Inspection Paradox

Renewal theory

Renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary holding times. Applications include calculating the expected time for a monkey who is randomly tapping at a keyboard to type the word Macbeth and comparing the long-term benefits of different insurance policies.

Renewal processes

Introduction

A renewal process is a generalisation of the Poisson process. In essence, the Poisson process is a continuous-time Markov process on the positive integers (usually starting at zero) which has independent identically distributed holding times at each integer i (exponentially distributed) before advancing (with probability 1) to the next integer:i+1. In the same informal spirit, we may define a renewal process to be the same thing, except that the holding times take on a more general distribution. (Note however that the IID property of the holding times is retained).

Formal definition

Let S_1 , S_2 , S_3 , S_4 , S_5, ldots be a sequence of independent identically distributed random variables such that

0 < mathbb{E}[S_i] < infty.

We refer to the random variable S_i as the "ith" holding time.

Define for each n > 0 :

J_n = sum_{i=1}^n S_i,

each J_n referred to as the "nth" jump time and the intervals

[J_n,J_{n+1}]

being called renewal intervals.

Then the random variable (X_t)_{tgeq0} given by

X_t = sum^{infty}_{n=1} mathbb{I}_{{J_n leq t}}=sup left{, n: J_n leq t, right}

is called a renewal process.

Interpretation

One may choose to think of the holding times { S_i : i geq 1 } as the time elapsed before a machine breaks for the "ith" time since the last time it broke. (Note this assumes that the machine is immediately fixed and we restart the clock immediately.) Under this interpretation, the jump times { J_n : n geq 1 } record the successive times at which the machine breaks and the renewal process X_t records the number of times the machine has so far had to be repaired at any given time t.

However it is more helpful to understand the renewal process in its abstract form, since it may be used to model a great number of practical situations of interest which do not relate very closely to the operation of machines.

Renewal-reward processes

Let W_1, W_2, ldots be a sequence of IID random variables (rewards) satisfying

mathbb{E}|W_i| < infty.,

Then the random variable

Y_t = sum_{i=1}^{X_t}W_i

is called a renewal-reward process. Note that unlike the S_i, each W_i may take negative values as well as positive values.

Interpretation

In the context of the above interpretation of the holding times as the time between successive malfunctions of a machine, the "rewards" W_1,W_2,ldots (which in this case happen to be negative) may be viewed as the successive repair costs incurred as a result of the successive malfunctions.

An alternative analogy is that we have a magic goose which lays eggs at intervals (holding times) distributed as S_i. Sometimes it lays golden eggs of random weight, and sometimes it lays toxic eggs (also of random weight) which require responsible (and costly) disposal. The "rewards" W_i are the successive (random) financial losses/gains resulting from successive eggs (i = 1,2,3,...) and Y_t records the total financial "reward" at time t.

Properties of renewal processes and renewal-reward processes

We define the renewal function:

m(t) = mathbb{E}[X_t].,

The elementary renewal theorem

The renewal function satisfies

lim_{t to infty} frac{1}{t}m(t) = 1/mathbb{E}[S_1].

Proof

Below, you find that the strong law of large numbers for renewal processes tell us that

lim_{t to infty} frac {X_t}{t} = frac{1}{mathbb{E}[S_1]}.

To prove the elementary renewal theorem, it is sufficient to show that left{frac{X_t}{t}; t geq 0right} is uniformly integrable.

To do this, consider some truncated renewal process where the holding times are defined by overline{S_n} = a mathbb{I}{S_n > a} where a is a point such that 0 < F(a) = p < 1 which exists for all non-deterministic renewal processes. This new renewal process overline{X_t} is an upper bound on X_t and its renewals can only occur on the lattice {na; n in mathbb{N} } . Furthermore, the number of renewals at each time is geometric with parameter p. So we have

begin{align} overline{X_t} &leq sum_{i=1}^{[at]} mathrm{Geometric}(p) mathbb{E}left[,overline{X_t},right]^2 &leq C_1 t + C_2 t^2 Pleft(frac{X_t}{t} > xright) &leq frac{Eleft[X_t^2right]}{t^2x^2} leq frac{Eleft[overline{X_t}^2right]}{t^2x^2} leq frac{C}{x^2}. end{align}

The elementary renewal theorem for reward renewal processes

We define the reward function:

g(t) = mathbb{E}[Y_t].,

The renewal function satisfies

lim_{t to infty} frac{1}{t}g(t) = frac{mathbb{E}[W_1]}{mathbb{E}[S_1]}.

