Why Holiday Traffic Jams Triple Travel Time: A Mathematical Model

The Mid‑Autumn moon is not yet full, but the National Day atmosphere is already strong. Today, the first day of the long holiday, is a moment of family reunion. Both high‑speed trains and crowded highways carry people's longing for home, yet travelers quickly discover they are not alone on the road. Netizens joke that "smart people got on the highway early, only to find it full of smart people..." Traffic congestion becomes a major headache during the holiday; a distance that is usually short feels stretched. This article attempts to explore the logic and patterns behind holiday traffic congestion through mathematical modeling, hoping to offer suggestions.

Overall Model Overview

Travel time can be divided into four main parts: preparation time, actual highway driving time, and the times for entering and exiting the highway.

Highway Driving Time Analysis

We consider a one‑way highway without any intersections or additional entrances, only a start and an end. Vehicles travel at a constant speed, but must decelerate when a preceding vehicle is too close.

Basic Assumptions

Each vehicle’s average speed on the highway is v , but if the distance to the vehicle ahead is less than a safety distance, the vehicle will decelerate.

The safety distance is proportional to speed, i.e., d = k·v , where k is a constant.

Vehicle arrival intervals are random; on average one vehicle enters the highway every τ seconds.

Model

The relationship between vehicle density ρ and vehicle number N is ρ = N / L , where L is the highway length.

The average speed v is affected by the vehicles ahead. As density increases, the available space per vehicle decreases, reducing average speed. Thus v can be expressed as a decreasing function of ρ (specific formula omitted for brevity).

Waiting time can be regarded as the extra time caused by deceleration, leading to a non‑linear relationship between waiting time and vehicle number.

Entry and Exit Time Model

Entry time includes queuing at toll stations, payment, and waiting for preceding vehicles to clear. Exit time is analogous, also involving toll‑station queuing and payment.

Queueing Time Model and Analysis

Queueing theory provides the mathematical framework for estimating average queue length.

Basic Parameters

Arrival rate λ : number of vehicles arriving per unit time.

Service rate μ : number of vehicles that can be processed per unit time.

M/M/1 Queue Model

In the M/M/1 model, arrivals follow a Poisson process, service times are exponentially distributed, and there is a single service channel (e.g., one toll booth). The average queue length is L_q = λ^2 / (μ (μ - λ)) , applicable only when λ < μ .

Finite‑Length Queue System

If the arrival rate exceeds the service rate but the queue length is limited to K , the system reaches a steady state with probabilities P_n for having n vehicles in the queue. The average queue length is computed from these probabilities.

Case Study and Summary

We first determine normal‑period and holiday‑period arrival rates. Assume the holiday arrival rate is four times the normal rate.

Typical highway design capacity is 1,500–2,200 vehicles per lane per hour. For a three‑lane highway, this yields 4,500–6,600 vehicles per hour. Assuming an average inter‑vehicle distance of 20 m gives a density of about 50 vehicles per kilometer.

Normal flow: 2,250 vehicles per hour ≈ 0.625 veh/s. Holiday flow: ≈ 2.5 veh/s.

During peak holiday periods, density may rise to 60–80 vehicles per kilometer.

Each vehicle spends about 1.25 seconds at a toll station, giving a service rate of 0.8 veh/s.

Waiting Time

Normal period: Using the M/M/1 model, the average queue length is about 2.79 vehicles.

Holiday period: Assuming a maximum queue capacity of 50 vehicles, the finite‑length model yields an average queue length of about 49.53 vehicles, nearly filling the capacity.

Highway Driving Time

Highway length assumed to be 200 km.

Base speed (no traffic) assumed to be 120 km/h.

Average toll‑station dwell time per vehicle: 30 seconds.

Using the earlier formulas, the total travel time (preparation time ignored) is approximately 4.89 hours under normal conditions and 15.37 hours during the holiday, i.e., more than three times longer.

Sensitivity Analysis

The following trends are observed:

Higher holiday arrival rates increase total travel time.

Higher service rates (faster toll processing) reduce total travel time.

Increasing the maximum queue length slightly raises travel time, but the effect plateaus after a certain point.

Longer highway length proportionally increases travel time.

Higher base vehicle speed reduces travel time.

Higher maximum vehicle density increases travel time.

The model clearly shows that holiday traffic congestion dramatically lengthens travel time due to increased vehicle density and prolonged queuing at entry and exit points. May each journey be safe and reunite families. Happy Mid‑Autumn and National Day!