In a M/M/1 system, the inter-arrival time of dumpers to a shovel follows exponential distribution with a mean arrival rate of 9 dumpers per hour. The service time of the shovel fol... In a M/M/1 system, the inter-arrival time of dumpers to a shovel follows exponential distribution with a mean arrival rate of 9 dumpers per hour. The service time of the shovel follows exponential distribution with a mean service rate of 12 dumpers per hour. The probability that exactly one dumper is available to the shovel is?
Understand the Problem
The question is asking for the probability that exactly one dumper is available to the shovel in an M/M/1 queueing system with specified arrival and service rates.
Answer
The probability that exactly one dumper is available is $\frac{3}{16}$.
Answer for screen readers
The probability that exactly one dumper is available to the shovel is $\frac{3}{16}$.
Steps to Solve
- Identify arrival and service rates
The arrival rate $\lambda$ is given as 9 dumpers per hour, and the service rate $\mu$ is 12 dumpers per hour.
- Calculate traffic intensity
Traffic intensity $\rho$ is calculated using the formula: $$ \rho = \frac{\lambda}{\mu} $$ Substituting the values gives: $$ \rho = \frac{9}{12} = 0.75 $$
- Use the M/M/1 queue probability formula
The probability of having exactly $n$ customers in an M/M/1 queue is given by: $$ P(n) = (1 - \rho) \cdot \rho^n $$ For exactly 1 dumper ($n = 1$): $$ P(1) = (1 - 0.75) \cdot (0.75)^1 $$
- Calculate the probability
Calculating $P(1)$: $$ P(1) = 0.25 \cdot 0.75 = 0.1875 $$
- Convert to a fraction
Convert $0.1875$ to a fraction: $$ 0.1875 = \frac{3}{16} $$
The probability that exactly one dumper is available to the shovel is $\frac{3}{16}$.
More Information
In an M/M/1 queueing system, the model assumes that arrivals are random (following Poisson distribution), and services are also random (following exponential distribution). This system is widely used in fields such as telecommunications, traffic engineering, and operations management.
Tips
- Forgetting to convert decimal probabilities into fractions when required.
- Misapplying the formula for M/M/1 queues, especially in determining $\rho$.
- Not ensuring that the arrival rate is less than the service rate before applying the formulas to ensure stability.
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