$24\mathrm{.}1$ Server Farm with Size$-$Interval$-$Task$-$Assignment

We are given a server farm with $2$ identical FCFS hosts. Arrivals into the system

occur according to a Poisson process with rate $.$ Job sizes follow a power$-$law

distribution, denoted by random variable $S,$ where $P\{S>x\}=x^{-2\mathrm{.}5},$ for

Small jobs $($those of size $<mn\; id="EditorElement\_2178500">1</mn><mn\; id="EditorElement\_2178501">0</mn><mtext\; id="EditorElement\_2178430"></mtext><mi\; id="EditorElement\_2178431">)</mi>$ are routed to the first server, and

large jobs $($those of size $10)$ are routed to the second server.

$($a$)$ Derive the mean response time, $E[T],$ for this system.

$($b$)$ What would $E[T]$ be if the job size distribution were changed to

$P\{S>x\}=x^{-1\mathrm{.}5},$ for $1$