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


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.


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