I need a MATLAB Expert help to get my MATLAB Code work to validate the MMCQ Multiserver finite call center queuing system tests all passed. I posted the project many times to get it right but no luck with chegg expert so far. Here is the code below: The issues to be fixed are : All validation tests should pass $,$ correct the p$0$ calcualtion, pN$,$ and N values and recomended the optimal number of servers and lambda for the all validation tests to pass: $\%$ Run this test case to check your code

$\%$ Test and validation code

lambda $=2$; $\%$ Arrival rate: $2$ calls per minute

mu $=0\mathrm{.}1$; $\%$ Service rate: $0\mathrm{.}1$ calls per minute per agent

$\%$ Assuming an initial number of agents for the calculation

c $=4$; $\%$ Placeholder, needs adjustment based on system requirements

$\%$ Nwait $=$ Nwait Maximum number of customers waiting in the queue. Placeholder, needs adjustment based on system requirements

$\%$ Calculate Nwait based on the arrival and service rate

$\%$ Nwait is maximum queue length.

Nwait $=$ lambda $/($mu $-$ lambda$)$;

$\%$ Calculate System Capacity$($N$)=$ c $+$ Queue Length

N $=$ c $+$ Nwait;

$\%$ 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$,2\mathrm{.}1557)<0\mathrm{.}005$

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$))$;

$\%$ 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

$\%$ Extra Tests

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

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

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

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

$\%$ Calculate rho: the traffic intensity per server, which is the ratio of the arrival rate to the combined service rate of all servers.

rho $=$ lambda $/($mu$)$; $\%$ updated for the MMCQ Finite Queuing System

$\%$ Calculate p$0,$ the probability of having zero customers in the system.

$\%$ 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$))))$;

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

$\%$ Nwait $=(($rho$^$c $*$ c $*$ rho $*$ P$0)/($factorial$($c$)*(1-$ rho$)^2))$;

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

$\%$ Calculate the total number of possible customers in the system.

$\%$ N $=(($c$*$lambda$/$mu$)^$N $/$ factorial$($c$))*(1-$ lambda$/($c$*$mu$))*$ p$0$;

$\%$ Calculate System Capacity$($N$)=$ c $+$ Queue Length

N $=$ c $+$ Nwait;

$\%$ disp$($N$)$;

$\%$ fprintf$(\text{'}$The total number of possible customers in the system, N $=\%$s

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

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

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

$\%$ Calculate the p$\_$state based on the number of customers relative to the number of servers $($c$)$:

$\%$ For states c to Nwait $($Customers Being Served or Waiting$)$:the system includes customers waiting in the queue. for i $=1$:Nwait

for i $=1$:N

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$)$;

$\%$ Calculate pN: the probability of dropping a customer $($the system being full$).$

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

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

$\%$ pN $=(($c$*$lambda$/$mu$)^$N $/$ factorial$($c$))*(1-$ lambda$/($c$*$mu$))*$ p$0$;

$\%$ Calculate the rate of lost customers due to the system being full.

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

$\%$ Calculates the effective arrival rate, considering the lost customers.

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

$\%$ Calculate Lq: the average number of customers in the queue, calculated using the state probabilities.

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

$\%$ Calculates Ls: the total average number of customers in the system, both in service and in the queue.

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

$\%$ Calculate c$\_$bar $($Average Busy Servers$)$ and c$\_$util $($Utilization Factor$)**$:

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

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

p$\_$drop $=$ pN;

$\%$ Calculate Ws:the average time a customer spends in the system.

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

$\%$ Calcualte Wq: the average time a customer spends in the queue.

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

end