Inter-arrival times Time that elapses between arrivals (A) P(A<=t)=1-P(A>t) =1-P(0 arrivals in time t =1-et This is known as the exponential distribution Inter-arrival CDF=FIA(t)=1-e-it Inter-arrival PDF d/dt Fa(t)=e-lt The exponential distribution is often used to model the service times (Le, the packet length distribution
Eytan Modiano Slide 6 Inter-arrival times • Time that elapses between arrivals (IA) P(IA <= t) = 1 - P(IA > t) = 1 - P(0 arrivals in time t) = 1 - e-λt • This is known as the exponential distribution – Inter-arrival CDF = FIA (t) = 1 - e-λt – Inter-arrival PDF = d/dt FIA(t) = λe-λt • The exponential distribution is often used to model the service times (I.e., the packet length distribution)
Markov property(Memoryless) P(T≤b+1|7>t)=P(T≤0 Proof P≤1+1> (t<T≤t0+) P(T>to he dt u to+I A(+0)⊥-A(o e he dt (0) e P(T≤D Previous history does not help in predicting the future! Distribution of the time until the next arrival is independent of when the last arrival occurred!
Eytan Modiano Slide 7 Markov property (Memoryless) • Previous history does not help in predicting the future! • Distribution of the time until the next arrival is independent of when the last arrival occurred! PT t t T t PT t oof PT t t T t Pt T t t PT t e dt e dt e e e e t t t t t t t t t t t t t t ( | )( ) Pr : ( |) ( ) ( ) | | ( ) ≤+ > = ≤ ≤+ > = <≤+ > = = − − = − + − + − ∞ − + − ∞ − + ∫ ∫ 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0 λ λ λ λ λ λ − − − =− = ≤ λ λ λ ( ) ( ) ( ) t t t e e PT t 0 0 1
Example Suppose a train arrives at a station according to a Poisson process with average inter-arrival time of 20 minutes When a customer arrives at the station the average amount of time until the next arrival is 20 minutes Regardless of when the previous train arrived The average amount of time since the last departure is 20 minutes Paradox: If an average of 20 minutes passed since the last train arrived and an average of 20 minutes until the next train, then an average of 40 minutes will elapse between trains But we assumed an average inter-arrival time of 20 minutes! What happened? Answer: You tend to arrive during long inter-arrival times If you don't believe me you have not taken the t
Eytan Modiano Slide 8 Example • Suppose a train arrives at a station according to a Poisson process with average inter-arrival time of 20 minutes • When a customer arrives at the station the average amount of time until the next arrival is 20 minutes – Regardless of when the previous train arrived • The average amount of time since the last departure is 20 minutes! • Paradox: If an average of 20 minutes passed since the last train arrived and an average of 20 minutes until the next train, then an average of 40 minutes will elapse between trains – But we assumed an average inter-arrival time of 20 minutes! – What happened? • Answer: You tend to arrive during long inter-arrival times – If you don’t believe me you have not taken the T