# Examples of Waiting Lines

$F\left( x \right)=\Pr \left( X\le x \right)=\int\limits_{-\infty }^{x}{\frac{1}{45}{{e}^{-s/45}}ds}=\left\{ \begin{array}{lc} & 1-{{e}^{-x/45}}\text{ }x\ge 0 \\ & \,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x<0 \\ \end{array} \right.$
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$F\left( x \right)=\left\{ \begin{array}{lc} & 1\,\,\,\,\,\,\,\,\,\,\,\,\,x\ge 5 \\ & x/5\,\,\,\,\,\,\,0\le x<5 \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,x<0 \\ \end{array} \right.$
 T-Shirts Expected Profit 10,000 $151,092.5 10,500$152,890.0 11,000 $154,265.0 11,500$153,167.5 12,000 $155,007.0 12,500$153,335.0 13,000 $152,445.0 13,500$149,700.0 14,000 $146,192.5 $R=\left\{ \begin{array}{ll} & 138X+98\times 25\text{ if }X<70 \\ & 138\times 70+98\times 25\,\,\,\,\,\,\,\,\,\text{if }X\ge 70 \\ \end{array} \right.$ $R = \left\{ \begin{array}{ll} & 138X+98\times \left( 95-F \right)\text{ if }X<F \\ & 138\times F+98\times \left( 95-F \right)\,\,\,\,\,\,\,\,\,\,\,\text{if }X\ge F \\ \end{array} \right.$  Full Fare Reservation Expected Revenue 30$10,493.44 40 $10,830.93 50$11,015.65 60 $10,931.54 70$10,494.71