Can you please help to fix the pN calculation with the users in the queue and being served. function $[$Ws$,$ Wq$,$ c$\_$util, p$\_$drop, p$\_$state$]=$ MMCQ$($lambda$,$ mu$,$ c$,$ Nwait$)$

$\%$ Calculate the utilization factor

$\%$ rho $=$ lambda $/$ mu;

rho $=$ lambda $/($c $*$ mu$)$;

$\%$ Calculate the average number of customers in the queue

$\%$ c$\_$bar $=$ rho $*(1+($c $*$ rho$)^($c $-1)/$ factorial$($c $-1)/(1-$ rho$)^2)$; $\%$ Fix c$\_$bar calculation

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

p$0\_$denom $=$ sum$(($c $*$ rho$).^(0$:$($c $-1))/$ factorial$(0$:$($c $-1)))+($c $*$ rho$)^$c $/$ factorial$($c$)/(1-$ rho$)$; $\%$ Fix p$0\_$denom calculation

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

$\%$ Calculate the probability of zero customers in the system

$\%$ P$0=1/($sum$(($c$*$rho$).^(0$:c$-1)/$ factorial$(0$:c$-1))+(($c$*$rho$)^$c $/($factorial$($c$)*(1-$ rho$))))$;

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

p$\_$state$(1)=$ p$0$;

$\%$ Calculate pN $($probability of having N customers in the system$)$

$\%$ for i $=1$:Nwait

$\%\%$ p$\_$state$(1)=($c$-$i$)/($c $-($c $-$ i$))$;

$\%$ if i $<=$ c

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

$\%$ else

$\%$ p$\_$state$($i $+1)=($rho$^$i $/$ factorial$($c$)*($c$^($i $-$ c$)))*$ p$0$; $\%$ Fix the p$\_$state calculation

$\%$ end

$\%$ end

for i $=1$:Nwait

ci $=$ min$($i$,$ c$)$;

if i $<=$ c

p$\_$state$($i $+1)=$ ci $/$ c $*$ p$\_$state$($i$)*$ rho $/$ factorial$($i$)$;

else

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

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 $*$ p$\_$state$($Nwait $+1)$;

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

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

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

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

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

$\%$ Calculate the utilization

c$\_$bar $=$ Ls $-$ Lq;

c$\_$util $=$ c$\_$bar $/$ c;

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

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

$\%$ Calculate Ws and Wq

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

Wq $=$ Lq $/$ lambda$\_$eff;

end$\%$ c $=$ Total service time $/$ Total time

lambda $=2$; $\%$ arrival rate

mu $=5$; $\%$ service rate

c $=3$; $\%$ number of servers

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