1 / 18

# IOE/MFG 543 - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' IOE/MFG 543' - floramaria-catalina

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### IOE/MFG 543

Chapter 11: Stochastic single machine models with 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

• 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

• 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

• 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

• 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?

• 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)

• 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)=…

• 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

• 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)