Proof of pollaczek-Khinchin Let Wi= waiting time in queue of itn arrival R: Residual service time seen by l( l.e., amount of time for current customer receiving service to be done) N;= Number of customers found in queue by i i arrives W ⅹ3X2Xx1 R Time R;+ E=E[R]+E[XEN]=R+N/μ Here we have used PaSta property plus independent service time property W=R+MW=>W=R/(1-) Using little' s formulaProof of Pollaczek-Khinchin • Let Wi = waiting time in queue of ith arrival Ri = Residual service time seen by I (I.e., amount of time for current customer receiving service to be done) Ni = Number of customers found in queue by i i arrives Wi Ri X i-3 X i-2 X i-1 Xi Time -> Ni = 3 i-1 W i = R i + � X j j=i￾N i • E[Wi] = E[Ri] + E[X]E[Ni] = R + N Q/µ – Here we have used PAST A prope rty plus independent service time property • W = R + λW/ µ => W = R/(1- ρ) Eytan Modiano – Using little’s form ula Slide 5