Best Practices
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March 13, 2023
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# Demystifying the Math in Erlang-C Traffic Model for Non-Mathematicians Alessandro Cardinali
Founder & Product Development, Soon The Erlang-C traffic model is a powerful mathematical tool used to predict the performance of call centers, customer service centers, and other service-oriented facilities. However, many non-mathematicians find it challenging to understand the math behind the model. In this essay, we will demystify the math in Erlang-C for non-mathematicians by explaining the concepts and providing examples.

## Traffic Intensity

The traffic intensity, denoted as ρ, is the ratio of the arrival rate (λ) to the service rate (μ) of the call center. Mathematically, ρ = λ / μ. The traffic intensity is a critical parameter that determines the behavior of the call center, as it indicates whether the system is underloaded, overloaded, or in a steady state.

For example, suppose a call center receives 50 calls per hour (λ) and can handle 60 calls per hour (μ). The traffic intensity is ρ = 50 / 60 = 0.83. This means that the call center is operating at a high traffic intensity, and there is a high probability of customers waiting in the queue.

## Average Waiting Time

The average waiting time, denoted as W, is the expected time that a customer spends waiting in the queue before being served. The Erlang-C traffic model calculates the average waiting time as W = (ρ^c / c!) * (c / (c * μ - λ)) * (1 - (ρ / c)), where c is the number of servers in the call center.

For example, suppose a call center has 10 servers (c) and receives 50 calls per hour (λ), and each server can handle six calls per hour (μ). The traffic intensity is ρ = 50 / (10 * 6) = 0.83. Using the Erlang-C formula, the average waiting time is W = (0.83^10 / 10!) * (10 / (10 * 6 - 50)) * (1 - (0.83 / 10)) = 4.58 minutes. This means that the average customer will wait for 4.58 minutes in the queue before being served.

## Average Queue Length

The average queue length, denoted as Lq, is the expected number of customers waiting in the queue at any given time. The Erlang-C traffic model calculates the average queue length as Lq = (ρ^(c+1) / c!(c * μ - λ)) * (1 - (ρ / (c + 1))).

For example, using the same call center as before, the traffic intensity is ρ = 0.83. Using the Erlang-C formula, the average queue length is Lq = (0.83^(10+1) / 10!(10 * 6 - 50)) * (1 - (0.83 / (10 + 1))) = 4.42 customers. This means that on average, 4.42 customers are waiting in the queue at any given time.

## Probability of Blocking

The probability of blocking, denoted as P_b, is the probability that a customer is unable to enter the queue due to the queue being full. The Erlang-C traffic model calculates the probability of blocking as P_b = (ρ^c / c!) / (Σ(i=0 to c)(ρ^i / i!)).

For example, suppose a call center has five servers (c) and receives 30 calls per hour (λ), and each server can handle eight calls per hour (μ). The traffic intensity is ρ = 30 / (5 * 8) = 0.75. Using the Erlang-C formula, the probability of blocking is P_b = (0.75^5 / 5!) / (Σ(i=0 to 5)(0.75^i / i!)) = 0.035. This means that there is a 3.5% chance that a customer will not be able to enter the queue due to queue overflow.

## Probability of Abandonment

The probability of abandonment, denoted as P_a, is the probability that a customer who enters the queue will abandon before being served. The Erlang-C traffic model calculates the probability of abandonment as P_a = ((c * ρ)^c / c!) / (Σ(i=0 to c-1)((c * ρ)^i / i!) + ((c * ρ)^c / c!)*(c * μ / (c * μ - λ))).

For example, suppose a call center has four servers (c) and receives 40 calls per hour (λ), and each server can handle 10 calls per hour (μ). The traffic intensity is ρ = 40 / (4 * 10) = 1. Using the Erlang-C formula, the probability of abandonment is P_a = ((4 * 1)^4 / 4!) / (Σ(i=0 to 3)((4 * 1)^i / i!) + ((4 * 1)^4 / 4!)*(4 * 10 / (4 * 10 - 40))) = 0.128. This means that there is a 12.8% chance that a customer who enters the queue will abandon before being served.

## Conclusion

The Erlang-C traffic model can seem daunting to non-mathematicians, but with some explanation, it becomes more accessible. The key concepts in the model include traffic intensity, average waiting time, average queue length, probability of blocking, and probability of abandonment. The calculations for each concept involve simple algebraic expressions and factorials, which can be computed using a calculator or spreadsheet. By understanding these concepts and calculations, non-mathematicians can use the Erlang-C traffic model to make informed decisions about call center performance and customer satisfaction.

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