Ws: $2\mathrm{.}8685$

Wq: $0\mathrm{.}86851$

c$\_$util: $0\mathrm{.}48042$

p$\_$drop: $0\mathrm{.}039156$

p$\_$state: $0\mathrm{.}52860\mathrm{.}26430\mathrm{.}132150\mathrm{.}0330380\mathrm{.}00275310\mathrm{.}039156$

Ws $=2\mathrm{.}869$ FAIL

Wq $0\mathrm{.}869$ FAIL

c$\_$util $0\mathrm{.}480$ FAIL

p$\_$drop $0\mathrm{.}03916$ FAIL

p$\_$state $0\mathrm{.}52860\mathrm{.}26430\mathrm{.}132150\mathrm{.}0330380\mathrm{.}00275310\mathrm{.}039156$

The probabilities sum to $1.$ Test PASSED

p$\_$state: $0\mathrm{.}52860\mathrm{.}26430\mathrm{.}132150\mathrm{.}0330380\mathrm{.}00275310\mathrm{.}039156$

The probabilities sum to $1.$ Test PASSED.$\%$ 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

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

$\%$ Placeholder, needs adjustment based on system requirements

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

$\%$ Calculate System Capacity $($N$)$

N $=$ c $+$ Nwait;

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

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

$\%$ Display the results

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

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

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

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

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

$\%$ 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$)$

$\%$ Calculate rho: the traffic intensity per server

rho $=$ lambda $/$ mu;

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

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

$\%$ Initialize p$\_$state

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

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

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

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$)$;

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

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

$\%$ 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$((0$:Nwait$).*$ 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;

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

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

end