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Chapter 4Simulation and analytical modelling pThus, the waiting time in the queue results in:Then, by substituting the utilisation factor2 2 2ρ + λ ⋅σXρ +2 2 22 ⋅ (1 − ρ)ρ + λ ⋅σXW = − X =(4.9)λ2 ⋅ λ ⋅ (1 − ρ)ρ = λ ⋅ Xformula for the mean waiting time in the system, as:The mean and the variance of the service time ( X and2in Equation 4.9, we can obtain the2λ ⋅ X + λ ⋅σXW = (4.10)2⋅(1− ρ)2σ respectively) are neededXalso. From analysing Figure 4.13, we can see that the mean service time has threevalues, represented by X 0 , X 1 and X 2 . A service time of X 0 is given to packets that findthe system idle. In other words, the packets start being processed immediately and donot have to wait in queue to be served. A service time of X 1 is given to packets that findthe system busy and have to wait in queue before they are processed.The service time X 2 (=X 1 ) is for special cases when packets find the system idle andhave to wait for a very short period of time before being processed. This is because suchpackets arrive at the end of the current MCI frame and the DVB/DAVIC reservationmechanism is unable to start service immediately, due to the time required for thepropagation and the slot transmission delay. In other words, when a packet arriveswithin the last two contention slots of the current MCI frame, in order to start serviceimmediately (by sending or scheduling a request) the following condition should besatisfied,Current _ time + D + D ≤ CSn(4.11)Sl _ TxpropTx _ timewhere Current_time is the time when the packet arrives and CSn Tx_time is the time atwhich the n th contention starts. If this condition is not satisfied, the DVB/DAVICreservation mechanism retries with the contention slots of the following MCI frame andthen selects randomly one contention slot from the set of contention slots that satisfyEquation 4.11.4-21