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Many call centres are subject to service level agreements that stipulate that they must achieve targets in terms of the proportion of calls that are answered within a specified time. In order to manage a centre so that targets like these are met, we need to have a method of calculating the waiting time distributions experienced by customers. In this paper, we provide such a method for small call centres that employ skills-based routing. We first build the methodology for the single-skill case and then extend it to a multi-skill case. We model the call centre system as a continuous-time Markov chain and then make use of the Laplace transform to calculate the relevant quantities. We later demonstrate the use of this method to find the optimal routing policy in a certain class of policies.

In this paper, we show how to calculate transient performance measures in models for small call centres that employ skills-based routing. In particular, we calculate the expected number of customer losses and call abandonments in a fixed time. We use the results to compare how call allocation policies can minimise the expected numbers of losses and abandonments, and make recommendations about which policies should be employed.

In traditional priority queues, we assume that every customer upon arrival has a fixed, class-dependent priority, and that a customer may not commence service if a customer with a higher priority is present in the queue. However, in situations where a performance target in terms of the tails of the class-dependent waiting time distributions has to be met, such models of priority queueing may not be satisfactory. In fact, there could be situations where high priority classes easily meet their performance target for the maximum waiting time, while lower classes do not. Here, we are interested in the stationary distribution at the times of commencement of service of this maximum priority process. Until now, there has been no explicit expression for this distribution. We construct a mapping of the maximum priority process to a tandem fluid queue, which enables us to find expressions for this stationary distribution. We derive the results for the stationary distribution of the maximum priority process at the times of the commencement of service., Comment: The Eleventh International Conference on Matrix-Analytic Methods in Stochastic Models (MAM11), 2022, Seoul, Republic of Korea

Malaria is a parasitic disease, transmitted by mosquito vectors. Plasmodium vivax presents particular challenges for disease control, in light of an undetectable reservoir of latent parasites (hypnozoites) within the host liver. Superinfection, which is driven by temporally proximate mosquito inoculation and/or hypnozoite activation events, is an important feature of P. vivax. Here, we present a model of hypnozoite accrual and superinfection for P. vivax. To couple host and vector dynamics, we construct a density-dependent Markov population process with countably many types, for which disease extinction is shown to occur almost surely. We also establish a functional law of large numbers, taking the form of an infinite-dimensional system of ordinary differential equations that can also be recovered under the hybrid approximation or a standard compartment modelling approach. Recognising that the subset of these equations that models the infection status of human hosts has precisely the same form as the Kolmogorov forward equations for a Markovian network of infinite server queues with an inhomogeneous batch arrival process, we use physical insight into the evolution of the latter to write down a time-dependent multivariate generating function for the solution. We use this characterisation to collapse the infinite-compartment model into a single integrodifferential equation (IDE) governing the intensity of mosquito-to-human transmission. Through a steady state analysis, we recover a threshold phenomenon for this IDE in terms of a bifurcation parameter $R_0$, with the disease-free equilibrium shown to be uniformly asymptotically stable if $R_0<1$ and an endemic equilibrium solution emerging if $R_0>1$. Our work provides a theoretical basis to explore the epidemiology of P. vivax, and introduces a general strategy for constructing tractable population-level models of malarial superinfection.

In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network, and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.

Malaria is a vector-borne disease that exacts a grave toll in the Global South. The epidemiology of Plasmodium vivax, the most geographically expansive agent of human malaria, is characterised by the accrual of a reservoir of dormant parasites known as hypnozoites. Relapses, arising from hypnozoite activation events, comprise the majority of the blood-stage infection burden, with implications for the acquisition of immunity and the distribution of superinfection. Here, we construct a hybrid transmission model for P. vivax that concurrently accounts for the accrual of the hypnozoite reservoir, (blood-stage) superinfection and the acquisition of immunity. We begin by analytically characterising within-host dynamics as a function of mosquito-to-human transmission intensity, extending our previous model (comprising an open network of infinite server queues) to capture a discretised immunity level. To model transmission-blocking and antidisease immunity, we allow for geometric decay in the respective probabilities of successful human-to-mosquito transmission and symptomatic blood-stage infection as a function of this immunity level. Under a hybrid approximation -- whereby probabilistic within-host distributions are cast as expected population-level proportions -- we couple host and vector dynamics to recover a deterministic compartmental model in line with Ross-Macdonald theory. We then perform a steady-state analysis for this compartmental model, informed by the (analytic) distributions derived at the within-host level. To characterise transient dynamics, we derive a reduced system of integrodifferential equations (IDEs), likewise informed by our within-host queueing network, allowing us to recover population-level distributions for various quantities of epidemiological interest. Our model provides insights into important, but poorly understood, epidemiological features of P. vivax.