PAT/source/html/a1014.html

Waiting in Line
<div class="ques-view"><p>Suppose a bank has <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> windows open for service. There is a yellow line in front of the windows which devides the waiting area into two parts. The rules for the customers to wait in line are:</p>
<ul>
<li>The space inside the yellow line in front of each window is enough to contain a line with <span class="katex"><span class="katex-mathml"><math><mrow><mi>M</mi></mrow>M</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">M</span></span></span></span> customers. Hence when all the <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> lines are full, all the customers after (and including) the <span class="katex"><span class="katex-mathml"><math><mrow><mo>(</mo><mi>N</mi><mi>M</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow>(NM+1)</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="mord mathit" style="margin-right:0.10903em;">M</span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span>st one will have to wait in a line behind the yellow line.</li>
<li>Each customer will choose the shortest line to wait in when crossing the yellow line. If there are two or more lines with the same length, the customer will always choose the window with the smallest number.</li>
<li><span class="katex"><span class="katex-mathml"><math><mrow><mi>C</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mi>i</mi></msub></mrow>Customer_i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> will take <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>T</mi><mi>i</mi></msub></mrow>T_i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.13889em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> minutes to have his/her transaction processed.</li>
<li>The first <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> customers are assumed to be served at 8:00am.</li>
</ul>
<p>Now given the processing time of each customer, you are supposed to tell the exact time at which a customer has his/her business done.</p>
<p>For example, suppose that a bank has 2 windows and each window may have 2 custmers waiting inside the yellow line. There are 5 customers waiting with transactions taking 1, 2, 6, 4 and 3 minutes, respectively. At 08:00 in the morning, <span class="katex"><span class="katex-mathml"><math><mrow><mi>c</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>1</mn></msub></mrow>customer_1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> is served at <span class="katex"><span class="katex-mathml"><math><mrow><mi>w</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>o</mi><msub><mi>w</mi><mn>1</mn></msub></mrow>window_1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="mord mathit">i</span><span class="mord mathit">n</span><span class="mord mathit">d</span><span class="mord mathit">o</span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> while <span class="katex"><span class="katex-mathml"><math><mrow><mi>c</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>2</mn></msub></mrow>customer_2</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> is served at <span class="katex"><span class="katex-mathml"><math><mrow><mi>w</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>o</mi><msub><mi>w</mi><mn>2</mn></msub></mrow>window_2</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="mord mathit">i</span><span class="mord mathit">n</span><span class="mord mathit">d</span><span class="mord mathit">o</span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>. <span class="katex"><span class="katex-mathml"><math><mrow><mi>C</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>3</mn></msub></mrow>Customer_3</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> will wait in front of <span class="katex"><span class="katex-mathml"><math><mrow><mi>w</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>o</mi><msub><mi>w</mi><mn>1</mn></msub></mrow>window_1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="mord mathit">i</span><span class="mord mathit">n</span><span class="mord mathit">d</span><span class="mord mathit">o</span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math><mrow><mi>c</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>4</mn></msub></mrow>customer_4</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> will wait in front of <span class="katex"><span class="katex-mathml"><math><mrow><mi>w</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>o</mi><msub><mi>w</mi><mn>2</mn></msub></mrow>window_2</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="mord mathit">i</span><span class="mord mathit">n</span><span class="mord mathit">d</span><span class="mord mathit">o</span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>. <span class="katex"><span class="katex-mathml"><math><mrow><mi>C</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>5</mn></msub></mrow>Customer_5</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">5</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> will wait behind the yellow line.</p>
<p>At 08:01, <span class="katex"><span class="katex-mathml"><math><mrow><mi>c</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>1</mn></msub></mrow>customer_1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> is done and <span class="katex"><span class="katex-mathml"><math><mrow><mi>c</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>5</mn></msub></mrow>customer_5</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">5</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> enters the line in front of <span class="katex"><span class="katex-mathml"><math><mrow><mi>w</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>o</mi><msub><mi>w</mi><mn>1</mn></msub></mrow>window_1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="mord mathit">i</span><span class="mord mathit">n</span><span class="mord mathit">d</span><span class="mord mathit">o</span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> since that line seems shorter now. <span class="katex"><span class="katex-mathml"><math><mrow><mi>C</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>2</mn></msub></mrow>Customer_2</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> will leave at 08:02, <span class="katex"><span class="katex-mathml"><math><mrow><mi>c</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>4</mn></msub></mrow>customer_4</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> at 08:06, <span class="katex"><span class="katex-mathml"><math><mrow><mi>c</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>3</mn></msub></mrow>customer_3</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> at 08:07, and finally <span class="katex"><span class="katex-mathml"><math><mrow><mi>c</mi><mi>u</mi><mi>s</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mi>r</mi><mn>5</mn></msub></mrow>customer_5</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">u</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord mathit">o</span><span class="mord mathit">m</span><span class="mord mathit">e</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">5</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> at 08:10.</p>
<h3 id="input-specification-">Input Specification:</h3>
<p>Each input file contains one test case. Each case starts with a line containing 4 positive integers: <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo><mn>2</mn><mn>0</mn></mrow>\le 20</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span><span class="mord mathrm">2</span><span class="mord mathrm">0</span></span></span></span>, number of windows), <span class="katex"><span class="katex-mathml"><math><mrow><mi>M</mi></mrow>M</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">M</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo><mn>1</mn><mn>0</mn></mrow>\le 10</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span></span></span></span>, the maximum capacity of each line inside the yellow line), <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">K</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow>\le 1000</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span></span></span></span>, number of customers), and <span class="katex"><span class="katex-mathml"><math><mrow><mi>Q</mi></mrow>Q</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">Q</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow>\le 1000</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span></span></span></span>, number of customer queries).</p>
<p>The next line contains <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">K</span></span></span></span> positive integers, which are the processing time of the <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">K</span></span></span></span> customers.</p>
<p>The last line contains <span class="katex"><span class="katex-mathml"><math><mrow><mi>Q</mi></mrow>Q</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">Q</span></span></span></span> positive integers, which represent the customers who are asking about the time they can have their transactions done. The customers are numbered from 1 to <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">K</span></span></span></span>.</p>
<h3 id="output-specification-">Output Specification:</h3>
<p>For each of the <span class="katex"><span class="katex-mathml"><math><mrow><mi>Q</mi></mrow>Q</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">Q</span></span></span></span> customers, print in one line the time at which his/her transaction is finished, in the format <code>HH:MM</code> where <code>HH</code> is in [08, 17] and <code>MM</code> is in [00, 59]. Note that since the bank is closed everyday after 17:00, for those customers who cannot be served before 17:00, you must output <code>Sorry</code> instead.</p>
<h3 id="sample-input-">Sample Input:</h3>
<pre><code class="lang-in">2 2 7 5
1 2 6 4 3 534 2
3 4 5 6 7
</code></pre>
<h3 id="sample-output-">Sample Output:</h3>
<pre><code class="lang-out">08:07
08:06
08:10
17:00
Sorry
</code></pre>
</div>