BUS606: Operations and Supply Chain Management

A Birth-Death process is a Markov process in which states are numbered by an integer and transitions are only permitted between two neighbouring states. Births are the cases when state variables are increased by one and deaths are the cases when state variables are decreased by one. When birth occurs, the state N moves to state N + 1 and when the death occurs, state N changes to state N - 1.

Figure 2 shows the simple birth-death process with which we can establish the balance equations as follows:

State $\quad$ Rate in $=$ Rate out

$\begin{array}{ll}0: & \mu_{1} P_{1}=\lambda_{0} P_{0} \\ 1: & \lambda_{0} P_{0}+\mu_{2} P_{2}=\left(\lambda_{1}+\mu_{1}\right) P_{1}\end{array}$

2: $\quad \lambda_{1} P_{1}+\mu_{3} P_{3}=\left(\lambda_{2}+\mu_{2}\right) P_{2}$

$\begin{array}{lll}\cdots & \cdots & \cdots\end{array}$

${ }^{N-1}{ }_{-1}={\lambda}_{N-2}{ }^{P}{ }_{N-2}+\mu_{N}{ }^{P}{ }_{N}=\left(\lambda_{N-1}+\mu_{N-1}\right) P_{N-1}$

$\mathrm{N}: \quad \lambda_{\mathrm{N}-1} \mathrm{P}_{\mathrm{N}-1}+\mu_{\mathrm{N}+1} \mathrm{P}_{\mathrm{N}+1}=\left(\lambda_{\mathrm{N}}+\mu_{\mathrm{N}}\right) \mathrm{P}_{\mathrm{N}}$

All the above balanced equations can be expressed in terms of $\mathrm{P}_{0}$ as follows:

$0: \quad \mathrm{P}_{1}=\frac{\lambda_{0}}{\mu_{1}} \mathrm{P}_{0}$

1: $\quad P_{2}=\frac{\lambda_{1}}{\mu_{2}} P_{1}+\frac{\left(\mu_{1} P_{1}-\lambda_{0} P_{0}\right)}{\mu_{2}}$

$\begin{aligned}&=\frac{\lambda_{1}}{\mu_{2}} P_{1}+\frac{\left(\mu_{1} P_{1}-\mu_{1} P_{1}\right)}{\mu_{2}} \\ &=\frac{\lambda_{1}}{\mu_{2}} \frac{\lambda_{0}}{\mu_{1}} P\end{aligned}$

Continuing this way

$\begin{aligned} N-1: \qquad P_{N} &=\frac{\lambda_{N-1}}{\mu_{N}} P_{N-1}+\frac{\left(\mu_{N-1} P_{N-1}-\lambda_{N-2} P_{N-2}\right)}{\mu_{N}} \\ &=\frac{{ }{\lambda}_{\mathrm{~N}-1}}{\mu_{N}} P_{N-1}+\frac{\left(\mu_{N-1} P_{N-1}-\mu_{N-1} P_{N-1}\right)}{\mu_{N}} \\ &=\frac{\lambda_{{N}-1}}{\mu_{N}} P_{N-1}\end{aligned}$

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