Can you please advise to fix the error for summation of p$\_$state not getting to equal to $1?$ this is for multiserver MMCQ Valdation test for finite MMCQ queuing system 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$)$;

$\%$ 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;

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

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

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

end

end

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

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

$\%\_=\_\_$

$\%$ Lq $=$ lambda$\_$eff $*$ Wq;

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

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

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

$\%$ 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$\%$ INSERT YOUR CODE HERE

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

$\%$ p$=$lamda.$/($c$.*$mu$)$; $\%$ p$=$ System Utilization

$\%$ for k$=0$:$20$

$\%$ approxvalue$($k$+1)=($lamda$/$mu$).^$k$./$factorial$($k$)$;

$\%$ end

$\%$ r $=15$; $\%$ The number of callers in the series$($hold lines$)$

$\%$ result$(1)=$approxvalue $(1)$; $\%$ Initializing the result

$\%$ for i$=2$:$5$

$\%$ result $($i$-1)=$ sum$($approxvalue$(1$:i$))$;

$\%$ end

$\%\%$ To calculate the smallest number of customer agents that meets the requirements

$\%\%$ Wq $=$ Waiting time in queue

$\%\%$ Ws $=$ Waiting time in system

$\%\%$ Lq $=$ # of items in queue

$\%\%$ ls $=$ # of items in system

$\%$ p$0=($result$(1,$:$)+(($lamda$./$mu$).^$c$(1,$:$))./($factorial$($c$(1,$:$)).*(1-($lamda$./($c$(1,$:$).*$mu$))))).^-1$;$\%$Probability system is empty

$\%\_=\_\_$

Lq $=$ lambda $*$ Wq;

Wa $=1/($c $*$ mu$-$lambda$)$; $\%$ Average waiting time

$\%$ disp$($Wa$)$;

Wq $=$ Lq $/$ lambda; $\%$ Waiting time in queue

P$\_$w $=$ Wq $/$ Wa;$\%$ Probability that arrival must wait

$\%=\_+1/$

$\%$ Ws $=$ Wq $+1/$mu; $\%$ Average time in System

$\%=\_\_\_=\_\_?$

Ls $=$ Ws $*$ lambda; $\%$ Average Number in the System$($I am someow getting errors with Ls so I have to change Ls $)$

$\%$ disp$($Ls $)$;

$\%$Figure$8$

figure

subplot$(3,1,2)$

xlim$([220])$

ylim$([0\mathrm{.}2018980\mathrm{.}201933])$

horzcat$($Wq$)$;

horzcat$($c$)$;

plot$([$Wq$,$c$])\%$ Probability that arrival must wait against thd number of servers

xlabel$(\text{'}$Number of Servers $($c$)\text{'})$

ylabel$(\text{'}$Average hold Lines'$)$

title$(\text{'}$Probability that arrival must wait'$)$;

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

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

$\text{'},$ Ws$)$; $\%$ indicating that waiting times meet the specified criteria.

else

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

$\text{'},$ Ws$)$; $\%$ Ws is compared with p$\_$state$(1).$

end

if rel$\_$error$($Wq$,$ p$\_$state$(2))<0\mathrm{.}005\%$ Wq is compared with $0\mathrm{.}1557.$

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, $0\mathrm{.}4986)<0\mathrm{.}005\%$ c$\_$util is compared with $0\mathrm{.}4986.$

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

if rel$\_$error$($p$\_$drop, $0\mathrm{.}00283)<0\mathrm{.}005$

fprintf$(\text{'}$p$\_$drop $\%6\mathrm{.}3$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$)$: