Queueing Formula L = λW

Properties

Little’s Formula (or Little’s Law) is a fundamental identity for queueing systems in steady state:

$L=\lambda W$

where:

• $L$: average number of customers in the system
• $\lambda$: average arrival rate of entering customers
• $W$: average time a customer spends in the system

Interpretation

Consider a long time period $T$. The average number of arrivals is $\lambda T$, and the average number of departures is $\lambda (T-W)$. The remaining customers in the system: $L=\lambda T-\lambda (T-W)=\lambda W$.

Variants

For the queue (waiting line) only:

$L_Q=\lambda W_Q$

For customers in service:

$\text{avg. number in service}=\lambda \mathbb{E}[S]$

where $\mathbb{E}[S]$ is the mean service time.

Cost Equation Derivation

The general cost identity underlying Little’s formula:

$\text{avg. rate system earns}=\lambda \times \text{avg. amount an entering customer pays}$

By choosing appropriate cost rules (e.g., each customer pays $1 per unit time in system/queue), the specific variants are obtained.

Validity

Little’s formula holds for almost all queueing models, regardless of:

• Arrival process
• Number of servers
• Queue discipline

It requires only that the system is in steady state.

Related

• Queueing System
• 1 Queueing System

Exercises

Back to Roadmap 📖 → 🃏 → ✏

Verifikasi Little’s Law. M/M/1/2: $\lambda =3$/jam, $\mu =4$/jam. Dari perhitungan sebelumnya: $L=30/49$, ${\pi }_0=16/49$, ${\pi }_1=12/49$, ${\pi }_2=9/49$. Hitung $W$ dan verifikasi $L={\lambda }_{a}W$ dengan ${\lambda }_a=\lambda (1-{\pi }_2)$.

Jawaban: ${\lambda }_a=3(1-\frac{9}{49})=\frac{120}{49}$. $W=\frac{L}{{\lambda }_a}=\frac{30/49}{120/49}=\frac{30}{120}=\frac{1}{4}$ jam = 15 menit. Verifikasi: ${\lambda }_{a}W=\frac{120}{49}\cdot \frac{1}{4}=\frac{30}{49}=L$ ✓.

M/M/1. Untuk M/M/1 dengan $\lambda =4$, $\mu =6$, $L=2$, $W=0.5$. Verifikasi $L_Q=\lambda W_Q$.

Jawaban: $W=W_Q+1/\mu \Rightarrow W_Q=0.5-1/6=1/3$. $L_Q=4\cdot 1/3=4/3$. Alternatif: $L_Q=\frac{{\rho }^2}{1-\rho }=\frac{4/9}{1/3}=4/3$ ✓.