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What It Is
What It Is
The exponential distribution is a continuous, single parameter distribution. It is defined by a value :
λ the average arrival rate per time
With this, the time between independent events can be derived. It considers all possible (non-negative) outcomes, with the lower bound being most likely, and decreasing likelihood for larger values. Usually, the lower bound is set to zero. (more shapes)
Classically, the distribution is used in queuing theory. In it, defining the value as : λ = time / number of arrivals, the time between arrivals can be defined. This approach is also used in simulation to model arrivals.
The example above is a sample of 1000 events with an average rate of occurrence, λ = 6 (one every six time units).
Variants
Variants
Where λ = t / n , we can assume that there are n number of arrivals per unit of time, t. For the following, assume that the unit of time measured is hours. Thus, we can plot 1000 samples of times between arrivals (interarrival times) per hour.
λ = 1 / 6
λ = 1 / 6
Where t = 1 and n = 6, we can assume there are on average six arrivals per hour. The expected interarrival time is 1 / 6 = 1.667 hours or ten minutes.
Notice that most values are less than 0.53 or about 32 minutes.
λ = 8 / 3
λ = 8 / 3
Where t = 8 and n = 3, three arrivals within eight hours, a slower process is described. The expected interarrival time is 8 / 3 = 2.667 or 160 minutes.
Notice that the range of observed values extends to more than fifteen hours.
📑 Note: This distribution will appear to have an identical shape in various cases. For example, these two variants look similar. However, the ranges indicate their differences.
In Real Life (IRL)
In Real Life (IRL)
This distribution's single parameter, λ, can be derived in a handful of ways. Here, we will look at a few types of records, and use them to generate exponential distributions to describe different processes.
Example A
After a moderate living as an accountant at a mid-tier paper supply company, you are released to pursue your other passions. You become part owner of a dining establishment. Your partners trust in your math abilities. They ask, "Is it likely that we serve more customers/parties than the Chili's across the street?"
Your colleagues inform you that your establishment has served approximately 2048 parties over the previous 8 weeks, based on receipts. The week is defined as Tuesday through Saturday. You determine:
mean arrivals per hour = 2048 / ( 8 hours/day * 5 days/week * 8 weeks ) = 6.4
For the next four weeks, on random days within Monday through Friday, you observe Chili's, estimate , and record the number of groups entering per hour:
M [12 7 3 4 14 12 3 4]
T [13 11 12 3 13 11 9 7]
F [ 6 3 7 14 9 11 14 13]
M [ 4 11 7 4 6 9 8 6]
R [12 9 12 4 12 7 14 5]
F [ 9 10 11 14 13 11 12 5]
W [ 3 9 10 11 14 4 10 14]
R [ 4 7 13 3 11 8 7 10]
F [11 11 5 9 5 11 11 9]
M [ 9 8 13 9 14 3 3 9]
T [12 4 11 13 12 14 4 5]
R [11 12 12 8 3 5 10 6]
sum: 850, mean : 8.9, std: 3.55
min : 3, max : 14
Your λ = 1 / 6.4 Chili's λ = 1 / 8.9
Your λ = 1 / 6.4 Chili's λ = 1 / 8.9
Where the Chili's model is larger than yours, this plot shows that it is more likely that the time between customers is lower, thus more customers per hour.
You return to your partners with these key observations, with the caveat that Chili's was only observed during the morning shift (8 hours), for twelve non consecutive days:
There were no hours where zero arrivals entered Chili's.
There were no hours where less than three arrivals entered Chili's.
The average number of arrivals to Chili's is greater than your restaurant.
The plot shows that it is more likely that Chili's will have more customers per hour (lower interarrival times).
Your restaurant sometimes has hours with no customers (large interrrival time).
The plot shows your restaurant is more likely to have fewer customers per hour (larger interarrival times).
You conclude that it is unlikely that your restaurant serves more customers than Chili's.
Example B
Your office hires a temporary employee. The temporary employee is not familiar with the process and wants to maximize the time between assignments to work on a personal endeavor, a social network messaging application. The temp asks you for historical data on the assignments usually completed by this role. You comply, delivering the day number of the fiscal year for all assignment last year :
3 4 8 11 13 14 15 15 16 18 18 23 31 33 40 41 42 44 45 51 53 54 54 62 62 71 71 73 74 77 77 79 79 85 89 90 93 94 112 112 118 122 123 125 128 133 134 135 140 141 143 145 150 156 156 158 165 177 178 179 182 185 200 201 203 216 220 221 232 233 234 234 235 237 239 243 243 246 248 250 256 265 268 268 271 272 273 273 283 284 285 286 286 289 290 292 306 306 313 313 320 321 337 338 340 342 344 344 344 344 344 348 352 353 356 356 360 363 364 364
sum: 22067 mean: 184.0 std: 114.049
The temp knows that to determine the interarrival time, the difference of each successive pair of values must be used, resulting in values of :
[ 1 4 3 2 1 1 0 1 2 ... 1 0]
A fancy plot was made.
λ = 3.03
λ = 3.03
There is a large proportion of same day or consecutive day assignments (i.e. λ =1, λ =2) . There are many days between assignments, historically as much as 18.
The temp concludes that it is most likely to receive multiple assignments in one day (intearrival time = 0) or on consecutive days (interarrival time = 1). This shows that about half of the assignments would be assigned on the same or consecutive days. The temp reasons that about half the year can be allotted to working on his application (assuming each assignment can be completed in a day or less).
Conclusion
Conclusion
We presented and explained the exponential distribution and some of its uses. In combination with other distributions, it is an important component in queuing theory and simulation. The logic of these topics can get cumbersome. Here, we attempted to use only the exponential distribution to characterize a common instances in business.
YHWH, Thank you for the time you have given us to do the things for You. Thank you for the time between to enjoy things for ourselves.
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