lnmb course advanced queueing theorysem/aqt/lecture02042012.pdf · 2012-04-02 · we will study...

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LNMB Course

Advanced Queueing Theory

Lecture 7, April 2, 2012

Onno Boxma, Sem Borst (TU/e)

http://www.win.tue.nl/˜sem/AQT/


Course overview

1. Product-form networks: Queue lengths2. Product-form networks: Sojourn times3. The M/G/1 queue; multi-class queues4. Polling systems I5. PS, symmetric disciplines, DPS, GPS, BS networks6. Achievable delay region, delay optimization7. Size-based scheduling, SRPT, FBPS/LAS8. Heavy tails; impact of the service discipline9. Polling systems II


Characterization of achievable mean waiting-time performance

Main result: A = A∗ = P

A = {E{Wπ} : π ∈ 5} is achievable performance region

(set of all achievable performance vectors)

A∗ := conv({E{Wπ} : π ∈ 5∗})

(convex hull of performance vectors achieved by priority strategies)provides geometric/structural description

P := {w ∈ RN: w satisfies cons. law & ineq.}

provides analytic/polyhedral characterization


Optimization

Let us first consider case of non-preemptive strategies

Consider problem of finding scheduling strategy which minimizes meanholding cost per unit of time:

minπ∈5

N∑j=1

λ jc jE{Wπj }

Since A = A∗ = conv({E{Wπ} : π ∈ 5∗}), there must exist priority strategy

which is optimal, i.e., above problem may be reduced to solving

minπ∈5∗

N∑j=1

λ jc jE{Wπj }


Optimization (cont’d)

To determine which strategy is optimal, suppose we swap two ad-jacent classes, i and i + 1, in (presumed) optimal priority ordering(1, 2, . . . , i, i + 1, . . . , N )

Then,E{Wi} < E{W′i},

E{Wi+1} > E{W′i+1},

while for all j 6= i, i + 1,E{W′j} = E{W j}



Because of work conservation property,

N∑j=1

ρ jE{W j} =

N∑j=1

ρ jE{W′j},

so that

ρiE{Wi} + ρi+1E{Wi+1} = ρiE{W′i} + ρi+1E{W′i+1},

or equivalently,

ρi(E{Wi} − E{W′i}) = ρi+1(E{W′i+1} − E{Wi+1}) =: 1 < 0



Thus,N∑

j=1

λ jc jE{W j} −

N∑j=1

λ jc jE{W′j} =

λiciE{Wi} + λi+1ci+1E{Wi+1} − λiciE{W′i} − λi+1ci+1E{W′i+1} =

λici(E{Wi} − E{W′i})− λi+1ci+1(E{W′i+1} − E{Wi+1}) =

ci

βiρi(E{Wi} − E{W′i})−

ci+1

βi+1ρi+1(E{W′i+1} − E{Wi+1}) =

1

(ci

βi−

ci+1

βi+1

)In order for strategy to be optimal, latter term must be non-positive, i.e.,since 1 < 0,

ci

βi≥

ci+1

βi+1



Thus, customer classes must be ranked in order of non-increasing value ofratio ci/βi , in particular

• “larger holding cost ci → higher priority”

• “larger mean service time βi → lower priority”

Recovers classical result [Fife (1965), Smith (1956)]

Conversely, that classical result shows that any linear combination of meanwaiting times is minimized by some priority strategy, which directly impliesthat A ⊆ A∗

Side-note: minimizing linear function over polyhedron P can apparently bedone efficiently using greedy algorithm, despite fact that P is described byas many as 2N

− 2 linear inequalities – reflects ‘special structure’ of P


Interpretation

Consider consecutive service of class-i customer and class- j customer

Order i , j yields expected cost ciwi + c j(wi + βi)

Reversed order j , i yields expected cost c jwi + ci(wi + β j)

Difference in expected cost equals

ciβ j − c jβi

which must be non-negative for order i , j to be optimal, i.e.,

ci

βi≥

c j

β j

Complication in proof: need to deal with possibility of additional arrivals


Optimization – preemptive strategies

Let us now turn to case of (possibly) preemptive strategies

As before, assume exponentially distributed service times, and considersojourn times as opposed to waiting times

Similar arguments as above then show that customer classes must beranked in order of non-increasing value of product ciµi (“cµ rule”)


Some special cases

Minimization of mean holding cost: βi ≡ β for all i = 1, . . . , N →Customer classes must be ranked in order of non-increasing value of ci

Minimization of mean waiting time: ci ≡ c for all i = 1, . . . , N →Customer classes must be ranked in order of non-decreasing value of βi :Shortest Expected Processing Time first (SEPT)


