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Consider the following data arrival rate = 6 per hour, departure rate = 10 per hour The probability that queue size is greater than 3 is
- a)0.2160
- b)0.1160
- c)0.1296
- d)0.3600
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Consider the following data arrival rate = 6 per hour, departure rate...
λ = 6/hour
μ = 10/hour
ρ(n > 3) = ρn+1
= 0.1296
Most Upvoted Answer
Consider the following data arrival rate = 6 per hour, departure rate...
Data:
Arrival rate = 6 per hour
Departure rate = 10 per hour
To find the probability that the queue size is greater than 3, we can use the M/M/1 queuing model.
M/M/1 Queuing Model:
The M/M/1 queuing model is a simple queuing model that assumes a single server and a Poisson arrival process and exponential service time distribution.
Arrival Rate (λ):
The arrival rate represents the average number of arrivals per unit time. In this case, the arrival rate is 6 per hour.
Departure Rate (μ):
The departure rate represents the average number of departures (or completed service) per unit time. In this case, the departure rate is 10 per hour.
Utilization (ρ):
Utilization is the ratio of the arrival rate to the departure rate. It gives an indication of how busy the system is. In this case, the utilization can be calculated as:
ρ = λ/μ = 6/10 = 0.6
Probability of Queue Size Greater than 3:
To find the probability that the queue size is greater than 3, we can use the formula for the steady-state probability of having n customers in the system, given by:
Pn = (1 - ρ) * ρ^n
In this case, we want to find P(Q > 3), which is equal to 1 - P(Q ≤ 3). Therefore, we need to calculate P(Q ≤ 3) and subtract it from 1.
Calculating P(Q ≤ 3):
P(Q ≤ 3) = P0 + P1 + P2 + P3
Using the formula Pn = (1 - ρ) * ρ^n, we can calculate the probabilities for each value of n:
P0 = (1 - ρ) = (1 - 0.6) = 0.4
P1 = (1 - ρ) * ρ^1 = 0.4 * 0.6^1 = 0.24
P2 = (1 - ρ) * ρ^2 = 0.4 * 0.6^2 = 0.144
P3 = (1 - ρ) * ρ^3 = 0.4 * 0.6^3 = 0.0864
P(Q ≤ 3) = P0 + P1 + P2 + P3 = 0.4 + 0.24 + 0.144 + 0.0864 = 0.8704
Calculating P(Q > 3):
P(Q > 3) = 1 - P(Q ≤ 3) = 1 - 0.8704 = 0.1296
Therefore, the probability that the queue size is greater than 3 is 0.1296, which corresponds to option C.
Arrival rate = 6 per hour
Departure rate = 10 per hour
To find the probability that the queue size is greater than 3, we can use the M/M/1 queuing model.
M/M/1 Queuing Model:
The M/M/1 queuing model is a simple queuing model that assumes a single server and a Poisson arrival process and exponential service time distribution.
Arrival Rate (λ):
The arrival rate represents the average number of arrivals per unit time. In this case, the arrival rate is 6 per hour.
Departure Rate (μ):
The departure rate represents the average number of departures (or completed service) per unit time. In this case, the departure rate is 10 per hour.
Utilization (ρ):
Utilization is the ratio of the arrival rate to the departure rate. It gives an indication of how busy the system is. In this case, the utilization can be calculated as:
ρ = λ/μ = 6/10 = 0.6
Probability of Queue Size Greater than 3:
To find the probability that the queue size is greater than 3, we can use the formula for the steady-state probability of having n customers in the system, given by:
Pn = (1 - ρ) * ρ^n
In this case, we want to find P(Q > 3), which is equal to 1 - P(Q ≤ 3). Therefore, we need to calculate P(Q ≤ 3) and subtract it from 1.
Calculating P(Q ≤ 3):
P(Q ≤ 3) = P0 + P1 + P2 + P3
Using the formula Pn = (1 - ρ) * ρ^n, we can calculate the probabilities for each value of n:
P0 = (1 - ρ) = (1 - 0.6) = 0.4
P1 = (1 - ρ) * ρ^1 = 0.4 * 0.6^1 = 0.24
P2 = (1 - ρ) * ρ^2 = 0.4 * 0.6^2 = 0.144
P3 = (1 - ρ) * ρ^3 = 0.4 * 0.6^3 = 0.0864
P(Q ≤ 3) = P0 + P1 + P2 + P3 = 0.4 + 0.24 + 0.144 + 0.0864 = 0.8704
Calculating P(Q > 3):
P(Q > 3) = 1 - P(Q ≤ 3) = 1 - 0.8704 = 0.1296
Therefore, the probability that the queue size is greater than 3 is 0.1296, which corresponds to option C.
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Consider the following data arrival rate = 6 per hour, departure rate = 10 per hour The probability that queue size is greater than 3 isa) 0.2160b) 0.1160c) 0.1296d) 0.3600Correct answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Consider the following data arrival rate = 6 per hour, departure rate = 10 per hour The probability that queue size is greater than 3 isa) 0.2160b) 0.1160c) 0.1296d) 0.3600Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following data arrival rate = 6 per hour, departure rate = 10 per hour The probability that queue size is greater than 3 isa) 0.2160b) 0.1160c) 0.1296d) 0.3600Correct answer is option 'C'. Can you explain this answer?.
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