Can you please help me to fix the MATLAB Code to fix the calculation of p$\_$state which is incorrect as it does not computer all p$\_$states which are in the queue and being served $?$

The MMCQ Validation failed except c$\_$util and the correct Ws value should be $2\mathrm{.}1557.$instead of $2\mathrm{.}140$ as shown the out put below. Thanks

$\%$ Define system parameters

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

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

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

Nwait $=15$; $\%$ maximum number of customers in the queue

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

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

$\%$ Run this test case to check your code

$[$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$,$ p$\_$state$(1))<0\mathrm{.}005$

$\%$ 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$($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, $0\mathrm{.}00283)<0\mathrm{.}005$

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$))$;

$\%$ Extra Tests

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

if rel$\_$error$($sum$($p$\_$state$),1)<0\mathrm{.}005$

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

else

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

end

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

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

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

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

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

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

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

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

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

pN $=$ p$\_$state$($Nwait$)$;

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

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

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

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

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

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

p$\_$drop $=$ pN;

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

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

End

Ws $=2\mathrm{.}140$ FAIL

Wq $0\mathrm{.}140$ FAIL

c$\_$util $0\mathrm{.}500$ PASS

p$\_$drop $0\mathrm{.}00002$ FAIL

p$\_$state $0\mathrm{.}875950\mathrm{.}109490\mathrm{.}0136870\mathrm{.}000855421\mathrm{.}7821$e$-052\mathrm{.}4752$e$-07$

The probabilities sum to $1.$ Test PASSED