Some reflection on assumptions/limitations

Admissible scheduling strategies

• work-conserving

• uses no knowledge of (remaining) service times

In case of preemptive strategies, characterization / optimization results lim-ited to exponentially distributed service times


Some reflection on assumptions/limitations(cont’d)

Work-conserving assumption is non-restrictive (for Poisson arrivals)

Knowledge of (remaining) service times may be modeled by introducing(possibly continuum of) artificial customer ‘classes’, each corresponding todifferent possible value of service time

May thus be interpreted as special case with (possibly continuum of) cus-tomer classes, each with different deterministic service time


Non-preemptive strategies

In order to minimize mean waiting time, previous results imply that artifi-cial customer ‘classes’ must be ranked in order of non-decreasing value ofmean service time, which now equals actual service time

Thus, customers must be served in order of non-decreasing value of servicetime: Shortest Processing Time first (SPT)

In case of preemptive strategies two (related) issues remain to be addressed

• Knowledge of (remaining) service times

• Generally distributed service times

Both issues will be revisited soon


Extensions

Multi-server queues [Federgruen & Groenevelt (1988)]

Issues

• Work conservation property does not hold, except for identical determin-istic service times

• Mean amount of work can only be evaluated for identically exponentiallydistributed service times


Extensions (cont’d)

General multi-class stochastic system (multi-armed bandit problems,branching bandit problems, Klimov’s model)

For conciseness, let I := {1, . . . , N } index set of classes

Let 5 be class of admissible scheduling strategies

xπ performance vector for strategy π ∈ 5



Performance vector xπ is said to satisfy generalized conservation laws ifthere exist set function b : 2I

→+ and matrix (ASi )i∈I,S⊆I such that

• ∑j∈S

ASj xξj = b(S)

for all strategies ξ that give priority to classes j ∈ S, S ⊆ I, over classesj ∈ I \ S

• For all strategies π ∈ 5, ∑j∈I

AIj xπj = b(I),

and ∑j∈S

ASj xπj ≥ b(S)

for all proper subsets S ⊂ I



Define P(A, b) as polyhedron specified by generalized conservation laws

Then,

• Performance vectors achieved by priority strategies are verticesof P(A, b)

• Achievable performance region is P(A, b)

Thus, optimization problem minπ∈5

N∑j=1

c j xπj

may equivalently be expressed as linear program minx∈P(A,b)

N∑j=1

c j x j

Due to special polymatroidal structure, above linear program can be effi-ciently solved by running greedy algorithm

Simple index rules are optimal


Preemptive strategies

Two (related) issues remain to be addressed

• Knowledge of (remaining) service times

• Generally distributed service times

We will study these two issues in context of G/G/1 queue

Specifically, we will identify strategies that minimize mean sojourn time inG/G/1 queue with possible preemption

Literature: D. Towsley (1995).Application of majorization to control problems in queueing systems.In: Scheduling Theory and its Applications, Chapter 14, Ph. Chrétienne, E.G.Coffman Jr., J.K. Lenstra, Z. Liu (eds.), John Wiley & Sons, Chichester.


Preemptive strategies (cont’d)

First suppose we may use knowledge of (remaining) service times

We will show that Shortest Remaining Processing Time first (SRPT) strat-egy minimizes mean number of customers in system

SRPT in fact minimizes number of customers at any point in time for given‘sample path’ (i.e. realization of arrival and service times)

Because of Little’s law, SRPT also minimizes mean sojourn time



It is not possible to prove directly that SRPT outperforms any other strategyby considering number of customers in system only

Instead, we will consider vector of remaining service times of all customers,and show that this vector under SRPT ‘dominates’ that under any otherstrategy in certain sense

Dominance implies that number of customers is lower

Two proof concepts/techniques

• Majorization

• Sample-path comparison


Majorization

Let x and y be n-dimensional (real-valued) vectors

Let x[i] and y[i] denote i -th largest component of x and y, respectively

DefinitionVector x is said to be majorized by vector y, denoted x ≺ y, if

j∑i=1

x[i] ≤j∑

i=1

y[i]

for all j = 1, . . . , n − 1, and

n∑i=1

x[i] =n∑

i=1

y[i]

Informally speaking, x ≺ y means that vector x is ‘more balanced’ thanvector y, average value of components being equal


Sample-path comparison

Let Zπ be performance vector for strategy π ∈ 5

Want to show that some strategy π∗ optimizes some performance mea-sure Z in stochastic sense

• either Zπ∗

≤st Zπ for all admissible strategies π , i.e., P{Zπ∗ ≤ z} ≥P{Zπ ≤ z} for all z

• or Zπ∗

≥st Zπ for all admissible strategies π i.e., P{Zπ∗ ≤ z} ≤ P{Zπ ≤z} for all z


Sample-path comparison (cont’d)

