Ihsan Ayyub Qazi

1
Ihsan Ayyub Qazi
Achievable sojourn times in M/M/1 and GI/GI/1
systems A Comparison between SRPT and Blind
Policies
D(t)
A(t), l
Q(t)
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Plan for the Presentation

• Some basics
• Motivation for minimizing sojourn times
• Intuition for the Optimality of SRPT
• Paper 1 On the average sojourn time under
  M/M/1/SRPT by Nikhil Bansal
• Paper 2 Achievable sojourn times by non-size
  based policies in a GI/GI/1 queue by Nikhil
  Bansal

3
Some Basics

• The sojourn time of a job is the time since a job
  arrives until the time it completes its service
  requirement.

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Why minimize sojourn time?

• Sojourn time is one of the most useful measures
  of
• user satisfaction and
• performance in a scheduling system.
• Keeping the average sojourn time low is often an
  important criteria in the choice of a scheduling
  policy.

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Optimizing the average sojourn time

• SRPT policy that at any works on the job with the
  smallest remaining service requirement, achieves
  the optimum possible average sojourn time for all
  possible instances of the problem 1, 2

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Optimality of SRPT
5

• Shortest Job First (SJF) is a non-preemptive
  policy that schedules the jobs in increasing
  order of their sizes.
• It is optimal for all non-preemptive polices
• However, the SJF policy is optimal only when all
  the jobs are present at the server at once.

Scheduler
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Total Completion Time 2025 45 Average
Completion Time 45/2 22.5
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Scheduler
5
Total Completion Time 525 30 Average
Completion Time 30/2 15
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Optimality of SRPT
However, one can consider the remaining
processing time (i.e. 15) as a separate job and
hence swap it with 5

• One can view SRPT as a preemptive version of SJF.
• Hence one can get an idea about the optimality of
  SRPT

5
Scheduler
15
5
20
Total Completion Time 520 25 Average
Completion Time 25/2 12.5
Total Completion Time 1520 35 Average
Completion Time 35/2 17.5
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Disadvantages of SRPT

• SRPT requires exact knowledge of the service
  requirement for each job
• It is unfair to Jobs with large service
  requirements and may even starve them.
• Blind policies have many attractive properties
  such as
• they are stateless and
• easier to implement in a real system.

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On the average sojourn time under M/M/1/SRPT by
Nikhil Bansal
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Plan

• Problem
• Results
• Background
• Analysis

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Problem

• How much more can the sojourn time improve if
  the knowledge of job sizes is used while
  scheduling?

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Classical Result

• It is a well-known result that the average
  sojourn time in an M/M/1 system is
• This holds for all scheduling policies that do
  not make use of the job sizes while scheduling
  3.

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Problem Definition (revisited)

• How much more can the sojourn time improve if
  the knowledge of job sizes is used while
  scheduling?
• In essence, this question reduces to finding the
  expression for the average sojourn time in a
  M/M/1/SRPT system

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Plan

• Problem
• Results
• Background
• Analysis

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Main result of the Paper

• The average sojourn time under M/M/1/SRPT with
  exponentially distributed job sizes varies as

Thus SRPT offers a factor improvement over
policies that ignore knowledge of job sizes while
scheduling
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Improvement Factor
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Contribution of the paper

• This result places a lower bound on the
  achievable average sojourn time under any
  scheduling policy in an M/M/1 system
• The result provides an analytical justification
  for the empirically observed improvement under
  SRPT at high loads.

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Plan

• Problem
• Results
• Background
• Analysis

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Some Previous work

• Schrage and Miller first obtained the expression
  for the expected sojourn time for a job of size x
  under SRPT in the more general M/G/1 queuing
  system. They showed that,

Mean Waiting Time
Mean Residence Time
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Mean Waiting and Residence times for a job of
size (Intuition)

• Average time a job of size x takes from when it
  arrives to when it receives service for the first
  time

Expected Waiting Time of a job of size x
corresponds to waiting for all jobs of sizes
less than or equal to x to complete

• Average time it takes for a job of size x to
  complete once it begins execution.

