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Chapter 4Simulation and analytical modelling pFirst we analyse the case when the service times of the data packets are either X 0 or X 1 ,and consider for the moment that packets that arrive at the end of the MCI cycle aregiven a service time of X 0 (instead of X 1 ). Once we have derived a formula for the meanservice time, we can easily obtain a similar equation, but now taking into account thethird case of service time X 2 (=X 1 ).The initial mean service time, let’s sayX ' , is given by the probability that a data packetfinds the system idle, P 0 ’, multiplied by the service time when the system is idle, X idle=X 0 , plus the probability that a data packet finds the system busy, P 1 ’, multiplied byX busy = X 1 . Thus the initial service time is given by:X' = P ' ⋅ Xidle+ P1' ⋅ X busy0(4.12)where the probability of finding the system busy or idle is given by Equations 4.13 and4.14, respectively.PλP ' 1=(4.13)µ0' P1= 1−'(4.14)The service time when the system is in idle state can be obtained by making a closeranalysis in Figure 4.13, as depicted in Figure 4.14. Here X idle consists of three differentdelays, as indicated by Equation 4.15.X idle = X t1 + X t2 + X t3 (4.15)The first delay, X t1 , is a variable delay but can be approached by using the mean interarrivaltime in the range [MCI 0_start – MCI 0_end ] given by Equation 4.16.MCIMCI 0_start MCI 0_end= MCI 2_end1_start MCI 1_end= MCI 2_startD Pk0MCI 0MCI 1 MCI 21 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9R D DX t1X t2X t3A Pk0X idleContention SlotRContention Slot withReservation RequestDData SlotFigure 4.14 – Mean service time when the system is in idle state.4-22