General principle (stochastic coupling): compare ‘guessed’ optimal strat-egy π∗ with with some arbitrary strategy π by constructing processes U andU∗ on common probability space such that

• U ≤ U∗ (or U ≥ U∗, depending on context)

• U =st Zπ

• U∗ =st Zπ∗

which implies Zπ ≤st Zπ∗

(or Zπ ≥st Zπ∗

)

Although existence of U and U∗ is guaranteed, constructing them is non-obvious and involves subtleties

No universal recipe or systematic framework

Requires combination of ingenuity, intuition and caution


Proof of SRPT optimality

We will now use notion of majorization and sample path comparison toprove that π∗ = SRPT minimizes number of customers in system at anypoint in time

As mentioned earlier, considering number of customers only is not suffi-cient

Instead, we will consider vector of remaining service times of all customers,and show that this vector under SRPT majorizes that under any otherstrategy

Majorization implies that vector contains maximum number of zero com-ponents, and hence number of customers is minimized


Proof of SRPT optimality (cont’d)

Notation

A(t): number of customers that arrive during [0, t]

Qπ(t): number of customers in system at time t under strategy π ∈ 5

Tπ(t) := (Tπ1 (t), . . . ,TπA(t)(t)): vector of remaining service times under

strategy π ∈ 5, possibly zero, arranged in non-increasing order

TheoremIf Qπ

∗

(0) = Qπ(0) = 0, then

{Qπ∗

(t)}t≥0 ≤st {Qπ(t)}t≥0

for all strategies π ∈ 5

Above inequality in fact holds sample-path wise



Condition on arrival times, service times and initial queue lengths

Let 0 = t0 < t1 < · · · < tn < . . . be event times under strategies π and π∗

(arrivals, service completions, service preemptions)

We will show by forward-induction argument that

Tπ(t) ≺ Tπ∗

(t)

for all t ≥ 0 over all sample paths

Observe that latter inequality implies that vector Tπ∗

(t) contains more zerocomponents than Tπ(t), i.e.,

Qπ∗

(t) ≤ Qπ(t)



Basis step

By assumption, inequality relations hold for t = t0

Induction step

Assume that inequality relation holds through t = tn

We prove relation for t ∈ (tn, tn+1]

By induction hypothesis,

k∑i=1

Tπ[i](tn) ≤

k∑i=1

Tπ∗

[i] (tn)

for k = 1, . . . ,A(tn)



Strategy π∗ serves customer with smallest remaining service time – letthat be m-th largest component of vector Tπ

∗

(tn) – so Tπ∗

[i] (tn) = 0 fori = m + 1, . . . ,A(t)

Let π serve customer with l-th largest remaining service time

We need to distinguish between two cases: l ≤ m and l > m


Case l ≤ m

π*

1 2 3 A(t) 1 2 3 A(t)

π

ml

•k∑

j=1

Tπ[ j](t) =

k∑j=1

Tπ[ j](tn) ≤

k∑j=1

Tπ∗

[ j](tn) =k∑

j=1

Tπ∗

[ j](t)

for k = 1, . . . , l − 1


•k∑

j=1

Tπ[ j](t) =

k∑j=1

Tπ[ j](tn)− (t − tn) ≤

k∑j=1

Tπ∗

[ j](tn) =k∑

j=1

Tπ∗

[ j](t)

for k = l, . . . ,m − 1

•k∑

j=1

Tπ[ j](t) =

k∑j=1

Tπ[ j](tn)− (t − tn) ≤

k∑j=1

Tπ∗

[ j](tn)− (t − tn) =k∑

j=1

Tπ∗

[ j](t)

for k = m, . . . ,A(t)


Case l > m

π*

1 2 3 A(t) 1 2 3 A(t)

π

ml

•k∑

j=1

Tπ[ j](t) =

k∑j=1

Tπ[ j](tn) ≤

k∑j=1

Tπ∗

[ j](tn) =k∑

j=1

Tπ∗

[ j](t)

for k = 1, . . . ,m − 1


•k∑

j=1

Tπ[ j](t) =

k∑j=1

Tπ[ j](tn) ≤

A(t)∑j=1

Tπ[ j](tn)− (t − tn) =

A(t)∑j=1

Tπ∗

[ j](tn)− (t − tn) =k∑

j=1

Tπ∗

[ j](t)

for k = m, . . . , l − 1

•k∑

j=1

Tπ[ j](t) =

k∑j=1

Tπ[ j](tn)− (t − tn) ≤

k∑j=1

Tπ∗

[ j](tn)− (t − tn) =k∑

j=1

Tπ∗

[ j](t)

for k = l, . . . ,A(t)



