Appendix 1: Poisson Distribution

Appendix 1:

Poisson distribution:

A Poisson process represents discrete events, like arrival of customers or phone calls at an exchange or a call center. As shown in the Fig. 5 below, N(t) = Number of arrivals in the time interval (0,t) is a random variable, which follows the Poisson distribution.

 
Mathematically, the probability of getting N(t) = x arrivals in a time window (0,t) can be given by,

The inter arrival times are independent and obey an exponential distribution, with mean interarrival time (1/ µ) where µ is the arrival rate per unit time. For example, for an arrival rate of 40 customers per hour, µ =40. Figure 6 demonstrates this graphically.

References:

1. A. M. Lee and P. A. Longton, “Queueing Processes Associated with Airline Passenger Check-in,” Oper. Res. Quart. 10: 56–71 (1959)