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\begin{aligned} & N: \quad P_{N+1} =\frac{\lambda_{N}}{\mu_{N-1}} P_{N}+\frac{\left(\mu_{N} P_{N}-\lambda_{N-1} P_{N-1}\right)}{\mu_{N-1}}\\ & \qquad \qquad \quad =\frac{\lambda_{N}}{\mu_{N-1}} P_{N}+\frac{\left(\mu_{N} P_{N}-\mu_{N} P_{N}\right)}{\mu_{N-1}}\\ &  \qquad \qquad \quad =\frac{\lambda_{N}}{\mu_{N-1}} P_{N}\\ &  \qquad \qquad \quad =\frac{\lambda_{N}{N}_{-1} \cdots \lambda_{0}}{\mu_{N-1} m_{k} \cdots \mu_{1}} P_{0}\\ & \text { If } \mathrm{C}_{\mathrm{N}}=\frac{\lambda_{\mathrm{N}-1}{ }^{\lambda} \mathrm{N}-2 \cdots \lambda_{0}}{\mu_{\mathrm{N}}+\mu_{\mathrm{N}-1} \ldots \mu_{1}}, \text { then } \mathrm{P}_{\mathrm{N}}=\mathrm{C}_{\mathrm{N}} \mathrm{P}_{0} \end{aligned}