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Plan

• Problem
• Results
• Background
• Analysis

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Analysis Expressions with exponentially
distributed job sizes
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Expression for the sojourn time under SRPT for
exponential distributed job sizes

• The average sojourn time under SRPT is given by

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Results

• Theorem 1 For all p between 0 and 1
• Theorem 2 (Heavy Traffic Case)

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Sketch of the proof

• An upper bound on ER is derived.
• Then an upper bound for EW is derived for the
  case when the utilization is more than 75.
• For other values of utilization, the result is
  shown to be true by considering the average
  sojourn time under FCFS as the upper bound.
• It is shown that that the contribution of ER to
  ET is not significant.
• Hence a good tight bound can be obtained for ET
  by lower bounding it by EW

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Upper Bounding the Sojourn Time

• Lemma 1 For any load p, such that it is between
  0 and 1
• For any load p, such that it is between 2/3 and 1

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Comparison between the bounds on ER and EW
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Lower Bounding the Sojourn Time

• Lemma 3

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Sketch of the proof (contd)

• For any load, the average sojourn time under SRPT
  is atleast equal to the average job size. Since
  SRPT is optimal ET is upper bounded by the
  average sojourn time under the policy FCFS. Thus
• And hence for p lt 2/3. The theorem equation is
  seen to be true in this case.

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Upper Bound for Theorem 1
Lemma 3 gives the required lower bound, hence
theorem 1 is proved
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Thankyou
32

• Achievable sojourn times by non-size based
  policies in a GI/GI/1 queuing system by Nikhil
  Bansal

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Plan

• Problem
• Results
• Background
• Analysis

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Problem

• How much worse can the average time under the
  best blind policy be as compared to SRPT (in a
  GI/GI/1 system)?

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Plan

• Problem
• Results
• Background
• Analysis

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Main result of the paper

• For a GI/GI/1 system, the average sojourn time
  under the best blind policy is atmost time
  worse (upto a constant factors) then the best
  average sojourn time possible under any arbitrary
  policy
• Thus in a sense, the lack of knowledge of actual
  job-sizes does not pose a serious problem if the
  blind policy is chosen carefully.

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Improvement Factor of SRPT against the Best Blind
Scheduling Policy
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Background and Motivation

• A GI/GI/1 queuing system
• Inter-arrival times are i.i.d rvs, A, taken from
  a General distribution Ga
• Job Sizes are i.i.d rvs, S, taken from a General
  distribution Gs
• Different from a G/G/1 system.
• Gs and Ga specify a GI/GI/1 system completely.
  Therefore, the optimum blind policy A (which
  minimizes the average sojourn time) only depends
  on the respective distributions.
• Opt(Ga, Gs) Average sojourn time under the
  optimum blind policy.

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Difference in the analyses of M/M/1 and G1/G1/1
queuing systems

• In a M/M/1 system all blind policies are
  identical as far as the average sojourn time is
  concerned. In particular any blind scheduling
  policy has average sojourn equal to
• So the question reduced to determining the
  average sojourn time under SRPT. Bansal showed
  that for exponential job sizes the average
  sojourn time under M/M/1/SRPT is

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So why is the analysis of a G1/G1/1 queuing
system different and more difficult?

• Unlike the case in M/M/1, all blind policies are
  no more identical !!!
• No single blind policy is optimal for e.g.
• In an M/G/1 system FCFS is optimum for job size
  distributions with increasing failure rates.
• Foreground Background (FB) that at time works on
  the job with the least attained service is
  optimum for job size distributions with
  decreasing failure rates

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So why is the analysis of a G1/G1/1 queuing
system different and more difficult?

• Same blind policy can have very different
  behaviours for different job sizes.
• While FB has average sojourn time
  when job sizes are exponentially distributed, the
• Average sojourn time under FB varies as
  when the job sizes have the pareto distribution
  with

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What to do? So why is the analysis of a G1/G1/1
queuing system so difficult?

