35  Erlang Distribution

The Erlang distribution is the Gamma distribution restricted to positive-integer shape parameters. Developed by A.K. Erlang for telephone traffic engineering, it models the accumulated waiting time across exactly \(k\) independent exponential stages — the standard model for multi-phase service times in queuing systems.

Formally, the random variate \(X\) defined for the range \(X > 0\), is said to have an Erlang Distribution (i.e. \(X \sim \text{Erlang}(k, \lambda)\)) with positive-integer shape \(k \geq 1\) and rate parameter \(\lambda > 0\). The Erlang\((k, \lambda)\) distribution equals the Gamma\((k, \lambda)\) distribution when \(k\) is a positive integer. In R, use dgamma(x, shape = k, rate = lambda) with integer shape.

35.1 Probability Density Function

\[ f(x) = \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}, \quad x > 0 \]

The figure below shows examples of the Erlang Probability Density Function for different shape values with \(\lambda = 1\).

Code 
par(mfrow = c(2, 2))
x <- seq(0, 12, length = 500)

plot(x, dgamma(x, shape = 1, rate = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(k == 1, ",  ", lambda == 1)))

plot(x, dgamma(x, shape = 2, rate = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(k == 2, ",  ", lambda == 1)))

plot(x, dgamma(x, shape = 3, rate = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(k == 3, ",  ", lambda == 1)))

plot(x, dgamma(x, shape = 5, rate = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(k == 5, ",  ", lambda == 1)))

par(mfrow = c(1, 1))
Figure 35.1: Erlang Probability Density Function for various shape values (rate = 1)

35.2 Purpose

The Erlang distribution was developed to model the total waiting time across a fixed number of exponential service stages in telephone switching systems. Its requirement of a positive-integer shape parameter makes it more interpretable than the Gamma distribution in queuing contexts: \(k\) is a literal count of stages. Common applications include:

  • Multi-stage service times in call centers and telecommunications networks
  • Total processing time across sequential stages in manufacturing
  • Aggregate customer service duration when each interaction has \(k\) steps
  • Waiting time before the \(k\)-th arrival in a Poisson process
  • Hypo-exponential distribution building block (sum of different Exponentials)

Relation to the discrete setting. The Erlang distribution is the continuous analog of the Negative Binomial distribution — the Negative Binomial counts discrete failures before the \(k\)-th success; the Erlang measures continuous total time until the \(k\)-th Poisson event.

35.3 Distribution Function

\[ F(x) = 1 - \sum_{n=0}^{k-1}\frac{(\lambda x)^n e^{-\lambda x}}{n!}, \quad x > 0 \]

In R: pgamma(x, shape = k, rate = lambda).

The figure below shows the Erlang Distribution Function for \(k = 3\) and \(\lambda = 1\).

Code 
x <- seq(0, 12, length = 500)
plot(x, pgamma(x, shape = 3, rate = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "F(x)", main = "Erlang Distribution Function",
     sub = expression(paste(k == 3, ",  ", lambda == 1)))
Figure 35.2: Erlang Distribution Function (k = 3, rate = 1)

35.4 Moment Generating Function

\[ M_X(t) = \left(\frac{\lambda}{\lambda - t}\right)^k, \quad t < \lambda \]

35.5 1st Uncentered Moment

\[ \mu_1' = \frac{k}{\lambda} \]

35.6 2nd Uncentered Moment

\[ \mu_2' = \frac{k(k+1)}{\lambda^2} \]

35.7 3rd Uncentered Moment

\[ \mu_3' = \frac{k(k+1)(k+2)}{\lambda^3} \]

35.8 4th Uncentered Moment

\[ \mu_4' = \frac{k(k+1)(k+2)(k+3)}{\lambda^4} \]

35.9 2nd Centered Moment

\[ \mu_2 = \frac{k}{\lambda^2} \]

35.10 3rd Centered Moment

\[ \mu_3 = \frac{2k}{\lambda^3} \]

35.11 4th Centered Moment

\[ \mu_4 = \frac{3k(k+2)}{\lambda^4} \]

35.12 Expected Value

\[ \text{E}(X) = \frac{k}{\lambda} \]

35.13 Variance

\[ \text{V}(X) = \frac{k}{\lambda^2} \]

35.14 Median

The median has no closed form and is computed numerically: qgamma(0.5, shape = k, rate = lambda).

35.15 Mode

\[ \text{Mo}(X) = \frac{k-1}{\lambda}, \quad k \geq 1 \]

For \(k = 1\) (Exponential), the mode is 0.

35.16 Coefficient of Skewness

\[ g_1 = \frac{2}{\sqrt{k}} \]

The skewness decreases as \(k\) increases; for large \(k\) the Erlang distribution approaches the Normal distribution.

35.17 Coefficient of Kurtosis

\[ g_2 = 3 + \frac{6}{k} \]

For \(k = 1\) (Exponential), \(g_2 = 9\).

