Properties of the Erlang formula

Knowing the Erlang formula is one thing, understanding it is another. The Erlang formula
has a number of properties with important managerial consequences. We will discuss these
in this section. Robustness One agent more or less can make a big difference in SL, even for big call centers. This is good news for call centers with a moderate SL: with a relatively limited
effort the SL can be increased to an acceptable level. On the other hand it means that
a somewhat higher load, necessitating an additional agent, can deteriorate the SL considerably.
In general we can say that the Erlang formula is very sensitive to small changes
in the input parameters, which are _, _ en s. This is especially the case if a is close to
s, as we can see in Figures 4.1 and 4.2. The figures get steeper when _ approaches s/_,
and thus small changes in the value of the horizontal axis give big changes at the vertical
axis. This sensitivity can make the task of a call center manager a very hard one: small
unpredictable changes in arrival rate or unanticipated absence of a few agents can ruin the
SL. In Chapter 7 we discuss in detail the consequences of this sensitivity.
In our small call center with _ = 1, _ and s = 8 we expect an ASA of around 17 seconds.
However, there are 10% more arrivals (i.e., _ = 1.1). The ASA almost doubles to over 30
seconds!
Stretching time A second property is related to the absolute and relative values of the
call characteristics, i.e., _ and _. Recall that the load is defined by a = _ × _. If either
_ or _ is doubled, and the other is divided by two, then the load remains the same. This
does not mean that the same number of agents is needed to obtain a certain service level.
A manager is working in a call center that merely connects calls, thus call durations are
short. As a rule she uses a load to agent ratio of 80%. From experience with the call center
she knows that this gives a reasonable service level. For parameters equal to _ = 32 seconds
and 15 calls per minute the load is a = 8 Erlang. Indeed, with 10 agents the average speed
of answer is 6.5 seconds. After a promotion she is responsible for a telephone help desk
with also a load of 8 Erlang, but with _ approximately 5 minutes, more than nine times as
much. She uses the same rule of thumb, to find out that the average waiting is now around
60 seconds! When _ is multiplied by the same number (bigger than 1) as _ is divided with, then
the load remains the same but it is like the system goes slower. Evidently, the waiting
time also increases. If AWT is multiplied by the same number then the TSF remains the
same. The relationship between the ASA and stretching time is more complicated.
It is like saying that the load is insensitive to the ”stretching” of time. Certain performance
measures depend only on s and a, but not on the separate values of _ and _.
The probability of delay, C(s, a), is a good example. It does not hold anymore for the
TSF, here the actual value of _ and _ play an important role. In fact, for given a and
the service level depends only on AWT/_. Thus if time is stretched, and the acceptable
waiting time is stretched with it, then the TSF remains the same. Of course, this is just
theory, although we often see that the AWT is higher in call centers with long talk times
compared to call centers with short talk times. For the ASA the effect of stretching time
is simple: the ASA is stretched by the same factor. Let us go back to the call center with _ = 1, _ = 5, and s = 8. Then TSF = 86% for AWT = 20 seconds, and ASA = 16.7 seconds. Now stretch time by a factor 2, i.e., _ = 0.5 and _ = 10. Then TSF = 83% for AWT = 20 seconds (a difference, but surprisingly small; the reason of this is explained below), TSF = 86% for AWT = 2 × 20 = 40 seconds, and
ASA = 2 × 16.7 = 33.4 seconds.
Economies of scale Another well known property is that big call centers work more
efficiently. This is the effect of the economies of scale: if we double s, then we can increase
_ to more than twice its value while keeping the same service level, assuming that _ and
AWT remain constant. A firm has two small decentralized call centers, each with the same parameters: _ = 1 and _ = 5 minutes. With 8 agents the average waiting time is approximately 17 seconds in each call center. If we join these call centers “virtually”, then we have a single call
center with _ = 2 and 16 agents. The average waiting time is now less than 3 seconds, and
employing only 14 agents gives a waiting time of only 13 seconds. An additional advantage
is that there is more flexibility in the assignment of agents to call centers, as there is
only a constraint on the total number of agents (although there will probably be physical
constraints, such as the number of work places in a call center).
