\[\begin{cases} 0 = \lambda_{10}P_1 + \lambda_{100}P_{10} - (\lambda_{00} + \lambda_{01})P_0 \\ 0 = \lambda_{01}P_0 + \lambda_{21}P_2 - (\lambda_{10} + \lambda_{11} + \lambda_{12})P_1 \\ 0 = \lambda_{12}P_1 + \lambda_{32}P_3 - (\lambda_{21} + \lambda_{22} + \lambda_{23})P_2 \\ 0 = \lambda_{23}P_2 + \lambda_{43}P_4 - (\lambda_{32} + \lambda_{33} + \lambda_{34})P_3 \\ 0 = \lambda_{34}P_3 + \lambda_{54}P_5 - (\lambda_{43} + \lambda_{44} + \lambda_{45})P_4 \\ 0 = \lambda_{45}P_4 + \lambda_{65}P_6 - (\lambda_{54} + \lambda_{55} + \lambda_{56})P_5 \\ 0 = \lambda_{56}P_5 + \lambda_{76}P_7 - (\lambda_{65} + \lambda_{66} + \lambda_{67})P_6 \\ 0 = \lambda_{67}P_6 + \lambda_{87}P_8 - (\lambda_{76} + \lambda_{77} + \lambda_{78})P_7 \\ 0 = \lambda_{78}P_7 + \lambda_{98}P_9 - (\lambda_{87} + \lambda_{88} + \lambda_{89})P_8 \\ 0 = \lambda_{89}P_8 + \lambda_{10,9}P_{10} - (\lambda_{98} + \lambda_{99} + \lambda_{9,10})P_9 \\ 0 = \lambda_{9,10}P_9 - (\lambda_{10,9} + \lambda_{10,10} + \lambda_{100})P_{10} \\ P_0 + P_1 + P_2 + P_3 + P_4 + P_5 + P_6 + P_7 + P_8 + P_9 + P_{10} = 1 \end{cases}\]