Difference between Erlang distribution and Poisson distribution

[ChuckLee]

Hello. I would like to understand the difference between Erlang distribution and Poisson. For example, here is a sample question. Should I use Erland or Poisson to solve the problem? And what is the reasoning?

Sample question:
The number of customers arriving at a sales counter is 1 customer per minute.
a. What is the probability that 5 or less customers will arrive within 10 minutes?
b. What is the probability that the time to 5th customer is more than 10 minutes?

[Sergey]

Hi! If time between customers arriving is distributed according Exponential Law (Poison flow of requests), the full time for 5 customers arriving will be distibuted according Erlang of 5 order with Mean = 5 and Variance = 5. But, generally speaking, you can approximate this "full time" of Normal Law (according Central Limit Theorem) and in this case you could not know exactly Law of arriving time (exponential, uniform, or some else). It is enough to know only mean and variance of this arriving time (e.g., for exponential law in your case Mean=Variance=1).
Regards, Sergey.
P.S. Caution. Don't use last approach for rare event estimation (e.g., time to 5 customers more than 20 min), in this case you have to use Large Deviation Aproach.

[Harry]

1. The counting process or the number of customers follows Poisson distribution.
2. The time between customer arrivals is exponential distribution.
3. For a given number of arrivals, say, the time by the nth arrival, is gamma or Erlang distribution.

Poisson distribution aslo can be calcualted using Normal approximation, like Sergey said. But only if lambda*t>5, it can be used.

For question
a:
Pr(n<=5 within 10 min)= Sum{(lamda*10)^n*exp(-lambda*10)/(n!)}=Sum{10^n*exp(-10)/(n!)}, n=0,1,2,3,4,5.
b:
Pr(S5 >10 min)= 1-Pr(S5<=10 min) Here you can use Gamma distribution directly or you can use:

1-Pr(S5<=10>=5 within 10 mins) = Pr(n<5 within 10 min)= Pr(n=0, 1,2,3,4 within 10 min). It is Poisson Distribution.

you can see the relationship between Gamma and Poisson distribution.