Queuing Models

How to Analyze a System*

*Simulation, Modeling & Analysis (3/e) by Law and Kelton, 2000, p. 4, Figure 1.1

Traffic Queues

• Queues form at intersections and roadway bottlenecks, especially during congested periods and are a source of considerable delay.

• Queuing theory is not unique to traffic analysis, (industrial plants, retail stores, service-oriented industries).

• The purpose of studying traffic queuing is to provide means to estimate important measures of highway performance including vehicular delay

Queuing models are derived from underlying assumptions about:

A. Arrival Patterns Equal time intervals – uniform or deterministic intervals Exponentially distributed time intervals- Poisson

B. Departure Characteristics Given average vehicle departure rate, the assumption of a

deterministic or exponential distribution is appropriate. Number of departure channels; example of multiple departure

channels in traffic?

C. Queue Discipline First-in-first-out (FIFO) Last-in-first-out (LIFO)Which one is realistic for traffic queues?

Queuing Theory

Arrival Rate Assumption

D = Deterministic or Uniform Distribution

M = Exponential Distribution

Departure Rate Assumption

D = Deterministic or Uniform Distribution

M = Exponential Distribution

Number of Departure Channels

Queuing Theory: Assumptions Defining Queuing Regime

EXAMPLE

Vehicles arrive at an entrance to a national park. There is a single gate (at which all vehicles must stop) where a ranger distributes a free brochure. The park opens at 8:00 a.m., at that time vehicles begin to arrive at the rate of 480 vehicles/hour. After 20 minutes, the flow rate declines to 120 vehicles/hour and continues at that level for the remainder of the day. If the time required to distribute the brochure is 15 seconds, determine:

Whether a queue will form, and if so, how long is the maximum queue?

How long is the maximum delay?

When will the queue dissipate?

D/D/1 Queuing Regime

Graphical Approach

EXAMPLE

A freeway has a directional capacity of 4000 vehicles/hr and a constant flow of 2900 vehicles/hr during the morning commute to work (i.e., no adjustments to traffic flow are produced by the incident). At 8:00 a.m. a traffic accident closes the freeway to all flow. At 8:12 a.m. the freeway is partially opened with a capacity of 2000 vehicles/hr. Finally, the wreckage is removed and the freeway is restored to full capacity (4000 vehicles/hr) at 8:31 a.m.. Assume a D/D/1 queuing regime to determine total delay, longest queue length, time of queue dissipation, and longest vehicle delay.

Traffic Analysis at Highway Bottlenecks

Use to determine delay on traffic signals

• Total Number of vehicles delayed

• V = 𝑣 𝑅 + 𝑡𝑐 = 𝑠𝑡𝑐

• 𝑡𝑐 =𝑣 𝐶−𝑔

𝑠−𝑣→ V =

𝑠𝑣 𝐶−𝑔

𝑠−𝑣

• Total Delay (veh-sec) = 1

2V 𝐶 − 𝑔 =

1

2

𝑠𝑣 𝐶−𝑔 2

𝑠−𝑣

• Average delay over a signal cycle:Total Delay

𝑣𝐶=

1

2

𝑠 𝐶−𝑔 2

𝑠−𝑣 𝐶

• 𝑠 =𝑐𝑔

𝐶

→ Average Delay =𝐶 1−

𝑔

𝐶

2

2 1−𝑔

𝐶𝑋

where 𝑋 =𝑣

𝑐

Highway bottlenecks can be generally defined as a section of highway with lower capacity than the incoming section of the highway.

Sources for the reduction in capacity:Decrease in number of through traffic lanes

Reduced shoulder widths

Presence of traffic signals

Traffic Analysis at Highway Bottlenecks

Recurring (e.g., physical reduction in number of lanes)

Incident-provoked (e.g., vehicle breakdown or accident): Incident bottlenecks are unanticipated, temporary, and have varying capacity over time.

Type of Highway Bottlenecks

Arrival Rate Assumption

D = Deterministic or Uniform Distribution

M = Exponential Distribution

Departure Rate Assumption

D = Deterministic or Uniform Distribution

M = Exponential Distribution

Number of Departure Channels

Queuing Theory: Assumptions Defining Queuing Regime

Markov Process

• A Markov process is a random process that undergoes transitions from one state to another on a state space.

• It is a memoryless process, i.e. the probability distribution of the next state depends only on the current state.

• Assume stability of distribution.

Method for assessing Independence

Method for assessing Independence

• Scatter Plots

Method for Assessing Stability of Distribution

Tests for Distributions

Tests for Distributions

• Chi-square test

• Kolmogorov-Smirnov test

• Anderson Darling Test

2

1

mk k

k k

O E

E

Little’s Law

• For a given arrival rate, the time in the system is proportional to packet occupancy

N = T

where

N: average # of vehicles

: vehicle arrival rate (packets per unit time)

T: average delay (time in the system) per vehicle

• Examples:• On rainy days, streets and highways are more crowded• Fast food restaurants need a smaller dining room than regular restaurants with the

same customer arrival rate

Arrival Pattern

• Most commonly is assumed to have a Poisson Distribution

• Holds kind of true in low-medium traffic

Poisson Distribution is Discrete

Headway Distribution of a Poisson Arrival

• If no vehicle arrives during a given time period (i.e. x=0)

M/D/1

• M stands for an exponential distribution of inter-arrival time or a poisson arrival process. Arrival rate is v, i.e. the flow

• D is a deterministic departure process. The departure rate is c, i.e. the capacity

• 1 is the number of channels

Steady State Conditions

• 𝑣𝑃𝑖−1 = 𝑐𝑃𝑖• Recursively solve this

Workshop Question

• Vehicles arrive at an entrance to a national park. There are five gates (at which all vehicles must stop) where a ranger distributes a free brochure. The park opens at 7:00 a.m., at that time vehicles begin to arrive at the rate of 100-2t vehicles/min (where t is the time elapsed after 7:00AM). After 35 minutes, the flow rate declines to 10 vehicles/minute and continues at that level for the remainder of the day. If the time required to distribute the brochure is 6 seconds, determine:

• Whether a queue will form, and if so, how long is the maximum queue?

• After one hour, how many vehicles will be let into the park, and what will be the length of the queue?

• How long is the maximum delay and at what time will the queue dissipate?