1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Science Advisor
    Homework Helper

    λ_A = 0.4, not 2.5.

  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

    User Avatar
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook