In the following:
Discrete models of combat are a rare part of the combat modelling literature. Our work introduces a stochastic version of a discrete ground combat based on Epstein theory featuring two adversarial sides, namely an attacker and a defender. Noticeably, the Epstein model of ground combat features an evolving battle front through a withdrawal mechanism to capture the connection between attrition and movement of the front historically prevalent in ground war. Our extension from the deterministic setting of the Epstein model to the stochastic setting is achieved by taking the exchange ratio of attackers lost to defenders to be a mean-reverting stochastic process. The extension of the exchange ratio to a stochastic process is interrupted to be the result of changing strategies and engagements by either side as well of the generally uncertainty of warfare known as the “Fog of War” upon the outcome of combat
In the deterministic setting of our model, our toy numerical example results in an attacker victory. In the extension of the exchange ratio to a stochastic process, the attackers are no longer assured victory. However, the variations in the exchange ratio can be of benefit to the attackers in that they may achieve victory in a shorter combat duration and as a consequence suffer less attrition. Thus we interpret the stochastic process as introducing a ``risk vs reward'' scenario for the attackers where the risk is quantified through the volatility of the process. Our numerical simulations explore the shift in the outcome of combat for the attackers as they take on additional risk and more uncertainty is introduced into combat
We observe the probability that the attacker is victorious, the time till victory when the attacker is victorious, and the remaining ground force strength of the attacking forces for varying volatility. Our results show that for increasing values of the volatility of the exchange ratio process, the probability of an attacker victory increases but the combat duration decreases and the remaining combat power of the attacker forces increases.
Keywords: Combat models, stochastic processes, discrete models.
T.A. McLennan-Smith, M. Nelson, Z. Jovanoski, M. Rodrigo, and H.S. Sidhu. Stochastic processes in a discrete model of ground combat. In S. Elsawah, editor, 23rd International Congress on Modelling and Simulation, pages 116-122. Modelling and Simulation Society of Australia and New Zealand, 2019. ISBN 978-0-9758400-9-2. https://doi.org/10.36334/modsim.2019.A5.mclennansmith.
he activated sludge process is the most widely used process for the biological treatment of domestic and industrial wastewaters. Wastewater treatment plants based on the activated sludge process are in widespread use in developed and developing countries. The activated sludge model number 1 (ASM #1) is an internationally accepted standard for activated sludge modeling. It describes nitrogen and chemical oxygen demand within suspended-growth treatment processes, including mechanisms for nitrification and denitrification.
We analyse the biological treatment of a wastewater when a cascade of four reactors is used. We assume that each reactor in the cascade has the same volume. Operating conditions are investigated in which the first reactor is not aerated whilst the last two reactors are aerated. The second reactor may either be aerated or not aerated. The process configuration includes one settling unit and one recycle unit. The settling unit is placed after the final reactor of the cascade. Part of its exit stream is wasted and the remainder is fed into the first reactor. The recycle unit is also placed after the final reactor of the cascade. The entirety of its exit stream is fed into the first reactor.
The performance of a wastewater treatment plant can be characterised by a number of process parameters. Here we consider the nitrogen concentration in the effluent stream leaving the treatment plant (TNe). When the reactor configuration includes a settling unit this is defined by
TNe = SNO +SNH +SND,
where the state variables on the right hand are the concentration of soluble nitrate and nitrite nitrogen (SNO), soluble ammonium nitrogen (SNH), and soluble biodegradable organic (SND) respectively.
A combination of direct numerical integration and continuation methods are used to investigate the steady-state behaviour of the system. The governing equations were solved using both matlab (ode15s) and maple (lsode[backfull]). For continuation XPPAUT was used. We take the hydraulic retention time (HRT) as the bifurcation parameter, primarily allowing it to vary over the range 0<HRT (days)≤ 1. We investigate how the nitrogen concentration in the effluent stream depends upon the operation of the recycle units and the state of aeration in the second reactor.
Our results are summarised as follows.
M.I. Nelson and F.I. Hai. Nitrogen removal in a cascade of four reactors employing the activated sludge process. In S. Elsawah, editor, 23rd International Congress on Modelling and Simulation, pages 130-136. Modelling and Simulation Society of Australia and New Zealand, 2019. ISBN 978-0-9758400-9-2. 2019 https://doi.org/10.36334/modsim.2019.A5.nelson.