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Utilization is Service Time times Arrival Rate, ie if requests arrive twice as fast utilization is doubled, and also if a service takes twice as long to process.

When the theoretical service is optimized to take half a long, the assumption is that the request rate doesn't actually change. So they use the above relationship to calculate the new utilization, given half the service time and the same arrival rate.

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so the unit for Utilization in your example is "active request" ?

I just wish the article didn't assume units are easily inferred which would imply most readers are familiar with the subject.

Having read it more than once now I think utilization is a ratio, maybe?

But that means the unit for S needs to be seconds/request such that:

utilization(ro) = S * arrival_rate(lambda) = seconds/request * request/seconds = bare ratio.

More coffee required.

Yes, that's normally how utilisation is understood - as the fraction of time used by requests, which is therefore a dimensionless quantity.

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