Question

Consider a customer arrival process that is a Poisson process. To find the probabilities described below, which of the following random variable selections (as Poisson, Exponential or k-Erlang) is correct?

		to find the probability that the time between the 2nd and 3rd customer arrivals is 5 minutes, use a k-Erlang random variable with k>1
		to find the probability that 10 customers arrive during a 30-minute period, use a k-Erlang random variable
		to find the probability that the total time elapsed until the 5th customer arrival is 30 minutes, use an Exponential random variable
		to find the probability that more than 7 minutes pass before the 3rd customer arrival, use a k-Erlang random variable

Answer 1

1. Here we will use an exponential distribution. When the arrival process is poisson, then the time between two arrivals follows an exponential distribution.

2. Here we will use a poisson distribution. In the customer arrival poisson process we will be told the average no. Of customers arriving in a particular time (say 1 hour). We will convert it to 30 minute time period and then find P(X=10) by putting it in the poisson PDF.

3. This will be found using k Erlang random variable. It is the sum of the exponential random variables.

4. This will be found using k Erlang random variable.