Modeling Slotted Aloha as a Stochastic Game with Random Discrete Power Selection Algorithms

<table>Bi-dimensional Markov chain for the game setting. The state of the system is the backlog vector; the first component corresponds to the number of backlogged packets for <svg height="7.1374998" id="M250" style="vertical-align:-0.10033pt" version="1.1" viewbox="0 0 11.225 7.1374998" width="11.225" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,7.1375)">
<g transform="translate(72,-66.29)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑚</tspan>
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</svg> first mobiles whereas the second component indicates the number of backlogged packets of the tagged mobile <svg height="11.325" id="M251" style="vertical-align:-0.51414pt" version="1.1" viewbox="0 0 36.6875 11.325" width="36.6875" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.325)">
<g transform="translate(72,-62.94)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑚</tspan>
<tspan style="font-size: 12.50px; " x="11.652796" y="0">+</tspan>
<tspan style="font-size: 12.50px; " x="22.993017" y="0">1</tspan>
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</g>
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</svg> (either 0 or 1).</table>

Journal of Computer Networks and Communications

fig3

Figure 3

Figure 3: Modeling Slotted Aloha as a Stochastic Game with Random Discrete Power Selection Algorithms 