PAGES 249 255 Khaoula Chnina INTRODUCTION Renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary interarrival holding times In ID: 1027448
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1. RENEWAL PROCESSES IN CONTINUOUS TIMEPAGES 249 – 255Khaoula Chnina
2. INTRODUCTION Renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary inter-arrival (holding) times. In the classical Poisson process, the intervals between successive occurrences are independently and identically distributed with a negative exponential distribution. Suppose that there is a sequence of events E such that the intervals between successive occurrences of E are distributed independently and identically but have a distribution not necessarily negative exponential; we have then a certain generalization of the classical Poisson process : the corresponding process is called a renewal process.
3. A renewal process is an idealized stochastic model for events that occur randomly in time. These temporal events are generically referred to as renewals or arrivals. Here are some typical interpretations and applications.The arrivals are customers arriving at a service station. Again, the terms are generic. A customer might be a person and the service station a store, but also a customer might be a file request and the service station a web server.A device is placed in service and eventually fails. It is replaced by a device of the same type and the process is repeated. We do not count the replacement time in our analysis; equivalently we can assume that the replacement is immediate. The times of the replacements are the renewalsThe arrivals are times of some natural event, such as a lightening strike, a tornado or an earthquake, at a particular geographical point.The arrivals are emissions of elementary particles from a radioactive source.
4. RENEWAL PROCESSS IN CONTINUOUS TIMELet be the time to the first renewal and let be the interarrival time (or waiting time) between (n − 1)-th renewal and n-th renewal. When have common exponential distribution, we get Poisson process as a particular case.The renewals where are i.i.d random variables, are called Palm flow of events.The renewals where are i.i.d exponential random variables, are called Poisson flow of events or ordinary.
5. Assume that , and are i.i.d. non-negative random variables with distribution function F(.) .Let : which can be infinite. When P(X = c) = 1 for some c > 0, {Sn = nc, n ≥ 1} is called a deterministic renewal process.
6. The time of the n-th renewal : If for some n, = t , then a renewal is said to occur at t gives the time of the nth renewal and is called the nth renewal (or regeneration ) time.The distribution function of
7. The number of renewals by time t (occurring in [0,t] : N(t) (2.1) Then, the counting process { N(t), t ≥ 0} will be a renewal process with distribution F (or generated or induced by F) . The sequence of random variables () and () constitute renewal processes with distribution F.
8. Simple Examples :
9. Renewal Function and Renewal DensityThe function M(t) = E[N(t)] is called the renewal function of the process with distribution F.the event { N(t) ≥ n} is equivalent to the event { ≤ t}. (2.2)or { N(t) ≥ n} if and only if { Sn ≤ t}Equivalently { N(t) < n} is equivalent to the event { Sn > t}. Theorem 6.3The distribution of N(t) is given by : And the expected number of renewlas by : (2.4)
10. Proof :
11. M(t) in relation (2.4) can be put in terms of Laplace transform as follows :Let F’(x)= f(x) be the density function of (p.d.f) of and denotes the Laplace transform of a function g(t).Then taking Laplace transform of both sides we get : (2.5)This equivalent to (2.6)
12. These show that M(t) and F(x) can be determined uniquely one from other.M(t) is a sure function and not a random function or stochastic process.Example :
13. The renewal density function
14. The function specifies the mean number of renewals to be expected in a narrow interval near t. is not a p.d.f.As It follows that : and
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17. Particular cases:
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19. Markovian case :When p=1, the distribution of reduces to negative exponential and then C=0 i.e. The second term of (2.10) vanishes, so that we get M(t)=at (bt).
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