Mathematics

PhD student in the School of Mathematics at The University of Edinburgh, Sep 2011 – Jul 2015

During my university studies, probability, measure theory and stochastic processes were my favourite areas of mathematics. Using tools from these areas, I developed novel queueing models, during my PhD research.

Queueing Theory is a branch of mathematics that studies waiting times and queue lengths. Its origins in research were laid by the Danish mathematician Agner Krarup Erlang in the 20th century, when he constructed mathematical models to predict waiting times for the Copenhagen telephone exchange, so that people could be connected with a receiver rather than hearing a busy signal.

“It’s when the world comes to an end that you don’t mind waiting your turn or if someone jumps the queue.” — Anthony T. Hincks.

We are now living in an highly efficient world. Food is delivered within minutes, financial trades are processed in milliseconds and opening a bank account can be accomplished almost instantly. Yet, we are made to pause routinely waiting, on the phone with customer services, on a job portal for a suitable position to show up or on hold with a busy signal for hours to make a medical appointment. We wish there were more service representatives and fewer queues.

However, simply adding more service representatives to a system is not the solution. Customers’ arrivals and departures in a system have random elements, and service providers cannot afford to pile up unlimited inventory and servers to meet unexpected bursts in customer demands. There is trade-off between the cost of running a system versus service quality provided. In some systems like a job portal, the system operators do not have direct control of the availability of the suitable resource.

To provide insights to constructing effective solutions for service operators to design their operating policies, mathematicians approach problems like these using tools including stochastic processes, probability theory, game theory and optimisation. With practical considerations of early and late arrivals, service cancellations and the probability of a customer not finding a suitable service, we develop mathematical models to predict customers arrival and departure times, and we seek solutions to maximise the service quality while minimising the cost.

In recent years there has been a great increase in the popularity of web portals as a meeting place for individuals who provide services and those who require services. Some commonly used examples are employment portals (LinkedIn, efinancialcareers), classified advertisement portals (Gumtree, eBay), rental portals (Rightmove, Zoopla) and dating websites. A common feature of these systems is that each pair of users has a probability to match with each other and the system operator has no control on who matches with whom. Since so many people use these systems, it is important to guarantee a certain level of service quality, and this raises lots of questions which are also mathematically interesting:

During my PhD study, I introduced a novel queueing model, which was defined as a probabilistic matching system, to model the traffic in these web portals. To answer many of these mathematically interesting questions, I explored the natures of probabilistic matching systems using sophisticated mathematical techniques including stochastic modelling, fluid and diffusion approximation and game theory. Two papers from my PhD study were published in a flagship journal in Queueing Theory.

My research revealed many important properties the system exhibits, some of which seem counter-intuitive at the beginning. Through stability analysis, I showed that if the system is uncontrolled, many users accumulate in the system without finding a match. I suggested four types of admission controls to remedy this problem and studied the performance measures of the system under these controls. One of the surprising results was that under a class of admission control policies with general conditions, a higher probability for individuals to match leads to fewer users finding a match. Further, I suggested analytical diffusion processes to approximate the mathematically intractable number-in-system processes, so that the waiting time of users could be estimated. Finally, I revealed properties of different pricing mechanisms that lead to benefit of the system operators and the users.