EEL6507sp09L36

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EEL6507 Spring 2009, Lecture 36, Wednesday 2009/04/15 (Notes created by Ajay Jain)

Topic :Finish M/G/1 Busy Time

From Last Lecture

G^*(s) = B^*(s + \lambda (1-G^*(s) ))\!


we have a recursive definition of the busy time.So if Distribution of service time is given for any queueing system Busy time can

be calculated.In general its very difficult to find solution for busy time distribution in term of service time distribution alone.


Some points :

  • Service rate does not depend on queue length.
  • Arrival rate does not depend on queue length.

As mentioned above if service time distribution is given we can find the busy time distribution ,Lets analyze for M/M/1

For M/M/1 case

B^*(s) = \frac{\mu}{(s + \mu)}
But,
s = s + \lambda - \lambda G^*(s)\!
G^*(s) = \frac{\mu}{\ mu + s + \lambda - \lambda G^*(s)}
G^*(s) = \frac{1}{ s+ \rho + 1 - \rho G^*(s)}

Making Quadratic in G * (s) and solving we get

G^*(s) = \frac{ s+ \rho + 1 - \sqrt( s+ \rho + 1 - 4 \rho)}{2 \rho}

Moments of Busy Time

First Moment

G^*(s) = B^*(s + \lambda (1-G^*(s) ))\!

Applying our regular technique of taking derivative and equating S=0

G^*(s) = \frac {B^'*(s + \lambda (1-G^*(s) ))}{1 + \lambda B^'*(s + \lambda (1-G^*(s) ))} \!


G^*(s) = \frac {B^'*(0)}{1 + \lambda B^'*(0)}


<Busy Time> = \frac{\bar{x}}{1-\lambda \bar{x}}

Second Moment

\bar{g^2}= \bar{x^2}(1+ \lambda \bar{g^2}) + \bar{x} \lambda \bar{g^2}


\bar{g^2}= \frac {\bar{x^2}}{(1 - \rho)^2}


\sigma_g ^2= \frac {\sigma_g ^2 + \rho \bar(x)^2}{(1 - \rho)^3}

Note:

  • Higher Variance in service give higher variance in busy time
  • Higher Traffic rate give higher variance in busy time

Lecture Vocie Recording

lecture_recording

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