Probability Distribution of Number of Arrivals in a Given Time

In a queuing problem where the arrivals are completely random, the probability distribution of the number of arrivals in a given time follows the binomial distribution. Let's understand why this is the case:

1. Definition of Binomial Distribution:
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. It is characterized by two parameters: the number of trials (n) and the probability of success (p). Each trial has only two possible outcomes: success or failure.

2. Arrivals in a Given Time:
In a queuing problem, the arrivals are random events. We can model each arrival as a Bernoulli trial, where success represents an arrival and failure represents no arrival. The probability of success (p) is the average arrival rate in a given time interval.

3. Independent Arrivals:
Assuming the arrivals are completely random, we can consider each arrival as an independent event. This means that the occurrence of one arrival does not affect the occurrence of another arrival. Therefore, the arrivals can be modeled as a sequence of independent Bernoulli trials.

4. Fixed Number of Trials:
To determine the probability distribution of the number of arrivals in a given time, we need to specify the number of trials (n). In the context of a queuing problem, the number of trials can be defined as the number of time intervals or slots in the given time period.

5. Probability of Success:
The probability of success (p) in the binomial distribution represents the probability of an arrival occurring in a single trial. It can be calculated by dividing the average arrival rate by the number of time intervals.

6. Probability Distribution:
By applying the binomial distribution formula, we can calculate the probability of obtaining each possible number of arrivals in the given time period. The probability distribution will give us the likelihood of observing 0 arrivals, 1 arrival, 2 arrivals, and so on.

7. Why Not Other Distributions?
- Poisson Distribution: The Poisson distribution is commonly used to model the number of arrivals in a fixed interval of time or space. However, it assumes that the arrivals occur at a constant average rate, which may not be the case in a queuing problem with completely random arrivals.
- Normal Distribution: The normal distribution is a continuous probability distribution, while the number of arrivals in a given time is a discrete variable. Therefore, the normal distribution is not suitable for modeling the number of arrivals.
- Exponential Distribution: The exponential distribution is often used to model the time between arrivals in queuing problems. It does not directly provide information about the number of arrivals in a given time period.

Hence, the correct answer is option 'C' - Binomial distribution.