Erlang C Calculator - Call Center Agent Sizing | Federico Calò
Calculate the optimal number of agents for your contact center using Erlang C formula. Enter daily call volume, average handling time (AHT), and service level target - immediate result, safe for large volumes.
The model Erlang C is the international standard for sizing contact center telephones. Developed by Danish engineer A.K. Erlang in 1917 for the Copenhagen Telephone Administration, it models the behavior of M/M/N queues (arrivals Poisson, exponential service times, N agents, infinite queue). Given the current volume of incoming calls, the average handle time (AHT) and target service level (e.g., "80% responses within 20 seconds"), the calculator returns the minimum number of agents required, utilization index ρ, and a sensitivity table to optimize the trade-off between cost and quality.
Implementation uses iterative Erlang B→C derivation (Jagerman formula, 1974), avoiding factorial calculation for N > 170 agents - a source of numerical overflow. The algorithm is numerically stable for any real contact center size (up to 1,000 agents). The tool has been validated on the reference Call Centre Helper: 100 calls/hour, AHT=300s, SL 80%/20s → 12-14 agents.
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Practical example
Scenario: contact center retail con 100 chiamate/ora nell'ora di punta, AHT = 300 secondi (5 min), obiettivo SL 80/20 (80% risposte in 20 s).
- Traffic intensity: a = (100 / 3600) × 300 = 8,33 Erlang
- Minimum agents for SL 80%/20s: 12-14 agents (reference Call Centre Helper)
- With 12 agents: ρ ≈ 69%, SL ≈ 80%
- With 14 agents: ρ ≈ 60%, SL ≈ 96% (hit buffer)
Erlang C Dictionary
- Erlang (E)
- Unit of measurement for telephone traffic. 1 Erlang = 1 continuously occupied channel per hour. Formula: a = (calls/hour) × AHT / 3600.
- Average Handle Time (AHT)
- Average call duration, including conversation time and post-call work (ACW), measured in seconds on the ACD or CRM system.
- Service Level (SL)
- Call response rate within the time target threshold. Example: SL 80/20 = 80% response in 20 seconds. Common industry standard for phone-based contact centers.
- Use agents
- Time fraction in which each agent is engaged: ρ = a / N. For ρ ≥ 1, the queue grows indefinitely (unstable system). Practical values: 65-80% for good balance between cost and quality.
- Erlang C Distribution - C(N, a)
- Probability that a call finds all N agents busy and must wait in line. Calculated using iterative Erlang B→C derivation (no factorial, numerically stable for any N).
Frequently Asked Questions
What is the Erlang C formula?
Erlang C is a mathematical model (M/M/N with infinite queue) that calculates how many agents are needed to ensure a certain level of service in a contact center. It considers the volume of incoming calls, average call duration, and response target. Unlike Erlang B, it assumes customers wait indefinitely in line without abandoning.
What is the service level of 80/20?
Service level 80/20 means that 80% of calls must be answered within 20 seconds, which is the most common standard in phone-based contact centers. Alternative standards include 90/20 (banking sector), 95/30 and 70/30. The Erlang C index returns the number of agents needed for each target.
What is the traffic intensity in Erlang?
Traffic unit (E symbol) is the unit of measurement for telephone traffic. It's calculated as: a = (calls/hour ÷ 3600) × AHT. Example: 100 calls/hour with AHT=300s → a = 8.33 Erlang. Indicates the average number of agents occupied simultaneously during peak hours.
What is the usage rate of agents (ρ)?
Rate of utilization ρ = a / N indicates the percentage of time each agent is occupied. With ρ = 0.69 (69%), each agent is engaged for 69% of the shift. Values above 0.85 indicate risk of increasing code; Erlang C assumes stable system only for ρ < 1.
Does Erlang C model work for real contact centers?
Erlang C is a theoretical model (M/M/N): assumes Poisson arrivals, exponential AHT and zero abandonments. In real contact centers, a portion of callers abandon the queue, so actual service level is often better than calculated. Erlang C provides a conservative estimate (upper bound) of the number of agents needed.
How is sensitivity table used?
Table shows service level and usage for N_min-2 to N_min+4 agents. Allows evaluating trade-off between cost (more agents) and quality (higher SL). Adding 1-2 more agents beyond minimum reduces usage and improves resilience to spikes.
How is it used?
- Insert call volume
Indicate the average number of calls arriving per hour (e.g. 100 calls/hour). Use the peak hour value for prudent scaling.
- Define average duration
Average Handle Time includes conversation time and post-call work. Measure from your CRM or ACD system. Example: 300 seconds = 5 minutes.
- Set Service Goal
Standard Service Level is "80% of calls responded to within 20 seconds" (SL 80/20). Enter desired target: level (%) and response time threshold (seconds).
- Interpret the result
The calculator returns the minimum number of agents, utilization rate (ρ), and sensitivity table. An utilization rate > 85% indicates risk of prolonged code: add at least 1-2 agent buffers.