Notes in Week 7 - Conceptual Models and Random Variables

To Subscribe, use this Key

Status	Last Update	Fields
Published	11/26/2024	A {{c1::conceptual model}} is a representation of a system using specialized concepts and terms.
Published	11/26/2024	Before building a conceptual model, a {{c1::mental image}} of the system under study must be developed.
Published	11/26/2024	Discrete-Event Systems (DESs) consist of entities that change their {{c1::state}} upon occurrence of events.
Published	11/26/2024	The five key elements of a conceptual model are: {{c1::entity}}, {{c2::attribute}}, {{c3::state variable}}, {{c4::event}}, and {{c5::activity}}.
Published	11/26/2024	Entities in a conceptual model represent {{c1::physical or logical objects}} that must be explicitly captured.
Published	11/26/2024	Attributes are local variables defined inside an entity to store {{c1::data about its state}}.
Published	11/26/2024	State variables track the {{c1::properties of static entities}} over time.
Published	11/26/2024	An event represents a {{c1::stimulus}} that causes the system to change its state.
Published	11/26/2024	An activity is defined as an action performed by the system over a finite {{c1::duration of time}}.
Published	11/26/2024	In the single-server queueing system (SSQS), the three entities are the {{c1::queue}}, {{c2::server}}, and {{c3::person}}.
Published	11/26/2024	The state variables in SSQS include \( Q \), which represents {{c1::the number of people in the queue}}, and \( S \), which represents {{c2::the statu…
Published	11/26/2024	In SSQS, the events driving the system are {{c1::Arrival}}, {{c2::Start_Service}}, and {{c3::End_Service}}.
Published	11/26/2024	The waiting time (WT) in SSQS is the difference between {{c1::arrival time}} and {{c2::start of service time}}.
Published	11/26/2024	The total time a person spends in SSQS is called the {{c1::response time (RT)}}.
Published	11/26/2024	State diagrams use {{c1::circles}} to represent states and {{c2::arrows}} to represent transitions between states.
Published	11/26/2024	A state variable in a DES evolves over time as a {{c1::piecewise constant function}}, also referred to as a {{c2::step function}}.
Published	11/26/2024	Inter-Arrival Time (IAT) is the time between {{c1::two consecutive arrival events}}.
Published	11/26/2024	The probability distributions for IAT and ST in SSQS are typically {{c1::exponential random variables}}.
Published	11/26/2024	Simulated time represents the time inside the {{c1::conceptual model}}, not the program execution time.
Published	11/26/2024	A queue in SSQS can be represented as a {{c1::state variable}} that tracks the number of customers waiting.
Published	11/26/2024	The status of the server in SSQS is a binary state variable that can be either {{c1::Free}} or {{c2::Busy}}.
Published	11/26/2024	The time evolution of state variables can be visualized as {{c1::sample paths}} in a graph.
Published	11/26/2024	In SSQS, when the server is busy, a newly arriving person is added to the {{c1::queue}}.
Published	11/26/2024	A simulation run corresponds to one realization of a system's {{c1::sample path}}.
Published	11/26/2024	State diagrams for SSQS combine the state variables for {{c1::queue length}} and {{c2::server status}}.
Published	11/26/2024	Transitions between states in a state diagram are caused by {{c1::events}}.
Published	11/26/2024	Simulated time advances only during {{c1::events}} in discrete-event systems.
Published	11/26/2024	State variables in DESs are updated only when an {{c1::event occurs}}.
Published	11/26/2024	A random experiment is an activity whose {{c1::outcome is not known}} in advance.
Published	11/26/2024	The sample space, denoted as \( \Omega \), represents the {{c1::set of all possible outcomes}} of a random experiment.
Published	11/26/2024	An event occurs if any of the {{c1::outcomes associated with it}} are observed.
Published	11/26/2024	In probability, all assigned probabilities must sum to {{c1::one}}.
Published	11/26/2024	The relative frequency of an event is the {{c1::ratio of occurrences}} of the event to the total number of trials.
Published	11/26/2024	For a {{c1::discrete}} sample space, the probability of each outcome is given by \( P(\omega_i) = \frac{1}{\|\Omega\|} \).
Published	11/26/2024	For {{c2::continuous}} sample spaces, probabilities are assigned to {{c1::regions}} rather than individual elements.
Published	11/26/2024	Kolmogorov's axioms include the requirement that \( P(\omega_i) {{c1::\in [0, 1] }}\) for all \( \omega_i \in \Omega \).
Published	11/26/2024	The sample mean converges to the true mean as the {{c1::number of trials increases}}.
Published	11/26/2024	The law of large numbers states that the sample mean converges to the {{c1::population mean}}.
Published	11/26/2024	Random numbers are generated using algorithms referred to as {{c1::random number generators (RNGs)}}.
Published	11/26/2024	Pseudo-random numbers mimic true randomness but are generated by {{c1::deterministic algorithms}}.
Published	11/26/2024	A good RNG should produce random numbers that exhibit {{c1::uniformity}} and {{c2::independence}}.
Published	11/26/2024	The Monte Carlo method uses {{c1::random sampling}} to compute estimates.
Published	11/26/2024	A random variable is a function that assigns a {{c1::numerical value}} to outcomes in a sample space.
Published	11/26/2024	Stochastic processes extend random variables by introducing {{c1::time}} as a dimension of variability.
Published	11/26/2024	A sample path is one possible {{c1::realization}} of a stochastic process.
Published	11/26/2024	The ensemble mean is calculated over all possible {{c1::sample paths}}.
Published	11/26/2024	In dynamic systems, the state can be described by {{c1::state variables}} such as queue length and server status.
Published	11/26/2024	The horizontal mean of a stochastic process is computed along {{c1::one sample path}}.
Published	11/26/2024	Each simulation run represents a unique {{c1::trajectory}} in the state space.
Published	11/26/2024	Dynamic systems evolve over time, with state transitions driven by {{c1::events}}.
Published	11/26/2024	A discrete random variable takes values from a {{c1::finite set}}, while a continuous random variable takes values from an {{c2::infinite set}}.
Published	11/26/2024	Stochastic processes can evolve in {{c1::multiple directions}}, each representing a different realization.
Published	11/26/2024	A state space diagram contains all possible {{c1::states of a system}} and their transitions.
Published	11/26/2024	The Monte Carlo method relies on generating {{c1::random samples}} to estimate probabilities or values.
Published	11/26/2024	Simulated systems with random behavior are modeled using {{c1::stochastic processes}}.
Status	Last Update	Fields