Page 82 - A First Course In Stochastic Models

EXERCISES 73

repair facilities. Both the running times and the repair times are sequences of independent
and identically distributed random variables. It is also assumed that these two sequences are
independent of each other. The running time has a positive density on some interval. Denote
by α the mean running time and by β the mean repair time.
(a) Prove that

n k n−k
lim P{k components are in repair at time t} = p (1 − p)
t→∞ k
for k = 0, 1, . . . , n, where p = β/(α + β).
(b) Argue that the limiting distribution in (a) becomes a Poisson distribution with mean
λβ when n → ∞ and 1/α → 0 such that n/α remains equal to the constant λ. Can you
explain the similarity of this result with the insensitivity result (1.1.6) for the M/G/∞
queue in Section 1.1.3?
2.12 A production process in a factory yields waste that is temporarily stored on the factory
site. The amounts of waste that are produced in the successive weeks are independent and
identically distributed random variables with finite first two moments µ 1 and µ 2 . Opportuni-
ties to remove the waste from the factory site occur at the end of each week. The following
control rule is used. If at the end of a week the total amount of waste present is larger than
D, then all the waste present is removed; otherwise, nothing is removed. There is a fixed
cost of K > 0 for removing the waste and a variable cost of v > 0 for each unit of waste
in excess of the amount D.
(a) Define a regenerative process and identify its regeneration epochs.
(b) Determine the long-run average cost per time unit.
(c) Assuming that D is sufficiently large compared to µ 1 , give an approximate expression
for the average cost.
2.13 At a production facility orders arrive according to a renewal process with a mean
interarrival time 1/λ. A production is started only when N orders have accumulated. The
production time is negligible. A fixed cost of K > 0 is incurred for each production set-up
and holding costs are incurred at the rate of hj when j orders are waiting to be processed.
(a) Define a regenerative stochastic process and identify its regeneration epochs.
(b) Determine the long-run average cost per time unit.
(c) What value of N minimizes the long-run average cost per time unit?
2.14 Consider again Exercise 2.13. Assume now that it takes a fixed set-up time T to start a
production. Any new order that arrives during the set-up time is included in the production
run. Answer parts (a) and (b) from Exercise 2.13 for the particular case that the orders arrive
according to a Poisson process with rate λ.
2.15 How do you modify the expression for the long-run average cost per time unit in
Exercise 2.14 when it is assumed that the set-up time is a random variable with finite first
two moments?
2.16 Consider Example 1.3.1 again. Assume that a fixed cost of K > 0 is incurred for each
round trip and that a fixed amount R > 0 is earned for each passenger.
(a) Define a regenerative stochastic process and identify its regeneration epochs.
(b) Determine the long-run average net reward per time unit.
(c) Verify that the average reward is maximal for the unique value of T satisfying the
equation e −µT (RλT + Rλ/µ) = Rλ/µ − K when Rλ/µ > K.
2.17 Passengers arrive at a bus stop according to a Poisson process with rate λ. Buses
depart from the stop according to a renewal process with interdeparture time A. Using
renewal-reward processes, prove that the long-run average waiting time per passenger equals
2
E(A )/2E(A). Specify the regenerative process you need to prove this result. Can you give