Question: Why does the effective service rate f(x) go to zero when x = 0 in the example on queuing systems?
| Chapter(s) | System Modeling |
|---|---|
| Author(s) | Richard Murray |
| Date | 2009/10/10 |
Q: The formula for is . This is zero when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x = 0} (no queue) and 1 when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} goes to infinity. Why is this the right model?
A: The model used by Agnew (1976) is that the rate at which jobs are processed is linear in the queue length when the length is small, and saturates and the maximum service rate . At the extreme where there are no jobs on the queue, there is no need to process incoming requests and Agnew's assumption was the more processing would be applied as the queue got longer, until it saturates for very large.
Notice that the term in the denominator means that the service rate very rapidly reaches it maximum as the queue increases. If there is 1 job on the queue then the service rate is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0.5 \mu_\text{max}} , 3 jobs gives a rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0.75 \mu_\text{max}} and 9 jobs gives a rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0.9 \mu_\text{max}} .