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Chapter 4Simulation and analytical modelling pThe transmission delay of a data slot and the propagation delay are given by Equations4.5 and 4.6, respectively.D512= 83 s(4.5)CCSl _ Tx= µlD prop= = 50µs(4.6)0.67 ⋅ cIn Equation 4.5, CC corresponds to the upstream channel capacity set to 6.176 Mbpsand l in Equation 4.6 corresponds to the maximum length of the HCF network, from theINA to the further NIU. For this analysis l was set to 10Km and c is the speed of lightconstant (≈3×10 8 m/s).The solution of this queuing system depends on the arrival and service distribution ofthe data packets. Assuming that the data packets follow a Poisson distribution (e.gexponential distributed inter-arrival times) and by obtaining the mean service time ( X )of Figure 4.13, we can approach the solution by using an M/G/1 queuing system. Suchsystem is characterised by a Poisson arrival process (at a mean rate of λ arrivals persecond with a mean inter-arrival rate of 1/λ) and with a general service time distribution(with a mean service time of X seconds).According to [45], a well-known result for the M/G/1 system is given by the Pollaczek-Khintchine (PK) formula presented by Equation 4.7.2 2 2ρ + λ ⋅σXL = ρ +(4.7)2 ⋅ (1 − ρ)This formula gives the average number of packets in the system L, where ρ is theutilisation factor given byρµλ= = λ ⋅ X , ρ < 1, andσ 2 is the variance of the service-Xtime distribution. From this formula the expected waiting time W in queue can beobtained by using the well-known Little’s theorem, L = λ·T [10], where T is the meanwaiting time in the system given by:T = X + W(4.8)4-20