3.1. Is it true that (a) N(I) 1? (b) N() snif and only if S, 21? (c) N() > n if and only if S, Note: this information below for answering the question above Renewal Theory 3.1 INTRODUCTION AND PRELIMINARIES In the previous chapter we saw that the interarrival times for the Poisson process are independent and identically distributed exponential random vari- ables. A natural generalization is to consider a counting process for which the interarrival times are independent and identically distributed with an arbitrary distribution Such a counting process is called a renewal process. Formally, let {X., n = 1, 2, - } be a sequence of nonnegative independent random variables with a common distribution F, and to avoid trivialities sup- pose that F(0) - P{X, = 0} 0, this means that S, must be going to infinity as n goes to infinity. Thus, S, can be less than or equal to 1 for at most a finite number of values of n Hence, by (311), N() must be finite, and we can write N(O) - max{n. S, St} 3.2 DISTRIBUTION OF N(t) The distribution of N(t) can be obtained, at least in theory, by first noting the important relationship that the number of renewals by time is greater than or equal to n if, and only if, the nth renewal occurs before or at time t That is, (321) N() 2n S, St. From (3.2 1) we obtain (322) P{N(1) = n} = P{N(1) 2n} - P{N(1) 2n+ 1} 3.1. Is it true that (a) N(I) 1? (b) N() snif and only if S, 21? (c) N() > n if and only if S, Note: this information below for answering the question above Renewal Theory 3.1 INTRODUCTION AND PRELIMINARIES In the previous chapter we saw that the interarrival times for the Poisson process are independent and identically distributed exponential random vari- ables. A natural generalization is to consider a counting process for which the interarrival times are independent and identically distributed with an arbitrary distribution Such a counting process is called a renewal process. Formally, let {X., n = 1, 2, - } be a sequence of nonnegative independent random variables with a common distribution F, and to avoid trivialities sup- pose that F(0) - P{X, = 0} 0, this means that S, must be going to infinity as n goes to infinity. Thus, S, can be less than or equal to 1 for at most a finite number of values of n Hence, by (311), N() must be finite, and we can write N(O) - max{n. S, St} 3.2 DISTRIBUTION OF N(t) The distribution of N(t) can be obtained, at least in theory, by first noting the important relationship that the number of renewals by time is greater than or equal to n if, and only if, the nth renewal occurs before or at time t That is, (321) N() 2n S, St. From (3.2 1) we obtain (322) P{N(1) = n} = P{N(1) 2n} - P{N(1) 2n+ 1}<><>