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### IOE/MFG 543

Chapter 11: Stochastic single machine models with release dates

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

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

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)

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

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

Two job classes times without preemptions (2)

- 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

Theorem 11.1.1 times without preemptions (2)

- 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

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

Work in the system releases

- 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

Work in the system (2) releases

- 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

Poisson releases and single job class releases

- 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

Poisson releases and PASTA releases

- 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?

Computing the expected amount of work in the system releases

- 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)

Computing the amount paid by a job releases

- 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

Computing the releasesexpected 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)=…

Computing the expected wait in queue releases

- 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

Computing the expected length of a busy period releases

- 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)

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