Markov Models Assignment 1
نویسندگان
چکیده
The caterers will be serving a ‘surprise drink’ at the upcoming AMSI BBQ on Friday. To make this concoction, they have to mix a special liquid, which is soluble in water, with large amounts of water. Specifically, they pump these two different liquids via two separate hoses into a large cylindrical container and these liquids are mixed together with an electric stirrer situated inside the container. At the bottom of the container is a nozzle to dispense the drink. In this system, we may be interested in modelling mathematically the rate of outward flow from the output nozzle over time or the distribution of the concentration of the liquid in the tank or the proportion of liquid in the tank over time. These quantities can vary and can be controlled via the flow-in rate of the two liquids from the hoses, the concentration of the special liquid, and the speed of the stirrer. Optimally, we would like the concentration of the drink to be spread uniformly (otherwise the drink would taste yukky!), and that the amount of flow-in and flow-out of the liquids are balanced such that there will be no overflows or an empty system at any point in time. A natural model to consider when modelling the flow rate out of the tank is differential equations as we consider the system to continually evolve over time. After developing a suitable mathematical model to describe our situation, we may solve the differential equation for a trajectory solution and derive quantities of interest, such as the fixed points and their stability, which would tell us the long term or equilibrium behaviour of our system. We may also extract insight of our system by graphing our model, for example plotting the vector field of the differential equation and its trajectories. Lastly, we may want to control and optimize the rate of flow-in from the two inputs and the speed of the mixer to get the best tasting refreshment, so everyone gets to have an enjoyable evening!
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