A queueing system has one single server workstation that admits an infinitely long queue. The rate of arrival of jobs to the queueing system follows the Poisson distribution with a... A queueing system has one single server workstation that admits an infinitely long queue. The rate of arrival of jobs to the queueing system follows the Poisson distribution with a mean of 5 jobs/hour. The service time of the server is exponentially distributed with a mean of 6 minutes. In steady state operation of the queueing system, the probability that the server is not busy at any point in time is?
Understand the Problem
The question is asking to determine the probability that a server in a queuing system is not busy at any point in time, given the arrival and service rates of jobs. This will involve analyzing the characteristics of the queuing system, typically applying the formulas for M/M/1 queues.
Answer
$0.5$
Answer for screen readers
The probability that the server is not busy at any point in time is $0.5$.
Steps to Solve
- Identify the arrival and service rates
The rate of arrival of jobs, $\lambda$, is given as 5 jobs/hour.
The service time is 6 minutes, which can be converted to hours:
$$ \text{Service time} = \frac{6 \text{ minutes}}{60 \text{ minutes/hour}} = 0.1 \text{ hours} $$
The service rate, $\mu$, is the reciprocal of the service time:
$$ \mu = \frac{1}{\text{Service time}} = \frac{1}{0.1} = 10 \text{ jobs/hour} $$
- Find the traffic intensity
The traffic intensity, denoted as $\rho$, is calculated using the formula:
$$ \rho = \frac{\lambda}{\mu} $$
Substituting the values:
$$ \rho = \frac{5}{10} = 0.5 $$
- Calculate the probability that the server is not busy
The probability that the server is not busy (idle), $P_0$, for an M/M/1 queue is given by:
$$ P_0 = 1 - \rho $$
Substituting the value of $\rho$:
$$ P_0 = 1 - 0.5 = 0.5 $$
The probability that the server is not busy at any point in time is $0.5$.
More Information
In an M/M/1 queue, the server's busy and idle probabilities are integral in analyzing performance. In this case, with a traffic intensity of 0.5, the server is busy half the time.
Tips
- Not converting time units: Ensure all time measurements are in the same unit (e.g., hours) before using in calculations.
- Wrong application of formulas: Verify correct use of the formulas for M/M/1 queues when calculating traffic intensity and idle probabilities.
AI-generated content may contain errors. Please verify critical information
