# Why do we always assume waiting time has exponential distribution?

In many continuous models, like waiting for a car, we always assume the waiting time $t$ to have an exponential distribution. Why is such an assumption appropriate?

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One possible answer would be to look at en.wikipedia.org/wiki/Maximum_entropy_probability_distribution – Tom Artiom Fiodorov May 17 '12 at 14:07
There are at least two reasons. First, the exponential distribution appears to be an adequate model for some, maybe many, and perhaps even most, real-life situations, and so the results of the analysis are useful in determining courses of action etc in real life. Second, the model is relatively easy to analyze (and to generalize). – Dilip Sarwate May 17 '12 at 14:10
Because it arises from the ubiquitous Poisson process. – bgins May 17 '12 at 14:10
The memorylessness property of exponential distribution plays a role too. – Tom Artiom Fiodorov May 17 '12 at 14:12
Because that is the distribution that they will have if the cars' arrival times are independent of one another. If all the cars have left from the same traffic-light-controlled intersection at the same time, then an exponential distribution is not appropriate. But in many cases it is a reasonable assumption. – MJD May 17 '12 at 14:12

## 1 Answer

My answer is that we don't. Assuming an exponential arrival time assumes the number of arrivals by time $t$ follows a Poisson process. It is just one of many possible rival time distributions and corresponding point processes that could be used. It is like saying "why do we 'always' assume a normal distribution for continuous random variables?" There too the answer is that we don't. In both cases the method is simple and convenient and there is a limit theorem that sometimes justifies its use.

Recognize that assuming exponential waiting times implies lack of memory. The lack of memory property states that if you are waiting for a bus or car to arrive, and have already waited five minutes, the remaining waiting time has the same exponential distribution that you had when you had just started waiting. This is not always a good assumption. Exponential waiting time/Poisson process are justified under some assumptions of the rarity of events over short time intervals, just like the normal distribution is justified for averages or sums of several observations from some population distribution.

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Got it, thank you~ – hxhxhx88 May 19 '12 at 1:30