Erlang B & Erlang C Voice Trunk & Queuing Capacity Calculator

1. The Foundations of Teletraffic Engineering: Agner Krarup Erlang

In the early 20th century, Danish mathematician and engineer Agner Krarup Erlang (1878–1929), working for the Copenhagen Telephone Company, published groundbreaking papers laying the mathematical foundations of teletraffic engineering and modern queuing theory. Erlang sought to solve a fundamental economic challenge: how many physical telephone lines or telephone switchboard operators are required to provide an acceptable standard of telephone service without incurring unnecessary capital expenditure?

The Erlang Unit (E): Standardized by the International Telegraph and Telephone Consultative Committee (CCITT, now ITU-T) in 1946, the Erlang is a dimensionless unit of traffic intensity. One Erlang represents the continuous, uninterrupted occupation of a single transmission channel or service resource over an observation interval of one hour (3,600 seconds of cumulative usage). Mathematically:

Where:

• A: Traffic intensity in Erlangs.
• λ: Call arrival rate (calls per unit time).
• h: Average holding time (call duration) in seconds.
• T: Observation period in seconds (typically the 3,600-second Busy Hour).

For example, if a company initiates 600 calls during the peak hour with an average holding time of 180 seconds (3 minutes), the total traffic volume is (600 × 180) / 3600 = 30.0 Erlangs. Sizing a network for 30 Erlangs means 30 voice circuits would be 100% occupied continuously during that hour.

The Time Consistent Busy Hour (TCBH): Telecommunications circuits cannot be sized for absolute instantaneous peak loads (which would result in massive idle infrastructure costs for 99% of the day), nor can they be sized for 24-hour daily averages (which would lead to severe trunk exhaustion and call drops during business hours). ITU-T Recommendations Q.543 and E.490 mandate sizing telecommunications infrastructure according to the Time Consistent Busy Hour—the continuous 60-minute period experiencing the highest average traffic demand across four consecutive weeks.

2. Erlang B: The M/M/m/m Loss Model (Blocked Calls Cleared)

The Erlang B model, represented in Kendall's queuing notation as M/M/m/m (Poisson arrivals, exponential service times, m parallel servers, system capacity of m), represents a loss system with zero queuing capability.

Core Assumptions of Erlang B:

1. Call arrival events follow a memoryless Poisson process with average arrival rate λ.
2. Call durations (holding times) are exponentially distributed with mean h = 1/μ.
3. There are m identical, parallel channels or trunks available.
4. Blocked Calls Cleared: If an incoming call arrives when all m channels are occupied, the call is blocked immediately, receives a fast-busy tone, and is discarded from the system without queue or retry.
5. The source population is infinitely large (traffic intensity does not diminish as trunks fill).

The probability that all m channels are occupied—defining the Grade of Service (GoS) or blocking probability B(m, A)—is expressed by the classical closed-form equation:

The Numerical Overflow Crisis & The Recursive Recurrence: In modern telecommunications networks with thousands of concurrent SIP trunks, direct computer evaluation of A^m and m! is computationally impossible. In IEEE 754 64-bit double-precision floating-point arithmetic, 171! overflows to infinity (∞). Attempting to calculate factorials directly crashes algorithms or outputs NaN.

To circumvent factorial overflow, telecommunications software implements the mathematically rigorous, numerically stable recursive recurrence relation:

Because each successive term B(k, A) strictly satisfies 0 ≤ B(k, A) ≤ 1, this algorithm operates without intermediate exponential expansion, providing 100% numerical stability for tens of thousands of channels.

3. Erlang C: The M/M/m Queuing Model (Blocked Calls Delayed)

In customer contact centers, technical help desks, and emergency dispatch centers, calls that arrive when all agents are busy are not disconnected; instead, callers hold in an automated FIFO queue listening to music and status announcements until an agent becomes available. This is modeled by Erlang C, known in queuing theory as the M/M/m model.

The Queuing Stability Criterion: For an Erlang C queue to remain in steady-state equilibrium, the traffic load must strictly be less than the number of agents:

If A ≥ m, the customer arrival rate exceeds the collective servicing rate of the agent pool. In a pure Erlang C model (with infinite caller patience), the queue length and wait time expand to infinity.

Deriving Erlang C Directly from Erlang B: Rather than computing complex infinite series summations, Erlang C probability of delay P_c = P(Wait > 0) can be derived directly and stably from the Erlang B result:

Key Contact Center Performance Metrics:

• Average Speed of Answer (ASA): The average time all incoming callers spend waiting in queue before reaching an agent:
  ASA = (Pc × h) / (m - A)
• Average Wait Time of Delayed Callers (W_d): The expected wait time experienced exclusively by callers who were actually placed in queue:
  Wd = h / (m - A)
• Service Level Agreement (SLA): The probability that a caller will be answered within a predefined target window of t seconds (e.g., the standard "80/20 Rule", where 80% of calls are answered within 20 seconds):
  P(Wait ≤ t) = 1 - Pc × e-(m - A) × t / h

4. SIP Trunk Dimensioning & PBX Over-Subscription Ratios

In legacy Time-Division Multiplexing (TDM) telephony, voice channels were tied to rigid physical copper or optical spans:

• North American T1 PRI: 24 channels (23 64-kbps B-channels for voice + 1 D-channel for Q.931 signaling) operating at 1.544 Mbps.
• European / International E1 PRI: 32 channels (30 64-kbps B-channels for voice + 1 framing channel TS0 + 1 D-channel TS16) operating at 2.048 Mbps.

In modern enterprise voice architectures, physical T1/E1 lines have been superseded by Session Initiation Protocol (SIP) Trunks running over IP/MPLS or internet connections. A SIP trunk channel represents a logical concurrent call session license negotiated via RFC 3261 INVITE transactions and carried via Real-time Transport Protocol (RTP).

Over-Subscription Ratios: Because employees do not make external PSTN calls simultaneously, enterprise telecom architects over-subscribe telephone extensions to external SIP trunk channels:

• 1:1 Ratio (Dedicated): Contact center agents, telesales queues, and financial trading desks where phone utilization is near 100%.
• 3:1 to 4:1 Ratio (Enterprise Standard): Standard corporate headquarters. A campus with 300 desktop extensions typically requires 75 to 100 concurrent SIP channels to guarantee an Erlang B blocking probability below 1% (P.01).
• 5:1 to 8:1 Ratio (Knowledge Workers): Organizations with high internal collaboration (Slack, Teams) and moderate outbound calling.
• 10:1 Ratio (Light Office / Manufacturing): Warehouses, schools, and manufacturing plants with low external calling volume.