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 ha appened?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? Eytan Modiano Slide 8