Page 342 - A First Course In Stochastic Models

BIBLIOGRAPHIC NOTES 337

the long-run average cost per time unit is given by
   −1
N−2 N−1
 N − 1 

T p k +  1 − p k 
µ
 
k=0 k=0
 
N−2 N−2 k+1 N−1 ∞
 
T T
× J 2 + (J 2 − J 1 ) p k + p k a i + a i p k ,
k + 1
 k 
k=0 k=0 i=1 i=1 k=N−1
−µT k
where p k = e (µT ) /k!. This problem is motivated by Luss (1976).
8.11 Consider a two-unit reliability model with one operating unit and one unit in warm
standby. The operating unit has a constant failure rate of λ 0 , while the unit in warm standby
has a constant failure rate of λ 1 . Upon failure of the operating unit, the unit in warm standby
is put into operation if available. The repair time of a failed unit has a general probability
distribution function G(x) with density g(x) and mean µ R . The system is down when both
units have failed. For the case of a single repair facility, prove that the long-run fraction of
time the system is down is as follows. This problem is based on Gaver (1963).
∞ −λ 0 x

µ R − {1 − G(x)}e dx
0 .
µ R + (λ 0 + λ 1 ) −1 0 ∞ −λ 0 x g(x) dx
e
8.12 Consider an unreliable production unit whose output is temporarily stored in a finite
buffer with capacity K. The buffer serves for the demand process as protection against
random interruptions in the production process. For the output there is a constant demand
at rate ν. When operating, the production unit produces at a constant rate P > ν if the
buffer is not full and produces at the demand rate ν otherwise. If demand occurs while
the unit is down and the buffer is empty then it is lost. The operating time of the unit is
exponentially distributed with mean 1/λ. If a failure occurs, the unit enters repair for an
exponentially distributed time with mean 1/µ. Determine the long-run fraction of demand
lost and determine the average inventory level in the buffer. (Hint: define the state of the
system as (1, x) and (0, x) respectively when the inventory in the buffer is x and the unit is
operating or down. The process regenerates itself each time the system enters state (0,0). Use
differential equations to get the desired performance measures). This problem is motivated
by Wijngaard (1979).

BIBLIOGRAPHIC NOTES
The key renewal theorem has a long history, and analytic proofs were given under
rather restrictive conditions. The reader is referred to the book of Feller (1971) for a
transparent proof under the weak condition of direct Riemann integrability; see also
Asmussen (1987). The results for the alternating renewal process in Section 8.3 are
proved in greater generality in Tak´ acs (1957). The material from Example 8.3.1 is
based on the paper of Van der Heijden (1987). The renewal-theoretic method used
in Section 8.4 to derive asymptotic estimates for ruin and waiting-time probabilities
comes from Feller (1971). Another application of this powerful method to a storage
problem for dams is given in De Kok et al. (1984). A good reference on heavy-
tailed distributions is Embrechts et al. (1997).