1. **Problem statement:** We analyze equilibrium costs and arrival times in a bottleneck model where downstream capacity is $s$. The goal is to determine the time window for arrivals, equilibrium cost, queueing delay function, and total system cost. 2. **Given:** - Downstream capacity $s$ - Total number of travelers $N$ - The equilibrium arrival times satisfy $t_E = t_S + \frac{N}{s}$. - The cost function at equilibrium for first and last travelers is equal: $f(t_E) = f(t_S)$. - Cost functions for early and late arrivals: $f(t) = \beta (t^* - t)$ for $t < t^*$ and $f(t) = \gamma(t - t^*)$ for $t > t^*$. 3. **Equations:** - $t_E = t_S + \frac{N}{s}$ - $\beta (t^* - t_S) = \gamma (t_E - t^*)$ 4. **Solve for $t_S$ and $t_E$: ** Substitute $t_E$ from first equation into second $$\beta (t^* - t_S) = \gamma (t_S + \frac{N}{s} - t^*)$$ Rearranging, $$\beta t^* - \beta t_S = \gamma t_S + \gamma \frac{N}{s} - \gamma t^*$$ Group terms, $$\beta t^* + \gamma t^* - \gamma \frac{N}{s} = \beta t_S + \gamma t_S = t_S(\beta + \gamma)$$ Solve for $t_S$: $$t_S = t^* - \frac{\gamma}{\beta + \gamma} \frac{N}{s}$$ Similarly, $$t_E = t_S + \frac{N}{s} = t^* + \frac{\beta}{\beta + \gamma} \frac{N}{s}$$ 5. **Equilibrium cost $c$ for all travelers:** Using $f(t_S)$ or $f(t_E)$, $$c = f(t_S) = \beta (t^* - t_S) = \beta \frac{\gamma}{\beta + \gamma} \frac{N}{s} = \frac{\beta \gamma}{\beta + \gamma} \frac{N}{s}$$ 6. **Queueing delay $Q(t)$ experienced by traveler arriving at time $t$:** $$Q(t) = c - f(t) = \frac{\beta \gamma}{\beta + \gamma} \frac{N}{s} - \beta \max(0, t^* - t) - \gamma \max(0, t - t^*)$$ 7. **Total system cost $TSC$ experienced by all travelers:** Multiply individual cost $c$ by number of travelers $N$: $$TSC = cN = \frac{\beta \gamma}{\beta + \gamma} \frac{N^2}{s}$$ This corresponds to the area of the rectangle shown in the figure, scaled by capacity $s$. 8. **Interpretation:** - Travelers choose arrival times within $[t_S, t_E]$ such that costs are equal. - Queueing delays reflect early and late arrival penalties adjusted by cost factors $\beta$ and $\gamma$. **Final answers:** $$t_S = t^* - \frac{\gamma}{\beta + \gamma} \frac{N}{s}, \quad t_E = t^* + \frac{\beta}{\beta + \gamma} \frac{N}{s}$$ $$c = \frac{\beta \gamma}{\beta + \gamma} \frac{N}{s}$$ $$Q(t) = \frac{\beta \gamma}{\beta + \gamma} \frac{N}{s} - \beta [t^* - t]_+ - \gamma [t - t^*]_+$$ $$TSC = \frac{\beta \gamma}{\beta + \gamma} \frac{N^2}{s}$$