Ferkans — Interactive Telecom Tutor

Beyond a Single Server

The M/M/1 queue has one server and infinite buffer — a useful starting point, but real systems differ in two important ways. First, most systems have multiple servers (think of $c$ parallel trunks in a telephony exchange, or $c$ frequency channels at a base station). Second, buffers are always finite: when the buffer fills, arriving customers are either lost (blocked) or balked. The M/M/c queue and the M/M/1/K queue model these two extensions. The Erlang formulas — Erlang-B for blocking and Erlang-C for waiting — are the engineering tools that emerge from these models and have been used to dimension telephone networks since the early 20th century.

Definition:
The $M/M/c$ Queue

The $M/M/c$ queue has Poisson arrivals (rate $\lambda$ ), $c$ identical servers each with exponential service time (rate $\mu$ ), and infinite buffer. It is a birth-death process with: $\lambda_n = \lambda \text{ for all } n, \qquad \mu_n = \begin{cases} n\mu & n \leq c, \\ c\mu & n > c. \end{cases}$ Define the offered load $A = \lambda/\mu$ (in Erlangs) and the server utilization $\rho = A/c = \lambda/(c\mu)$ . The system is stable iff $\rho < 1$ .

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Definition:
The $M/M/1/K$ Queue

The $M/M/1/K$ queue has Poisson arrivals (rate $\lambda$ ), one server (rate $\mu$ ), and a buffer of capacity $K$ (including the customer in service). Arrivals finding $K$ customers in the system are blocked (lost). It is a birth-death process on $\{0, 1, \ldots, K\}$ with: $\lambda_n = \begin{cases} \lambda & n < K, \\ 0 & n = K, \end{cases} \qquad \mu_n = \mu \text{ for } n \geq 1.$ Since the state space is finite, a stationary distribution always exists (no stability condition on $\rho$ ).

The blocking probability $P_K = \pi_K$ is the key performance metric: it gives the fraction of arrivals that are lost. This is the starting point for Erlang-B.

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Theorem: Erlang-C Formula ( $M/M/c$ Waiting Probability)

For the $M/M/c$ queue with offered load $A = \lambda/\mu$ and $\rho = A/c < 1$ , the probability that an arriving customer must wait (all servers busy) is: $C(A, c) = \frac{\dfrac{A^c}{c!}\cdot\dfrac{1}{1-\rho}}{\displaystyle\sum_{k=0}^{c-1}\frac{A^k}{k!} + \frac{A^c}{c!}\cdot\frac{1}{1-\rho}}.$ The mean sojourn time is: $W = \frac{1}{\mu} + \frac{C(A,c)}{c\mu - \lambda}.$

$C(A,c)$ is the probability of finding all $c$ servers busy upon arrival. With probability $1-C(A,c)$ the customer goes directly to a free server (zero wait). With probability $C(A,c)$ , the customer waits; the mean wait given waiting is $1/(c\mu-\lambda)$ (the excess service capacity acts as a single fast server on the queue).

Proof

Stationary distribution of M/M/c

From the birth-death formula with $\lambda_n = \lambda$ , $\mu_n = \min(n,c)\mu$ : $\pi_n = \begin{cases} \pi_0 \dfrac{A^n}{n!} & 0 \leq n \leq c, \\[6pt] \pi_0 \dfrac{A^c}{c!}\rho^{n-c} & n > c. \end{cases}$ Normalizing: $\pi_0 = \left(\sum_{k=0}^{c-1}\frac{A^k}{k!} + \frac{A^c}{c!}\frac{1}{1-\rho}\right)^{-1}$ .

Waiting probability

A customer waits iff $N \geq c$ upon arrival (PASTA: Poisson arrivals see time averages). $C(A,c) = \sum_{n=c}^{\infty}\pi_n = \pi_0\frac{A^c}{c!}\frac{1}{1-\rho}$ , which gives the formula. $\blacksquare$

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Theorem: Erlang-B Formula (Blocking Probability)

Consider $c$ servers, Poisson arrivals at rate $\lambda$ , exponential service at rate $\mu$ , and no buffer (blocked customers are lost). This is the $M/M/c/c$ queue (Erlang loss system). The blocking probability is: $B(A, c) = \frac{A^c / c!}{\displaystyle\sum_{k=0}^{c} A^k / k!},$ where $A = \lambda/\mu$ is the offered load in Erlangs. This is the Erlang-B formula. It is insensitive to the service time distribution: it holds for any distribution with mean $1/\mu$ .

