Difference between Renewal Process and Poisson Processes

[thrillhouse86]

Hey All,

Can someone please explain to me the difference between a Poisson Process and a Renewal Process ? is it just that the Holding times for Poisson processes are exponential and Holding times for Renewal Processes are any kind of probability distribution (as the wiki page seems to imply)

If this is the case I don't understand why on the wiki page when reffering to the "Proof of the renewal equation" they invoke the Markov property - which I thought only held for memoryless (i.e. exponential ) distributions

Thanks

 

[Valkarie]

in advance!</code>A:The major difference between the two processes is that a Poisson process is a continuous-time process, while a renewal process is a discrete time process. A Poisson process is characterized by the probability of an arrival occurring at any given instant, while a renewal process is characterized by the probability of an arrival happening at any discrete point in time.In a Poisson process, arrivals of events occur randomly and independently of each other, and the interarrival times follow an exponential distribution. This means that the probability of an event occurring at any given time is the same no matter how many events have already occurred.In contrast, a renewal process follows a different probability distribution for interarrival times, and the probability of an event occurring at any given time is dependent on how many events have already occurred. This means that the probability of an event occurring at any given time is not constant, but rather changes depending on the history of past events. This is why the Markov property is necessary to prove the renewal equation - because the probability of an event occurring at any given time is dependent on the history of past events.