Birth and Death Process

Definition

Consider a system whose state at any time is the number of people in the system. A Birth and Death Process is a CTMC with states $\{0,1,2,\dots \}$ where:

• When $n$ people are in the system, new arrivals enter at exponential rate ${\lambda }_n$ (birth rate)
• People leave at exponential rate ${\mu }_n$ (death rate)
• Birth and death occur independently

Transitions can only go from state $n$ to $n+1$ (birth) or $n-1$ (death):

$\mathrm{Pr}\{X(t+h)=n+1\mid X(t)=n\}={\lambda }_{n}h+o(h)$ $\mathrm{Pr}\{X(t+h)=n-1\mid X(t)=n\}={\mu }_{n}h+o(h)$ $\mathrm{Pr}\{X(t+h)=n\mid X(t)=n\}=1-({\lambda }_n+{\mu }_n)h+o(h)$

Interpretation

Think of a population: births increase the count, deaths decrease it. The time between events is exponential, and the next event is either a birth or a death, depending on competing exponentials.

Transition Parameters

The state transition rates and probabilities:

\nu_0 &= \lambda_0, & \nu_i &= \lambda_i + \mu_i \quad (i > 0) \\ P_{0,1} &= 1 \\ P_{i,i+1} &= \frac{\lambda_i}{\lambda_i + \mu_i}, & P_{i,i-1} &= \frac{\mu_i}{\lambda_i + \mu_i} \quad (i > 0) \end{aligned}$$ The probability of birth before death at state $i$ follows from the minimum of independent exponentials: $\Pr\{T_i^{(b)} < T_i^{(d)}\} = \lambda_i / (\lambda_i + \mu_i)$. ## Special Cases - **Pure Birth Process**: $\mu_n = 0$ for all $n$ - **Pure Death Process**: $\lambda_n = 0$ for all $n$ - **Poisson Process**: $\lambda_n = \lambda$, $\mu_n = 0$ - **Yule Process**: $\lambda_n = n\lambda$, $\mu_n = 0$ ## Related - [[3 Reference/continuous-time-markov-chain_202605080515\|Continuous-Time Markov Chain]] - [[3 Reference/pure-birth-process_202605080516\|Pure Birth Process]] - [[3 Reference/yule-process_202605080517\|Yule Process]] - [[3 Reference/birth-and-death-queueing-models_202605080521\|Birth and Death Queueing Models]] ## Exercises > [!NOTE] Back to [[4 Projects/kuis-2_202605211907#roadmap\|Roadmap 📖 → 🃏 → ✏]] > **Kuis 2 2025 No. 2.** Reaksi kimia: $N$ molekul A → B irreversibel. Jika ada $j$ molekul, setiap molekul berubah dengan laju $q$. Klasifikasikan $\{X(t)\}$ (banyaknya molekul A) sebagai proses stokastik. > **Jawaban:** Proses kematian murni (pure death process). Hanya transisi $j \to j-1$ yang mungkin, dengan laju kematian $\mu_j = qj$ dan laju kelahiran $\lambda_j = 0$ untuk semua $j$. **Kuis 2 2024 No. 3.** Taksi dan pelanggan tiba di stasiun — taksi dengan rate 1/menit, pelanggan dengan rate 2/menit. Taksi selalu menunggu, pelanggan pergi jika tidak ada taksi. Tentukan rate kelahiran $\lambda_n$ dan rate kematian $\mu_n$. > **Jawaban:** State $n$ = jumlah taksi menunggu (bisa negatif = pelanggan menunggu, tapi pelanggan pergi jika taksi kosong). $\lambda_n = 1$ (taksi tiba), $\mu_n = 2$ untuk $n > 0$ (pelanggan mengambil taksi), tapi $\mu_0$ = hanya kedatangan pelanggan yang langsung pergi. Lebih tepat: model sebagai BD dengan state taksi.