Queueing Theory Calculator

What Is Queueing Theory?

Queueing theory is the mathematical study of waiting lines, congestion, and operational delays. Developed in 1909 by Danish engineer A.K. Erlang to analyze telephone switchboard capacities, queueing models are widely applied across telecommunications network packet routing, cloud server autoscaling, call centers, healthcare emergency triage, manufacturing production lines, airport security checkpoints, and supermarket cash registers.

In Kendall's notation, queueing systems are categorized using shorthand format $A/S/c$, where:

• A (Arrival process): Distribution of time intervals between customer arrivals (e.g., $M$ for Markovian or Poisson exponential distribution).
• S (Service process): Distribution of customer service duration (e.g., $M$ for memoryless exponential service times).
• c (Servers): Number of parallel service channels operating simultaneously.

The Single-Server M/M/1 Model

The M/M/1 queue is the foundational model with Poisson arrivals at mean rate $\lambda$, exponential service durations at rate $\mu$, and a single server ($c = 1$).

Traffic Intensity and Stability Condition

The ratio between arrival rate and service rate is the traffic intensity or server utilization $\rho$:

$$\rho = \frac{\lambda}{\mu}$$

For a waiting line to achieve steady-state equilibrium without exploding towards infinity, the service capacity must exceed the arrival rate ($\rho < 1$, meaning $\mu > \lambda$). If $\rho \ge 1$, the arrival rate matches or overwhelms the server, causing infinite queue growth.

M/M/1 Key Performance Formulas

Under steady-state conditions ($\rho < 1$), the key system metrics are:

• Probability system is idle ($P_0$): $P_0 = 1 - \rho$
• Average number of customers in system ($L$): $L = \frac{\lambda}{\mu - \lambda} = \frac{\rho}{1 - \rho}$
• Average number of customers in queue ($L_q$): $L_q = L - \rho = \frac{\lambda^2}{\mu(\mu - \lambda)} = \frac{\rho^2}{1 - \rho}$
• Average total time in system ($W$): $W = \frac{1}{\mu - \lambda}$
• Average waiting time in queue ($W_q$): $W_q = W - \frac{1}{\mu} = \frac{\rho}{\mu - \lambda}$

Little's Law

A central theorem in operations research and queueing theory is Little's Law, formulated by John Little in 1961. It states that the long-term average number of items in a stationary queueing system ($L$) equals the long-term average arrival rate ($\lambda$) multiplied by the average time an item spends in the system ($W$):

$$L = \lambda W, \quad L_q = \lambda W_q$$

Remarkably, Little's Law holds true regardless of the arrival distribution, service distribution, or queue discipline (FIFO, LIFO, or priority queueing).

The Multi-Server M/M/c Model (Erlang-C)

When $c$ identical servers operate in parallel, the total system service capacity expands to $c\mu$. The overall utilization is:

$$\rho = \frac{\lambda}{c\mu}$$

The probability $P_0$ that all servers are simultaneously idle is given by:

$$P_0 = \left[ \sum_{n=0}^{c-1} \frac{(\lambda / \mu)^n}{n!} + \frac{(\lambda / \mu)^c}{c!(1 - \rho)} \right]^{-1}$$

The probability that an arriving customer must wait in line before reaching an available server is given by the Erlang-C formula:

$$C(c, \lambda/\mu) = P(\text{Wait}) = \frac{\frac{(\lambda/\mu)^c}{c!(1 - \rho)}}{\sum_{n=0}^{c-1} \frac{(\lambda/\mu)^n}{n!} + \frac{(\lambda/\mu)^c}{c!(1 - \rho)}} = P_0 \frac{(\lambda/\mu)^c}{c!(1 - \rho)}$$

From Erlang-C, the average queue length and waiting time are:

$$L_q = \frac{P(\text{Wait}) \cdot \rho}{1 - \rho}, \quad W_q = \frac{L_q}{\lambda}, \quad W = W_q + \frac{1}{\mu}$$

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