# resultant demand rate in central depot from K local depots

assume I have 1 disposable product (a pump for example). It fails according to exponential distribution.

plant 1 <----- local depot 1 <---
|---- central depot
plant 2 <----- local depot 2 <---


local depot 1 & 2 are supplied from central depot. plant 1 inventory is supplied from local depot 1 and same is true for local depot 2 and plant 2 .

Each failure of pump creates a demand at a rate of $\lambda$.

My opinion is that even rates at plant 1 & 2 are exponential and each rate is same as $\lambda$, one can NOT say that result demand rate at central depot is also exponential and rate is 2$\lambda$

my reason: distribution of demand of pump in central inventory will be something different than exponential

Am I correct? If I am not correct can you please explain at least basically why i am not correct? (for the details I can read sources over web)

thanks regards

## 1 Answer

You are correct. The sum of identically distributed independent exponentially distributed variables is Gamma (or Erlang) distributed. See the Wikipedia article, or this earlier question: Sum of exponential random variables follows Gamma, confused by the parameters.