Operations and Supply Chain Management The Core

214 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
exhibit 7.7
Exponential Distribution
f(t)
t
The cumulative area beneath the curve in Exhibit 7.7 is the summation of equation 7.1
over its positive range, which is e −λt . This integral allows us to compute the probabilities
of arrivals within a specified time. For example, for the case of one arrival per minute to
a waiting line (λ = 1), the following table can be derived either by solving e −λt or by using
Appendix D. Column 2 shows the probability that it will be more than t minutes until the
next arrival. Column 3 shows the probability of the next arrival within t minutes (computed
as 1 minus column 2).
(1)
t
(MINUTES)
0
0.5
1.0
1.5
2.0
(2)
PROBABILITY THAT THE NEXT
ARRIVAL WILL OCCUR IN
t MINUTES OR MORE (FROM
APPENDIX D OR SOLVING e −t )
100%
61%
37%
22%
14%
(3)
PROBABILITY THAT THE NEXT
ARRIVAL WILL OCCUR IN t
MINUTES OR LESS
[1 − COLUMN (2)]
0%
39%
63%
78%
86%
Poisson distribution
Probability
distribution for the
number of arrivals
during each time
period.
Poisson Distribution. In the second case, where one is interested in the number of
arrivals during some time period T, the distribution appears as in Exhibit 7.8 and is
obtained by finding the probability of exactly n arrivals during T. If the arrival process
is random, the distribution is the Poisson, and the formula is
n e
−λT ​
​(λT )​
P T (n) = ​ ________ ​ [7.2]
n !
Equation 7.2 shows the probability of exactly n arrivals in time T. For example, if the mean
arrival rate of units into a system is three per minute (λ = 3) and we want to find the probability
that exactly five units will arrive within a one-minute period (n = 5, T = 1), we have
​(3 × 1)​ 5 e −3×1 ​
P 1 (5) = ​ ____________ ​ = _____ ​3​ 5 e −3 ​
5 ! 120 ​ = 2.025​e −3 ​ = 0.101​