Littles theorem Network(system) λ packet per second (N,T) N=average number of packets in system T= average amount of time a packet spends in the system n= arrival rate of packets into the system (not necessarily Poisson) Littles theorem NEnT Can be applied to entire system or any part of it Crowded system - long delays On a rainy day people drive slowly and roads are more congested!
Little’s theorem Network (system) λ packet per second (N,T) • N = average number of packets in system • T = average amount of time a packet spends in the system • λ = arrival rate of packets into the system (not necessarily Poisson) • Little’s theorem: N = λT – Can be applied to entire system or any part of it – Crowded system -> long delays On a rainy day people drive slowly and roads are more congested! Eytan Modiano Slide 11
Proof of little,s Theorem at) 14 a(t),β(t) t1 t2 t3 t4 a(t=number of arrivals by time t B(t=number of departures by time t s arrival time of ith customer Ti= amount of time ith customer spends in the system N(t=number of customers in system at time t=a(t)-B(t) Similar proof for non First-come-first-serve
Proof of Little’s Theorem α(t), β(t) t1 t2 t3 t4 • α(t) = number of arrivals by time t • β(t) = number of departures by time t • ti = arrival time of ith customer • Ti = amount of time ith customer spends in the system • N(t) = number of customers in system at time t = α(t) - β(t) • Similar proof for non First-come-first-serve α(t) T1 T2 T3 T4 β(t) Eytan Modiano Slide 12