Queuing Assumptions for Call Centers




There are some pretty strong assumptions that are made when the queuing formulas are used that should be clearly understood. This came to me as I was reading an article from the Wharton Financial Institutions Center about retail banking call centers. (The Wharton Financial Institutions Center is affiliated with the Wharton Business School of the University of Pennsylvania). Even though retail banking phone centers have little to do with the high technology service industry, the basics of call center operations are universal. While the article was long and quite rigorous, I believe the authors overlooked two very basic assumptions used in queuing, which I think leads them to answers that may have little value. I have chosen to write about these two very important assumptions because, I believe, there are many call center managers who take the calculations from their telecommunications systems without fully understanding the underlying assumptions made and that are imbedded in the calculation of the call center statistics.

Clarity on Why Your Call Center Statistics Don't Make Sense!

The reason this article could be very beneficial to you is that it may give you some new insights into why your call center statistics don't make sense. Perhaps you are one of those call center managers who scrupulously review the call center statistics in order to understand and better manage your operation. Unfortunately, you can never seem to relate the call center statistics with your own observation of the call center. It seems that every time you observe the operation everybody seems very busy, yet, the statistics keep telling you that there is plenty of capacity to take on additional call volume. Likewise, it seems that customers are on hold for periods much longer than the statistics would indicate. You think you need additional resources to handle the workload because all your supervisors indicate their people are at their maximum yet your boss points out the average workload is nowhere near maximum. If you are in any of these situations, the following explanation may open your eyes to the non linearity of call center queues and that might impact the queuing assumptions used to compute those call center operations statistics.

The Constant Call Center Arrival Rate Assumption

Most of the standard queuing formulas assume that the arrival of calls into the call center follows a Poisson probability distribution. This is usually a very reasonable assumption since the Poisson probability distribution is often used to characterize the occurrence of independent random events. However, within this assumption of independent random events is the assumption that the rate of these random occurrences is relatively constant. Thus, the queuing formulas would use the average arrival rate of calls into the call center for the time period being evaluated (such as first shift). The problem with this assumption is now obvious; namely, the average rate of arrival of calls into the call center is not relatively constant throughout the time period being evaluated. In fact, the average rate of arrival of calls may vary as much as 100 percent (or more) during the course of an eight hour shift.

The telecommunications software accumulates the actual call center performance and displays generally accurate statistics regarding the phone system. For example, the telecommunications software will generally compute the average wait time by taking the arithmetic average of every call that waited; thus providing an accurate picture of the queues. There is no problem until the accurate picture is blurred by the expanded time horizon of the calculation. While the average wait time calculated from the actual phone system performance is accurate, it loses its meaning because the waiting time is very sensitive to the arrival rate of calls into the queue.

Consider a simple example of a call center with 25 personnel taking customer calls. Further consider that each person can complete 12 calls per hour. Further consider a phone system that has 10 extra lines for people on hold. Thus, the phone system has 35 lines; 25 for personnel and 10 for customers waiting. The queuing formulas would yield the following statistics for call arrival rates of 250, 260 and 275 calls per hour.

Average Arrival Rate 250/hour 260/hour 275/hour
Average Waiting Time 12 seconds 17 seconds 25 seconds
Number of Customers Per Hour Who Balk 2/hour 3.5/hour 7/hour
Percent Who Have to Wait 26 percent 33 percent 46 percent
Percent Utilization of the Call Center 82.7 percent 85.5 percent 89.3 percent


Some obvious points of interest can be seen in the call center statistics. For example, if the arrival rate increases from 250 to 275 per hour (a 10 percent change), the average waiting time doubles. An increase in the arrival rate of less than 5 percent yields an increase in average waiting time of about 50 percent. Furthermore, the number of customers who balk will more than triple when the arrival rate increases by 10 percent.

The Call Center Steady State Assumption

The second assumption imbedded in the queuing formulas that compute the various queue statistics is that the queue is in a steady state condition. The queuing formulas all rely on the queue not being in any kind of a transition mode such as start up, shut down or dramatic change from arrival rate to another. An easily understood example is that of a branch banking operation. The branch opens in the morning and there is the morning rush when local businesses come in to get the cash to start their day. Following the morning rush there is a period of relatively low teller activity until the noon hour when local workers come in to do their personal banking. After the noon rush there is another relatively slow period until late afternoon when local businesses close and come in to make their nightly deposits. There will be an average arrival rate at the teller windows for the day but the operation of the queue of customers at the teller windows will not appear anything like the calculations from the queuing equations because the call center performance measures, such as average time in the queue, average time in the system, average number in the queue, etc., are based on the queuing system being in a steady state condition.

A steady state queuing system is one in which the average rate of arrivals is relatively constant during the period of analysis (even though the occurrence of a specific arrival is random). Thus, call centers, like branch banking teller operations, probably never achieve a steady state condition and hence never reach a point where the queuing formulas will accurately represent the call center operation.

Resolving the Dilemma

It seems clear that constant rate of arrivals of calls is unlikely and that steady state cannot be achieved over an entire shift. That does not mean that the call center operation cannot be analyzed. It just means the analysis has to accommodate the assumptions of the queuing models being used. For example, the constant arrival rate may occur for period within a given time period (shift). Just as the branch bank has obvious peak and slow times, a call center operation may also have peak and slow times.

During these times, the arrival rate may be relatively constant so that the call center queuing formulas will provide representative results for these shorter periods and accurately reflect the customer's perception of the call center queuing operation. The intermediate step required of the call center manager in order to understand the actual utilization of his resources is to determine the magnitude and duration of the peaks and slow times.

With regard to the condition for steady state in order to apply the queuing formulas, the answer is similar to the previous paragraph. While steady state may never be achieved for the call center operation, it may be very close to steady state once the system has started up. While the call center operation may move to various steady state conditions throughout the day as call volumes rise and fall, each steady state condition should provide a basis for the application of the queuing equations and the accumulation of the queuing statistics. Of course, as the system moves toward shutdown, steady state will no longer be viable.

The Quick Answer

Don't believe the call center queuing statistics if they are computed over a time period that has varying call center arrival rates. In order to accurately measure the queue performance of the call center, get the queuing statistics for each period during which the arrival rate is relatively constant. You can combine these statistics to get an overall picture but don't allow the telecommunications calculations to cover periods with arrival rates that are very different. The call center is not the place to go for a leisurely time. There always appears to be a rise and fall in the activity. Every one working in the call center knows that as soon as a crisis is over it is just a matter of time before the next crisis occurs. The call center is not a steady state environment so don't use steady state equations to explain it.

First published on Call Center IQ.