A machine in use is replaced by a new machine either when it fails or when it reaches the age of T years. If the lifetimes of successive machines are independent with a common distribution F having density f , show that
(a) the long-run rate at which machines are replaced equals

Question

[latex]\left[\int_{0}^{T}{}xf (x) dx +T (1 −F(T )) \right]^{-1} [/latex]

(b) the long-run rate at which machines in use fail equals

[latex]\frac{F(T)}{\int_{0}^{T}{}xf (x) dx +T [1 −F(T )]}[/latex]

Accepted Answer

(a) The number of replaced machines by time t constitutes a renewal process. The time between replacements equals T, if lifetime of new machine is [latex]\geqslant T ; x[/latex], if lifetime of new machine is x, x [latex]E ext { [time between replacements }]=\int_0^T x f(x) d x+T[1-F(T)][/latex]
and the result follows by Proposition 3.1.
(b) The number of machines that have failed in use by time t constitutes a renewal process. The mean time between in-use failures, E[F], can be calculated by conditioning on the lifetime of the initial machine as E[F]= E[E[F [latex]\mid[/latex] lifetime of initial machine ]]. Now
[latex]E[F \mid ext { lifetime of machine is } x]= \begin{cases}x, & ext { if } x \leqslant T \ T+E[F], & ext { if } x>T\end{cases}[/latex]
Hence,
[latex]E[F]=\int\limits_0^T x f(x) d x+(T+E[F])[1-F(T)][/latex]
or
[latex]E[F]=\frac{\int_0^T x f(x) d x+T[1-F(T)]}{F(T)}[/latex]
and the result follows from Proposition 3.1.

Question 7.EX.8: A machine in use is replaced by a new machine either when it......

This Question has been Answered.

Already have an account?

Login

Related Answered Questions

For an interarrival distribution F having mean μ, we defined the equilibrium distribution of F, denoted Fe, by Fe(x) =1/μ∫0^x[1-F(y)]dy (a) Show that if F is an exponential distribution, then F = Fe. (b) If for some constant c, F(x)={0, x<c 1, x ≥ c show that Fe is the uniform distribution ...

Verified Answer:

Verified Answer:

Verified Answer:

Compute the renewal function when the interarrival distribution F is such that 1−F(t) = pe^−μ1t + (1 −p)e^−μ2t ...

Verified Answer:

If the mean-value function of the renewal process {N(t), t ≥ 0} is given by m(t) = t/2, t ≥ 0, what is P{N(5) = 0}? ...

Verified Answer:

Compute E[T], the expected time until the pattern h, h, h, t , h, h, h appears, when a coin that comes up heads with probability p and tails with probability q = 1− p is continually flipped. ...

Verified Answer:

Consider a machine that can be in one of three states: good condition, fair condition, or broken down. Suppose that a machine in good condition will remain this way for a mean time μ1 and then will go to either the fair condition or the broken condition with respective probabilities 3/4 and 1/4 ...

Verified Answer:

The rate a certain insurance company charges its policyholders alternates between r1 and r0. A new policyholder is initially charged at a rate of r1 per unit time. When a policyholder paying at rate r1 has made no claims for the most recent s time units, then the rate charged becomes r0 per unit ...

Verified Answer:

Verified Answer:

Although a system needs only a single machine to function, it maintains an additional machine as a backup. A machine in use functions for a random time with density function f and then fails. If a machine fails while the other one is in working condition, then the latter is put in use and, ...

Verified Answer: