Real-Time Delay Estimation Based on Delay History

Abstract

Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative real-time delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the last customer to enter service (LES), (ii) the delay experienced so far by the customer at the head of the line (HOL), and (iii) the delay experienced by the customer to have arrived most recently among those who have already completed service (RCS). We compare these delay-history estimators to the standard estimator based on the queue length (QL), commonly used in practice, which requires knowledge of the mean interval between successive service completions in addition to the QL. We characterize performance by the mean squared error (MSE). We do an analysis and conduct simulations for the standard GI/M/s multiserver queueing model, emphasizing the case of large s. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation to its mean value serves as a refined estimator. For all three candidate delay estimators, the MSE relative to the square of the mean is asymptotically negligible in the many-server and classical heavy-traffic (HT) limiting regimes.delay estimation, real-time delay estimation, delay prediction, delay announcements, many-server queues, call centers, heavy traffic

Waiting time predictors for multi-skill call centers

“…A second branch of research focuses exclusively on delay predictions, without considering the impact on the behavior of customers. The work of Whitt (1999b), Ibrahim and Whitt (2008, 2009a, 2009b, 2010, 2011a, 2011b, and Ibrahim, Armony, and Bassamboo (2015), as well as the present paper, fall into this category.…”

“…Nakibly (2002), Armony, Shimkin, andWhitt (2009), andWhitt (2009a) propose simple heuristic predictors that return the delay times experienced by previous customers. As examples of DH-based predictors, one may return the waiting time of the last customer to enter service (LES), the customer at the head of the line (HOL), the last customer to complete service (LCS), or the most recently arrived customer to complete service (RCS).…”

“…As examples of DH-based predictors, one may return the waiting time of the last customer to enter service (LES), the customer at the head of the line (HOL), the last customer to complete service (LCS), or the most recently arrived customer to complete service (RCS). Ibrahim and Whitt (2009a, 2011a, 2011b compare QL-and DH-based predictors in the context of a single queue, by means of simulation and analytical comparisons. They conclude that QL-based predictors are generally more accurate and preferable to DH-based ones, if all the information that goes into the formula is available.…”

Delay predictors for customer service systems with time-varying parameters

“…The RRASE of QL a ranges from about 14% for s = 100 to about 4% when s = 1000. From Section 4 of Ibrahim and Whitt (2009a) and Section 5 of Ibrahim and Whitt (2009b), we have theoretical results that provide useful perspective for the more complicated models we consider here. For example, we anticipate that the ASE of QL a and HOL a should be inversely proportional to the number of servers.…”

“…In Ibrahim and Whitt (2009a), we studied the performance of QL s and HOL in the GI/M/s queueing model, which has a renewal arrival process and no abandonment. We showed that QL s is the most effective predictor, under the MSE criterion, in the GI/M/s model.…”