## Queues and Capacity Calculations in Telecom Systems

How to know how many agents are necessary for a contact center serving given number of calls per time frame?

In one of the previous posts I have discussed a simple formula which helps to calculate number of ports required for a given number of minutes passing through the system in a time period (a day, hour etc). The calculation was based of very simplified assumption of averages, i.e. that in an average one minute of call fills one channel (or port) of the system for a one minute. From there you can easily multiply and see that, for example 30 ports can carry 30 minutes every minute or 1800 minues every hour (30ports x 60 minutes in hour) and so on. I have even published a simple calculator to quickly do it when needed ( I use it myself a lot).

However, this model is very simplified, and assumes that voice calls always arriving at the exact time, i.e. whenever the previous one has just ended. The reality is far more complex: while you can always have an average call duration and average call arrival rate, the the variation of real values quickly defeats the simplified approach as there will be always calls which take longer than average to complete and arrival times can be shorter than the average.

To work out the solution, one can take Queueing Theory at help. It was first formulated in very beginning of 20th century and for the exact purpose: calculating loads in the telephone exchange. Back then it required physical circuits to interconnect callers, and physical operators who would manually interconnect them. Nowadays it is still important, as it helps to calculate number channels in trunking and switching systems, and number of operators in contact centers.

Let's just take one simple example: an operator in the contact center, serving incoming calls. Let's say it takes on average 5 minutes to serve a customer, and calls arrive at slightly larger intervals on average: 5 minutes and 15 seconds. The average should be fine for one operator: There is 15 seconds guard time between each call of 5 minutes.

However, queue theory shows a different picture: the expected waiting time for one operator queue is calculated by formula:

t = λ / (μ * (μ – λ))

where μ is service rate and λ is arrival rate. Lets fill the numbers in: arrival rate λ= 60/5 =12 calls per hour and service rate μ = 60/5.25 = 11.43 calls per hour. Put the numbers in the formula and we get waiting time t = 1.666, i.e. 1 hour and 40 minutes! The number is suprisingly high: the queue would grow so long and stabilize around that number if all callers would have patience to wait so long. Adding a second operator would drop waiting times almost to few minutes (against intuition) but this is more complex to calculate and I will touch it in some of the later posts.

To summarize, the Queueing theory is important tool when building telecom systems, including contact centers. It allows to correctly plan for resources and avoid congestions when underestimating system capacity.