If another mechanic is hired, fill up the table below and answer the following questions (1pt):
a. Which configuration has the lowest Ls?
b. Which configuration has the lowest Ws?
c. Which configuration has the lowest Wq?
d. Which configuration has the lowest Total Cost (please refer question 4)?
e. What would be your recommendations on which system to use?
System
Single Queue with Team Service
(M/M/1 with 2 µ service rate)
Multiple Queue and Multiple Servers
(M/M/1, M/M/1 with arrival rate = λ/2)
Single Queue with Multiple Servers
(M/M/2)
To determine the answers, we need to calculate the various performance metrics for each configuration. Let's calculate the values for each configuration step-by-step:
Configuration 1: Single Queue with Team Service (M/M/1 with 2 µ service rate)
Performance Metrics:
- Ls (Average number of customers in the system)
Formula: Ls = λ / (µ - λ)
Given: λ = arrival rate = ? (unknown), µ = service rate = 2
Since we don't have the arrival rate, we cannot calculate Ls for this configuration.
Configuration 2: Multiple Queue and Multiple Servers (M/M/1, M/M/1 with arrival rate = λ/2)
Performance Metrics:
- Ls (Average number of customers in the system)
Formula: Ls = (λ^2) / (2µ(µ - λ))
Given: λ = arrival rate = ? (unknown), µ = service rate = 1
Since we don't have the arrival rate, we cannot calculate Ls for this configuration.
Configuration 3: Single Queue with Multiple Servers (M/M/2)
Performance Metrics:
- Ls (Average number of customers in the system)
Formula: Ls = λ / (µ - λ)
Given: λ = arrival rate = ? (unknown), µ = service rate = 2
Since we don't have the arrival rate, we cannot calculate Ls for this configuration.
Now, let's try to answer the questions using the available information:
a. Which configuration has the lowest Ls?
We cannot determine the configuration with the lowest Ls since we don't have the arrival rate for any of the configurations.
b. Which configuration has the lowest Ws?
We cannot determine the configuration with the lowest Ws since we don't have the arrival rate for any of the configurations.
c. Which configuration has the lowest Wq?
We cannot determine the configuration with the lowest Wq since we don't have the arrival rate for any of the configurations.
d. Which configuration has the lowest Total Cost (please refer question 4)?
Since we don't have the total cost equation provided in question 4, we cannot determine which configuration has the lowest total cost.
e. What would be your recommendations on which system to use?
Based on the information provided, it is not possible to make any recommendations on which system to use since we don't have the arrival rate for any of the configurations. Without the arrival rate, we cannot calculate the performance metrics needed for comparison.
To answer the questions, we need to compare the performance of each configuration in terms of Ls, Ws, Wq, and Total Cost.
1. Ls: Ls refers to the average number of customers in the system, including those in service and those waiting in the queue. To determine which configuration has the lowest Ls, we need to calculate the average number of customers for each system.
- Single Queue with Team Service (M/M/1 with 2 µ service rate): Since this is an M/M/1 system, the formula to calculate Ls is Ls = λ/ (µ - λ). Since µ = 2 and λ is the arrival rate, we need to know the arrival rate of customers to calculate Ls for this configuration.
- Multiple Queue and Multiple Servers (M/M/1, M/M/1 with arrival rate = λ/2): This configuration consists of multiple queues and multiple servers. The formula for calculating Ls in this system is Ls = λ / (µ - λ) + (λ/2)* λ / (µ - λ)². Again, we need to know the arrival rate λ to calculate Ls.
- Single Queue with Multiple Servers (M/M/2): The formula for calculating Ls in this system is Ls = λ / (µ - λ)². We need to know the arrival rate λ to calculate Ls for this configuration.
To compare the Ls values for each system and identify the one with the lowest value, we need the arrival rate λ for each configuration.
2. Ws: Ws refers to the average time a customer spends in the entire system, including both service time and waiting time. To determine which configuration has the lowest Ws, we need to calculate the average time spent for each system.
- Single Queue with Team Service (M/M/1 with 2 µ service rate): The formula to calculate Ws in an M/M/1 system is Ws = 1 / (µ - λ). We need to know the arrival rate λ for this configuration.
- Multiple Queue and Multiple Servers (M/M/1, M/M/1 with arrival rate = λ/2): The formula for calculating Ws in this system is Ws = 1 / (µ - λ) + λ / (µ - λ)². We need to know the arrival rate λ for this configuration.
- Single Queue with Multiple Servers (M/M/2): The formula for calculating Ws in this system is Ws = 1 / (µ - λ)². We need to know the arrival rate λ for this configuration.
To compare the Ws values for each system and identify the one with the lowest value, we need the arrival rate λ for each configuration.
3. Wq: Wq refers to the average time a customer spends waiting in the queue before being served. To determine which configuration has the lowest Wq, we need to calculate the average waiting time for each system.
- Single Queue with Team Service (M/M/1 with 2 µ service rate): The formula to calculate Wq in an M/M/1 system is Wq = λ / (µ*(µ - λ)). We need to know the arrival rate λ for this configuration.
- Multiple Queue and Multiple Servers (M/M/1, M/M/1 with arrival rate = λ/2): The formula for calculating Wq in this system is Wq = λ / (µ*(µ - λ)) + (λ/2)*(λ / (µ*(µ - λ))). We need to know the arrival rate λ for this configuration.
- Single Queue with Multiple Servers (M/M/2): The formula for calculating Wq in this system is Wq = λ / (2µ*(µ - λ))). We need to know the arrival rate λ for this configuration.
To compare the Wq values for each system and identify the one with the lowest value, we need the arrival rate λ for each configuration.
4. Total Cost: The total cost for each configuration should be provided in question 4. We can compare the total costs and identify the configuration with the lowest value.
Once we have the necessary information, we can determine the configuration with the lowest Ls, Ws, Wq, and Total Cost, and make recommendations based on the results.