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Thread: Find the right stochastic process

  1. #1
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    Find the right stochastic process

    Hello everyone

    I am given a problem and asked to formulate a model describing the system. I have problems determining the right process to use and was hoping someone had an idea.

    We consider a system of finite capacity $\displaystyle N$, and at a fixed time every day, a number of new units arrive according to Poisson process with mean $\displaystyle \mu$. Also, units
    leave the system before the next arrival time with probability $\displaystyle \mathcal{P}$. If the remaining units in the system and the new arrivals exceeds the capacity $\displaystyle N$,
    the exceeding units are just disregarded. The question is then what process can be used to model the number of units just before new arrivals?

    Thanks in advance
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  2. #2
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    Re: Find the right stochastic process

    Each arrival interval is non overlapping so all these arrivals are independent with distribution

    $P[k] = \dfrac{\mu^k e^{-\mu}}{k!}$

    Departures are binomial with distribution

    $P[n] = \dbinom{M_t}{n} \mathcal{P}^n(1-\mathcal{P})^{M_t-n}$

    where $M$ is the number of items in the system at discrete time $t$. $M \in 0,1, \dots, N$

    and we'll have to assume that departures during two of the non overlapping intervals are also independent.

    Finally we have to assume that arrivals are independent of departures.

    We can now come up with a model of $M_t$ given $M_{t-1}$

    at $t-1$ there are $M_{t-1}$ items. The arrival at $t$ occurs and $k_t$ new items are added, saturating the system at $N$ if enough new arrivals.

    Then, for all intents and purposes immediately, there are $n_t$ departures.

    So $M_t = \min(M_{t-1}+k_t,N)-n_t$ with the associated probabilities on $k_t$ and $n_t$

    the next step is to convert this to

    $M_t = M_{t-1} + m_t$ and come up with a probability distribution of $m_t$

    This is the sum of all the probabilities of all the possible combinations of $k_t,~n_t$ such that $\min(M_{t-1}+k_t,N)-n_t = M_{t-1}+m_t$

    This will take some doing and I'm going to leave you here to digest this and think about it.
    Thanks from Krisly
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  3. #3
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    Re: Find the right stochastic process

    you might want to take a look at this

    You've essentially got a birth death process where the births and deaths come in chunks rather than discretely.
    Thanks from Krisly
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    Re: Find the right stochastic process

    Thank you very much, i just don't understand why $\displaystyle n_t$ has to leave immediately, or how to manipulate with $\displaystyle \min \left( M_{t-1} + k_t,N \right)$. But i'll figure it out.
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  5. #5
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    Re: Find the right stochastic process

    Quote Originally Posted by Krisly View Post
    Thank you very much, i just don't understand why $\displaystyle n_t$ has to leave immediately, or how to manipulate with $\displaystyle \min \left( M_{t-1} + k_t,N \right)$. But i'll figure it out.
    they don't, but it doesn't affect the problem one way or the other and there is no information given about the rate at which they are removed so you might as well model it as them leaving in a chunk.
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