Hello Expert I am getting failed MMCQ Validation results $.$ Can you please help me to correct the MATALAB Code. The equations for c$\_$bar, p$\_$state, c$\_$util, p$0\_$denom, and Lq are wrong. I need to get them corrected and all tests passed. Here is the matalb code.function $[$Ws$,$ Wq$,$ c$\_$util, p$\_$drop, p$\_$state$]=$ MMCQ$($lambda$,$ mu$,$ c$,$ Nwait$)$

$\%$ INPUTS:

$\%$ lambda $=$ arrival rate

$\%$ mu $=$ service rate

$\%$ Nwait $=$ Number of queue elements, not including servers

$\%$ c $=$ Number of servers

$\%$ OUTPUTS:

$\%$ Ws $=$ Average wait in the system

$\%$ Wq $=$ Average wait in the queue

$\%$ c$\_$util $=$ Average percentage of servers busy

$\%$ p$\_$drop $=$ probability the queue is full

$\%$ p$\_$state $=$ probability of each state

$\%$ Calculate traffic intensity rho

rho $=$ lambda $/($c $*$ mu$)$; $\%$ Calculate the traffic intensity

$\%$ Calculate p$0($probability of no customers in the system$)$

p$0\_$denom $=$ sum$(($rho$.^$linspace$(0,$ c $-1,$ c$))/$ factorial$(0$:c$-1))+($rho$^$c $/$ factorial$($c$))*$ sum$($rho$.^$linspace$($c$,$ Nwait, Nwait $-$ c $+1)/$ factorial$($c$))$;

p$0\_$denom $=$ p$0\_$denom $+(($c$*$rho$)^$c $/($factorial$($c$)*(1-$rho$)))*(1-($rho$^$Nwait $/(1-$ rho$)))$;

p$0=1/$ p$0\_$denom;

$\%$ n $=0$:$($c$-1)$;

$\%$ term$1=$ sum$(($crho$.^$n$)./$ factorial$($n$))$;

$\%$ term$2=(($c$*$crho$)^$c $/($factorial$($c$)*(1-$crho$)))*(1-($crho$^$Nwait $/(1-$ crho$)))$;

$\%$ p$0=1/($term$1+$ term$2)$;

$\%$ Calculate the utilization factor

c$\_$bar $=($rho$^$c $/$ factorial$($c$))*$ p$0/$ sum$(($rho$.^$linspace$(0,$ c $-1,$ c$))/$ factorial$(0$:c$-1))+($rho$^$c $/$ factorial$($c$))*$ sum$($rho$.^$linspace$($c$,$ Nwait, Nwait $-$ c $+1)/$ factorial$($c$))$;

$\%$ System Utilization for finite call center queue systems with some probability of filling up $($and rejecting users$).$

c$\_$util $=$ c$\_$bar $/$ c; $\%$ Calculate system utilization

$\%$ Calculate p$\_$state

p$\_$state $=$ zeros$(1,$ Nwait $+1)$;

for n $=0$:Nwait

if n $<$ c

p$\_$state$($n $+1)=($rho$^$n $/$ factorial$($n$))*(($c $*$ rho$)^$n $/$ factorial$($n$))*$ p$0$;

else

p$\_$state$($n $+1)=($rho$^$n $/$ factorial$($n$))*(($c $*$ rho$)^$n $/$ factorial$($c$))*$ p$0$;

end

end

$\%$ Normalize p$\_$state to ensure probabilities sum to $1$

p$\_$state $=$ p$\_$state $/$ sum$($p$\_$state$)$;

pN $=$ p$\_$state$($Nwait $+1)$; $\%$ Probability of having N customers in the system

$\%$ Calculate lambda$\_$loss and lambda$\_$eff

lambda$\_$loss $=$ lambda $*$ pN;

lambda$\_$eff $=$ lambda $-$ lambda$\_$loss;

$\%$ Calculate Lq $($average number of customers in the queue$)$

$\%$ Lq $=$ sum$((0$:Nwait$).*$ p$\_$state$)$;

Lq $=$ sum$((($c$*$rho$)^$c $/($factorial$($c$)*(1-$rho$)))*$ p$0*(($rho$^($Nwait$-$c$+1)*($Nwait$-$c$+1))/(1-$ rho$))^$n $/$ factorial$($n$))$;

