Page 84 - A First Course In Stochastic Models

EXERCISES 75

probability that a repair involves the high repair cost R c ? Give the long-run average cost
per time unit.
(c) Verify that the average cost is minimal for the unique solution z to the equation
αz exp[−α(L − z)] = R p /(R c − R p ) when αL > R p /(R c − R p ).
2.23 A group of N identical machines is maintained by a single repairman. The machines
operate independently of each other and each machine has a constant failure rate µ. Repair
is done only if the number of failed machines has reached a given critical level R with
1 ≤ R ≤ N. Then all failed machines are repaired simultaneously. Any repair takes a
negligible time and a repaired machine is again as good as new. The cost of the simultaneous
repair of R machines is K + cR, where K, c > 0. Also there is an idle-time cost of α > 0
per time unit for each failed machine.
(a) Define a regenerative process and identify its regeneration epochs.
(b) Determine the long-run average cost per time unit.
2.24 The following control rule is used for a slow-moving expensive product. No more than
one unit of the product is kept in stock. Each time the stock drops to zero a replenishment
order for one unit is placed. The replenishment lead time is a positive constant L. Customers
asking for the product arrive according to a renewal process in which the interarrival times
are Erlang (r, λ) distributed. Each customer asks for one unit of the product. Each demand
occurring while the system is out of stock is lost.
(a) Define a regenerative process and identify its regeneration epochs.
(b) Determine the long-run fraction of demand that is lost.
(c) Determine the long-run fraction of time the system is out of stock. (Hint: use part (b)
of Exercise 2.5.)
2.25 Jobs arrive at a station according to a renewal process. The station can handle only one
job at a time, but has no buffer to store other jobs. An arriving job that finds the station busy
is lost. The handling time of a job has a given probability density h(x). Use renewal-reward
theory to verify for this loss system that the long-run fraction of jobs that are rejected is

given by ∞ ∞ M(x)h(x) dx, where M(x) is the renewal
0 M(x)h(x) dx divided by 1 + 0
function in the renewal process describing the arrival of jobs. What is the long-run fraction
of time that the station is busy? Simplify the formulas for the cases of deterministic and
Poisson arrivals.
2.26 Use the renewal-reward theorem to prove relation (2.3.3) when customers arrive accord-
ing to a renewal process and the stochastic processes {L(t)} and {U n } regenerate themselves
each time an arriving customer finds the system empty, where the cycle lengths have finite
expectations. For ease assume the case of an infinite-capacity queue. Use the following
relations:
(i) the long-run average reward earned per time unit = (the expected reward earned in
one cycle)/(expected length of one cycle),
(ii) the long-run average amount paid per customer = (the expected amount earned in
one cycle)/(expected number of arrivals in one cycle),
(iii) the long-run average arrival rate = (expected number of arrivals in one cycle)/(expec-
ted length of one cycle).
2.27 Let {X(t), t ≥ 0} be a continuous-time regenerative stochastic process whose state
space is a subset of the non-negative reals. The cycle length is assumed to have a finite
expectation. Denote by P(y) the long-run fraction of time that the process {X(t)} takes on
a value larger than y. Use the renewal-reward theorem to prove that
1 t ∞
lim X(u) du = P (y) dy with probability 1.
t→∞ t 0 0
2.28 Consider a queueing system in which the continuous-time process {L(t)} describing
the number of customers in the system is regenerative, where the cycle length has a finite