$L=\frac{-}{-}$

$5=\frac{30}{-30}$

$=36$

b$.$ For the service rate in part $($a$),$ what is the probability that more than five customers are in line and being served?

The probability that more than five customers are in the system is one minus the probability of five or fewer customers in the system.

$P=1-(P_0+P_1+P_2+P_3+P_4+P_5)$

Here $P_n=(1-)({)}^n$ is the probability that $n$ customers are in the system.

Simplify the expression for $P$

$P=1-\{[(1-)({)}^0]+[(1-)({)}^1]+[(1-)({)}^2]+[(1-)({)}^3]+[(1-)({)}^4]+[(1-)({)}^5]\}$

$=1-\{(1-)[({)}^0+({)}^1+({)}^2+({)}^3+({)}^4+({)}^5]\}$

$=1-(1++{}^2+{}^3+{}^4+{}^5)+(1++{}^2+{}^3+{}^4+{}^5)$

$=1-1--{}^2-{}^3-{}^4-{}^5++{}^2+{}^3+{}^4+{}^5+{}^6$

$={}^6$

Calculate average utilization of the drive$-$in window

$=0\mathrm{.}833$

Use the obtained value of $$ to evaluate $P$

$P={0\mathrm{.}833}^6=0\mathrm{.}3349$

Consequently, there is about a $33$ percent chance of more than five customers in the system

c$.$ For the service rate in part $($a$),$ what is the average waiting time in line for each customer?

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