This is a supplement to the main paper, having the same title. In this work we develop methods to study the impact upon steady-state performance of delay announcements made to arriving customers in a many-server queue with customer abandonment. We assume that the queue is not visible to waiting customers, as in most customer contact centers, when contact is made by telephone, email or instant messaging. We propose simple robust announcement schemes: (i) the delay of the last served (DLS) customer and (ii) a fixed delay announcement (FDA) based on an appropriate long-run average delay. For any single-number delay announcement made immediately upon arrival, customers may balk or have new abandonment behavior as a function of the announced delay. To perform a rough-cut performance analysis, prior to detailed simulation, we introduce a fluid model, which provides an approximate and highly simplified description for large systems in an overloaded regime. In the fluid model, all customers are faced with the same delay and consequently can be given the same delay announcement. That property motivates considering a second approximation scheme: an equilibrium fixed delay announcement (FDA) in the stochastic model. We show that these two approximate descriptions of aggregate performance are effective by comparing to simulations of systems with state-dependent DLS announcements. Here we present additional material, supplementing the main paper.

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Motivated by interest in making delay announcements in service systems, we study real-time delay estimators in many-server service systems, both with and without customer abandonment. Our main contribution here is to consider the realistic feature of time-varying arrival rates. We focus especially on delay estimators exploiting recent customer delay history. We show that time-varying arrival rates can introduce significant estimation bias in delayhistory-based delay estimators when the system experiences alternating periods of overload and underload. We then introduce refined delay-history estimators that effectively cope with time-varying arrival rates together with non-exponential service-time and abandonment-time distributions, which are often observed in practice. We use computer simulation to verify that our proposed estimators outperform several natural alternatives.

Motivated by interest in making delay announcements in service systems, we develop new real-time delay predictors that effectively cope with customer abandonment and time-varying parameters. First, we focus on delay predictors exploiting recent customer delay history. We show that time-varying arrival rates can introduce significant prediction bias in delay-history-based predictors when the system experiences alternating periods of overload and underload. We then introduce a new delay-history-based predictor that effectively copes with time-varying arrival rates. Second, we consider a time-varying number of servers. We develop two new predictors which exploit an established deterministic fluid approximation for a many-server queueing model with time-varying demand and capacity. The new predictors effectively cope with those features, often observed in practice. Throughout, we use computer simulation to quantify the performance of the alternative delay predictors. 1

Motivated by interest in making delay announcements in service systems, we study real-time delay estimators in many-server service systems, both with and without customer abandonment. Our main contribution here is to consider the realistic feature of time-varying arrival rates. We focus especially on delay estimators exploiting recent customer delay history. We show that time-varying arrival rates can introduce significant estimation bias in delayhistory-based delay estimators when the system experiences alternating periods of overload and underload. We then introduce refined delay-history estimators that effectively cope with time-varying arrival rates together with non-exponential service-time and abandonment-time distributions, which are often observed in practice. We use computer simulation to verify that our proposed estimators outperform several natural alternatives.

We develop new, improved real-time delay predictors for many-server service systems with a time-varying arrival rate, a time-varying number of servers, and customer abandonment. We develop four new predictors, two of which exploit an established deterministic fluid approximation for a many-server queueing model with those features. These delay predictors can be used to make delay announcements. We use computer simulation to show that the proposed predictors outperform previous predictors. Subject classifications: delay prediction; delay announcements; simulation; time-varying arrival rates; time-varying number of servers; nonstationary queues.