lt;br> | $\frac{\bar{N}}{\lambda}$ $\frac{1}{\mu-\lambda}$ faster service rate and smaller arrival rate lowers wainging time | | | | | | | $\bar{T_q}$ average waiting time in a queue | $\frac{\lambda}{\mu(\mu-\lambda)}$ | | | | | | | $P_0$ | | | $\frac{1-\frac{\lambda}{2 \mu}}{1+\frac{\lambda}{2 \mu}}$ | $\left[\frac{(\lambda / \mu)^c}{c !(1-\rho)}+\sum_{k=0}^{c-1} \frac{(\lambda / \mu)^k}{k !}\right]^{-1}$ | | | | $P_k$ | | | $2\left(\frac{\lambda}{2 \mu}\right)^k P_0$ | $P_k= \begin{cases}\frac{(\lambda / \mu)^k}{k !} P_0, & k=1,2, \ldots, c-1 \\ \frac{(\lambda / \mu)^k}{c ! c^{k-c}} P_0, & k=c, c+1, \ldots\end{cases}$ | $\frac{1-\rho}{1-\rho^{K+1}} \rho^nlt;br><br>$\lambda (1-P_k)$arrival at state k | | | diagram | | ![[Pasted image 20240310162654.png\|300]] | | | ![[Pasted image 20240310162431.png\|300]] | | $P_0$ is the only state that doesn't transition to -1 (point mass) departure serve -> because of stochasticity, we have distribution along [P_0, ...,] (not P_0=1) if rho = 1, won't change from initialize $\begin{aligned} & \\ &\end{aligned}$ B1, BG, BO, B4, BP, B6, (5) G1, G2, G3, GR, G6, G7, G8, (7) P1, P2, PR, (3) O1, OR, O4, O5, O6, O7, O8 (7) R1, R2, R3, R6, R7, R8 (6) [time space diagram](https://web.mit.edu/1.041/www/lectures/L2-time-space-diagram-2024sp.pdf), [cumulative diagram](https://web.mit.edu/1.041/www/lectures/L3-cumulative-diagram-2024sp.pdf) + facility dynamics, [traffic flow model](https://web.mit.edu/1.041/www/lectures/L5-traffic-flow-theory-2024sp.pdf) + control, [vehicle dynamics](https://web.mit.edu/1.041/www/lectures/L4-vehicle-motion-2024sp.pdf), numerical integration, lab1 traffic jam [uncertainty](https://web.mit.edu/1.041/www/lectures/L6-probabilistic-concepts-2024sp.pdf), poisson process, queueing model, cumulative diagram + facility dynamics, stochastic + simulation, makov chains, discrete event simulation, lab2 seattle transit queuing | | UNIT_DOT | | --- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | R | [time space diagram](https://web.mit.edu/1.041/www/lectures/L2-time-space-diagram-2024sp.pdf), [cumulative diagram](https://web.mit.edu/1.041/www/lectures/L3-cumulative-diagram-2024sp.pdf)<br>+ facility <br>dynamics, [traffic flow model](https://web.mit.edu/1.041/www/lectures/L5-traffic-flow-theory-2024sp.pdf) + control, [vehicle dynamics](https://web.mit.edu/1.041/www/lectures/L4-vehicle-motion-2024sp.pdf), numerical integration, lab1 traffic jam | | DOT | Chapter | | --- | ----------------------------------------------------------------------------------------------------------------- | | B1 | [Time Space Diagram](https://web.mit.edu/1.041/www/lectures/L2-time-space-diagram-2024sp.pdf) | | BG | [Cumulative Diagram ](https://web.mit.edu/1.041/www/lectures/L3-cumulative-diagram-2024sp.pdf)+ Facility Dynamics | | BO | [Traffic Flow Model + Control](https://web.mit.edu/1.041/www/lectures/L5-traffic-flow-theory-2024sp.pdf) | | B4 | [Vehicle Dynamics](https://web.mit.edu/1.041/www/lectures/L4-vehicle-motion-2024sp.pdf) | | BP | Numerical Integration | | B6 | Lab1 Traffic Jam | | G1 | [Uncertainty](https://web.mit.edu/1.041/www/lectures/L6-probabilistic-concepts-2024sp.pdf) | | G2 | Poisson Process | | G3 | Queueing Model | | BG | Cumulative Diagram + Facility Dynamics | | GR | Stochastic + Simulation | | G6 | Markov Chains | | G7 | Discrete Event Simulation | | G8 | Lab2 Seattle Transit Queuing | | O1 | Sequential Decision Problems | | BO | Traffic Flow Model + Control | | OR | Dynamic + Program | | O4 | Markov Decision Process | | O5 | Value Iteration | | O6 | Q Learning | | O7 | Deep Q Network | | O8 | Lab3 AI Agent Optimizing Traffic | | R1 | Linear Model | | R2 | Simplex Method | | R3 | Math Program | | OR | Dynamic + Program | | GR | Stochastic + Simulation | | R6 | Integer Program | | R7 | Branch & Bound | | R8 | Lab4 TSP Program | | P1 | Uncertainty | | P2 | Stochastic Process | | BP | Numeric Integration | | PR | Continuous Simulations | | | B | G | O | R | P | | -------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ----------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------- | | chapter (dot) | [time space diagram](https://web.mit.edu/1.041/www/lectures/L2-time-space-diagram-2024sp.pdf), [cumulative diagram](https://web.mit.edu/1.041/www/lectures/L3-cumulative-diagram-2024sp.pdf)<br>+ facility <br>dynamics, [traffic flow model](https://web.mit.edu/1.041/www/lectures/L5-traffic-flow-theory-2024sp.pdf) + control, [vehicle dynamics](https://web.mit.edu/1.041/www/lectures/L4-vehicle-motion-2024sp.pdf), numerical integration, lab1 traffic jam | [uncertainty](https://web.mit.edu/1.041/www/lectures/L6-probabilistic-concepts-2024sp.pdf), poisson process, queueing model, cumulative diagram + facility dynamics, stochastic + simulation, makov chains, discrete event simulation, lab2 seattle transit queuing | sequential decision problems, traffic flow model + control, dynamic<br>+ program, markov dec. process, value iteration, Q learning, Deep Q network, lab3 ai agent optimizing traffic | linear model, simplex method, math program, dynamic + program, stochastic + simulation, integer program, branch & bound, lab4 tsp program | uncertainty, stochastic process, numeric integration , continuous simulations | | number of chapter | 6 | 8 | 8 | 8 | 3 | | intersecting chapter | cumulative diagram + facility dynamics with green unit, traffic flow model control with orange unit | cumulative diagram + facility dynamics with green unit, stochastic simulation with red unit | traffic flow model + control with blue unit, dynamic + program with red unit | dynamic program with orange unit, stochastic simulation with green unit | numeric integration with blue unit | | theme | model transportation systems | model transportation systems' agent | optimize transportation systems' agent in multi-stage | optimize transportation systems in single-stage | model transportation systems' environment |