13 years, 1 days

## About the figure...

The figure foloows to state infinity by the same manner of rates.

## Urgent Help for Prove of \lambda=1/E[T]...

Dear All,

I am working on network modeling. Now I have a problem to prove the relationship between  "mean arrival rate"  and "average interarrival time". I know that they are reverse of each other based on "The art of computer systems performance analysis" page 521.
It is mentioned there that if T is interarrival time and \lambda = mean arrival rate then
\lambda = 1/ E[T] is correct for all G/G/m queues.
Now I should prove it. I want to know that is there any formal proof in probabilistic books.

Thanks a lot

Best Regards,
Saeid

## Thanks a lot...

Dear Alec,

It was very helpfull. Thanks a lot.

Best Regards,

Saeid

## Thanks a lot...

Dear Alec,

It was very helpfull. Thanks a lot.

Best Regards,

Saeid

## Symbolic format...

Dear Alec,

Thanks a lot for your reply and guidances.

I am quite new in maple. I want to add something. p and k are between 0 and 1.

Also my goal is solve it in symbolic manner. at least based on one of the n , c, p, alpha. However alpha is not important. Also f can not work for some parameters like f(10, 1/3, 2, 1) or f(1000, 1/3, 2, 1) or  f(1000, 1/70, 2, 100), f(1000, 1/70, 1.2, 100).

For example  f(10, 1/3, 2, 1) returns

fsolve(1-1/(423360/1968329+m)-1/(423360/1968329+4*m)-1/(423360/1968329+9*m)-1/(423360/1968329+16*m)-1/(423360/1968329+25*m)-1/(423360/1968329+36*m)-1/(423360/1968329+49*m)-1/(423360/1968329+64*m)-1/(423360/1968329+81*m)-1/(423360/1968329+100*m), m = 1, 0 .. 1)

Saeid

## Symbolic format...

Dear Alec,

Thanks a lot for your reply and guidances.

I am quite new in maple. I want to add something. p and k are between 0 and 1.

Also my goal is solve it in symbolic manner. at least based on one of the n , c, p, alpha. However alpha is not important. Also f can not work for some parameters like f(10, 1/3, 2, 1) or f(1000, 1/3, 2, 1) or  f(1000, 1/70, 2, 100), f(1000, 1/70, 1.2, 100).

For example  f(10, 1/3, 2, 1) returns

fsolve(1-1/(423360/1968329+m)-1/(423360/1968329+4*m)-1/(423360/1968329+9*m)-1/(423360/1968329+16*m)-1/(423360/1968329+25*m)-1/(423360/1968329+36*m)-1/(423360/1968329+49*m)-1/(423360/1968329+64*m)-1/(423360/1968329+81*m)-1/(423360/1968329+100*m), m = 1, 0 .. 1)

Saeid

## Not in this format...

Dear Alec

Thnks a lot for your replu. However I want to solve the equation sum(m/(k*p*c+m*i^alpha), i = 1 .. n)-m = 0

in a way tha find m based on n , p , alpha and c. After that I can set the value for them.

I need to evaluate for example that if I increase the alpha what happens to the m. I mean solve m based on other parameters.

Best Regards,

Saeid

## Not in this format...

Dear Alec

Thnks a lot for your replu. However I want to solve the equation sum(m/(k*p*c+m*i^alpha), i = 1 .. n)-m = 0

in a way tha find m based on n , p , alpha and c. After that I can set the value for them.

I need to evaluate for example that if I increase the alpha what happens to the m. I mean solve m based on other parameters.

Best Regards,

Saeid

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