IOE/MFG 543
Chapter 11:
Stochastic single machine models
with release dates
1
Random release dates

Jobs (or orders) come in at different
unknown times
– The release date of a job is unknown


Random release dates are similar to
customer arrivals to a queuing system
Jobs have different priorities
– Not necessarily optimal to have a FIFO
policy
– Priority queues
2
Total weighted flow time


Since jobs are released at different
times it makes sense to minimize the
total weighted time a job spends in the
system, or flow time
Flow time
– Let the release date of job j be Rj
– The flow time is Cj-Rj
3
Minimizing expected total
weighted flow time


The objective function is E(Swj(Cj-Rj))
Taking the expected value inside the
sum we get
E(Swj(Cj-Rj))=...

So minimizing E(Swj(Cj-Rj)) is equivalent
to minimizing E(SwjCj)
4
Section 11.1 Arbitrary releases and
arbitrary processing times without
preemptions

The problems 1 | rj | SwjCj is NP-hard
– It may be optimal to keep the machine
idle until a job is released

Example 11.1.3
job j
pj
rj
wj
1
4
0
1
2
1
1
100
5
Section 11.1 Arbitrary releases and
arbitrary processing times without
preemptions (2)


WSPT for available jobs may not be
optimal even if we do not allow
unforced idleness
Example 11.1.2
job j
pj
rj
wj
1
1
0
1
2
4
0
5
3
1
1
100
6
Two job classes

Suppose there are only two types of jobs
– All jobs in the same class have the same
distribution
– The mean processing times of jobs in classes 1
and 2 are 1/l1 and 1/l2, respectively
– The weight of jobs in classes 1 and 2 are w1 and
w2, respectively

The release dates can have any distribution
7
Theorem 11.1.1

Assume that
– Unforced idleness is not allowed
– There are only two job classes

Under the optimal nonpreemptive
dynamic policy, the decision maker
follows the WSEPT rule whenever the
machine is freed
8
Section 11.2 Priority queues,
work conservation and Poisson
releases



Suppose jobs (an unknown number)
arrive randomly to the machine
Each job requires a random amount of
processing time Xj on the machine
If a job is being processed at time t let
xr(t) be the remaining processing time
9
Work in the system



At any time t there may be a number
of jobs waiting to be processed on the
machine (excluding the one in
process)
Let V(t) be the total processing time of
those jobs plus xr(t)
V(t) is referred to as the amount of
work in the system
10
Work in the system (2)



Any time a job j arrives V(t) jumps by
Xj
Between jumps V(t) decreases at rate
1 as long as the machine is processing
jobs
We can use the stochastic process V(t)
to analyze the system
11
Poisson releases and
single job class


To simplify the discussion we assume
that the time between release dates
are exponentially distributed at rate n
We also assume that there is only a
single job class
– The processing time of job j is X where X
is a random variable with distribution F
12
Poisson releases and
PASTA



PASTA=Poisson Arrivals See Time Averages
This is a very useful property that Poisson
releases have
Example 11.2.1
– Poisson releases at rate 1 per 10 minutes
– Processing times equal 4 minutes
– What is the time average number of jobs being
processed?
– What is the probability that a job can
immediately start processing when released?
– What if the time between releases is
deterministic and equal to 10 minutes?
13
Computing the expected
amount of work in the system


Let E(V)=limt∞E(V(t)) be the expected
amount of work in the system when the
system is in steady state
Suppose the jobs pay $1 per unit processing
time left for each time unit they spend in
the system
– How much money does the system earn per unit
time?

The average amount of money the system
earns per unit time is
E(V)=n E(amount paid by a job)
14
Computing the amount
paid by a job




Let Wq be the time a job spends in the
queue
Then Ws=Wq+X is the total time spent
in the system
The job pays at a constant rate X
while it is in the queue and the total
payout while in service is X2/2
Amount paid by a job = XWq+X2/2
15
Computing the expected
amount paid by a job

If the dispatching rule is independent
of X then Wq and X are independent
and
E(amount paid by a job)=…
16
Computing the expected
wait in queue



By the PASTA and if a FCFS rule is used
E(Wq)= …
This gives the equation
E(Wq)=nE(X)E(Wq)+nE(X2)/2
or
E(Wq)=nE(X2)/[2(1-nE(X))]
This is known as the PollaczekKhintchine (or simply P-K) formula
17
Computing the expected
length of a busy period




Let B be the length of a busy period
Let I be the length of an idle period
Then B+I is a cycle
The (long run) proportion of time the
machine is busy is
E(B)/(E(B)+E(I))=l/n

It is clear that for Poisson releases
E(I)=1/n

Then
E(B)=1/(l-n)
18