The renewal equation

The renewal function satisfies

m(t) = F_S(t) + int_0^t m(t-s) f_S(s), ds

where F_S is the cumulative distribution function of S_1 and f_S is the corresponding probability density function.

Proof of the renewal equation

We may iterate the expectation about the first holding time:

m(t) = mathbb{E}[X_t] = mathbb{E}[mathbb{E}(X_t mid S_1)].

But by the Markov property

mathbb{E}(X_t mid S_1=s) = mathbb{I}_{{t geq s}} left(1 + mathbb{E}[X_{t-s}] right).

So

begin{align} m(t) & {} = mathbb{E}[X_t] & {} = mathbb{E}[mathbb{E}(X_t mid S_1)] & {} = int_0^infty mathbb{E}(X_t mid S_1=s) f_S(s), ds & {} = int_0^infty mathbb{I}_{{t geq s}} left(1 + mathbb{E}[X_{t-s}] right) f_S(s), ds & {} = int_0^t left(1 + m(t-s) right) f_S(s), ds & {} = F_S(t) + int_0^t m(t-s) f_S(s), ds, end{align}

as required.

Asymptotic properties

(X_t)_{tgeq0} and (Y_t)_{tgeq0} satisfy

lim_{t to infty} frac{1}{t} X_t = frac{1}{mathbb{E}S_1} (strong law of large numbers for renewal processes)

lim_{t to infty} frac{1}{t} Y_t = frac{1}{mathbb{E}S_1} mathbb{E}W_1 (strong law of large numbers for renewal-reward processes)

almost surely.

Proof

First consider (X_t)_{tgeq0}. By definition we have:

J_{X_t} leq t leq J_{X_t+1}

for all t geq 0 and so

frac{J_{X_t}}{X_t} leq frac{t}{X_t} leq frac{J_{X_t+1}}{X_t}

for all t ≥ 0.

Now since 0< mathbb{E}S_i < infty we have:

X_t to infty

as t to infty almost surely (with probability 1). Hence:

frac{J_{X_t}}{X_t} = frac{J_n}{n} = frac{1}{n}sum_{i=1}^n S_i to mathbb{E}S_1

almost surely (using the strong law of large numbers); similarly:

frac{J_{X_t+1}}{X_t} = frac{J_{X_t+1}}{X_t+1}frac{X_t+1}{X_t} = frac{J_{n+1}}{n+1}frac{n+1}{n} to mathbb{E}S_1cdot 1

almost surely.

Thus (since t/X_t is sandwiched between the two terms)

frac{1}{t} X_t to frac{1}{mathbb{E}S_1}

almost surely.

Next consider (Y_t)_{tgeq0}. We have

frac{1}{t}Y_t = frac{X_t}{t} frac{1}{X_t} Y_t to frac{1}{mathbb{E}S_1}cdotmathbb{E}W_1

almost surely (using the first result and using the law of large numbers on Y_t).

The inspection paradox

A curious feature of renewal processes is that if we wait some predetermined time t and then observe how large the renewal interval containing t is, we should expect it to be typically larger than a renewal interval of average size.

Mathematically the inspection paradox states: for any t > 0 the renewal interval containing t is stochastically larger than the first renewal interval. That is, for all x > 0 and for all t > 0:

mathbb{P}(S_{X_t+1} > x) geq mathbb{P}(S_1>x) = 1-F_S(x)

where FS is the cumulative distribution function of the IID holding times Si.

