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A curious renewal process average
Stochastic Processes and their Applications 11 (1981) 293-295 North-Holland Publishing Company
SHORT COMMUNICATION
CESS AV
A CU William S. JEWELL
Departrnen;mof Industrial Engineering and Operations Research, University of Californi~m, Berkeley, CA 94700, USA.
Received 7 July 1978
The lifetimes of a renewal process observed during a fixed interval (0, tj are smaher, on the average, than the process mean lifetime; it is shown that the mean observed lifetime has ia particularly simple form. I‘
Renewal processes MTBF
Conside;. an ordinary renewal process consisting of nonnegative and independent random variab’les {Xk, k 2 1) with common distribution F, and density fi The random number of renewals in a fixed interval (0, t], N(t), has the distribution P,(t) :=P{N(t) = n} = F”“(t) - F(“+*)*(t) (n a 0). Where F”“(t) is the n -fold convolution of F with itself. Let M(t) = EN(t). It is known that E{&)J+~} = E(X) [M(t) + 11. However, simple results for Z&), the sum of completed intervals in (0, t], do nolt seem to be available; for exampRe we have l
j;x[l-F(t-x)]dM E{SN,t)1N(t) > 01= F(t)
(x) ’
The mean value of such an interval has the ffollowing form which is unexpectedly simple-this was discovered while investigating biases in renewal testing [I].
* This research was supported by the Air Force Ofhoe of Scientific Researf:.h (AFSC), USAF, undid Grant AFOSR-77-3179 with the University of Californi,:a.
0~~~-4~49/1;l/OOQO-0000/$02.50
@ North-Holland
Publishing Company
W.S. Jewel1 / A curious renewal process average
294
POOL We have
E(g-p(t)>O] S=ln-’ j; [yf"*(y)l[lF(t)
-F(t-
~11 dy
.
fis) :=1; e-“’ dF (t) is the transform off, then the integral above is the convolution of 1 -F(t) and tf”*(t), which have transforms [1-&)1/s and If
J
00
e?fn*(t)
Jo
d dt = -ds
*
0
e-“‘f”*(t) dt = -$
[f(S)]” = -n~(s)[~(s)]n-l,
respectively. Therefore, the transform of the above sum is simply
(-f(s))[l-.fCs)lf S
1
n[f(s)ln-l
_I
n=ln
=- - f()’ s s
'
which is recognized as the transform of ji x dF (x). This leads to the desired result. Rem~~rk.A simple proof which perhaps explains the theorem better was suggested by II. Gerber (University of Michigan) along the following lines: It is easily verified that, given N(t) = n > 0, Xl, X1, . . . , Xn are exchangeable random variables. Therefore, @ven N(t) = n > 0, E{$/n} = E{Xl}; but, cases in which n = 1,2, . . . are exactly those cases in which X1 s t. The average completed interval is approximately it for small values of t, and increases monotonically to E(Xl}, usually slowly. For example, if f is exponential with parameter A, then
Unfortunately, the simplicity of the above result does not generalize to other funr:tions of W(t) = S&N(t)(N(t) > 0). For example, the density of W = w consists of dr’ifFerentsums of convolutions over different intervals. In the exponential case, this density, [k/WI n n (hw)“-’
p(w)=h
nS~
(n-l)! (eA’-1)
is a real wonder, being level over
’ (it, t],
linear over
(it, it],
I would like to thank the referee for a correction.
quadratic over
(it, it],
etc.
W.S. Jewel1 / A curious renewal process average
295
[I] W.S. Jewell, ‘Reliability growth’ as an artifact of renewal testing, ORC 78-9, Operations Research Center, University ot California, Berkeley (1978).
SHORT COMMUNICATION
CESS AV
A CU William S. JEWELL
Departrnen;mof Industrial Engineering and Operations Research, University of Californi~m, Berkeley, CA 94700, USA.
Received 7 July 1978
The lifetimes of a renewal process observed during a fixed interval (0, tj are smaher, on the average, than the process mean lifetime; it is shown that the mean observed lifetime has ia particularly simple form. I‘
Renewal processes MTBF
Conside;. an ordinary renewal process consisting of nonnegative and independent random variab’les {Xk, k 2 1) with common distribution F, and density fi The random number of renewals in a fixed interval (0, t], N(t), has the distribution P,(t) :=P{N(t) = n} = F”“(t) - F(“+*)*(t) (n a 0). Where F”“(t) is the n -fold convolution of F with itself. Let M(t) = EN(t). It is known that E{&)J+~} = E(X) [M(t) + 11. However, simple results for Z&), the sum of completed intervals in (0, t], do nolt seem to be available; for exampRe we have l
j;x[l-F(t-x)]dM E{SN,t)1N(t) > 01= F(t)
(x) ’
The mean value of such an interval has the ffollowing form which is unexpectedly simple-this was discovered while investigating biases in renewal testing [I].
* This research was supported by the Air Force Ofhoe of Scientific Researf:.h (AFSC), USAF, undid Grant AFOSR-77-3179 with the University of Californi,:a.
0~~~-4~49/1;l/OOQO-0000/$02.50
@ North-Holland
Publishing Company
W.S. Jewel1 / A curious renewal process average
294
POOL We have
E(g-p(t)>O] S=ln-’ j; [yf"*(y)l[lF(t)
-F(t-
~11 dy
.
fis) :=1; e-“’ dF (t) is the transform off, then the integral above is the convolution of 1 -F(t) and tf”*(t), which have transforms [1-&)1/s and If
J
00
e?fn*(t)
Jo
d dt = -ds
*
0
e-“‘f”*(t) dt = -$
[f(S)]” = -n~(s)[~(s)]n-l,
respectively. Therefore, the transform of the above sum is simply
(-f(s))[l-.fCs)lf S
1
n[f(s)ln-l
_I
n=ln
=- - f()’ s s
'
which is recognized as the transform of ji x dF (x). This leads to the desired result. Rem~~rk.A simple proof which perhaps explains the theorem better was suggested by II. Gerber (University of Michigan) along the following lines: It is easily verified that, given N(t) = n > 0, Xl, X1, . . . , Xn are exchangeable random variables. Therefore, @ven N(t) = n > 0, E{$/n} = E{Xl}; but, cases in which n = 1,2, . . . are exactly those cases in which X1 s t. The average completed interval is approximately it for small values of t, and increases monotonically to E(Xl}, usually slowly. For example, if f is exponential with parameter A, then
Unfortunately, the simplicity of the above result does not generalize to other funr:tions of W(t) = S&N(t)(N(t) > 0). For example, the density of W = w consists of dr’ifFerentsums of convolutions over different intervals. In the exponential case, this density, [k/WI n n (hw)“-’
p(w)=h
nS~
(n-l)! (eA’-1)
is a real wonder, being level over
’ (it, t],
linear over
(it, it],
I would like to thank the referee for a correction.
quadratic over
(it, it],
etc.
W.S. Jewel1 / A curious renewal process average
295
[I] W.S. Jewell, ‘Reliability growth’ as an artifact of renewal testing, ORC 78-9, Operations Research Center, University ot California, Berkeley (1978).