Average and spread of call inter-arrival time

Question

Calls arrive at a call center at rate \lambda = 3 per hour, with exponential gaps between them. What are the mean and variance of the time (in hours) between two consecutive calls?

Accepted Answer

For an exponential with rate \lambda, mean = 1/\lambda and variance = 1/\lambda^2. With \lambda = 3 per hour: 	ext{mean} = \frac{1}{3} 	ext{ hour} = 20 	ext{ minutes}, \qquad \operatorname{Var} = \frac{1}{3^2} = \frac{1}{9} 	ext{ hour}^2. Standard deviation \sqrt{1/9} = 1/3 hour, again equal to the mean, the exponential's signature. Faster arrivals (bigger \lambda) mean shorter and less variable gaps. Takeaway. At rate 3 per hour, the gap has mean 1/3 hour and variance 1/9; both mean and sd equal 1/\lambda.

Average and spread of call inter-arrival time

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