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### Renewal Processes

Generalization of Poisson Process

Poisson Process

- In the classical Poisson Process, the interval between successive occurrences is i.i.d with a negative exponential distribution.

Exponential Dist.

T1

T2

T3

T4

T5

Poisson Process

In the other word, If N(t) = Poisson then the interval time duration between each occurrence is exponentially Distributed.

Now, Suppose that there is a sequence of events E such that the interval between successive events is i.i.d but the distribution is not exponential, say distributed with F distribution, in this case we have a certain generalization of the Poisson process namely Renewal Process.

Renewal Process in continues time

- Let {Xn , n=1,2,3, …} be a sequence of non negative independent random variables. Assume that Pr{Xn = 0} < 1and the random variables are i.i.d with a distribution function F(.). Since Xnis nonnegative it follows that E(Xn) exist and is:
- E(Xn) =
- where 𝞵 may be infinite. When ever is infinite, 1/ shall be interpreted as 0.

Renewal Process

- Let S0 = 0, Sn = X1 + X2 + X3 + … + Xn , n≥1
- And let Fn(x) = Pr{Sn ≤ x} be the distribution function of Sn, n≥1;
- F0(x) = 1 if x ≥ 0 and F0(x) = 0 if x < 0
- Definition. Define the random variable N(t) = sup{n: Sn ≤ t}.
- The process {N(t), t≥0} is called a renewal process with distribution of F (or Generated or Inducted by F).

Renewal Process

- If, for some n, Sn = t, then a renwalproces is said to occur at t;Sn gives the time (epoch) of the nth renewal epoch. The randome variable N(t) gives the number of renewals accuring in [0,t]. The randome variable xn gives the interval time (waiting time) between n-1th and nth renewals, the intervals are i.i.d, when they have common exponential Dist. We get a poisson process at a particular case.

A Simple Example

- Consider an electric bulb which works or fails completely. Suppose that the detection of the failure of the bulb and it’s replacement take place instantaneously, also suppose the life time of the bulbs are i.i.d random variable with distribution of F. we then will have a renewal process with Dist. F.

Renewal Function and Renewal density

- The function M(t) = E{N(t)} is called the renewal function of process with distribution F. It is crystal clear that;
- {N(t) ≥ n} {Sn ≤ t} Or {N(t) < n} {Sn> t}
- Theorem;
- The distribution of N(t) is given by;
- Pn(t) = Pr {N(t) = n} = Fn(t) – Fn+1 (t)
- And the expected number of renewals is:
- M(t) =

Proof

- We have;
- Pr{N(t) = n} = Pr{N(t)≥n} – Pr{N(t) ≥ n+1}
- = Pr{Sn≤ t} – Pr{Sn+1≤ t} = Fn(t) – Fn+1(t)

Again,

M(t) = E{N(t)} =

=

M(t) =

Calculating this function can be put in terms of laplas transformation as follows:

Let F’(x) = f(x) be the p.d.f of Xn and g*(s) denote the Laplas transform of a function g(t). Then taking laplas transformation of both sides of the equations leads to:

M*(s) = = 1/S =

1/S = f*(s)/s{1-f*(s)},

it is equivalent to :

f*(s) = s M*(s)/ 1 + s M*(s)

Renewal Density

The derivative m(t) of M(t) is called the renewal density. We have:

m(t) =

=

= {provided that F(x) is absolutely continues and F’n(t) = fn(t)} == = M’(t).

The function m(t) specifies the mean number of renewals to be expected in a narrow interval near t.

- Note that m(t) is not a probability Density function as m*(s)

Renewal Equation

As we know, M(t) = E{N(t)}, which gives the expected number of renewals in [0,t],

Theorem;

The renewal function M satisfies the equation ;

M(t) = F(t) + ,

this equation can be generalized as

V(t) = g(t) +

Then;

V(t) = g(t) + , M(t) =

Method of Substitution:

In this method some of the terms in the integrand of the renewal equation are substituted. Some of the notable contributions came from Bartholomew (1993), Deligonul (1985), Smeitink and Dekker (1990), Politis and Pitts (1998).Kambo (2012).

•Methods based on Riemann-Stieltjes integral:

In this method researchers approximate the integral on the right hand side of (1.1) by an infinite sum. The major contributions came from Xie (1989), Ayhan (1999), Xie (2003).

•Bounds:

These methods analyses the asymptotic nature and bounds to the solution of the renewal equations. Some important references using this method are Marshall (1973), Deley (1976), Li and Luo (2005) and Ran (2006).

•Method of moment matching:

In these methods the distribution function F(x) is approximated by mostly phase type distributions such that first few moments of the two distributions match. Notable literature using this method include Marie (1980), Whitt (1982), Altiok (1985), Lindsay (2000), Cui and Xie (2003) and Bux and Herzog (1997).

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