Page 83 - A First Course In Stochastic Models

74 RENEWAL-REWARD PROCESSES

a heuristic explanation of why the answer for the average waiting time is the same as the
average residual life in a renewal process?
2.18 Consider a renewal process in which the interoccurrence times have a positive density
on some interval. For any time t let the age variable δ t denote the time elapsed since the
last occurrence of an event. Use the renewal-reward model to prove that lim t→∞ E(δ t ) =
µ 2 /2µ 1 , where µ k is the kth moment of the interoccurrence times. (Hint: assume a cost at
rate x when a time x has elapsed since the last occurrence of an event.)
2.19 A common car service between cities in Israel is a sheroot. A sheroot is a seven-seat cab
that leaves from its stand as soon as it has collected seven passengers. Suppose that potential
passengers arrive at the stand according to a Poisson process with rate λ. An arriving person
who sees no cab at the stand goes elsewhere and is lost for the particular car service. Empty
cabs pass the stand according to a Poisson process with rate µ. An empty cab stops only at
the stand when there is no other cab.
(a) Define a regenerative process and identify its regeneration epochs.
(b) Determine the long-run fraction of time there is no cab at the stand and determine
the long-run fraction of customers who are lost. Explain why these two fractions are equal
to each other.
2.20 Big Jim, a man of few words, runs a one-man business. This business is called upon by
loan sharks to collect overdue loans. Big Jim takes his profession seriously and accepts only
one assignment at a time. The assignments are classified by Jim into n different categories
j = 1, . . . , n. An assignment of type j takes him a random number of τ j days and gives
a random profit of ξ j dollars for j = 1, . . . , n. Assignments of the types 1, . . . , n arrive
according to independent Poisson processes with respective rates λ 1 , . . . , λ n . Big Jim, once
studying at a prestigious business school, is a muscleman with brains. He has decided to
∗
accept those type j assignments for which E(ξ j )/E(τ j ) is at least g dollars per day for a
∗
carefully chosen value of g (in Exercise 7.4 you are asked to use Markov decision theory
to determine g ). Suppose that Big Jim only accepts type j assignments for j = 1, . . . , n 0 .
∗
An assignment can only be accepted when Big Jim is not at work on another assignment.
Assignments that are refused are handled by a colleague of Big Jim.
(a) Define a regenerative process and identify its regeneration epochs.
(b) Determine the long-run average pay-off per time unit for Big Jim.
(c) Determine the long-run fraction of time Big Jim is at work and the long-run fraction
of the assignments of the types 1, . . . , n 0 that are not accepted. Explain why these two
fractions are equal to each other.
2.21 Consider the (S − 1, S) inventory model with back ordering from Section 1.1.3. What
is the long-run fraction of customer demand that is back ordered? What is the long-run
average amount of time a unit is kept in stock?
2.22 Consider a machine whose state deteriorates through time. The state of the machine is
inspected at fixed times t = 0, 1, . . . . In each period between two successive inspections
the machine incurs a random amount of damage. The amounts of damage accumulate. The
amounts of damage incurred in the successive periods are independent random variables
having a common exponential distribution with mean 1/α. A compulsory repair of the
machine is required when an inspection reveals a cumulative amount of damage larger than
a critical level L. A compulsory repair involves a fixed cost of R c > 0. A preventive repair
at a lower cost of R p > 0 is possible when an inspection reveals a cumulative amount of
damage below or at the level L. The following control limit rule is used. A repair is done
at each inspection that reveals a cumulative amount of damage larger than some repair limit
z with 0 ≤ z < L. It is assumed that each repair takes a negligible time and that after each
repair the machine is as good as new.
(a) Define a regenerative process and identify its regeneration epochs.
(b) What is the expected number of periods between two successive repairs? What is the