Occupancy Distribution as Seen by a Poisson Arrival 2.3 2002/07/19 2003/03/05 Bart Sinclair bs@rice.edu Charlet Reedstrom charlet@rice.edu Bart Sinclair bs@rice.edu arrival times Bayes' rule occupancy distribution open system Poisson process (Blank Abstract) The occupancy distribution of a system is the steady state probability distribution of the number of jobs in the system. We can also look at the system only at arrival instants (or to be precise, just before each arrival instant) and talk about the occupancy distribution of the system as seen by each arrival. When the arrivals are the results of a Poisson process and the system is open (the rate of arrivals is independent of the number of jobs in the system), we can show that the arrival time occupancy distribution is the same as the occupancy distribution over all time. In effect, there is nothing special about the arrival instants of a Poisson process. This can be shown as follows: Define N t to be the number of jobs in an (open) system with Poisson arrivals at time t. Assume that the following limits exist. a n t Pr N t n | arrival just after time t p n t Pr N t n Then a n p n . Proof: A t t δ event that an arrival occurs in t t δ p n t Pr N t n a n t Pr N t n | arrival just after t By the definition of conditional probability, a n t δ 0 Pr N t n | A t t δ δ 0 Pr N t n A t t δ Pr A t t δ δ 0 Pr A t t δ | N t n Pr N t n Pr A t t δ For Poisson arrivals in an open system, we know that the number of arrivals in any interval t t δ where δ 0 is independent of N t . Pr A t t δ | N t n Pr A t t δ a n t Pr N t n p n t Hence, a n t a n t t p n t p n