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Chapter 4Simulation and analytical modelling pgiven a service time of X idle , the probability P 0 that a service time of X idle is actuallygiven, needs to calculated. This probability is obtained by the following equation:P0( MCIt− DSl_ Tx− Dprop)= P0'(4.27)MCItThus, the probability P 1 that a data packet is given a service time of X busy is thenP1 = 1−P 0(4.28)Therefore, the real mean service time results in:X= P X0idle+ P X1busy= (1 −+λµ) ⋅ (MCIt−DSl_ Tx −DpropMCItλ MCIt−DSl_ Tx −Dprop( 1- (1 − ) ⋅ ()) ⋅ (1 + MCI+ Pkmci) ⋅ MCItµMCIt) ⋅ (0.5 + MCI+ Pkmci) ⋅ MCIt+Hence,Xλ= ( DSl_ Tx+ Dprop) ⋅ (0.5 −2⋅ µ) + (0.5 + MCI+ Pkmci+2⋅µ)λ⋅ MCI (4.29)tOnce we have calculated the mean service time ( X ), we can now obtain the variance2( σ ), needed in Equation 4.10. It is well-defined that the variance is the averageXsquared deviation from the mean, given by the following formula:∑( X− X )2jj== 12σ (4.30)XIf we consider that all packets receive a service time either of X idle or X busy , then thevariance can be re-calculated as:= ⋅ ( X − X ) + (1 − ρ)⋅ ( X − X )222σ ρXbusyidle(4.31)In this new equation, the variance only depends on the value of the utilisation factorλ( ρ = µ< 1). Thus, the variance follows the distribution depicted in Figure 4.16.4-27