Erlang

Owned by Luna Gil (Unlicensed)
Last updated: Dec 27, 2021
7 min read

Simulation

It helps us carry out Staffing, that is, to know how many agents I need in my Contact Center to answer calls. It is also helpful to carry out ForeCasting estimating the growing volumes of the Contact Center to know how many agents I will need to reach the Service level metrics or Desired abandonment.

Erlang

In statistics, the Erlang distribution is a continuous probability distribution with two parameters and whose density function for values is:

The Erlang distribution is equivalent to the gamma distribution with the parameter y . For that is the exponential distribution. The Erlang distribution is used to describe the wait time until the number in a Poisson process. 

Properties: 

  • Its hope is given by

  • Its variance is given by:

  • The moment-generating function responds to the expression:

Erlang C

It assumes that the customers have infinite patience, that is, that there are no dropouts. If there are dropouts, we will use Erlang A. It evaluates the service level, wait probability, and average waiting time in a hold system. Considering the classic model M/M/c, Poisson process, exponential without dropouts call time. With infinite or fixed queue capacity.

 λ number of calls 1/μ  of average wait time, then the load is ρ=λ/μ.



Performance measurements

  • Service Level (SL):  Amount of customers that wait equal to or less than the acceptable average wait time. t               
    SL(y,t )=𝔼[X (y,t )]/𝔼[A] 

  • Where y is the number of agents X(y,t) is the number of customers that is the number of customers who waited for at least t as function of y and t, A is the total number of customers and 𝔼 the statistical expected value.

  • Waiting Probability: Probability that a new customer enters the queue to wait.

  • Standard customer wait time.

 

Parameters

  1. Amount of calls: the average amount of calls by timestamp. For example, 5.7 customers per minute. 

  2. Average talking time: average call time. For example, 2.5 minutes per customer. 

  3. Expected service level (Maximum desired wait time): it is the value of the service level. For example, 20 seconds.

  4. Queue size: Maximum queue size. When the queue is full, customers are not able to log in. -1 for infinite. 

  5. Unity to deploy the average wait time: it is for the outcome of the results. 

  6. Agents range: it allows evaluation measurements for multiple agents. For example, five and it is evaluated for ten agents, then we will have data from about 5 to 15 agents. 

  7. Evaluate the number of agents: for a specific number of agents. 

  8. Minimum number of agents for service level % searches for the minimum number of agents required to obtain a Service level (SL) bigger or equal to the selected target. For example, 80 and the acceptable wait time is 20 seconds, search the minimum number of agents needed to, at least, have an 80% service level using 20 seconds as maximum wait time. 

Example:

Graphics

1) Service level and waiting probability  

The graph shows the probability that a customer has to enter the waiting queue, as well as the behavior of the Service Level for these amounts of agents.

 

2) Wait

The graph shows the customers’ average waiting, including dropouts and answers, for that number of agents. The time unit is selected before generating the results.  

 

Erlang A

Erlang C extension adds queue dropouts. The customer's wait is molded as an exponential distribution.  

It evaluates the service level, waiting probability, dropout rate, and average wait time of a queue system. The difference with C is that it includes dropouts. Each customer has patience with exponential random distribution. It is a stable queue system because of dropouts. 

The formula used for the calculation is approximation, SLD based on diffusion equations, it is used from the paper by Garnett, Mandelbaum, and Reiman (2002), the queue is infinite in capacity.

Performance measurements

  • Service Level(SL): the ratio of customers who waited t or less to be served:


  •  

    • SL1: Customers who dropped out but waited at most t are excluded from the calculation (used in the campaign dashboard) SL1(y,t )=𝔼[X (y,t )] / 𝔼[A−G(y,t )]  

    • SL2: Customers that dropped out but waited at most t are counted as OK. SL2(y,t)=E[X(y,t)+G(y,t)] / E[A].

    • SLD: Approximation based on Diffusion equations. Formula from the paper by Garnett, Mandelbaum and Reiman (2002).

SL1 ≤ SL2.

For high traffic queues, use SLD.

  • Wait probability: the probability of a new Probability that a new customer will wait in line. 

  • Abandonment rate: Proportion of customers that hang up without having a response.

  • Wait: Customers’ waiting time (including dropouts and answers).



Parameters

  • Amount of calls: the average amount of calls by timestamp. For example, 5.7 customers per minute. 

  • Average talking time: the average call time. For example, 2.5 minutes per customer. 

  • Average wait time: the average time a customer is open to wait. For example, 2.3 minutes per customer. 

  • Expected service level (Maximum desired wait time): it is the value of the service level. For example, 20 seconds.

  • Queue size: Maximum queue size. When the queue is full, customers are not able to log in. -1 for infinite. 

  • Unity to deploy the average wait time: it is for the outcome of the results. 

  • Agents range: it allows evaluation measurements for multiple agents. For example, five and it is evaluated for ten agents, then we will have data from about 5 to 15 agents. 

  • Evaluate the number of agents: for a specific number of agents. 

  • Minimum number of agents for service level % searches for the minimum number of agents required to obtain a Service level (SL) bigger or equal to the selected target. For example, 80 and the acceptable wait time is 20 seconds, search the minimum number of agents needed to, at least, have an 80% service level using 20 seconds as maximum wait time. 

  • Service level type: SL1, SL2, SLD.

Example:

Graphics

1) Service level and wait probability 

The graph shows the probability that a customer has to enter the waiting queue, as well as the behavior of the Service Level for these amounts of agents.

2) Abandonment rate

The proportion of customers that hang up without having a response for that number of agents.

3) Wait

The graph shows the customers’ average waiting, including dropouts and answers, for that number of agents. The time unit is selected before generating the results.  

 

Erlang B

A calculation without queue evaluates rejection, does not enter if all agents are busy. It gives us the probability that a new customer will not enter if the agents are busy when the call arrives.

A classic model of M/M/c queues with Poisson input model, exponential average spoke time without queuing capacity (queues with maximum = number of agents).



Dropout probability 

It is defined as the probability that all agents are busy when a new customer arrives.

Parameters

  1. Maximum abandonment rate: Finds the minimum number of agents required to obtain a probability of abandonment of Y or less, where B (y) is the probability of abandonment as a function of agents y and T the maximum threshold. Y = min {y: B (y) ≤T, y∈ℕ}. Example, 5 to find the probability of abandonment of a maximum of 5% of Abandonment

  2. Amount of calls: the average amount of calls by timestamp. For example, 5.7 customers per minute. 

  3. Average talking time: average call time. For example, 2.5 minutes per customer. 

  4. Average wait time: average time a customer is open to wait. For example, 2.3 minutes per customer. 

  5. Agents range: it allows evaluation measurements for multiple agents. For example, five and it is evaluated for ten agents, then we will have data from about 5 to 15 agents. 

Example:

Graphics

1) Dropout probability 

It shows the probability that all agents are busy when a new customer arrives.