Sourav Ghosh (CSE-01406272)
Mr. Muhtadir Rahman / https://github.com/muhtadir
Lecturer, Department of CSE
Port City International University
Discrete variables are countable in a finite amount of time. For example, we can count the change in our pocket. we can count the money in your bank account. we could also count the amount of money in everyone’s bank accounts. It might take us a long time to count that last item, but it’s still countable.
Continuous Variables would take forever to count. In fact, you would get to “forever” and never finish counting them. For example, take age. You can’t count “age”. Why not? Because it would literally take forever. For example, you could be: 25 years, 10 months, 2 days, 5 hours, 4 seconds, 4 milliseconds, 8 nanoseconds, 99 picosends and so on. Time is a continuous variable. Anyone's age can be turned into a discrete variable and then you could count it. For example: A person’s age in years. A baby’s age in months.
A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume. In other words, the values of the variable vary based on the underlying probability distribution. Poisson Distribution and Exponential Distribution are among the other types of probability distribuiton.
A single-channel service system consisting of a service facility and queue, in which the possible times of arrival of units and the possible service times are discrete, is analyzed. A method of calculating the moments of the total service time of units in the system is developed. This total service time is related to the delay caused by the system. When arrivals at different times are assumed to be independent, the “values” of the resulting Markov process can be calculated. These values lead to information about the transient behavior, autocorrelation function, expected first passage time, and expected extra delay that arises if another unit is inserted into the system. An expression for the geometric transform or moment generating function of the probability distribution of the total service time of units in the system is determined. The results are derived for arbitrary arrival and service time distributions.
A Poisson distribution is a tool that helps to predict the probability of certain events from happening when you know how often the event has occurred. It gives us the probability of a given number of events happening in a fixed interval of time.
Poisson Distribution
Poisson distributions, valid only for integers on the horizontal axis. λ (also written as μ) is the expected number of event occurrences. If we want to calculate the poisson Distribution
pmf is: P(x; μ) = (e-μ * μx) / x!
Where: • The symbol “!” is a factorial. • μ (the expected number of occurrences) is sometimes written as λ. Sometimes called the event rate or rate parameter.
In general, Poison distribution works with a discrete value. As we know inter-arrival time does not occur in sequential manner, it works randomly. Thus poisson distribution are used for that.
The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.
Exponential Distribution deals with the time between occurrences of successive events as we know time flows continuously and in this project the starting and the ending of service can be considered as two successive events.