Determining Trunk Lines in Call Centers with Nonstationary Arrivals and Lognormal Service Times

Two important resources in a call center are the number of staff and the number of trunk lines required. In this paper, we focus on the decision of the number of trunk lines that a call center should have. The current practice is to use the Erlang B or the M/M/s/0 queueing model which assumes Poisson arrivals, exponential service times, s servers and no places in queue, i.e. no customers can wait. In this paper, we improve on the state of practice in determining the required number of trunk lines, by including two realistic features present in call centers. The first realistic feature is to consider nonstationarity of arrivals. The second feature is to consider the lognormal service time distribution instead of the exponential distribution. There is extensive empirical evidence for both features. In order to carry out our computations we use the results of a paper by Massey and Whitt, Operations Research, 44(6), 1996. We have two main findings. Firstly, we find numerically that in our nonstationary Erlang loss model, Mt/G/s/0, an insensitivity result holds. The blocking probability of arrivals at the call center depends only on the mean of the lognormal service time distribution and not on its variance. Our second finding is that current practice is quite robust. In particular, we find the number of trunk lines required using a stationary Poisson approximation. This approximation assumes stationary Poisson arrivals with an appropriately chosen arrival rate and exponential service times. The approximation does quite well in predicting the number of trunk lines required.