How then can the actual
arrival rate of calls in your call center be predicted? While we cannot predict
the exact time that each call will arrive, we can predict the probability
of when the next call will arrive. This probability follows a well known
distribution called the Poisson distribution.
Figure 1: Poisson Distribution of Call
Center Call Arrivals
It's a bell-shaped
distribution skewed out to the right. It says that at any given moment, the
probability of what the time will be to the next call arrival is
"scrunched up" in front of the average time to the next call arrival,
and tapers off slowly thereafter. The "hump" of the curve is before
the average time to the next call. If the hump was right at the average time to
the next call arrival (a normal distribution), then call center staff planning
would be far easier than it is.
So what does this mean in
practice? The mathematics behind the Poisson distribution tell us that call
arrivals will always tend to be clumped together and will not arrive in an even
manner. This is the reason why a graph of observed data for % Agent
Utilization has a gentle slope, but graphs for Average Speed to Answer and
Average Time in Queue have very steep slopes.
Figure 2: Example Call Center Measurements
Thus, the relationship
between how busy your agents are and the service your callers get is not a
simple linear relationship. In other words, you can’t gauge the level of service your callers are
getting be peeking over the cubicles and checking out how busy your agents are
over some given time period. And, there are break off points where the
addition or subtraction of one or two agents will result in dramatic
differences in customer service level as evidenced in the average speed to
answer and average queue time in queue graphs above. This runs counter to our
common intuition … but it is true.
The lengths of calls
are not uniform either. Call lengths nearly always closely follow what is
called the exponential distribution:
Figure 3: Exponential Distribution of Call
Center Call Lengths
This distribution says that
the most probable call lengths are those that are less than the average call
length, but that there are going to be some that are a whole lot longer. This
phenomenon also contributes to a "clumping effect": the length of
calls in progress at any given time may all be rather short or all be rather
long, but will very seldom be close to the average length of all the calls
experienced in a day. To complicate things even further, the average time that
callers spend in queue is also exponentially distributed around the average
queue time primarily because a large percentage of callers hang up very quickly
when they get the "please hold" message.
The Two Solutions
Using the assumptions of
the above distributions for call arrivals and call length, and
the customer/server/queuing nature of an inbound call center, there are
two mathematical methods that can be used to optimize a call center’s
method that calculates a statistical equation to predict queuing times,
agent workloads, and optimal agent levels, or
method that imitates the flow of calls into a call center and their
handling by agents, with queuing, blocking, and overflow.
The "Formula" Solution
The most common method used
for call center staffing is formula using the famous Erlang equations
(Erlang C and Erlang B) for staffing and trunking. They compute the
probability of delay in queue for calls when using a given agent level with a
given amount of inbound call traffic.
The software in the various
call center staffing and scheduling products that calculate the Erlang C
formula calculate it for several agent levels and then pick out and tell the
user the one agent level that gives a delay probability closest to that of
their desired service level. These products will do this for each hour, half
hour, or quarter hour time period throughout a workday.
So, in the formula method,
the user inputs number of calls, talk times, and wrap-up times for each period
of time in the workday, along with a desired service level (e.g., answer 80% of
calls within 20 seconds). The software calculates the Erlang equation, and pops
out optimum agent levels for all time periods in the day. It can also be used
to observe the effects on queuing and agent workload when using different agent
levels in any one time period.
Modified versions of these
formulas, with the ability to specify desired queue levels, are used in Portage
Communications’ Call Center Designer™ Day Planner, Staffing, and Trunking
software modules. In fact, most all of the staffing and scheduling software for
call centers use the Erlang formulas or some minor modification of them.
The Erlang analytical
method has done a good job over the years in helping call center managers
determine what agent levels they should use. However, it does not account for
some of the unique dynamics of inbound call centers using an ACD, and it tends
to over-staff many call centers to some degree.
The most talked about
shortcoming of the standard Erlang C formula is that it assumes calls will queue
up infinitely and wait forever to be answered by an agent. For centers with
limited inbound line capacity and callers with little queuing tolerance, the
Erlang method may not be accurate enough. Also, what about caller abandonment
(hanging up after being queued), or callers retrying after abandoning or
getting a busy, or the ACD overflowing calls to another agent group? And what
about that ACD ring delay feature (the number of rings the caller can be given
before the ACD picks up the line)?
Some companies and
mathematicians have derived "special versions" of the Erlang method
that claim to at least partially account for some of these issues especially
the infinite queuing problem. However, no tweaking of the equation will fully
take into account any of the issues mentioned above, much less all of them.
Some of the variants out there to the Erlang C equation are highly suspect,
having no published analysis of their mathematics, and may actually give less
reliable staffing predictions.
If a call center manager
finds that in practice Erlang C is not allowing them to maintain their desired
service levels to the degree they want, or that it calls for many more agents
than are actually needed, then they should probably just skip all the Erlang
variants and analytical equation methods and try the second method: simulation
The Simulation Solution
So what is computer
simulation? Most people have heard of weather and climate forecasts made by
computer simulations, or maybe they have played computer games like SimCity.
