Lecture
The mathematical model that has been given the name renewal process in the mathematical literature [1,2,4,5] is a special case of a random process (a random flow of homogeneous events) ξ(t), for which the range of values is the set of non-negative integers E={0,1,...,n,...}, ξ(t)∈E, and all trajectories are non-decreasing step functions. Such a random process can be specified by giving the joint distribution of the random sequence {tn=tn(ω), 1<=n<∞}, which determines the jump epochs; the number n gives the index of the jump of the random process ξ(t) [5, p.27]. Note that this distribution may be such that, with positive probability, the epochs tn(ω) coincide for different n. Consequently, the case is not excluded in which the trajectories of the random process ξ(t) have jumps greater than one (a group of unit jumps), and the numbering of the jump within the group is not essential.
Thus, let us define the random process ξ(t) as the number of jumps that have occurred up to time t (with this definition, the trajectories of the process are continuous from the left). Obviously, the process ξ(t) can be specified by giving the joint distribution of the random sequence
{ξn=tn-tn-1, t0=0, 1<=n<∞}
of the intervals between the epochs of successive jumps. Here, taking into account the previous remark about the size of the jumps, the case is not excluded in which, with positive probability, the random variable ξn equals zero, P{ξn=0}>0, that is, the distribution function of the random variable ξn may have a positive jump at zero.
Let us now turn to the specific definitions.
The step random process ξ(t), defined by the sequence {ξn=tn-tn-1, t0=0, 1<=n<∞}, is called a flow with limited aftereffect, if the random variables {ξn, 1<=n<∞} are mutually independent.
It follows from this definition that, at the jump moment of the random process ξ(t), if the jump index is known, the future behavior of this process in the probabilistic sense does not depend on the past trajectory. It also follows from the definition that, in order to specify a flow with limited aftereffect, it is sufficient to specify the sequence of distribution functions Fk(t)=P{ξk0, for which Fk(t)=0 for t<=0 (by virtue of the non-negativity of the intervals ξk).
A flow with limited aftereffect for which, when t>0
Fk(t)=F(t), k=2,3,..., F1(t)≠ F(t) (2.1)
is called a recurrent flow with delay or a delayed renewal process. It follows from the definition of the delayed renewal process that it is specified by a pair of distribution functions {F1(t),F(t)} - the distribution of the interval up to the first jump and the distribution of all subsequent intervals.
A flow with limited aftereffect for which Fk(t)=F(t), k=1,2,….. , is called a recurrent flow or a simple renewal process. Thus, a simple renewal process can often be defined as a sequence of independent, positive, identically distributed random variables specified by the distribution F(t), F(0)=0.
We can now explain the accepted terminology. Suppose there is a set of identical elements whose lifetimes ξk are distributed according to the same law F(t). At time t0=0 the first element is put into operation, and at the moment of its failure t1=ξ1 it is instantly replaced (renewed) by a new identical element. The new element then operates up to the calendar moment t2=ξ1+ξ2 – the moment of failure of the second element, then an instant replacement by the next element, and so on. In this way we obtain a model of replacements (renewals), which is called the renewal process, and the sequence {tn=tn(ω), 1<=n<∞} is called the sequence of renewal epochs.
Using the terminology introduced above, we shall henceforth study the random processes ξ(t) and ξ1(t), defined as the number of renewals that have occurred up to time t, t>=0, of the simple renewal process and the delayed renewal process, respectively.
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