• Furthermore, As the analytic expression for
  average sojourn time under an arbitrary GI/GI/1
  system is not known and moreover no analytic
  expression for Opt(Ga, Gs) for general Ga, Gs is
  known, therefore we cannot adopt the approach
  that we used in the case of M/M/1

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Hmmm
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Plan

• Problem
• Results
• Background
• Analysis

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Competitive Analysis

• Let A(I) Total sojourn time when the instance is
  executed according to the algorithm A.
• We say that a deterministic algorithm has a
  competitive ratio c(n) if
• The definition of competitive ratio is quite
  strict,
• A more useful notion is that of a randomized
  algorithm.

Worst Case Ratio over all input instances of size
atmost n achieved by A and the optimum cost on
that instance
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Competitive Analysis

• The competitive ratio of a randomized algorithm
  is defined as
• The crucial thing to observe is that there is no
  probabilistic assumption on the input instance.
• The input instance is still chosen adversarially
  to maximize the ratio.

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Competitive Analysis

• Randomized algorithms can have significantly
  better competitive ratios than deterministic
  algorithms.
• NOTE The notion of the performance of a
  randomized algorithm is dual to the notion of the
  average case performance of a system like
  GI/GI/1
• While the former deals with the performance over
  a distribution over algorithms on a fixed input
  instance, the latter deals with the performance
  of a fixed algorithm on a distribution over input
  instances. Can we relate the two?

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Yaos Minimax Theorem (Contrapositive)
Minimization Problem

• Suppose a cost minimization problem P has a
  c(R,n) competitive randomized online algorithm R
  for request sequences of length atmost n. Let
  distribution y(j) be any distribution over
  request sequences of length atmost n. Then,
• In particular, for any distribution over the
  input instances, if we consider the algorithm Ai
  that has the best average case performance on
  this distribution, then this performance is no
  worse than c(R,n) times the performance of the
  optimum algorithm.

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Plan

• Problem
• Results
• Background
• Analysis

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Some known results for competitive ratio of blind
scheduling algorithm

• For the problem of minimizing the average sojourn
  time
• Motwani, Phillips and Torng showed that no blind
  deterministic scheduling algorithm can have a
  competitive ratio better than
• The same people showed that any randomized
  algorithm has a competitive ratio atleast
  

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BreakThrough.

• In a significant breakthrough, Pruhs and
  Kalayanasundaram gave a non-trivial randomized
  algorithm that they called RMLF 5 and proved
  that it has a competitive ratio of
• Later it was shown that RMLF is infact an
• competitive randomized algorithm and hence the
  best possible upto constant factors. In other
  words,
• There is a universal constant r, such that for
  any scheduling instance with atmost n jobs, the
  expected total sojourn time under RMLF is atmost
  rlogn times that under SRPT

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Analysis

• Theorem 1 GI/GI/1 system with inter-arrival
  distribution Ga and service distribution Gs,
  there exists a universal constant d such that
• Where C is a bound on the sixth coefficient of
  variation of Ga and Gs, that is

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Analysis

• Lemma 1 Given any probability distribution on
  the input instances with atmost n jobs, there is
  some deterministic algorithm the expected total
  sojourn time of which is atmost r(logn) times
  that under SRPT
• If we take each busy period in a GI/GI/1 system
  as a separate instance then we can think of a
  GI/GI/1 system as defining a probability
  distribution on input instances.
• Can we apply Lemma1 ?
• No!
• Because we cannot get a bound on the number of
  jobs a busy period might contain !!

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Analysis (Basic Idea)

• For every n, there is a non-zero fraction of busy
  periods that contain more than n jobs.
• Can we come up with a n such that most of the
  action happens with those many jobs in a busy
  period?
• The analysis is based on this premise.
• A busy period has about 1/1-p jobs on the
  average, hence the logn factor in lemma 1 should
  essentially be log(1/1-p)
• Most of the action essentially happens in busy
  periods with atmost C6/(1-p)6 jobs and hence
  lemma 1 can be replaced by this number.

