Fluid Approximation of a Call Center Model with Redials and Reconnects

In many call centers, callers may call multiple times. Some of the calls are re-attempts after abandonments (redials), and some are re-attempts after connected calls (reconnects). The combination of redials and reconnects has not been considered when making staffing decisions, while ignoring them will inevitably lead to under- or overestimation of call volumes, which results in improper and hence costly staffing decisions. Motivated by this, in this paper we study call centers where customers can abandon, and abandoned customers may redial, and when a customer finishes his conversation with an agent, he may reconnect. We use a fluid model to derive first order approximations for the number of customers in the redial and reconnect orbits in the heavy traffic. We show that the fluid limit of such a model is the unique solution to a system of three differential equations. Furthermore, we use the fluid limit to calculate the expected total arrival rate, which is then given as an input to the Erlang A model for the purpose of calculating service levels and abandonment rates. The performance of such a procedure is validated in the case of single intervals as well as multiple intervals with changing parameters

On the estimation of the true demand in call centers with redials and reconnects

In practice, in many call centers customers often perform redials (i.e., reattempt after an abandonment) and reconnects (i.e., reattempt after an answered call). In the literature, call center models usually do not cover these features, while real data analysis and simulation results show ignoring them inevitably leads to inaccurate estimation of the total inbound volume. Therefore, in this paper we propose a performance model that includes both features. In our model, the total volume consists of three types of calls: (1) fresh calls (i.e., initial call attempts), (2) redials, and (3) reconnects. In practice, the total volume is used to make forecasts, while according to the simulation results, this could lead to high forecast errors, and subsequently wrong staffing decisions. However, most of the call center data sets do not have customer-identity information, which makes it difficult to identify how many calls are fresh and what fractions of the calls are redials and reconnects. Motivated by this, we propose a model to estimate the number of fresh calls, and the redial and reconnect probabilities, using real call center data that has no customer-identity information. We show that these three variables cannot be estimated simultaneously. However, it is empirically shown that if one variable is given, the other two variables can be estimated accurately with relatively small bias. We show that our estimations of redial and reconnect probabilities and the number of fresh calls are close to the real ones, both via real data analysis and simulation

A method for estimation of redial and reconnect probabilities in call centers

In practice, many call center forecasters use the total inbound volume to make forecasts. In reality, besides the fresh calls (initial call attempts), there are many redials (re-attempts after abandonments) and reconnects (re-attempts after answered calls) in call centers. Neglecting redials and reconnects will inevitably lead to inaccurate forecasts, which eventually leads to inaccurate staffing decisions. However, most of the call center data sets do not have customer-identity information, which makes it difficult to identify how many calls are fresh. Motivated by this, the goal of this paper is to estimate the number of fresh calls, and the redial and reconnect probabilities. To this end, we propose a model to estimate these three variables. We formulate our estimation model as a minimization problem, where the actual redial and reconnect probabilities lead to the minimum objective value. We validate our estimation results via real call center data and simulated data

Call centers with a postponed callback offer

We study a call center model with a postponed callback option. A customer at the head of the queue whose elapsed waiting time achieves a given threshold receives a voice message mentioning the option to be called back later. This callback option differs from the traditional ones found in the literature where the callback offer is given at customer’s arrival. We approximate this system by a two-dimensional Markov chain, with one dimension being a unit of a discretization of the waiting time. We next show that this approximation model converges to the exact one. This allows us to obtain explicitly the performance measures without abandonment and to compute them numerically otherwise. From the performance analysis, we derive a series of practical insights and recommendations for a clever use of the callback offer. In particular, we show that this time-based offer outperforms traditional ones when considering the waiting time of inbound calls

Robust estimation of bacterial cell count from optical density

Optical density (OD) is widely used to estimate the density of cells in liquid culture, but cannot be compared between instruments without a standardized calibration protocol and is challenging to relate to actual cell count. We address this with an interlaboratory study comparing three simple, low-cost, and highly accessible OD calibration protocols across 244 laboratories, applied to eight strains of constitutive GFP-expressing E. coli. Based on our results, we recommend calibrating OD to estimated cell count using serial dilution of silica microspheres, which produces highly precise calibration (95.5% of residuals <1.2-fold), is easily assessed for quality control, also assesses instrument effective linear range, and can be combined with fluorescence calibration to obtain units of Molecules of Equivalent Fluorescein (MEFL) per cell, allowing direct comparison and data fusion with flow cytometry measurements: in our study, fluorescence per cell measurements showed only a 1.07-fold mean difference between plate reader and flow cytometry data

Multi-class Fork-Join queues & The stochastic knapsack problem

Multi-class Fork-Join queues are extension of single-class Fork-Join queues. In a multi-class Fork-Join queuing system, different types of jobs arrive, and then split into several sub-jobs. Those sub-jobs go to parallel processing queues. Then, the synchronization is required before the departure of a job. We found very few scientific efforts in analyzing the multi-class Fork-Join queues. In this thesis, we analyzed the expected sojourn time and the expected synchronization time. Since it is difficult to derive analytical expressions for them, we used approximation methods to approximate them. Through extensive numerical results, we showed that the approximations provide a close approximation for the expected sojourn time and an accurate approximation of the optimal synchronization buffer