Thus, we conclude Tπ(t) ≺ Tπ∗

(t) for all t ∈ [tn, tn+1]

By forward induction, Tπ(t) ≺ Tπ∗

(t) for all t ≥ 0

Removal of conditioning on arrival times, service phases and initial queuelengths completes proof


Preemptive strategies

Now suppose we have generally distributed service times with distributionfunction F(x) = P{B < x}, density f (·) = dF(x)/dx , and no longerknowledge of actual values

Failure (or hazard) rate function of B is h(x) = f (x)/(1− F(x))



Results for monotone failure rate functions and Poisson arrivals

• If B has increasing failure rate function (IFR), then serving customersin any non-preemptive manner (e.g. in order of arrival (FCFS)) stochas-tically minimizes number of customers in system, and hence mean so-journ time

• If B has decreasing failure rate function (DFR), then serving customer(s)with Least Attained Service Time (LAST) stochastically minimizes num-ber of customers in system, and hence mean sojourn time

Note that under LAST service capacity is generally shared among severalcustomers simultaneously

LAST is also known under name Foreground Background Processor Shar-ing (FBPS)



• If B has increasing failure rate function (IFR), then LAST stochasticallymaximizes number of customers in system, and hence mean sojourntime

• If B has decreasing failure rate function (DFR), then serving customersin any non-preemptive manner stochastically maximizes number of cus-tomers in system, and hence mean sojourn time

In case B is exponentially distributed, failure rate function is constant (bothDFR and IFR), and it follows that any strategy gives rise to same distributionof number of customers, and hence same mean sojourn time (but not samesojourn time distribution)


References

D. Bertsimas (1995). The achievable region method in the optimal controlof queueing systems; formulations, bounds and policies. Queueing Systems21, 337–389.D. Bertsimas, J. Niño-Mora (1996). Conservation laws, extended polyma-troids and multiarmed bandit problems; a polyhedral approach to indexablesystems. Math. Oper. Res. 21, 257–306.P.P. Bhattacharya, L. Georgiadis, P. Tsoucas (1995). Problems of adaptiveoptimization in multiclass M/GI/1 queue with Bernoulli feedback. Math.Oper. Res. 20, 355–380.E.G. Coffman, Jr., I. Mitrani (1980). A characterization of waiting-time per-formance realizable by single-server queues. Oper. Res. 28, 810–821.G. Fayolle, I. Mitrani, R. Iasnogorodski (1980). Sharing a processor amongmany job classes. J. ACM 27, 519–532.A. Federgruen, H. Groenevelt (1988). Characterization and optimization ofachievable performance in general queueing systems. Oper. Res. 36, 733–741.


References (cont’d)

A. Federgruen, H. Groenevelt (1988). M/G/C queueing systems with mul-tiple customer classes: characterization and control of achievable perfor-mance under nonpreemptive priority rules. Mgmt. Science 34, 1121–1138.D.W. Fife (1965). Scheduling with random arrivals and linear loss functions.Mgmt. Science 11, 429–437.L. Kleinrock (1965). A conservation law for a wide class of queueing disci-plines. Naval Res. Log. Quart. 12, 181–192.G.P. Klimov (1974). Time-sharing service systems I. Th. Prob. Appl., 532–551.A.W. Marshall, I. Olkin (1979). Inequalities: Theory of Majorization and itsApplications, Academic Press, New York.I. Mitrani, J.H. Hine (1977). Complete parameterized families of jobscheduling strategies. Acta Informatica 8, 61–73.R. Righter (1994). Scheduling. In: Stochastic Orders and their Applications,eds. M. Shaked and J.G. Shanthikumar, Academic Press.


References (cont’d)

R. Righter, J.G. Shantikumar (1989). Scheduling multi-class single-serversystems to stochastically maximize the number of successful departures.Prob. Eng. Inf. Sc. 3, 323–333.L.E. Schrage (1968). A proof of the optimality of the SRPT discipline. Oper.Res. 16, 687–690.L.E. Schrage, L.W. Miller (1966). The queue M/G/1 with the shortest re-maining processing time discipline. Oper. Res. 14, 670–683.J.G. Shanthikumar, D.D. Yao (1992). Multiclass queueing systems: polyma-troidal structure and optimal scheduling control. Oper. Res. 40, S293–S299.W.E. Smith (1956). Various optimizers for single stage production. NavalRes. Log. Quart. 3, 59–66.P.D. Sparaggis, D. Towsley, C.G. Cassandras (1994). Sample path criteriafor weak majorization. Adv. Appl. Prob. 26, 155–171.D. Stoyan (1983). Comparison Methods for Queues, John Wiley & Sons, Berlin.

lnmb course advanced queueing theorysem/aqt/lecture02042012.pdf · 2012-04-02 · we will study...

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