Q.What is The main Difference Between Poisson and Exponential Distributions in a Traffic generation model ? ..... And from you experience ... Which one is the easier and the more accurate to apply? Can you give me some M2M applications that support either Poisson or Exponential distributions ?
The Poisson distribution is a discrete one. This means that the outcome is a number. In traffic engineering, you can use the Poisson distribution to find the probability that a number of vehicle come to point within a certain period of time. For example you can select a 15-min interval and find the probability that zero, one, two, three, etc cars come during that period. On the other hand, the exponential distribution is a continuous one. This means that the outcome could be any real number. With a continuous distribution, you calculate the probability that a variable be smaller (or greater) than a given number or the probability that a variable be between two numbers (interval). The probability that the variable equal a number is zero with a continuous distribution. In traffic engineering, you can use the exponential distribution to find the probability that time headway be smaller (greater) than a number or between 2 numbers.
The negative exponential distribution can be derived from the Poisson count distribution. Consider the case when no vehicles arrive in a time interval, t , then the probability of zero vehicles arriving using the Poisson formula is P(0) = exp(-m) where m is the average number of vehicles arriving in the time interval t. P(0) means that the headway, h, between vehicles is more than the interval t. P(0) = P(h>t) = exp(-m) = exp(-V*t/3600). where V is the hourly flow rate.
Google this:
Poisson arrivals exponential interarrival times
Many of the results will explain the elegant & strong relationship between the two distributions.
Dear Hassan,
Both distribution may refer to the same Poisson Process, and, in this case, from one of them you can derive the other.
The difference is that the Poisson Distribution gives the probability of having n events during a period of time, to say, for example, the probability of 5 arrivals during the period of 1 minute; the exponential gives the interval of time between two consecutive arrivals.
Considering Queuing Theory, if both the arrivals and the service times follow Poisson Processes, it is more practical to consider the arrivals as following a Poisson Distribution and the service times as following an exponential distribution, given that, to collect data and adjust a distribution to arrivals you can count easily the number of arrivals at intervals of, say, one minute, but it is almost impossible to collect the intervals of times between arrivals; on the other hand, it is easy to collect the amount of time for each attendance and practically impossible to collect the number of attendances in an interval of time of, say, 1 minute.
The Poisson distribution deals with the number of occurrences in a fixed period of time, and the exponential distribution deals with the time between occurrences of successive events as time flows by continuously. The Poisson distribution is discrete, defined in integers x=[0,inf]. The Exponential distribution is continuous, defined on x=[0,infinity], with one parameter. The Exponential distribution also describes the time between events in a Poisson process.For example, the incoming stream of passengers in metro station is Poison, and the time of service of passengers is exponential. For example Intensity of inflow of passengers -3000passengers/hour Poisson distribution; average intensity of service ticket office 10 s– exsponential distributionon. The both distribution are used in queuing systems - for example M/M/s. For example of metro station It is meen the incoming stream of passengers is Poison, and the time of service is exponential, the number of devices is "s".
Poisson is for discrete data and is based on arrivals or occurrence of events.
Exponential talks about the time between occurrence of events and is a Continuous Probability Distribution Function.
You can look at these two models as they are telling the same story but from different perspectives.
Poisson distribution is discrete distribution and exponential distribution is continuous both fall under the umbrella of Poisson process. Their application can be considered as deaths occurred in a Super store due to dengue in a month and treatment waiting time. The number of deaths due to dengue in a super store in a day is Poisson distribution and treatment waiting time is exponentially distributed.
Poisson: discrete distribution & deals with the number of occurrences in a fixed period of time.
Exponential: continuous distribution & deals with the time between occurrences of successive events as time flows by continuously. Moreover, it is memoryless. It's a special case of the Gamma distribution.
Poisson, Exponential & Gamma model different aspects in a Poisson process. More precisely:
Poisson models the # of events in the future.
Exponential models the wait time until the very 1st event.
Gamma predicts the wait time until the kth event.
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