Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers

Abstract

Both large and small firms maintain call centers to establish contact between themselves and their customers. A call center is staffed by Customer Service Representatives (CSRs). In addition to CSRs, a call center needs computers and telecommunication equipment such as an Automatic Call Distributor (ACD). If calls arrive according to a Poisson arrival process and the service times have an exponential distribution, the call center can be modeled as an M/M/s queue where s is the number of CSRs. Typical calculations include finding the number of CSRs required and finding the number of trunks lines required. However, if call center models ignore the abandonment behavior of customers, they distort information that is relevant to management. Typically, ignoring abandonment would lead to overstaffing, as fewer CSRs are actually needed, because some of the callers abandon the system. Also, nonstationarity of arrivals is highly prevalent in call centers. Green, Kolesar and Whitt plot hourly arrival rates for a financial services call center. There is significant variation in arrivals by time of day. In this paper, we model call centers as multiserver Markovian queues with both nonstationarity and abandonment. Nonstationarity is modeled by having a Poisson arrival stream with time dependent mean, which varies according to a sinusoid. Abandonment is defined in terms of an exponential patience random variable, which is the amount of time the caller would be on hold before abandoning the call. We numerically study the performance of this nonstationary M(t)/M/s+M queue with abandonment and compare its performance measures with those of the stationary M/M/s+M queue. We find that approximating a nonstationary system by a stationary system, even at low levels of nonstationarity, can lead to significant errors. Similar results in a system without abandonment, have been obtained by Green, Kolesar and Svoronos. Additionally, we find that abandonment dampens the effect of nonstationarity. Since abandonment and nonstationarity are both present together in real call centers, a real call center with a high level of abandonment behaves closer to an ideal stationary system.