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Analysis

• The busy periods are independent of the actual
  scheduling policy involved.
• Let
• B set of all possible busy periods
• M measure induced by GI/GI/1 system on B
• TA(B) Total sojourn time incurred when A is
  executed during the busy period B
• n(B) number of jobs in B
• Average sojourn time of a job under an algorithm
  A can be expressed as

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Analysis

• For any GI/GI/1 system, En(B) is identical for
  every work-conserving scheduling policy. Thus it
  suffices to compare the expected total sojourn
  time of a busy period for two scheduling
  policies, in order to compare the average sojourn
  time under them.
• A busy period is called bad if it contains more
  than N024C6/(1-p)6.
• Define a process P as follows Whenever there is
  a bad busy period of length B, we replace it with
  another busy period with n(B) dummy jobs of
  length 0.
• Let M be the measure induced by the new process
  on the busy periods. By construction EMn(B)
  EMn(B).
• Now lemma 1 can be applied..

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Analysis

• Key point The only contribution to the expected
  total sojourn time under the measure M is due to
  the busy periods that contain atmost N0 jobs.
• Let Opt blind algorithm A that minimizes
  EMTA(B)

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Few Lemmas

• Lemma 2 (follows easily from lemma 1)
• Lemma 3 For any work-conserving algorithm A,
  uses lemma 4,5
• Lemma 4 Let SnX1X2.Xn where Xi are
  independent and identically distributed according
  to the random variable X. Then, for epsilon gt 0,

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Few Lemmas

• Lemma 5 Given a GI/GI/1 system, let A and S be
  the random variables representing arrived time
  and service time respectively. Let
  and

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Proof of lemma 3

• Since M and M differ only on bad busy periods,
  therefore for any work-conserving algorithm
• To show the other side of the inequality we need
  to upper bound the contribution due to bad busy
  periods in EMTA(B)
• In a busy period of length l and consisting of n
  jobs, the total sojourn time of the jobs involved
  can be atmost nl, irrespective of the choice of
  the algorithm.
• Thus we need to upper bound

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Proof of Theorem 1

• Consider the optimum blind policy Opt that
  minimizes EMTOpt(B). It follows by lemma 3
• By definition
• By lemma 2
• Which by lemma 3 implies
• Combining the above we get

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Summary

• The paper proves that for any GI/GI/1 queuing
  system, the performance of the best blind policy
  for that system achieves average sojourn close to
  the best possible by any other (non-blind)
  policy.
• The result, however, does not give a construct
  way to produce the optimum (deterministic) blind
  scheduling policy for an arbitrary GI/GI/1
  system.
• The dependence on the sixth coefficient of
  variations of inter-arrival times and sizes is
  the artifact of the analysis. The finiteness of
  the sixth moments allows to show that the
  probability a busy period has more than n jobs
  decays at least as 1/n3.

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References

• 1 On the average sojourn time under M/M/1/SRPT,
  Nikhil Bansal, to appear in Operations Research
  Letters, Volume 33, 2, March 2005, 195-200.
• 2 Achievable sojourn times by non-size based
  policies in a GI/GI/1 queue, Nikhil Bansal,
  submitted for publication.
• 3 D.R. Smith. A new proof of the optimality of
  the shortest remaining processing time
  discipline. Operations Research, 26197199,
  1976.
• 4 L.E. Schrage. A proof of the optimality of
  the shortest processing remaining time
  discipline. Operations Research, 16678690,
  1968.
• 5 R. W. Conway, W. L. Maxwell, and L. W.
  Miller. Theory of Scheduling. Addison-Wesley
  Publishing Company, 1967.
• 6 R. Motwani, S. Phillips, and E. Torng.
  Nonclairvoyant scheduling. Theoretical Computer
  Science,130(1)1747, 1994.
• 7 B. Kalyanasundaram and K. Pruhs. Minimizing
  flow time nonclairvoyantly. In IEEE Symposium on
  Foundations of Computer Science, pages 345352,
  1997.
• 8 A. C-C. Yao. Probabilistic computations
  Toward a unified measure of complexity (extended
  abstract). In IEEE Symposium on Foundations of
  Computer Science (FOCS), 1977.

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Questions
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Thankyou