### Queuing Theory Problem Description and Solution#### Problem Statement:Vehicles begin to arrive at a parking lot at 7:00 AM at a rate of 2000 vehicles per hour (veh/hour). The demand reduces to 1000 veh/hour at 7:30 AM and continues at that rate. The ticketing booth to enter the parking lot can only serve vehicles at a rate of 1000 veh/hour until 7:15 AM, after which the service rate increases to 2000 veh/hour. Assuming D/D/1 queuing, we are required to:1. Draw a queuing diagram for this situation.2. Determine: a) The time at which the queue clears, b) The total delay, c) The longest queue length, d) The wait time of the 500th and 2000th vehicles to arrive at the parking lot (assuming FIFO conditions).#### Solution:1. **Queuing Diagram**: - Create a time vs. cumulative vehicles graph. - Plot the arrival rate initially at 2000 veh/hour from 7:00 AM to 7:30 AM. - Change the arrival rate to 1000 veh/hour from 7:30 AM onwards. - Plot the service rate initially at 1000 veh/hour from 7:00 AM to 7:15 AM. - Change the service rate to 2000 veh/hour from 7:15 AM onwards.2. **Calculations**: - **Time at which the queue clears**: By analyzing the intersection points of arrival and service curves. - **Total delay**: Calculate the area between the arrival and service rate plots before they intersect. - **Longest queue length**: Determine the maximum vertical distance between the arrival and service rate lines. - **Wait time of specific vehicles (500th and 2000th)**: Use the cumulative arrival plot to find these times based on FIFO queuing.---These detailed steps guide you through understanding the queuing scenario, constructing a helpful diagram, and calculating required metrics to analyze the given situation effectively.

D/D/1 queu analysis is done for linear arrival and service rate . To analyse this problem we have…

Vehicles begin to arrive to a parking lot at 7:00 AM at a rate of 2000 veh/hour, but the demand reduces to 1000 veh/hour at 7:30 AM and continues at that rate. The ticketing booth to enter the parking lot can only serve the vehicles at 1000 veh/hour until 7:15 AM, after which the service rate increases to 2000 veh/hour. Assuming D/D/1 queuing, draw a queuing diagram for this situation. Find: a) the time at which the queue clears, b) the total delay, c) the longest queue length, and d) wait time of the 500th and 2000th vehicles to arrive to the parking lot (assuming FIFO conditions)?

Vehicles begin to arrive to a parking lot at 7:00 AM at a rate of 2000 veh/hour, but the demand reduces to 1000 veh/hour at 7:30 AM and continues at that rate. The ticketing booth to enter the parking lot can only serve the vehicles at 1000 veh/hour until 7:15 AM, after which the service rate increases to 2000 veh/hour. Assuming D/D/1 queuing, draw a queuing diagram for this situation. Find: a) the time at which the queue clears, b) the total delay, c) the longest queue length, and d) wait time of the 500th and 2000th vehicles to arrive to the parking lot (assuming FIFO conditions)?

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Question

### Queuing Theory Problem Description and Solution

#### Problem Statement:
Vehicles begin to arrive at a parking lot at 7:00 AM at a rate of 2000 vehicles per hour (veh/hour). The demand reduces to 1000 veh/hour at 7:30 AM and continues at that rate. The ticketing booth to enter the parking lot can only serve vehicles at a rate of 1000 veh/hour until 7:15 AM, after which the service rate increases to 2000 veh/hour.

Assuming D/D/1 queuing, we are required to:
1. Draw a queuing diagram for this situation.
2. Determine:
a) The time at which the queue clears,
b) The total delay,
c) The longest queue length,
d) The wait time of the 500th and 2000th vehicles to arrive at the parking lot (assuming FIFO conditions).

#### Solution:

1. **Queuing Diagram**:
- Create a time vs. cumulative vehicles graph.
- Plot the arrival rate initially at 2000 veh/hour from 7:00 AM to 7:30 AM.
- Change the arrival rate to 1000 veh/hour from 7:30 AM onwards.
- Plot the service rate initially at 1000 veh/hour from 7:00 AM to 7:15 AM.
- Change the service rate to 2000 veh/hour from 7:15 AM onwards.

2. **Calculations**:
- **Time at which the queue clears**: By analyzing the intersection points of arrival and service curves.
- **Total delay**: Calculate the area between the arrival and service rate plots before they intersect.
- **Longest queue length**: Determine the maximum vertical distance between the arrival and service rate lines.
- **Wait time of specific vehicles (500th and 2000th)**: Use the cumulative arrival plot to find these times based on FIFO queuing.

---

These detailed steps guide you through understanding the queuing scenario, constructing a helpful diagram, and calculating required metrics to analyze the given situation effectively.

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Step 3: Calculating Total Delay.

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