The jobs arrive at a facility, for service, in a random manner. The probability distribution of number of arrivals of jobs in a fixed time interval is
- ANormal
- BPoisson
- CErlang
- DBeta
Solution & Step-by-step Explanation
Understanding Job Arrival Probability
When jobs arrive at a facility for service in a random manner, it implies that the exact time of each arrival is unpredictable, but there might be an average rate at which they occur. To model the "number of arrivals of jobs in a fixed time interval", we need a specific type of probability distribution that handles discrete counts of events happening randomly over a continuous period.
Poisson Distribution for Random Arrivals
The correct probability distribution for the scenario described – "jobs arrive at a facility, for service, in a random manner" and we are interested in the "number of arrivals of jobs in a fixed time interval" – is the Poisson distribution.
- The Poisson distribution is a discrete probability distribution that models the probability of a given number of events happening in a fixed interval of time or space. This is applicable when these events occur with a known constant mean rate and independently of the time since the last event.
- In the context of job arrivals, this means that if jobs arrive one by one, randomly, and at an average rate, the Poisson distribution is the ideal tool to predict the likelihood of seeing a certain number of arrivals (e.g., 5 jobs, 10 jobs) within a specific hour or day.
- Key characteristics that make the Poisson distribution suitable for modeling random job arrivals are:
- The events (job arrivals) occur randomly and independently of each other.
- The average rate of events (job arrivals per unit time) is constant over the fixed interval.
- The probability of two or more events occurring at exactly the same instant is negligible.
- It is specifically designed for counting the number of occurrences of an event.
Why Other Distributions are Not Suitable
Let's consider why the other options are not appropriate for modeling the number of random job arrivals in a fixed time interval:
- Normal Distribution: The Normal distribution is a continuous probability distribution, meaning it's used for variables that can take any value within a range (like height or temperature). It's also symmetric around its mean. Since the "number of arrivals" is a discrete count (you can have 1, 2, or 3 arrivals, but not 1.5 arrivals), and typically skewed for arrival processes, the Normal distribution is not suitable.
- Erlang Distribution: The Erlang distribution is also a continuous probability distribution. It is often used to model the waiting time until a certain number of events occur in a Poisson process, or the time between events. For example, it could describe the time you have to wait until the 3rd job arrives. It does not describe the number of arrivals within a fixed time interval.
- Beta Distribution: The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. It is primarily used to model probabilities, proportions, or rates, where the variable is constrained between two limits. It is not designed to model the discrete count of random events like job arrivals.
Therefore, given the nature of random, independent events occurring over a fixed interval, the Poisson distribution is the most fitting model for the "number of arrivals of jobs in a fixed time interval".
When jobs arrive at a facility for service in a random manner, it implies that the exact time of each arrival is unpredictable, but there might be an average rate at which they occur. To model the "number of arrivals of jobs in a fixed time interval", we need a specific type of probability distribution that handles discrete counts of events happening randomly over a continuous period.
Poisson Distribution for Random Arrivals
The correct probability distribution for the scenario described – "jobs arrive at a facility, for service, in a random manner" and we are interested in the "number of arrivals of jobs in a fixed time interval" – is the Poisson distribution.
- The Poisson distribution is a discrete probability distribution that models the probability of a given number of events happening in a fixed interval of time or space. This is applicable when these events occur with a known constant mean rate and independently of the time since the last event.
- In the context of job arrivals, this means that if jobs arrive one by one, randomly, and at an average rate, the Poisson distribution is the ideal tool to predict the likelihood of seeing a certain number of arrivals (e.g., 5 jobs, 10 jobs) within a specific hour or day.
- Key characteristics that make the Poisson distribution suitable for modeling random job arrivals are:
- The events (job arrivals) occur randomly and independently of each other.
- The average rate of events (job arrivals per unit time) is constant over the fixed interval.
- The probability of two or more events occurring at exactly the same instant is negligible.
- It is specifically designed for counting the number of occurrences of an event.
Why Other Distributions are Not Suitable
Let's consider why the other options are not appropriate for modeling the number of random job arrivals in a fixed time interval:
- Normal Distribution: The Normal distribution is a continuous probability distribution, meaning it's used for variables that can take any value within a range (like height or temperature). It's also symmetric around its mean. Since the "number of arrivals" is a discrete count (you can have 1, 2, or 3 arrivals, but not 1.5 arrivals), and typically skewed for arrival processes, the Normal distribution is not suitable.
- Erlang Distribution: The Erlang distribution is also a continuous probability distribution. It is often used to model the waiting time until a certain number of events occur in a Poisson process, or the time between events. For example, it could describe the time you have to wait until the 3rd job arrives. It does not describe the number of arrivals within a fixed time interval.
- Beta Distribution: The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. It is primarily used to model probabilities, proportions, or rates, where the variable is constrained between two limits. It is not designed to model the discrete count of random events like job arrivals.
Therefore, given the nature of random, independent events occurring over a fixed interval, the Poisson distribution is the most fitting model for the "number of arrivals of jobs in a fixed time interval".