## 3 Materials and Methods  
### Advantages of Simulation Modeling  
Researchers use simulation modeling in situations where experiments are not possible because either the process simply does not exist or it is not cost effective to perform in a real-world setting [17]. In some cases, traditional mathematical methods such as queuing theory, differential Equations, and linear programming have been and can be used successfully. However, the complexity and randomness found in healthcare make such methods inadequate because these methods cannot represent the system by just deriving an analytical model [17]. Crisis management and the healthcare model call for simulation modeling, as it is the preferred method for handling the complexity found in real-world settings.  
An event history is a record of when events occurred to a sample of individuals [18]. While event histories are ideal for studying causes of events, they have two varying explanatory variables--censoring and time--that create difficulties for standard statistical practices [18]. As most methods assume that time is measured as a continuous variable (having non-negative values), in some situations discrete-time models and methods may be more appropriate [18]. Some events can occur at any point in time, but available data only records the interval of time within which each event actually occurs [18].  
Computer models can be dynamic or static, depending on the criteria set for the event. In case studies involving emergency room visits, as the state changes, countable points in time can be measured,such as arrival, completion of exam/triage, departure, etc. This falls under discrete event simulation (DES) [4]. DES is the process where behavior found in a complex system is coded as an ordered sequence of clearly defined events [19]. In this context, the event includes a change in the system's state at a specific point in time (i.e., arrival to triage). When a simulation model has random inputs, the output is also random; so assumptions cannot be based on one simulation, but rather replications that include multiple runs. This modeling can be explained as follows: \(X_{1}\), \(X_{2}\), \(X_{i}\)... are each a random process representing the output from a single simulation run. By conducting a fixed amount of simulation runs or replications with a length of \(m\) observations, the parameter of interest is the average across each simulation. As each measurement of \(X_{i}\) is not independent from the next and each are identically distributed, this will give a sample from different random variables. Since the set of averages across these simulations are also independent of each other and identically distributed, the mean and variance of the sample can be found.  
Current studies that includes DES include [PERSON], et al. [4] which demonstrated the use of DES in healthcare for the reduction of unplanned hospital readmissions. In their study, discharged patients and care providers were inputs into the system. Symptom communication was key to the successful use of the DES, and it allowed for better symptom monitoring and behavior analysis [4]. More recently [PERSON], et al. [7] used a DES combining models of epidemiology of dementia with models of health and aged care sectors to estimate accurate future demands of dementia related illness on their healthcare system. This study highlighted the growing demand for resources as well as provided support that individual patient simulations (IPS) methods such as DES allowed for greater realistic depiction of the complex process of unique individuals with compounding complex health histories [7].