Birth and Death Queueing Models

Definition

A Birth and Death Queueing Model is a queueing system where the number of customers $X(t)$ follows a birth and death process.

The model is characterized by:

• Arrival (birth) rates: ${\lambda }_n$, $n\ge 0$
• Departure (death) rates: ${\mu }_n$, $n\ge 1$

Balance Equations

In steady state, the rate at which the process leaves a state equals the rate at which it enters:

${\lambda }_0{P}_0={\mu }_1{P}_1$ $({\lambda }_n+{\mu }_n)P_n={\lambda }_{n-1}{P}_{n-1}+{\mu }_{n+1}{P}_{n+1},n\ge 1$

This yields the recurrence:

$P_{n+1}=\frac{{\lambda }_n}{{\mu }_{n+1}}{P}_n$

General Solution

Define auxiliary quantities:

${\theta }_0=1,{\theta }_j=\frac{{\lambda }_0{\lambda }_1\cdots {\lambda }_{j-1}}{{\mu }_1{\mu }_2\cdots {\mu }_j},j\ge 1$

The steady-state probabilities are:

${\pi }_0=\frac{1}{{\sum }_{j=0}^{\infty }{\theta }_j}$ ${\pi }_k={\theta }_k{\pi }_0=\frac{{\theta }_k}{{\sum }_{j=0}^{\infty }{\theta }_j},k\ge 1$

Steady state exists when ${\sum }_{j=0}^{\infty }{\theta }_j<\infty$.

Common Birth-Death Queueing Models

Model	${\lambda }_n$	${\mu }_n$
1	$\lambda$	$\mu$
s	$\lambda$	$\min (n,s)\mu$
∞	$\lambda$	$n\mu$
M/M/1 with Balking	$\lambda {\alpha }_n$	$\mu$

Related

• Birth and Death Process
• Queueing System
• 1 Queueing System

Exercises

Back to Roadmap 📖 → 🃏 → ✏

Kuis 2 2025 No. 6. Server menangani maks 2 tugas sekaligus (1 dilayani + 1 menunggu). Tugas datang Poisson($\lambda =3$/jam), layanan $\text{Exp}(\mu =4$/jam). Tugas yang datang saat penuh tetap menunggu (tidak ada balking). Berapa rata-rata jumlah tugas dalam sistem?

Jawaban: State $\{0,1,2\}$. ${\lambda }_0={\lambda }_1=3$, ${\lambda }_2=0$ (penuh tapi soal mengatakan tetap diterima dan menunggu — sebenarnya kapasitas tak terbatas? Tidak: “menangani paling banyak dua tugas”. Jadi ${\lambda }_2=0$). ${\mu }_1=4$, ${\mu }_2=4$. ${\theta }_1=3/4$, ${\theta }_2=9/16$. ${\pi }_0=\frac{1}{1+3/4+9/16}=\frac{16}{49}$, ${\pi }_1=\frac{12}{49}$, ${\pi }_2=\frac{9}{49}$. $L=0\cdot \frac{16}{49}+1\cdot \frac{12}{49}+2\cdot \frac{9}{49}=\frac{30}{49}$.

Kuis 2 2025 No. 7. Untuk sistem yang sama, berapa probabilitas sistem sedang melayani tepat satu tugas?

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