Page 75 - A First Course In Stochastic Models

66 RENEWAL-REWARD PROCESSES

2.6 A CONTROLLED QUEUE WITH REMOVABLE SERVER ∗

Consider a production facility at which production orders arrive according to a
Poisson process with rate λ. The production times τ 1 , τ 2 , . . . of the orders are
independent random variables having a common probability distribution function
F with finite first two moments. Also, the production process is independent of the
arrival process. The facility can only work on one order at a time. It is assumed
that E(τ 1 ) < 1/λ; that is, the average production time per order is less than
the mean interarrival time between two consecutive orders. The facility operates
only intermittently and is shut down when no orders are present any more. A
fixed set-up cost of K > 0 is incurred each time the facility is reopened. Also a
holding cost h > 0 per time unit is incurred for each order waiting in queue. The
facility is only turned on when enough orders have accumulated. The so-called
N-policy reactivates the facility as soon as N orders are present. For ease we
assume that it takes a zero set-up time to restart production. How do we choose
the value of the control parameter N such that the long-run average cost per time
unit is minimal?
To analyse this problem, we first observe that for a given N-policy the stochastic
process describing jointly the number of orders present and the status of the facility
(on or off) regenerates itself each time the facility is turned on. Define a cycle as
the time elapsed between two consecutive reactivations of the facility. Clearly,
each cycle consists of a busy period B with production and an idle period I with
no production. We deal separately with the idle and the busy periods. Using the
memoryless property of the Poisson process, the length of the idle period is the
sum of N exponential random variables each having mean 1/λ. Hence
N
E(length of the idle period I) = .
λ
Similarly,

N − 1 1

E(holding cost incurred during I) = h + · · · + .
λ λ
To deal with the busy period, we define for n = 1, 2, . . . the quantities
t n = the expected time until the facility becomes empty given that
at epoch 0 a production starts with n orders present,
and
h n = the expected holding costs incurred until the facility becomes empty
given that at epoch 0 a production starts with n orders present.

These quantities are independent of the control rule considered. In particular, the
expected length of a busy period equals t N and the expected holding costs incurred

∗ This section contains specialized material and can be skipped at first reading.