Final answer:
The average time between two successive customer arrivals, when 30 customers arrive per hour, is 2 minutes. It will take on average 6 minutes for three customers to arrive. The exponential distribution is used to calculate the probability of time intervals between customer arrivals.
Step-by-step explanation:
The question pertains to a system where there is an exponential distribution of customer arrivals and service times. In such a system, inter-arrival times are generally expected to be exponentially distributed.
a. The average inter-arrival time is determined by taking the inverse of the average arrival rate. Since 30 customers are expected per hour, we calculate the average inter-arrival time as follows:
- Divide the number of minutes in an hour (60) by the average number of customers arriving in that hour (30).
- The average inter-arrival time is thus 60 / 30 = 2 minutes between successive arrivals.
b. For three customers to arrive, we simply multiply the average inter-arrival time by three. So, 2 minutes/customer × 3 customers = 6 minutes on average for three customers to arrive.
c. The probability of the next customer arriving in less than one minute can be found by integrating the exponential probability density function from 0 to 1 minute. This calculation typically involves using the exponential cumulative distribution function (CDF).
d. Conversely, the probability that it takes more than five minutes for the next customer to arrive is found by subtracting the CDF evaluated at 5 minutes from 1, as this would give us the upper tail of the distribution.
e. Seventy percent of the customers arriving within a certain number of minutes of the previous customer implies finding the time point at which the CDF equals 0.70.
f. The exponential distribution is reasonable for modeling the time between arrivals because the events (customer arrivals) are independent and occur at a constant average rate.