. Arrival of customers to a payment counter in a bank follow a Poisson distribution with an average of
10 per hour. The service time follows negative exponential distribution with an average of 4 minutes.
(i) What is the average number of customers in the queue?. (ii) The bank will open one more counter
when the waiting time of a customer is at least 10 minutes. By how much the flow of arrivals should
increase in order to justify the second counter.
Answers
Answer:The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.
Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.
The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts.
EXAMPLE
Let X = amount of time (in minutes) a postal clerk spends with his or her customer. The time is known to have an exponential distribution with the average amount of time equal to four minutes.
X is a continuous random variable since time is measured. It is given that μ = 4 minutes. To do any calculations, you must know m, the decay parameter.
m
=
1
μ
. Therefore,
m
=
1
4
=
0.25
The standard deviation, σ, is the same as the mean. μ = σ
The distribution notation is X ~ Exp(m). Therefore, X ~ Exp(0.25).
The probability density function is f(x) = me–mx. The number e = 2.71828182846… It is a number that is used often in mathematics. Scientific calculators have the key “ex.” If you enter one for x, the calculator will display the value e.
The curve is:
f(x) = 0.25e–0.25x where x is at least zero and m = 0.25.
For example, f(5) = 0.25e−(0.25)(5) = 0.072. The postal clerk spends five minutes with the customers. The graph is as follows:
Exponential graph with increments of 2 from 0-20 on the x-axis of μ = 4 and increments of 0.05 from 0.05-0.25 on the y-axis of m = 0.25. The curved line begins at the top at point (0, 0.25) and curves down to point (20, 0). The x-axis is equal to a continuous random variable.
Notice the graph is a declining curve. When x = 0,
f(x) = 0.25e(−0.25)(0) = (0.25)(1) = 0.25 = m. The maximum value on the y-axis is m.