Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By romsek

Thread: Adding exponential waiting time

  1. #1
    Newbie
    Joined
    Sep 2018
    From
    Under a rock
    Posts
    1

    Adding exponential waiting time

    Hello all!
    I'm having some difficulty understanding how to solve this problem. I rewrote it a bit for better understanding as the real problem has over 50 people and also to practice myself after I know how it should be done.

    There are 10 people waiting in the dentist office. The average waiting time is 5 minutes. What is the chance that the total waiting time is more than 60 minutes?
    It is exponentially distributed.

    This is how far I got:
    I understand that it is not the chance of everyone waiting 6 minutes or more, someone waiting 40 minutes while the rest is waiting just 5 will have the same result. So I can't multiply all the chances of someone waiting more than 6 minutes.
    On the internet I found a formula that says to substract the time someone waited from the 60 minutes, but this leaves me with over 50 variables in my problem. This can hardly be the solution?
    Can someone give me a hint?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    6,269
    Thanks
    2678

    Re: Adding exponential waiting time

    The sum of $n$ i.i.d. exponentially distributed rvs has an Erlang distribution.

    let $X_k \sim \lambda e^{-\lambda t},~k=1,~\dots,~n$

    $W = \sum \limits_{k=1}^n~X_k \sim \dfrac{\lambda^n t^{n-1} e^{-\lambda t}}{(n-1)!}$

    and thus, with $n=10,~\lambda = \dfrac 1 5$ we have

    $P[W > 60] = 1 - \displaystyle \int_0^{60}~\dfrac{5^{-10} t^9 e^{-t/5}}{9!}~dt \approx 0.2424$
    Thanks from MacstersUndead
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Calculating system/queue waiting time
    Posted in the Business Math Forum
    Replies: 0
    Last Post: Nov 13th 2012, 03:01 AM
  2. Adding and Subtracting Time ? Confused?
    Posted in the Business Math Forum
    Replies: 5
    Last Post: Jul 1st 2012, 05:05 AM
  3. Linear birth-death process waiting time problem
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Jun 18th 2012, 08:41 AM
  4. Jack and Jill Waiting Time question.
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Nov 8th 2009, 04:34 AM

/mathhelpforum @mathhelpforum