To give further insight in economies of scale, we plotted the two situations of the
example above in a single figure, Figure 4.3. We consider the TSF, and take 7 and 14
agents. To make comparisons possible we put _×_/s (the productivity) on the horizontal
axis, and the TSF on the vertical axis. Because _ × _ < s, TSF gets 0 as soon as _ × _/s
gets 1, no matter what call center we are considering.
In Figure 4.3 we see that the dashed line is more to the right: for the same productivity
we see that a bigger call center has a higher TSF. Stated otherwise: to obtain a target SL,
a big call center obtains a higher productivity. This is related to the steepness of the curve
for productivity values close to 1, which is the sensitivity of the Erlang formula to small
changes of the parameters, as discussed earlier in this section.
It is important to note that the relative gain of merging call centers (i.e., relative to
the size of the call center) decreases as the size increases. The absolute advantage however
(slowly) increases. Consider four call centers, each with _ = 10 and _ = 2 minutes, and 80% of the calls should be served within 20 seconds. If all call centers are separate then we need 24 agents in each call center, 45 in each when they are merged two by two, and 86 when we have one single call center. Merging two centers with arrival rate 10 saves 3 agents, merging two
0 0.5 1 1.5
0
20
40
60
80
100
Productivity _×_
s
TSF in %
Figure 4.3: The TSF for _ = 5, s = 7 (solid) and s = 14 (dashed), AWT = 0.33, and
varying _.
with arrival rate 10 saves 4. But divided by the arrival (that is, relative to the size), the
economies are higher when the small centers are merged.
Variations in waiting times Consider two different call centers: one has parameters
_ = 1, _ = 5, and s = 8, the other has _ = 20, _ = 0.333, and also s = 8. Both call centers
have a TSF of around 86% for AWT = 20 seconds. Does this mean that the waiting times
of both call centers are comparable? This is not the case. To make this clear, we plotted
histograms of waiting times of both call centers in Figure 4.4. The level at the right of 100
denotes the percentage of callers that has a waiting time exceeding 100 seconds. We see
that in the first call center, represented by the solid line, callers either do not wait at all
or wait very long, there are hardly any callers that wait between 10 and 100 seconds. In
the second call center (the dashed line) fewer calls get an agent right away, but very few
have to wait very long. There are two conclusions to be drawn from this example. In the first place: the TSF does not say everything. But more importantly, we see that depending on the characteristics of a call center there can be more or less variations in waiting times. Only a thorough investigation of for example the TSF for various AWTs can reveal the characteristics of a particular call center. The remaining waiting time* When we enter a queue that we can observe (as in the post office or the supermarket) then we can estimate our remaining service time on the Chapter 4 — The Erlang C formula 23
0 20 40 60 80 100 120
0
20
40
60
80
Waiting times in 10-second intervals % of calls basis of the number of customers in front of us. Usually our extimation of the remainingwaiting will decrease while we are waiting as we see customers in front of us leaving.But how about the remaining waiting time in an invisible queue as we encounter in call centers? The mathematics show that under the Erlang C model the remaining waitingtime is constant. Thus, no matter how long we have been waiting, the average remaining waiting time is always the same. How can this at first sight counterintuitive phenomenon be explained? As we enter the queue, we expect a certain number of calls to be waiting in front of us. As we wait a while, then we conclude that apparantly the queue was longer than expected. From the Erlang formula it follows that, as long as we are waiting, the expected number of customers remains always the same. A possible consequence of this fact for customers is that one should not abandon while waiting: why hang up after 1 minute if your remaining waiting time is as long as when you started waiting? In practice however there are good reasons to hang up after a while, and good reasons to stay in line. A reason to hang up is the fact that customers do not know the call center’s parameters, and therefore they do not know the average waiting time in the call center. The longer you wait, the more likely you entered a call center with unfavorable parameters, and thus your remaining waiting time does increase! On the other hand, the Erlang C formula does not account for abandonments. If your patience is longer than that of the customers ‘in front of you’, then they will abandon before you and you
will eventually be served. In a system where calls abandon the average remaining waiting time decreases while waiting. For call center managers it should be clear that, unless customers abandon quickly, very long waiting times can and will occur exceptionally. Theoretically there is no upper limit to the waiting time. To protect customers against unexpectedly long waiting times I think that it is good to inform customers on expected waiting times or numbers of customers
waiting in the queue. Together with this the waiting customer could be pointed towards other channels to make contact such as internet