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M/M/1 Systems

  1. Apr 17, 2012 #1
    1. The problem statement, all variables and given/known data
    For an M/M/1 queuing system with the average arrival rate of 0.4 min^−1 and the average service time of 2 minutes, compute
    A- the expected response time in minutes;
    B- the fraction of time when there are more than 5 jobs in the system;
    C- the fraction of customers who don't have to wait until their service begins


    2. Relevant equations



    3. The attempt at a solution
    For A I got that λ_A= 2.5 and that λ_S= 1/2 and then r would be 5. Obviously that's wrong since r has to be less than 1 I just don't understand where I messed up.
     
  2. jcsd
  3. Apr 17, 2012 #2

    Ray Vickson

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    λ_A = 0.4, not 2.5.

    RGV
     
  4. Apr 17, 2012 #3
    Thank you!
    so for A-10 and for C-.2 which were correct

    I'm still stuck on part B though. I know I'm supposed to find out P(X<5) but I don't know how I'm supposed to find it
     
  5. Apr 18, 2012 #4

    I like Serena

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    Hi lina29! :smile:

    Did you know that P(X<5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)?

    Actually, if you know what the sum of an geometric series is, it turns out that it's even easier to calculate P(X≥5)=P(X=5)+P(X=6)+...
     
  6. Apr 18, 2012 #5
    Thank you!
     
  7. Feb 16, 2013 #6
    A slight extension to this question:

    Given we know the arrival rate (λ)= 0.4 /min and the average service time (ε) = 0.5/min

    How would one go about finding the proportion of customers who are in the shop more than 10min?

    I am really unsure about how to approach this.
    The probability of a customer being in the shop (ie queueing) for more than 10mins can be solved P(W>10)=exp(-(ε-λ)x10))

    But what of the proportion of customers who spend more than 10 mins in the shop?

    Thank you for any help!
     
  8. Feb 16, 2013 #7

    Ray Vickson

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    For a customer named John Smith, what is the probability he spends > 10 min in the system? For a customer named Jennifer Jones, what is the probability she spends > 10 mins in the system? For any customer you can name, what is the probability that he/she spends > 10 mins in the system? So, what proportion of customers spend > 10 mins in the system?
     
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