Queueing System

About

A Queueing System is a system where customers arrive at random times to a service facility, wait in queue if necessary, receive service, and then leave.

Key Components

1. Input Process / Arrival Process: Probability distribution governing the pattern of customer arrivals (e.g., Poisson process, general distribution)
2. Service Distribution: Probability distribution of service times (e.g., exponential, deterministic)
3. Queue Discipline: Number of servers, order of service (FIFO, LIFO, priority), system capacity, population size

Notation (Kendall’s Notation)

Queueing models are described using the notation:

$A/S/r/K/N$

where:

• $A$: arrival process distribution
• $S$: service time distribution
• $r$: number of servers
• $K$: system capacity (optional, default $\infty$)
• $N$: population size (optional, default $\infty$)

Symbol	Meaning
$M$	Markovian / Memoryless (Poisson arrivals, exponential service)
$G$	General distribution
$D$	Deterministic
$E_k$	Erlang-$k$ distribution
$H$	Hyperexponential distribution
$PH$	Phase-type distribution

Commonly abbreviated as $A/S/r$ when capacity and population are infinite (FIFO discipline implied).

Performance Metrics

• $L$: average number of customers in the system
• $L_Q$: average number of customers waiting in queue
• $W$: average time a customer spends in the system
• $W_Q$: average time a customer spends waiting in queue
• $\rho$: traffic intensity (server utilization)

Common Assumptions

• Customer arrivals are random
• Service times are i.i.d. and independent of arrivals
• FIFO (first-in first-out) queue discipline
• System has reached steady-state equilibrium

Related

• Queueing Formula L = λW
• 1 Queueing System
• s Queueing System
• Birth and Death Queueing Models

Exercises

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Klasifikasi sistem antrian. Sebuah loket tiket memiliki 1 petugas, melayani maksimal 2 pelanggan sekaligus (1 dilayani + 1 menunggu). Pelanggan datang Poisson($\lambda =2$/menit), waktu layanan $\text{Exp}(\mu =3$/menit). Pelanggan yang datang saat penuh pergi. Tuliskan sistem ini dalam Kendall notation.

Jawaban: M/M/1/2. M = Poisson arrivals, M = exponential service, 1 = satu server, 2 = kapasitas sistem (termasuk yang dilayani).

Metrik dasar. Jika $L=30/49$ dan ${\lambda }_{\text{eff}}=2\cdot (1-{\pi }_2)=2\cdot \frac{45}{49}$, hitung $W$ menggunakan Little’s Law.

Jawaban: ${\lambda }_{\text{eff}}=\lambda (1-{\pi }_2)=2(1-\frac{4}{19})=\frac{30}{19}$ (pakai hasil 2025 Q10). $W=L/{\lambda }_{\text{eff}}=\frac{30/49}{30/19}=\frac{19}{49}$ menit.