Industrial and natural processes may often be simulated by a computer program.
The program imitates the flow of people, material, or events and is used
to experiment and see what the effects that alternative configurations of the
people, material or events would have on a real-world situation. In the past
few years simulation programs have become an increasingly popular method for
call center staffing and performance prediction.
A call center or ACD
simulator compresses a day’s worth of incoming call traffic into a short
period of time by representing each second of a day with a few microseconds of
computer time. Calls are made to randomly arrive in a "virtual" call
center created in the computer’s memory and are answered by agents, queued, met
with busy signals, overflowed, or abandoned depending on the parameters,
agent/trunk combinations, and call volumes defined for the call center.
The simulator doesn’t just
calculate equations in the way that a formula does, rather, it actually acts as
an experimental call center that reacts to the random arrival of inbound calls
with the agent and inbound line levels, inbound call volumes, caller
abandonment and retry behavior, and ACD overflow the call center manager wants
to test and try out. Simulation can then accurately predict the levels and
effects on service level of call blockage, abandonment, caller retries and
overflow when these are entered as parameters to the simulation. Also, a
simulation program can make the random call arrivals and call lengths follow
the Poisson and exponential distributions that are seen in the real world.
At the end of the
simulation, you will see how many calls were handled, abandoned, given a busy,
queued, and overflowed along with queue lengths, average speed to answer, and
the actual service levels that would be given to your callers. More simulations
with different combinations of agents and trunks are usually performed until
the manager is satisfied with the service levels, agent work load, and call
A good way to state the
difference between the Erlang formula method and the simulation method is this:
with the formula method, you enter your call volumes and what you want
to happen in the form of the desired service level you want to maintain, and
you are told what number of agents and lines you need. With the simulation
method you enter the number of agents and lines you want to use along with your
call volumes, and you see what will happen in the form of service levels
One drawback of simulation
is that it can take longer to determine your optimal agent levels. The Erlang
formula’s output is an optimal agent level: you only need to run it once. With
simulation you have to test and try out different agent levels and see what
happens. Are the resulting queue levels acceptable? Are the agents busy enough?
Is the service level sufficient? No? … then adjust the agent and line levels
and run another simulation, and another, and maybe another. This extra work
pays back in what will be more accurate and practical predictions of optimal
agent levels and call center performance. Also you can see values for calls
queued and overflowed, the number of callers who abandon or get busies, and the
calls that retry later.
makes SimACD an automatic call distributor simulator that takes into account
such dynamics as ring delay, call abandonment and retry behavior, and overflow.
It simulates a full workday by hour, half hour, or quarter hour, and displays
both numeric and graphic predictions for such items as calls, handled, queued,
overflowed, busy, retried and more. It also gives average speed to answer
times, queue times, and percentages of calls answered in less than ten, twenty,
thirty, and sixty second intervals. It is the call center industry’s first
affordable simulator designed for smaller call centers or agent groups.
What Method Is Best For Your Call Center?
If you require staffing
predictions to do simple scheduling and just need reasonably accurate levels to
on which to base your agent schedules, then staying with Erlang C or an Erlang
variant will probably fit your needs. Practically all of the existing workforce
management systems use an Erlang formula for calculating the required number of
agents, and these requirements are then fed into a scheduling algorithm and
agent database to create work schedules. Erlang C also works well with most of
the home grown spreadsheet based scheduling systems that many call centers have
Problems occur if your call
center is more complex. If you have multiple agent groups, skills based routing
of calls, overflow of calls from one group to another, or auto attendant and
IVR systems that handle some of the calls, then simulation becomes the better
option since these dynamics cannot be represented in a mathematical formula.
However, at present, there are hardly any simulation software products that
handle these and are combined with a scheduling module in the same package.
Many companies are working on one (including Portage Communications), so look
for them in the near future.
Most call center managers
use Erlang and simulation staffing methods for call center performance analysis
in addition to agent scheduling. They want to see what they’re predicted queue
levels and agent work load will be at varying agent levels. Analysis of call
center performance is especially helpful in taking a close look at your call
center’s busiest hours, least busy hours, and other problem periods.
For performance analysis,
you don’t want an Erlang formula tool that just spits out a required numbers of
agents based on a given service level goal. You want a tool that will also tell
you what your average speed to answer, percent queued, average queue length,
agent occupancy will be. These and other performance measures are important to
study in order to determine what your service level goal should be in the first
Simulation software will
also give these measures for call center analysis and is more accurate. Also,
as already stated, simulation can take into account caller abandonment, retry
behavior, ACD ring delay, and overflow. On the minus side, call center
simulators can be very expensive primarily because this type of software
is difficult to write, and the market for them is still very new.
Determining the optimum
number of agents and the effects of different agent levels is the key to a
smoothly running call center and happy callers. The dynamics of modern call
centers require managers to explore all the new formula and simulation software
to meet this challenge.
Stuart Harris is the President of Portage Communications,
Inc. He has worked in the telecommunications and call center industries for
twenty years and holds a BA in Mathematics and Computer Science from the
University of Colorado, and an MS in Software Engineering from Seattle