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Can you explain the answer of this question below:
Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line capacity of the yard is limited to 4 trains, find the average number of trains in the system.
The answer is 4.
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Can you explain the answer of this question below:Trains arrive at the...
Problem and Given Information:
- Trains arrive at the yard every 15 minutes
- Service time is 33 minutes
- Line capacity of the yard is limited to 4 trains
Analysis:
- The system consists of the waiting line and the service facility (yard)
- The arrival rate of trains (λ) is 4 trains per hour since a train arrives every 15 minutes
- The service rate of the yard (μ) is 2.18 trains per hour since the service time is 33 minutes (0.55 hours) per train
- The system can accommodate a maximum of 4 trains at a time
- The utilization factor (ρ) of the yard is 0.5 since μ/λ = 2.18/4 = 0.545
Solution:
- The average number of trains in the system (L) can be calculated using Little's Law as L = λW where W is the average time a train spends in the system
- The average time a train spends in the system (W) can be calculated using the formula W = Wq + 1/μ where Wq is the average time a train spends waiting in the queue
- Since the system is in equilibrium (i.e., the number of arrivals equals the number of departures), the average number of trains in the queue (Lq) is Lq = λWq
- Using the M/M/4 queuing model, the average number of trains in the queue can be calculated as Lq = ρ^4(1-ρ)/(1-ρ^5) = 0.206
- Therefore, Wq = Lq/λ = 0.206/4 = 0.0515 hours
- Also, W = Wq + 1/μ = 0.0515 + 1/2.18 = 0.541 hours
- Finally, L = λW = 4 * 0.541 = 2.164 trains
Conclusion:
- The average number of trains in the system is 2.164, which is less than the maximum capacity of 4 trains
- However, since the system is operating at a utilization factor of 0.5, it may experience congestion during peak periods
- Therefore, the line capacity of the yard may need to be increased to accommodate higher arrival rates or reduce the service time.
- Trains arrive at the yard every 15 minutes
- Service time is 33 minutes
- Line capacity of the yard is limited to 4 trains
Analysis:
- The system consists of the waiting line and the service facility (yard)
- The arrival rate of trains (λ) is 4 trains per hour since a train arrives every 15 minutes
- The service rate of the yard (μ) is 2.18 trains per hour since the service time is 33 minutes (0.55 hours) per train
- The system can accommodate a maximum of 4 trains at a time
- The utilization factor (ρ) of the yard is 0.5 since μ/λ = 2.18/4 = 0.545
Solution:
- The average number of trains in the system (L) can be calculated using Little's Law as L = λW where W is the average time a train spends in the system
- The average time a train spends in the system (W) can be calculated using the formula W = Wq + 1/μ where Wq is the average time a train spends waiting in the queue
- Since the system is in equilibrium (i.e., the number of arrivals equals the number of departures), the average number of trains in the queue (Lq) is Lq = λWq
- Using the M/M/4 queuing model, the average number of trains in the queue can be calculated as Lq = ρ^4(1-ρ)/(1-ρ^5) = 0.206
- Therefore, Wq = Lq/λ = 0.206/4 = 0.0515 hours
- Also, W = Wq + 1/μ = 0.0515 + 1/2.18 = 0.541 hours
- Finally, L = λW = 4 * 0.541 = 2.164 trains
Conclusion:
- The average number of trains in the system is 2.164, which is less than the maximum capacity of 4 trains
- However, since the system is operating at a utilization factor of 0.5, it may experience congestion during peak periods
- Therefore, the line capacity of the yard may need to be increased to accommodate higher arrival rates or reduce the service time.
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Community Answer
Can you explain the answer of this question below:Trains arrive at the...
Arrival rate=M= 60/15= 4/min
service rate= ¢=60/33= 1.81/min
Ang length of system= M^2/[¢×(M-¢)]
= 4^2/[1.8(4-1.81)]
=4.02
service rate= ¢=60/33= 1.81/min
Ang length of system= M^2/[¢×(M-¢)]
= 4^2/[1.8(4-1.81)]
=4.02
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Can you explain the answer of this question below:Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line capacity of the yard is limited to 4 trains, find the average number of trains in the system.The answer is 4.
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