Fault-Tolerant Control of a Distributed Database System

<table class="table-group" id="tab2"><tr><td><table class="table"><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr class="thead"><td align="left">Current state <svg height="12.165" id="M337" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 16.411249 12.165" width="16.411249" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.165)">
<text transform="matrix(1,0,0,-1,0.498,3.72)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑥</tspan>
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<text transform="matrix(1,0,0,-1,7.375,0.594)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑘</tspan>
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</svg></td><td align="center">One-step cost <svg height="15.8375" id="M338" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 59.080002 15.8375" width="59.080002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,15.8375)">
<text transform="matrix(1,0,0,-1,0.498,3.72)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐶</tspan>
<tspan style="font-size: 12.50px; " x="8.3895931" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="12.553132" y="0">𝑥</tspan>
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<text transform="matrix(1,0,0,-1,19.928,0.594)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑘</tspan>
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<text transform="matrix(1,0,0,-1,25.184,3.72)">
<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="5.2012978" y="0">𝐮</tspan>
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<text transform="matrix(1,0,0,-1,37.346,0.594)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑘</tspan>
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</svg></td></tr><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr><td align="left">The database is fully functional</td><td align="center">0</td></tr><tr><td align="left">One unavailable server has a queue</td><td align="center"><span style="width: 22.453751px;"><svg height="14.67" id="M339" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 22.453751 14.67" width="22.453751" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.67)">
<text transform="matrix(1,0,0,-1,0.498,2.786)">
<tspan style="font-size: 12.50px; " x="0" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="6.2515602" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="9.7274275" y="0">𝜇</tspan>
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</g>
</svg></span></td></tr><tr><td align="left">Two unavailable servers have a queue</td><td align="center"><span style="width: 22.453751px;"><svg height="14.67" id="M340" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 22.453751 14.67" width="22.453751" xmlns="http://www.w3.org/2000/svg">
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<text transform="matrix(1,0,0,-1,0.498,2.786)">
<tspan style="font-size: 12.50px; " x="0" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="6.2515602" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="9.7274275" y="0">𝜇</tspan>
</text>
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</svg></span></td></tr><tr><td align="left">All servers are unavailable and have a queue</td><td align="center"><span style="width: 22.453751px;"><svg height="14.67" id="M341" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 22.453751 14.67" width="22.453751" xmlns="http://www.w3.org/2000/svg">
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<text transform="matrix(1,0,0,-1,0.498,2.786)">
<tspan style="font-size: 12.50px; " x="0" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="6.2515602" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="9.7274275" y="0">𝜇</tspan>
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Discounted service rate with service demand consideration.

Journal of Control Science and Engineering

tab2

Table 2

Table 2: Fault-Tolerant Control of a Distributed Database System 