1. An Optimal Design of the M/M/C/K Queue for Call Centers
William A. Massey
Department of Operations Research
and Financial Engineering,
Princeton University

2. Acknowledgements This is joint work with Rodney B. Wallace of IBM and
George Washington University.
Related paper to appear in QUESTA.

3. Telephone Call Centers

4. Our Research Goal We have a call center manager who serves a given
customer base. The average calling rate and times
in service are known for these customers. Moreover,
they expect a specific level of service performance.
Our goal is to develop formulas and algorithms to
assist the manager in finding the minimal number of
call agents and telephone lines needed to provide
a given service level target.

5. Call Centers with Instant Answering: The M/M/∞ Queueing Model1 2 ∞ i
m Q∞ l
We have a Poisson arrival rate l, mean service time 1/m, as well as an infinite number of call center agents and telephone lines.

6. Steady State Properties of Q∞
Setting r = l/m, the number of customers in the system Q∞ has the Poisson distribution
The mean number of customers in the system Q∞ equals its variance and is called the mean offered load or the mean number of requested call agents.

7. Call Centers with “Music” (Delays): The M/M/C/∞ Queueing Model1 2 C
iii 1 2 ∞ l
m min(QC/ ∞, C) i
We have a Poisson arrival rate l, mean service time 1/m, C call center agents and an infinite number of telephone lines.

8. Steady State Properties of QC/∞
Let the random variable DC / ∞ be the delay for the
customer arriving to the M/M/C/∞ queue in steady-
state, which holds if and only if r < C (i.e. the mean
number of requested call agents is less than the
number of available call agents). We define the
probability of delay to be
This is called the Erlang C-Formula.

9. Applications of the Erlang C-Formula
It can be shown that the probability of the delay exceeding t is
The mean queue length and delay are

10. How to Compute the Erlang C-Formula
where r < C and bC(r) is the Erlang B formula,
which solves the recursion relation

11. Call Centers with Music and Busy Signals: The M/M/C/K Queue1 2 C
iii 1 2 K
l 1{Q < C+K}
C/K
m min(QC/K, C) i
We have a Poisson arrival rate l, mean service time 1/m, C call center agents, and C + K telephone lines.

12. The M/M/C/K Design Problem
The Design
Problem
(C,K) = ?
Given: l = customer calling rate,1/m = conversation
time, e = blocking probability, t = delay threshold,
d = conditional delay probability for exceeding t.
Find: The minimal number of agents C and phone lines C+K needed for the e, d and t requirements.

13. Ad-Hoc Erlang Design Solution
Let the effective arrival rate be l*=l(1-e).
Using an arrival rate l* and a service rate m, find the smallest C such that P(DC / ∞ > t) < d.
Compute E[DC / ∞].
Let the effective service time be 1/m*=1/m+ E[DC / ∞].
Using an arrival rate l and a service rate m*, find the smallest L such that P(QL / 0 = L) < e.
Now let K = (L-C)+.
Now we replace this ad-hoc method with two
new approaches (exact and asymptotic).

14. Performance Variables
QC/K = random number of customers in the M/M/C/K
system in steady-state. This is also called the carried
load or the number of customers that gain access to
a call agent.
DC/K = random conditional delay for the M/M/C/K
queue in steady-state.

15. Performance Measures bC/K(r) = P(QC/K = C+K)
= blocking probability (call center availability).
gC/K(r,m,t) = P( DC/K>t | QC/K < C+K ).
= conditional probability of delay exceeding t (speed to answer).
gC/K(r) = P( DC/K>0 | QC/K < C+K )
= conditional delay probability.

16. M/M/C/K Steady State Results
and L’Hopital’s rule applies when r = C.

17. Blocking Probability Limit
It follows from our blocking probability formula that
and this is a decreasing limit.
Hence for all 0<e<1, we can say that bC/K(r) < e
holds for some finite K if and only if r (1-e) < C.

18. Delay and Blocking Monotonicity Properties for (C,K)
If C1 < C2 and C1 + K1 < C2 + K2 , then
so the blocking probability for the first system is larger.
If C1 < C2 and C1 + K1 > C2 + K2 , then
and the delay for the first system is larger.

19. Defining a Minimal (C,K) Pair
The cost of an additional telephone line is insignificant
compared to the cost of an additional call agent. We
then define the pair (C1,K1) to be “smaller than” the
pair (C2,K2) if C1 < C2 or we have C1 = C2 and K1 < K2.

20. Exact Search Design AlgorithmK
3. First (C,K) Pair with
Blocking < e (and Delay < d?).
Delay > d or
C < r(1-e)
Region
Smaller Blocking
Larger Delay
2. Last (C,K) Pair with
Delay < d and r(1-e) < C.
Larger Blocking
Larger Delay
Blocking > e
Region
0. Blocking=1
and Delay=0.
1. First (C,0) Pair with
Blocking < e and Delay = 0. C
Smaller Blocking

21. Square Root Rule of Thumb for the M/M/C/K Design Problem
For instant answering (infinite number of agents), the
number of customers in the system has a Poisson
distribution with mean and variance equal to r.
The optimal number of agents required should be close to
the mean number in this system plus some constant x
times its standard deviation, which is C = r + x √r.
A well-designed system should only have people waiting as
an exceptional case so the amount of waiting spaces
should equal the standard deviation of this system times
some constant y or K = y √r.

22. Asymptotic Behavior of the Erlang B-Formula
Taking the limit as the mean offered load becomes
large, while using the square root rule of thumb, we
have
where

23. Defining the Special Function y
If we define the function y(y) for all positive y to be
then it is the unique solution to the differential equation

24. Blocking Asymptotics We can show that where
and setting x = y (z) gives us

25. Asymptotic Blocking Level Curve
The (x,y) pairs where
can be parameterized by
when y (z) > 0, and

26. Delay Asymptotics We can show that where
and setting x = y (z) gives us

27. Asymptotic Optimal Design Solution
Find x and y such that
Using the blocking level curve, this problem reduces to solving a fixed point equation for some z

28. Numerical Design Comparisons
Let 1/m = 10.0 minutes, t = 0.5 minutes, e = and d = 0.2.

29. Design Performance Comparisons
Let 1/m = 10.0 minutes, t = 0.5 minutes, e = and d = 0.2.

30. Summary of Numerical Results
All three methods do a good job in estimating the number of call agents C.
The ad-hoc method performs worst in estimating the number of additional telephone lines K.
As a result, ad-hoc performs worst in reaching the target blocking level and can be off by a factor from 2 to 12.
The asymptotic method gives answers close to the target blocking and delay levels.
Given a pre-computed table of values for y, the fixed point iteration for the asymptotic method is the easiest to compute and requires the least number of steps.

31. M/M/∞ State Space Diagram
iii 1
n-1 n l m 2m
(n-1)m nm
(n+1)m
n m · P(Q∞ = n) = l · P(Q∞ = n-1)

32. M/M/C/∞ State Space Diagram
iii 1 C
C+1 l m 2m Cm

33. M/M/C/K State Space Diagram
iii 1 C
C+1 l m 2m Cm
C+K