Optimality

On Competing risk and degradation processes 231
here is that since the failure of any component of the system leads to the failure
of the system, the system experiences multiple risks, each risk leading to failure.
Thus if Ti denotes the lifetime of component i, i = 1, . . . , k, say, then the cause
of system failure is that component whose lifetime is smallest of the k lifetimes.
Consequently, if T denotes a system’s lifetime, then
(3.1) Pr(T≥ t) = P(H1(t)≤X1, . . . , Hk(t)≤Xk),
where Xi is the hazard potential of the i-th component, and Hi(t) its cumulative
hazard (or the risk to component i) at time t. If the Xi’s are assumed to be
independent (a simplifying assumption), then (3.1) leads to the result that
(3.2) Pr(T≥ t) = exp[−(H1(t) +··· + Hk(t))],
suggesting an additivity of cumulative hazard functions, or equivalently, an additivity
of the risks. Were the Xi’s assumed dependent, then the nature of their
dependence will dictate the manner in which the risks combine. Thus for example
if for some θ, 0≤θ≤1, we suppose that
Pr(X1≥ x1, X2≥ x2|θ) = exp(−x1− x2− θx1x2),
namely one of Gumbel’s bivariate exponential distributions, then
Pr(T≥ t|θ) = exp[−(H1(t) + H2(t) + θH1(t)H2(t))].
The cumulative hazards (or equivalently, the risks) are no longer additive.
The series system model discussed above has also been used to describe the
failure of a single item that experiences several failure causing agents that compete
with each other. However, we question this line of reasoning because a single item
posseses only one unknown resource. Thus the X1, . . . , Xk of the series system model
should be replaced by a single X, where X1 = X2 =··· = Xk = X (in probability).
To set the stage for the single item case, suppose that the item experiences k
agents, say C1, . . . , Ck, where an agent is seen as a cause of failure; for example,
the consumption of fatty foods. Let Hi(t) be the consequence of agent Ci, were Ci
be the only agent acting on the item. Then under the simultaneous action by all of
the k agents the item’s survival function
(3.3)
Pr(T≥ t;h1(t), . . . , hk(t))
= P(H1(t)≤X, . . . , Hk(t)≤X)
= exp(−max(H1(t), . . . , Hk(t))).
Here again, the cumulative hazards are not additive.
Taking a clue from the fact that dependent hazard potentials lead us to a
non-additivity of the cumulative hazard functions, we observe that the condition
P P
X1 = X2 =··· P P
P
= Xk = X (where X1 = X2 denotes that X1and X2 are equal in
probability) implies that X1, . . . , Xk are totally positively dependent, in the sense
of Lehmann (1966). Thus (3.2) and (3.3) can be combined to claim that in general,
under the series system model for competing risks, P(T≥ t) can be bounded as
(3.4) exp(−
k�
Hi(t))≤P(T≥ t)≤exp(−max(H1(t), . . . , Hk(t))).
1
Whereas (3.4) above may be known, our argument leading up to it could be new.