Solving BD Process Balance Equations

Procedure

For a birth and death process with birth rates ${\lambda }_n$ and death rates ${\mu }_n$, the balance equations simplify to a recursive solution.

Balance Equations for B/D Process

State	Balance Equation
0	${\lambda }_0{\pi }_0={\mu }_1{\pi }_1$
$n\ge 1$	$({\lambda }_n+{\mu }_n){\pi }_n={\mu }_{n+1}{\pi }_{n+1}+{\lambda }_{n-1}{\pi }_{n-1}$

By adding each equation to the preceding one, we obtain the simplified form:

${\lambda }_n{\pi }_n={\mu }_{n+1}{\pi }_{n+1},n\ge 0$

Recursive Solution

Step 1: Express in terms of ${\pi }_0$

$\begin{array}{rl}{\pi }_0& ={\pi }_0\\ {\pi }_1& =\frac{{\lambda }_0}{{\mu }_1}{\pi }_0\\ {\pi }_2& =\frac{{\lambda }_1}{{\mu }_2}{\pi }_1=\frac{{\lambda }_1{\lambda }_0}{{\mu }_2{\mu }_1}{\pi }_0\\ {\pi }_n& =\frac{{\lambda }_{n-1}}{{\mu }_n}{\pi }_{n-1}=\frac{{\lambda }_0{\lambda }_1\cdots {\lambda }_{n-1}}{{\mu }_1{\mu }_2\cdots {\mu }_n}{\pi }_0\end{array}$

Step 2: Normalize

Using ${\sum }_{n=0}^{\infty }{\pi }_n=1$:

${\pi }_0=\frac{1}{1+{\sum }_{n=1}^{\infty }\frac{{\lambda }_0{\lambda }_1\cdots {\lambda }_{n-1}}{{\mu }_1{\mu }_2\cdots {\mu }_n}}$

Step 3: Final Solution

${\pi }_n=\frac{\frac{{\lambda }_0{\lambda }_1\cdots {\lambda }_{n-1}}{{\mu }_1{\mu }_2\cdots {\mu }_n}}{1+{\sum }_{k=1}^{\infty }\frac{{\lambda }_0{\lambda }_1\cdots {\lambda }_{k-1}}{{\mu }_1{\mu }_2\cdots {\mu }_k}},n\ge 1$

Existence Condition

Limit probabilities exist if and only if:

${\sum }_{n=1}^{\infty }\frac{{\lambda }_0{\lambda }_1\cdots {\lambda }_{n-1}}{{\mu }_1{\mu }_2\cdots {\mu }_n}<\infty$

NOTE

For irreducible B/D processes, the embedded MC can be argued to be positive recurrent. However, in practice, we solve the balance equations directly and determine the convergence condition.

Related

• CTMC Balance Equations
• Birth and Death Process
• Birth and Death Queueing Models

Exercises

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Kuis 2 2025 No. 6-7. M/M/1/2: $\lambda =3$/jam, $\mu =4$/jam. Tentukan:

(a) Rata-rata jumlah tugas dalam sistem.

Jawaban: $\rho =3/4$. ${\pi }_0=\frac{1}{1+\rho +{\rho }^2}=\frac{16}{49}$, ${\pi }_1=\frac{12}{49}$, ${\pi }_2=\frac{9}{49}$. $L=0\cdot \frac{16}{49}+1\cdot \frac{12}{49}+2\cdot \frac{9}{49}=\frac{30}{49}$.

(b) Probabilitas sistem sedang melayani tepat satu tugas.

Jawaban: $P(\text{tepat satu})={\pi }_1=\frac{12}{49}$.

Kuis 2 2024 No. 3. Taksi (rate 1/menit) dan pelanggan (rate 2/menit). Pelanggan pergi jika taksi kosong. Tentukan proporsi pelanggan yang langsung dapat taksi.

Jawaban: Misal state = jumlah taksi menunggu. $\lambda =1$, $\mu =2$. ${\pi }_n=(\frac{1}{2}{)}^n{\pi }_0$. $\sum {\pi }_n=1\Rightarrow {\pi }_0=\frac{1}{2}$. Proporsi pelanggan dapat taksi = $1-{\pi }_0=\frac{1}{2}$ (pelanggan hanya pergi saat state 0).