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 wmassey@princeton.edu

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

Telephone Call Centers

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.

Call Centers with Instant Answering: The M/M/∞ Queueing Model 1 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.

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.

Call Centers with “Music” (Delays): The M/M/C/∞ Queueing Model 1 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.

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.

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

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

Call Centers with Music and Busy Signals: The M/M/C/K Queue 1 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.

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.

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).

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.

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.

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

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.

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.

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.

Exact Search Design Algorithm K 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

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.

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

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

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

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

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

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

Numerical Design Comparisons Let 1/m = 10.0 minutes, t = 0.5 minutes, e = 0.001 and d = 0.2.

Design Performance Comparisons Let 1/m = 10.0 minutes, t = 0.5 minutes, e = 0.001 and d = 0.2.

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.

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)

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

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