Real-Time Delay Estimation Based on Delay History

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 for the delay before entering service of an arriving customer based on recent customer delay experience. We characterize performance by the bias and the mean squared error (MSE) We do analysis and conduct simulations for the standard GI/M/s multi-server queueing model, emphasizing the case of large s. 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 estimator based on the queue length (QL), which requires knowledge of the mean interval between successive service completions in addition to the queue length. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation for the mean value of that conditional distribution, which is not the observed delay, serves as a refined estimator, yielding lower MSE than the direct estimator. We show that the MSE relative to the square of the mean is asymptotically negligible for all three candidate delay estimators in the many-server and classical heavy-traffic limiting regimes.