35.18 Parameter Estimation

Since \(k\) must be a positive integer, estimation uses method of moments followed by rounding:

\[ \tilde k = \text{round}\!\left(\frac{\bar x^2}{s^2}\right), \qquad \tilde\lambda = \frac{\tilde k}{\bar x} \]

set.seed(42)
k_true <- 3; lambda_true <- 2
x_obs <- rgamma(100, shape = k_true, rate = lambda_true)

# Method of moments
xbar <- mean(x_obs); s2 <- var(x_obs)
k_hat <- round(xbar^2 / s2)
lambda_hat <- k_hat / xbar
cat("MoM k:     ", k_hat, "\n")
cat("MoM lambda:", round(lambda_hat, 4), "\n")
cat("True k:", k_true, "  True lambda:", lambda_true, "\n")
MoM k:      3 
MoM lambda: 2.0898 
True k: 3   True lambda: 2

35.19 R Module

35.19.1 RFC

The Erlang Distribution module is available in RFC under the menu “Distributions / Erlang Distribution”.

35.19.3 R Code

The following code demonstrates Erlang probability calculations using pgamma (which accepts integer shape):

k <- 3; lambda <- 2

# Distribution function: P(X <= x)
pgamma(1.5, shape = k, rate = lambda)

# Quantile (median)
qgamma(0.5, shape = k, rate = lambda)

# Mean and mode
cat("Mean:", k / lambda, "\n")
cat("Mode:", (k - 1) / lambda, "\n")
cat("SD:  ", sqrt(k) / lambda, "\n")

# Generate random Erlang numbers
set.seed(42)
rgamma(10, shape = k, rate = lambda)
[1] 0.5768099
[1] 1.33703
Mean: 1.5 
Mode: 1 
SD:   0.8660254 
 [1] 2.5687787 0.8434274 1.2881428 0.5292550 1.1675090 0.7842371 3.3549583
 [8] 1.2009118 0.5387973 0.3931260

35.20 Example

A call center requires customers to complete \(k = 3\) sequential service stages, each with an independent Exponential service time at rate \(\lambda = 2\) per minute. Total service time \(X \sim \text{Erlang}(3, 2)\) has mean \(k/\lambda = 1.5\) minutes.

k <- 3; lambda <- 2

# P(service time <= 1.5 min)
cat("P(service <= 1.5 min):", pgamma(1.5, shape = k, rate = lambda), "\n")

# Mean, mode, median
cat("Mean (min):", k / lambda, "\n")
cat("Mode (min):", (k - 1) / lambda, "\n")
cat("Median (min):", qgamma(0.5, shape = k, rate = lambda), "\n")
P(service <= 1.5 min): 0.5768099 
Mean (min): 1.5 
Mode (min): 1 
Median (min): 1.33703
Interactive Shiny app (click to load).
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35.21 Random Number Generator

Erlang random variates are generated as the sum of \(k\) i.i.d. Exponential variates. Using the inverse-CDF form for each:

\[ X = -\frac{1}{\lambda}\ln\!\left(\prod_{i=1}^k U_i\right) = -\frac{1}{\lambda}\sum_{i=1}^k \ln U_i, \quad U_i \overset{\text{i.i.d.}}{\sim} \text{U}(0,1) \]

set.seed(123)
n <- 1000; k <- 3; lambda <- 2

# Sum of k Exponential variates
x_inv <- replicate(n, -sum(log(runif(k))) / lambda)

cat("Simulated mean:", round(mean(x_inv), 4), "\n")
cat("Theoretical mean:", k / lambda, "\n")
cat("Simulated var:", round(var(x_inv), 4), "\n")
cat("Theoretical var:", k / lambda^2, "\n")
Simulated mean: 1.5069 
Theoretical mean: 1.5 
Simulated var: 0.7542 
Theoretical var: 0.75
Interactive Shiny app (click to load).
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35.22 Property 1: Special Case of Gamma with Integer Shape

The Erlang\((k, \lambda)\) distribution is identical to the Gamma\((k, \lambda)\) distribution when \(k\) is a positive integer. The Gamma distribution allows non-integer shapes (see Chapter 29).

35.23 Property 2: Sum of Independent Exponentials

If \(X_1, \ldots, X_k \overset{\text{i.i.d.}}{\sim} \text{Exp}(\lambda)\) then:

\[ X_1 + \cdots + X_k \sim \text{Erlang}(k, \lambda) \]

This is the defining construction and provides the clearest interpretation: \(X\) is the total time until the \(k\)-th event in a Poisson process.

35.24 Property 3: Special Case \(k = 1\)

When \(k = 1\), the Erlang reduces to the Exponential distribution with rate \(\lambda\):

\[ \text{Erlang}(1, \lambda) = \text{Exp}(\lambda) \]

See Chapter 27.