$14\mathrm{.}5$ M$/$M$/2$ with Heterogeneous Servers

Consider a variant of the M$/$M$/2$ queue where the service rates of the two

processors are not identical. Denote the service rate of the first processor by ${}_1$

and the service rate of the second processor by ${}_2,$ where ${}_1>{}_2.$ In the case

of heterogeneous servers, the rule is that when both servers are idle, the faster

server is scheduled for service before the slower one. Define the utilization, $,$

for this system to be $=\frac{}{{}_1+{}_2}.$

Set up a CTMC and determine the mean number of jobs in the system and the

mean response time. You should get

$E[N]=\frac{1}{A(1-{)}^2}$

where

$A=\frac{{}_1{}_2(1+2)}{(+{}_2)}+\frac{1}{1-}$