Understanding Renewal Processes in Continuous Time
Renewal theory is a branch of probability theory that extends Poisson processes for various inter-arrival times. A renewal process models randomly occurring events over time, such as customer arrivals at a service station or natural phenomena like earthquakes. This article delves into the concept of renewal processes, their interpretations, applications, and mathematical representations. Explore the characteristics and applications of deterministic renewal processes with detailed examples and illustrations.
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RENEWAL PROCESSES IN CONTINUOUS TIME PAGES 249 255 Khaoula Chnina
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.
Arenewal 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 renewals The 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.
RENEWAL PROCESSS IN CONTINUOUS TIME Let ?1be the time to the first renewal and let ??(n = 2,3,...) 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.
Assume that Pr(Xn= 0) < 1 , and Xn(n = 2,3,...) are i.i.d. non- negative random variables with distribution function F(.) .Let : xdF(x) = E Xn = 0 which can be infinite. Whenever = ,1 shall be interpreted as 0. When P(X = c) = 1 for some c > 0, {Sn = nc, n 1} is called a deterministic renewal process.
The time of the n-th renewal : n S0= 0 and Sn= Xn i=1 If for some n, Sn= t , then a renewal is said to occur at t Sngives the time of the nth renewal and is called the nth renewal (or regeneration ) time. The distribution function of ??,? ? ??? = Pr(?? ? )
The number of renewals by time t (occurring in [0,t] : N(t) (2.1) N t = max { n Sn t } 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 (Sn,n = 1,2, ) and (Xn,n = 1,2, ) constitute renewal processes with distribution F.
Renewal Function and Renewal Density The 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 { Sn 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 6.3 The distribution of N(t) is given by : pnt = Pr N t = n = Fnt Fn+1t (2.3) And the expected number of renewlas by : (2.4) M t = n=1 Fnt
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 Knand g (x) denotes the Laplace transform of a function g(t). Then taking Laplace transform of both sides we get : M t = Fnt n=1 Fn s M s = n=1 =1 s s n=1 1 s n=1 f s s[1 f s ] fn [f s ]n = (2.5) = This equivalent to sM s 1+sM s f s = (2.6)
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 :
The renewal density function ? ? ? ? = ? (?)
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 : = ?? ? ? ? 1 ? ? ? ? 1+? ? ? ? = and ? ? =
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).
