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Statistical Probabilities and Distributions - Lesson 10
Queuing Theory and the Poisson Distribution
Lesson Overview
In many retail stores and banks, management has tried to reduce the
frustration of customers by somehow increasing the speed of the checkout
and cashier lines. Although most grocery stores seem to have retained the
multiple line/multiple checkout system, many banks, credit unions,
and fast food providers
have gone in recent years to a queuing system where customers wait for
the next available cashier. The frustrations of "getting in a slow line"
are removed because that one slow transaction does not affect the throughput
of the remaining customers. Walmart and
McDonald's are other examples
of companies which open up additional lines when there are more
than about three people in line. In fact, Walmart has roaming clerks
now who can total up your purchases and leave you with a number which
the cashier enters to complete the financial aspect of your sale.
Disney
is another company where they face thousands of people a day.
One method to ameliorate the problem has been to use queuing theory.
It has been proved that throughput improves and customer satisfaction
increases when queues are used instead of separate lines.
Queues are also used extensively in computing---web servers
and print servers are now common.
Banks of 800 service phone numbers are a final example I will cite.
Queuing theory leads one directly to the
Poisson
Distribution, named
after the famous French mathematician Simeon Denis
Poisson (1781-1840)
who first studied it in 1837. He applied it to such morbid results as
the probability of death in the Prussian army resulting from
the kick of a horse and suicides among women and children.
As hinted above, operations research has applied it to
model random arrival times.
The Poisson distribution is the continuous limit of the discrete
binomial distribution. It depends on the following four assumptions:
- It is possible to divide the time interval of interest into
many small subintervals (like an hour into seconds).
- The probability of an occurrence remains constant
thoughout the large time interval (random).
- The probability of two or more occurrences in a
subinterval is small enough to be ignored.
- Occurrences are independent.
Clearly, bank arrivals might have problems with assumption number
four where payday, lunch hour, and car pooling may affect independence.
However, the Poisson Distribution finds applicability in a
surprisingly large variety of situations.
The equation for the Poisson Distribution is:
The number e in the equation above is the
base of the
natural logarithms or approximately 2.71828182845904523...
The variance is equal to the mean.
In fact, this can be a quick check to see if this distribution can be applied.
Traditionally the Greek letter lambda (
) is
often used instead of µ.
The differences between the Poisson distribution and the
binomial distribution are:
- The binomial distribution is affected by the sample size and the
probability while the Poisson distribution is ONLY affected by the mean.
- The binomial distribution has values from x = 0 to n
but the Poisson distribution has values from x = 0 to infinity.
Example: On average there are three babies born a day with hairy backs.
Find the probability that in one day two babies are born hairy.
Find the probability that in one day no babies are born hairy.
Solution: a. P(2) = 32 · e-3 ÷ 2 = .224
b. P(0) = 30 · e-3 = .0498
Example: Suppose a bank knows that on average 60 customers arrive
between 10 A.M. and 11 A.M. daily.
Thus 1 customer arrives per minute. Find the probability that
exactly two customers arrive in a given one-minute time interval between
10 and 11 A.M.
Solution: Let µ = 1 and x = 2.
P(2)=e-1/2!=0.3679÷2=0.1839.
Example: Other examples include,
the number of typographical errors on a page,
the number of white blood cells in a blood suspension,
or the number of imperfections in a surface of wood or metal.
I assume one could apply it to finding four-leaf clovers,
but a corresponding class activity has not yet been developed.
For various Java applets, include one for the Poisson Distribution, visit
this location.