IOE/MFG 543

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IOE/MFG 543

• Chapter 11 Stochastic single machine models
  with release dates

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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

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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

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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)

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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 1 2
pj 4 1
rj 0 1
wj 1 100
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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 1 2 3
pj 1 4 1
rj 0 0 1
wj 1 5 100
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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

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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

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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

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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

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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

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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

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Poisson releases and PASTA

• PASTAPoisson 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?

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Computing the expected amount of work in the
system

• Let E(V)limt?8E(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)

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Computing the amount paid by a job

• Let Wq be the time a job spends in the queue
• Then WsWqX 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 XWqX2/2

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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)

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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 Pollaczek-Khintchine (or
  simply P-K) formula

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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 BI 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)