What is r? (Time Average Residual Service Time) Residual service Time R(t) X X X time→> Let M(t=Number of customers served by time t E[R(t]=1/t (sum of area in triangles) M(t)、2 M(t)、2 (r)dτ 10口 2 t M(t) Ast→ Infini M(t average departure rate average arrival rate M(t)、2 Mt E2]=>R=ME2]2 MtWhat is R? (Time Average Residual Service Time) Residual Service Time R(t) X X X X 1 2 3 4 X1 X2 X3 X4 time -> Let M(t) = Number of customers served by time t E[R(t)] = 1/t (sum of area in triangles) t 2 M(t) X i M(t) X i 1 M(t) � 2 R t = 1 t 0 R( τ )d τ = 1 � = 2 t M(t) t i=1 i=1 As t -> Infinity M(t) = average departure rate = average arrival rate t M(t) X i 2 M(t) � M(t) = E[X2] => R = λE[X2]/2 t Eytan Modiano Slide 6 i=1