The system state is the number of busy servers, $n \in \{0, \ldots, c\}$ . The stationary distribution is a truncated Poisson: $\pi_n \propto A^n/n!$ . Blocking occurs in state $c$ .

Proof

Birth-death with finite state space

The $M/M/c/c$ system is a birth-death process on $\{0, \ldots, c\}$ with $\lambda_n = \lambda$ for $n < c$ , $\lambda_c = 0$ , and $\mu_n = n\mu$ . The stationary distribution: $\pi_n = \pi_0 \frac{A^n}{n!}$ , $n = 0, \ldots, c$ . Normalizing: $\pi_0 = 1/\sum_{k=0}^c A^k/k!$ . The blocking probability is $\pi_c = B(A,c)$ . $\blacksquare$

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Example: Dimensioning Voice Channels (Erlang-B)

A cellular base station receives voice calls at rate $\lambda = 20$ calls/hour. Each call lasts on average $1/\mu = 3$ minutes. How many channels $c$ are needed to keep the blocking probability below 2%?

Solution

Compute the offered load

$A = \lambda/\mu = 20 \times (3/60) = 1$ Erlang.

Wait — let us be careful with units. $\lambda = 20$ calls/hour, $1/\mu = 3$ minutes $= 1/20$ hours, so $\mu = 20$ /hour. $A = \lambda/\mu = 20/20 = 1$ Erlang.

Evaluate Erlang-B for increasing $c$

$B(1, 1) = \frac{1}{1+1} = 0.5$ (50% blocking). $B(1, 2) = \frac{1/2}{1+1+1/2} = \frac{0.5}{2.5} = 0.2$ (20%). $B(1, 3) = \frac{1/6}{1+1+1/2+1/6} = \frac{0.1667}{2.6667} = 0.0625$ (6.25%). $B(1, 4) = \frac{1/24}{1+1+1/2+1/6+1/24} = \frac{0.0417}{2.7083} = 0.0154$ (1.54%).

So $c = 4$ channels suffice for $B < 2\%$ .

Example: Call Center Staffing (Erlang-C)

A customer service center receives calls at rate $\lambda = 30$ /hour. Mean service time is $1/\mu = 4$ minutes. The target is that no more than 10% of callers wait. How many agents $c$ are needed?

Solution

Offered load

$A = \lambda/\mu = 30 \times (4/60) = 2$ Erlangs.

Evaluate Erlang-C

$\rho = A/c$ . We need $C(A,c) \leq 0.10$ .

$c = 3$ : $\rho = 2/3$ . Numerator: $\frac{2^3}{3!}\frac{1}{1-2/3} = \frac{8}{6}\cdot 3 = 4$ . Denominator: $1 + 2 + 2 + 4 = 9$ . $C(2,3) = 4/9 \approx 0.444$ .

$c = 4$ : $\rho = 1/2$ . Numerator: $\frac{2^4}{4!}\frac{1}{1/2} = \frac{16}{24}\cdot 2 = 4/3$ . Denominator: $1 + 2 + 2 + 4/3 + 4/3 = 7.667$ . $C(2,4) = (4/3)/7.667 \approx 0.174$ .

$c = 5$ : $\rho = 2/5$ . Numerator: $\frac{32}{120}\frac{1}{3/5} = \frac{32}{120}\cdot\frac{5}{3} = \frac{4}{9}$ . Denominator: $1+2+2+4/3+2/3+4/9 = 7.444$ . $C(2,5) = (4/9)/7.444 \approx 0.060$ .

So $c = 5$ agents suffice for $C(A,c) < 10\%$ .

Erlang-B and Erlang-C vs. Offered Load

Plot the Erlang-B blocking probability $B(A,c)$ and the Erlang-C waiting probability $C(A,c)$ as functions of offered load $A$ for different numbers of servers $c$ . Compare the two formulas to see the effect of buffering.

