arXiv subjects

Mark Fackrell

Publications and source records attributed to Mark Fackrell.

5 recordsLinked to original sources

arXiv2024

Performance Evaluation of Small Call Centres in a Transient Regime

This paper addresses a fundamental and practically significant problem in call centre operations -- determining optimal call allocation policies that meet client service targets while minimising staffing costs. Motivated by a problem presented by an industry partner, we examine a real-world setting involving a relatively small call centre with hierarchical structure among agents. It is natural to model the operation of such a centre as a continuous-time Markov chain. To gain insight into the structure of optimal policies, we first (i) apply backward induction based on Bellman's equation to a finite-horizon discrete-time model, and (ii) derive stationary policies for an infinite-horizon continuous-time model with discounting. Subsequently, we evaluate the performance of these policies in the original finite-horizon continuous-time setting by computing the expected number of abandonments and the waiting time distributions of customers. This is achieved using first-step analysis combined with Laplace transform methods. The effectiveness of the proposed approach is illustrated through numerical examples.

math.PR↗

arXiv2023

Minimising Numbers of Losses and Abandonments in Small Call Centres Under a Transient Regime

We consider the problem of allocating customers to agents in small call centres so that transient performance indicators in terms of expected numbers of losses and abandonments are optimised. To gain insight into the general structure of optimal policies, we start by (i) using backward induction based upon Bellman's equation for a finite-horizon discrete-time model, and (ii) deriving stationary policies for an infinite-horizon continuous-time model with discounting. We then compare the performance of such policies in our original finite-horizon continuous-time model using a first-step analysis applied to the Laplace transforms of the relevant measures.

math.PR↗

arXiv2021

The Value of Communication and Cooperation in a Two-Server Service System

In 2015, Guglielmi and Badia discussed optimal strategies in a particular type of service system with two strategic servers. In their setup, each server can either be active or inactive and an active server can be requested to transmit a sequence of packets. The servers have varying probabilities of successfully transmitting when they are active, and both servers receive a unit reward if the sequence of packets is transmitted successfully. Guglielmi and Badia provided an analysis of optimal strategies in four scenarios: where each server does not know the other's successful transmission probability; one of the two servers is always inactive; each server knows the other's successful transmission probability; and they are willing to cooperate. Unfortunately the analysis in Guglielmi and Badia contained errors. In this paper we correct these errors. We discuss three cases where both servers (I) communicate and cooperate; (II) neither communicate nor cooperate; (III) communicate but do not cooperate. In particular, we obtain the unique Nash equilibrium strategy in Case II through a Bayesian game formulation, and demonstrate that there is a region in the parameter space where there are multiple Nash equilibria in Case III. We also quantify the value of communication or cooperation by comparing the social welfare in the three cases, and propose possible regulations to make the Nash equilibrium strategy the socially optimal strategy for both Cases II and III.

cs.GT↗

arXiv2021

Strategic Customer Behavior in an $M/M/1$ Feedback Queue

We investigate the behavior of equilibria in an $M/M/1$ feedback queue where price and time sensitive customers are homogeneous with respect to service valuation and cost per unit time of waiting. Upon arrival, customers can observe the number of customers in the system and then decide to join or to balk. Customers are served in order of arrival. After being served, each customer either successfully completes the service and departs the system with probability $q$, or the service fails and the customer immediately joins the end of the queue to wait to be served again until she successfully completes it. We analyse this decision problem as a noncooperative game among the customers. We show that there exists a unique symmetric Nash equilibrium threshold strategy. We then prove that the symmetric Nash equilibrium threshold strategy is evolutionarily stable. Moreover, if we relax the strategy restrictions by allowing customers to renege, in the new Nash equilibrium, customers have a greater incentive to join. However, this does not necessarily increase the equilibrium expected payoff, and for some parameter values, it decreases it.

math.OC↗

arXiv2016

Modelling modal gating of ion channels with hierarchical Markov models

Many ion channels spontaneously switch between different levels of activity. Although this behaviour known as modal gating has been observed for a long time it is currently not well understood. Despite the fact that appropriately representing activity changes is essential for accurately capturing time course data from ion channels, systematic approaches for modelling modal gating are currently not available. In this paper, we develop a modular approach for building such a model in an iterative process. First, stochastic switching between modes and stochastic opening and closing within modes are represented in separate aggregated Markov models. Second, the continuous-time hierarchical Markov model, a new modelling framework proposed here, then enables us to combine these components so that in the integrated model both mode switching as well as the kinetics within modes are appropriately represented. A mathematical analysis reveals that the behaviour of the hierarchical Markov model naturally depends on the properties of its components. We also demonstrate how a hierarchical Markov model can be parameterised using experimental data and show that it provides a better representation than a previous model of the same data set. Because evidence is increasing that modal gating reflects underlying molecular properties of the channel protein, it is likely that biophysical processes are better captured by our new approach than in earlier models.

q-bio.QM↗