Network Model Of Commuters' Choices In The Case Shown In Fig

Network Model Of Commuters Choicesin The Case Shown In Figure P63 A

Network model of commuter's choices In the case shown in Figure P6.3, a commuter wishes to travel from his residence near station A to his place of employment at D. The commuter's transportation choices are: Ride a bus from A to D; ride on subway line 1 to station B, then transfer to subway line 2 to D; or ride on subway line 1 to station C, then transfer to subway line 3 to D. Ride times for each leg are provided, with headways between vehicles being exactly 10 minutes, and schedules coordinated so that transfers occur with minimal waiting. Transfers are considered to have negligible time. The problem involves two scenarios: one where the commuter times his arrival to coincide with vehicle departures, and another where arrivals are random due to stochastic headways modeled by exponential distributions. The goal is to develop network models suitable for shortest path algorithms to determine static and expected travel times, and to identify the optimal route and mode choice.

Paper For Above instruction

The scenario presented involves a commuter choosing among multiple transit options to travel from station A to station D, with considerations of schedule coordination and stochastic variability. This problem lends itself to network modeling, which facilitates the application of shortest-path algorithms for route optimization under both deterministic and stochastic conditions.

Deterministic Network Model (Part a)

In the initial deterministic case, where the commuter is aware of schedules and times arrivals to coincide perfectly, the network model simplifies to a static shortest-path problem. Nodes in the network represent transit points (A, B, C, D), while edges serve as links denoting transportation segments and transfers, each associated with known travel times.

The network includes the following nodes: the starting point A, transfer nodes B and C, and the destination D. Edges include: A to B through subway line 1, B to D via subway line 2, A to C via subway line 1, and C to D via subway line 3, as well as the direct bus from A to D.

The travel times are assigned to each edge based on the data provided: 25 minutes for the bus from A to D, specific times for subway legs, and negligible transfer times. The schedule coordination ensures that transfers at B and C occur with minimal or no waiting, hence making the transfer times effectively zero. The network model thus consists of weighted edges with deterministic costs, suitable for algorithms like Dijkstra's for shortest path determination.

The shortest path algorithm applied to this network yields the optimal route—either directly via bus or via the subway lines with transfers—that minimizes total travel time, assuming the commuter aligns arrivals to vehicle departures.

Stochastic Network Model (Part b)

In the stochastic scenario, the headways are random variables following exponential distributions with a mean of 10 minutes. This reflects real-world variability in transit arrival times. The random nature of headways affects the total expected travel time, necessitating a probabilistic network model.

Here, the wait times at each transit node are stochastic, modeled as the residual waiting time for a random arrival. For exponential distributions, the expected waiting time for an arriving passenger is equal to the mean headway divided by 2 due to the memoryless property, which results in an expected waiting time of 5 minutes (half of 10 minutes). The total expected travel time on each route now comprises the fixed travel times (as previously obtained) plus the expected waiting times at each transfer point.

To model this, each transfer and segment is represented by an edge with a stochastic travel time: the fixed travel time plus the expected waiting time at transfer points and boarding. These are aggregated to compute the total expected travel time for each route, accounting for the independence and exponential distribution of headways. The underlying network becomes a probabilistic graph where edge weights are expected travel times, enabling the use of modified shortest path algorithms designed for stochastic optimization (e.g., risk-aware shortest path algorithms).

Comparison of Routes and Expected Travel Times

The deterministic model favors routes that align with scheduled vehicle arrivals, often resulting in the fastest possible trip when timing is ideal. In contrast, the stochastic model accounts for variability and generally results in longer expected travel times, especially if waiting times at transfers are significant.

In the deterministic case, the minimal total travel time might favor transferring via the subway lines if their combined schedule and travel times are less than the direct bus. In the stochastic case, the expected travel times might shift preferences towards routes with higher frequency but longer fixed travel times, because waiting times are reduced statistically.

Quantitative results depend on actual data; based on schedule coordination, the shortest deterministic route could be via the subway system given the precise schedule alignment, with a total of, for example, approximately 20 minutes (assuming initial times), whereas the shortest expected travel time in the stochastic model could be slightly longer, say around 22-25 minutes, due to average waiting times at transfers.

Conclusion

The development of both deterministic and stochastic network models provides valuable insights into commuter route choices under different information scenarios. The deterministic model emphasizes schedule coordination, facilitating straightforward shortest-path calculations. The stochastic model recognizes real-world variability, leading to a probabilistic approach that captures average travel times and uncertainties. Commuters and transit planners can utilize these models to optimize routing strategies and improve schedule coordination, thereby reducing overall travel time and increasing reliability of urban transit systems.

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