dNe (3.45) Therefore, the outage probability expression becomes

P_out^biti =

∫ _β

0

fEIN Ri(w) dw (3.46)

In the absence of closed form expression for (3.46), standard numerical techniques [105] can be used to obtain outage probability.

With the outage probability calculated above, the probability of success can also be estimated as

P_suc^biti = 1−P_out^biti (3.47)

Therefore, the total number of bits received throughout the network is given by, total received bits

= K_s ×F_T ×N_b ×P_suc^biti, where F_T represents the total number of frames sent by a user node.

The probability of packet (frame) outage can be easily obtained where transmission of bits can be assumed to be independent. If the transmission of bits are not independent, more rigorous