Proof of the inspection paradox

Observe that the last jump-time before t is J_{X_t}; and that the renewal interval containing t is S_{X_t+1}. Then

begin{align} mathbb{P}(S_{X_t+1}>x) & {} = int_0^infty mathbb{P}(S_{X_t+1}>x mid J_{X_t} = s) f_S(s) , ds & {} = int_0^infty mathbb{P}(S_{X_t+1}>x | S_{X_t+1}>t-s) f_S(s), ds & {} = int_0^infty frac{mathbb{P}(S_{X_t+1}>x , , , S_{X_t+1}>t-s)}{mathbb{P}(S_{X_t+1}>t-s)} f_S(s) , ds & {} = int_0^infty frac{ 1-F(max { x,t-s }) }{1-F(t-s)} f_S(s) , ds & {} = int_0^infty min left{frac{ 1-F(x) }{1-F(t-s)},frac{ 1-F(t-s) }{1-F(t-s)}right} f_S(s) , ds & {} = int_0^infty min left{frac{ 1-F(x) }{1-F(t-s)},1right} f_S(s) , ds & {} geq 1-F(x) & {} = mathbb{P}(S_1>x) end{align}

as required.

Example applications

Example 1 - use of the strong law of large numbers

Eric the entrepreneur has n machines, each having an operational lifetime uniformly distributed between zero and two years. Eric may let each machine run until it fails with replacement cost €2600; alternatively he may replace a machine at any time while it is still functional at a cost of €200.

What is his optimal replacement policy?

Solution

We may model the lifetime of the n machines as n independent concurrent renewal-reward processes, so it is sufficient to consider the case n=1. Denote this process by (Y_t)_{t geq 0}. The successive lifetimes S of the replacement machines are independent and identically distributed, so the optimal policy is the same for all replacement machines in the process.

If Eric decides at the start of a machine's life to replace it at time 0 < t < 2 but the machine happens to fail before that time then the lifetime S of the machine is uniformly distributed on [0, t] and thus has expectation 0.5t. So the overall expected lifetime of the machine is:

mathbb{E}S = mathbb{E}[S mid mbox{fails before } t] cdot mathbb{P}[mbox{fails before } t] + mathbb{E}[S mid mbox{does not fail before } t] cdot mathbb{P}[mbox{does not fail before } t]

= frac{t}{2}left(0.5tright) + frac{2-t}{2}left(t right)

and the expected cost W per machine is:

mathbb{E}W = mathbb{E}(W mid mbox{fails before } t) cdot mathbb{P}(mbox{fails before } t) + mathbb{E}[W mid mbox{does not fail before } t).mathbb{P}(mbox{does not fail before } t)

= frac{t}{2}(2600 ) + frac{2-t}{2} (200 ) = 1200t + 200.,

So by the strong law of large numbers, his longterm average cost per unit time is:

lim_{t to infty} frac{1}{t} Y_t = frac{mathbb{E}W}{mathbb{E}S} = frac{ 4(1200t + 200) }{ t^2 + 4t - 2t^2 }

then differentiating with respect to t:

frac{partial}{partial t} frac{ 4(1200t + 200) }{ t^2 + 4t - 2t^2 } = 4frac{ (4t - t^2)(1200) - (4 - 2t)(1200t + 200) }{ (t^2 + 4t - 2t^2)^2 },

this implies that the turning points satisfy:

0 = (4t - t^2)(1200) - (4 - 2t)(1200t + 200)
= 4800t - 1200t^2 -4800t - 800 + 2400t^2 + 400t 

= -800 + 400t + 1200t^2,

and thus

0 = 3t^2 + t - 2 = (3t -2)(t+1).

We take the only solution t in [0, 2]: t = 2/3. This is indeed a minimum (and not a maximum) since the cost per unit time tends to infinity as t tends to zero, meaning that the cost is decreasing as t increases, until the point 2/3 where it starts to increase.

See also

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