Parameters

c

(number of servers)10

A_{\max}

(max offered load)20

Formula to plot

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Comparison of Standard Queueing Models

Property	$M/M/1$	$M/M/c$	$M/M/1/K$	$M/M/c/c$
Servers	1	$c$	1	$c$
Buffer	Infinite	Infinite	Finite ( $K$ )	None (loss)
Stability condition	$\rho < 1$	$\rho = \lambda/(c\mu) < 1$	Always stable	Always stable
Stationary $\pi_n$	$(1-\rho)\rho^n$	Truncated Poisson + geometric tail	$(1-\rho)\rho^n/(1-\rho^{K+1})$	$\dfrac{A^n/n!}{\sum_{k=0}^c A^k/k!}$
Key metric	$L = \rho/(1-\rho)$	Erlang-C $C(A,c)$	Loss $P_K$	Erlang-B $B(A,c)$
Telecom use	Packet buffer	Call center, multi-channel	Finite buffer router	Circuit-switched trunk

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Historical Note: Agner Krarup Erlang: Father of Queueing Theory

early 20th century

Agner Krarup Erlang (1878–1929) was a Danish mathematician and engineer who worked for the Copenhagen Telephone Company. In his seminal 1917 paper, he derived the blocking probability for a telephone exchange with $c$ trunks and Poisson arrivals — the formula now known as Erlang-B. He also analyzed the case with a waiting queue (Erlang-C). The unit of telecommunications traffic, the Erlang, is named in his honor: 1 Erlang corresponds to one continuously occupied server. Erlang's work established queueing theory as a mathematical discipline and remains the foundation for dimensioning cellular networks, VoIP systems, and cloud computing resources over a century later.

Historical Note: Kendall's Notation

20th century

The $A/B/c$ notation for queueing systems was introduced by David George Kendall in 1953. The first letter specifies the arrival process ( $M$ = Markovian/Poisson, $D$ = deterministic, $G$ = general), the second the service distribution, the third the number of servers. Extensions add buffer capacity ( $A/B/c/K$ ), population size, and scheduling discipline. This compact notation became the universal language for queueing models in telecommunications, operations research, and computer science.

🎓CommIT Contribution(2016)

Stochastic Geometry for Heterogeneous Networks

M. Haenggi, J. G. Andrews, G. Caire — Cambridge University Press

The Poisson point process model analyzed in this chapter is the starting point for stochastic geometry analysis of heterogeneous wireless networks. The CommIT group has contributed to applying these tools for modeling interference, coverage, and capacity in HetNets and cell-free architectures. The superposition and thinning properties of the Poisson process (Section 18.2) are essential: thinning models the decomposition of base stations into tiers, while superposition justifies the Poisson assumption when many independent operators are present.

stochastic geometryPoissonHetNets

Common Mistake: Erlang-B vs. Erlang-C: Loss vs. Delay

Mistake:

Using Erlang-B when the system has a buffer (customers wait), or Erlang-C when blocked customers are lost. Students sometimes compute Erlang-B for a call center with a hold queue, getting an optimistically low "blocking" probability.

Correction:

Erlang-B = no buffer, customers lost when all servers busy (circuit-switched model). Erlang-C = infinite buffer, customers wait when all servers busy (call center model). For the same $A$ and $c$ , Erlang-B blocking < Erlang-C waiting probability (since Erlang-B customers leave immediately, freeing capacity faster).

Quick Check

An $M/M/1/K$ queue with $\rho = \lambda/\mu = 1$ (arrival rate = service rate). Does a stationary distribution exist?

Yes — the finite buffer guarantees existence

No — $\rho = 1$ means the queue is unstable

Only if $K$ is even

Correction:

Yes — the finite buffer guarantees existence

With a finite state space $\{0,1,\ldots,K\}$ , the irreducible CTMC always has a unique stationary distribution. When $\rho = 1$ : $\pi_n = 1/(K+1)$ (uniform).

Erlang (unit)

A unit of telecommunications traffic intensity. One Erlang corresponds to one server being continuously busy. An offered load of $A$ Erlangs means that $A$ servers would be needed if there were no queueing or loss.

Related: Traffic intensity

Erlang-B formula

The blocking probability for an $M/M/c/c$ (loss) system: $B(A,c) = (A^c/c!) / \sum_{k=0}^c (A^k/k!)$ . Used to dimension circuit-switched trunk groups.

Related: Erlang (unit)

Erlang-C formula

The probability of waiting in an $M/M/c$ (delay) system: $C(A,c) = \frac{(A^c/c!)(1/(1-\rho))}{\sum_{k=0}^{c-1}A^k/k! + (A^c/c!)(1/(1-\rho))}$ . Used to staff call centers and dimension packet buffers.