$\%$ Calculate Ls $($average number of customers in the system$)$

Ls $=$ Lq $+$ lambda$\_$eff $/$ mu;

$\%$ Calculate p$\_$drop $($probability of a customer being dropped$)$

p$\_$drop $=$ pN $/(1+$ pN$)$;

$\%$ Calculate Ws and Wq

$\%$ Calculate Ws $($average time a customer spends in the system$($Ws$))$

Ws $=$ Ls $/$ lambda$\_$eff;

$\%$ Calculate Wq $($average time a customer spends waiting in the queue$($Wq$))$

Wq $=$ Lq $/$ lambda$\_$eff;$\%$ Run this test case to check your code

c $=2$; $\%$ Number of servers

lambda $=2$; $\%$ Average Arrival Rate per minute

mu $=0\mathrm{.}1$; $\%$ Service rate

Nwait $=15$;

$\%$ Call the MMCQ function with the given parameters

$\%[$Ws$,$ Wq$,$ c$\_$util, p$\_$drop, p$\_$state$]=$ MMCQ$($lambda$,$ mu$,$ c$,$ Nwait$)$;

$[$Ws$,$ Wq$,$ c$\_$util, p$\_$drop, p$\_$state$]=$ MMCQ$(1,0\mathrm{.}5,4,5)$;

$\%$ Define a relative error function

rel$\_$error $=$ @$($x$,$ y$)$ abs$($x $-$ y$)/$ y;

$\%$ Validate the results and print "PASS" or "FAIL" for each metric

if rel$\_$error$($Ws$,2\mathrm{.}1557)<0\mathrm{.}005$

fprintf$(\text{'}$Ws $=\%6\mathrm{.}3$f PASS

$\text{'},$ Ws$)$;

else

fprintf$(\text{'}$Ws $=\%6\mathrm{.}3$f FAIL

$\text{'},$ Ws$)$;

end

if rel$\_$error$($Wq$,0\mathrm{.}1557)<0\mathrm{.}005$

fprintf$(\text{'}$Wq $=\%6\mathrm{.}3$f PASS

$\text{'},$ Wq$)$;

else

fprintf$(\text{'}$Wq $=\%6\mathrm{.}3$f FAIL

$\text{'},$ Wq$)$;

end

if rel$\_$error$($c$\_$util, $0\mathrm{.}4986)<0\mathrm{.}005$

fprintf$(\text{'}$c$\_$util $=\%6\mathrm{.}3$f PASS

$\text{'},$ c$\_$util$)$;

else

fprintf$(\text{'}$c$\_$util $=\%6\mathrm{.}3$f FAIL

$\text{'},$ c$\_$util$)$;

end

if rel$\_$error$($p$\_$drop, $0\mathrm{.}00272)<0\mathrm{.}005$

fprintf$(\text{'}$p$\_$drop $=\%8\mathrm{.}5$f PASS

$\text{'},$ p$\_$drop$)$;

else

fprintf$(\text{'}$p$\_$drop $=\%8\mathrm{.}5$f FAIL

$\text{'},$ p$\_$drop$)$;

end

fprintf$(\text{'}$p$\_$state: $\%$s

$\text{'},$ num$2$str$($p$\_$state$))$;

fprintf$(\text{'}$sum$($p$\_$state$)$: $\%$f

$\text{'},$ sum$($p$\_$state$))$;

$\%$ Extra Tests: p$\_$state

$\%$ p$\_$state should have Nwait$+1$ probabilities

$\%$ disp$([\text{'}$The probability of being in state $\text{'},$ num$2$str$($Nwait$),\text{'}$ is $\text{'},$ num$2$str$($p$\_$state$($Nwait $+1))])$;

disp$([\text{'}$The probability of being in state $\text{'},\text{'}$ is $\text{'},$ num$2$str$($p$\_$state$)])$;

$\%$ Check that the probabilities sum to $1$

if abs$($sum$($p$\_$state$)-1)<1$e$-5$

disp$(\text{'}$The probabilities sum to $1.$ Test PASSED.$\text{'})$;

else

disp$(\text{'}$The probabilities do not sum to $1.$ Test FAILED.$\text{'})$;

end

end