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\lambda
\mu
0
1
2
0
1
2
0
1
2
P01
P12
P10
P21
0
1
2
P01
P12
P10
P21
= 1
= 1
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\displaystyle
P=\left[ {\begin{array}{ccc}
P_{00}&P_{01}&P_{02}\\
P_{10}&P_{11}&P_{12}\\
P_{20}&P_{21}&P_{22}\\
\end{array} } \right
]
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\displaystyle
P=\left[ {\begin{array}{ccc}
P_{00}&P_{01}&P_{02}\\
P_{10}&P_{11}&P_{12}\\
P_{20}&P_{21}&P_{22}\\
\end{array} } \right
]
\displaystyle
\Pi=\left[ {\begin{array}{ccc}
\pi_{0}&
\pi_{1}&
\pi_{2}
\end{array} } \right
]
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\displaystyle
P=\left[ {\begin{array}{ccc}
P_{00}&P_{01}&P_{02}\\
P_{10}&P_{11}&P_{12}\\
P_{20}&P_{21}&P_{22}\\
\end{array} } \right
]
\displaystyle
\Pi=\left[ {\begin{array}{ccc}
\pi_{0}&
\pi_{1}&
\pi_{2}
\end{array} } \right
]
\sum_n\Pi_n=1
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\displaystyle
P*\Pi=\left[ {\begin{array}{ccc}
P_{00}*\pi_0&P_{01}*\pi_0&P_{02}*\pi_0\\
P_{10}*\pi_1&P_{11}*\pi_1&P_{12}*\pi_1\\
P_{20}*\pi_2&P_{21}*\pi_2&P_{22}*\pi_2\\
\end{array} } \right
]
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\displaystyle
P*\Pi=\left[ {\begin{array}{ccc}
P_{00}*\pi_0&P_{01}*\pi_0&P_{02}*\pi_0\\
P_{10}*\pi_1&P_{11}*\pi_1&P_{12}*\pi_1\\
P_{20}*\pi_2&P_{21}*\pi_2&P_{22}*\pi_2\\
\end{array} } \right
]
P*\Pi=\Pi
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\displaystyle
P*\Pi=\left[ {\begin{array}{ccc}
0 & \pi_0 & 0 \\
P_{10}*\pi_1& 0 &P_{12}*\pi_1\\
0 &\pi_2&0\\
\end{array} } \right
]
P*\Pi=\Pi
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\displaystyle
P*\Pi=\left[ {\begin{array}{ccc}
0 & \pi_0 & 0 \\
P_{10}*\pi_1& 0 &P_{12}*\pi_1\\
0 &\pi_2&0\\
\end{array} } \right
]
\pi_0=\displaystyle\frac{\mu \pi_1}{\lambda+\mu}
\pi_1=\pi_0+\pi_2
\pi_2=\displaystyle\frac{\lambda \pi_1}{\lambda+\mu}
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu \pi_1}{\lambda+\mu}
\pi_1=\pi_0+\pi_2
\pi_2=\displaystyle\frac{\lambda \pi_1}{\lambda+\mu}
\pi_0+\pi_1+\pi_2=1
\pi_0+\pi_2=\displaystyle\frac{1}{2}=\pi_1
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu \pi_1}{\lambda+\mu}
\pi_1=\pi_0+\pi_2
\pi_2=\displaystyle\frac{\lambda \pi_1}{\lambda+\mu}
\pi_0+\pi_2=\displaystyle\frac{1}{2}=\pi_1
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
\lambda=\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{4\mu}=\displaystyle\frac{1}{4}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{4\lambda}=\displaystyle\frac{1 }{4}
\lambda=\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
\lambda=!\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
\lambda=!\mu
\lambda>\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
\lambda=!\mu
\lambda>>\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
\lambda>>\mu
\lambda+\mu\approx\lambda
\pi_0+\pi_2=\displaystyle\frac{1}{2}
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}\approx 0
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}\approx\displaystyle\frac{1}{2}
\lambda>>\mu
\lambda+\mu\approx\lambda
\pi_0+\pi_2=\displaystyle\frac{1}{2}
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
\lambda=!\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
\lambda=!\mu
\lambda<\mu
\lambda<<\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0=\displaystyle\frac{\mu}{2(\lambda+\mu)}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}
\lambda<<\mu
\lambda+\mu\approx\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\pi_0\approx\displaystyle\frac{\mu}{2\mu} =\displaystyle\frac{1}{2}
\pi_1=\displaystyle\frac{1}{2}
\pi_2=\displaystyle\frac{\lambda }{2(\lambda+\mu)}\approx 0
\lambda<<\mu
\lambda+\mu\approx\mu
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\lambda=\mu=3
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{3}{6}=0.5
=\displaystyle \frac{3}{6}=0.5
\lambda=\mu=3
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{3}{6}=0.5
=\displaystyle \frac{3}{6}=0.5
\lambda=\mu=3
\pi_1=0.5
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{3}{6}=0.5
=\displaystyle \frac{3}{6}=0.5
\lambda=\mu=3
\pi_1=0.5
\pi_0=0.5\pi_1
\pi_2=0.5\pi_1
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{3}{6}=0.5
=\displaystyle \frac{3}{6}=0.5
\lambda=\mu=3
\pi_1=0.5
\pi_0=0.25
\pi_2=0.25
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\lambda=11\mu=3
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{3}{14}
=\displaystyle \frac{11}{14}
\lambda=11\mu=3
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{3}{14}
=\displaystyle \frac{11}{14}
\lambda=11\mu=3
\pi_1=\displaystyle\frac{1}{2}\\
\pi_0=\displaystyle\frac{3}{14}\pi_1=\displaystyle\frac{3}{28}
\\
\\
\pi_2=\displaystyle\frac{11}{14}\pi_1=\displaystyle\frac{11}{28}\\
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{\mu}{\lambda+\mu}
=\displaystyle \frac{\lambda}{\lambda+\mu}
\lambda=2\\\mu=13
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{13}{15}
=\displaystyle \frac{2}{15}
\lambda=2\\\mu=13
0
1
2
P01
P12
P10
P21
= 1
= 1
= \displaystyle \frac{13}{15}
=\displaystyle \frac{2}{15}
\lambda=2\\\mu=13
\pi_1=\displaystyle\frac{1}{2}\\
\pi_0=\displaystyle\frac{13}{15}\pi_1=\displaystyle\frac{13}{30}
\\
\\
\pi_2=\displaystyle\frac{2}{15}\pi_1=\displaystyle\frac{2}{30}\\
deck
By salvador_moreno_carrillo
