LNMB Course Advanced Queueing Theorysem/AQT/lecture23042012.pdf · department of mathematics and...

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LNMB Course

Advanced Queueing Theory

Lecture 9, April 23, 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


Introduction

Optimization of polling systems

• has received relatively limited attention compared to analysis of pollingsystems for given policies and system parameters

• yet is broad and somewhat fragmented area


Introduction (cont’d)

Will focus on:

• Performance objective: minimize weighted sum of mean waiting timescaptures both efficiency and fairness

• Parametrized / structured policies for arbitration between queuesassuming FCFS within queues

– optimization of service policy (visit length): when to switch

– optimization of routing policy (visit frequency and order): where toswitch to

– joint optimization of visit length and frequency

• Scheduling policies for priorization within queuesassuming given service and routing policies


Introduction (cont’d)

Will not consider:

• Polling systems with zero switch-over times

– correspond to ‘ordinary’ single-server multi-class systems

– polyhedral characterization of achievable mean-waiting time perfor-mance (Lecture 6)

– index-type policies (e.g. cµ-rule) minimize weighted mean-waitingtime objective (Lecture 7)

• Specific mean-waiting approximations

• Joint optimization of arbitration between queues (service and routingpolicies) and scheduling within queues


Optimization of service policy / visit length

Assume routing policy / visit order is strictly cyclic

Aim to determine optimal parameters (x1, . . . , xN ) for structured servicepolicy, specifying visit lengths at various queues

• ki -limited

• τi -limited

• Bernoulli(pi )

None of these service policies belongs to class of disciplines with branchingproperty!!


Optimization of service policy / visit length (cont’d)

Starting point is (approximate) expression for mean waiting time as functionof design parameters (x1, . . . , xN ):

E{Wi} ≈ Fi(x1, . . . , xN )

Function Fi(·) further depends on traffic-related parameters, such as arrivalrate λi , service time moments, and switch-over time moments

Problem is then to minimize∑N

i=1 ciλi Fi(x1, . . . , xN )

subject to possible constraints on (x1, . . . , xN )

Depending on specific form of Fi(x1, . . . , xN ), optimization problem mayallow closed-form solution or require numerical solution procedure



In case of ‘limited’ service policies, typical approximation is of the form

Fi(x1, . . . , xN ) =ai + bi xi

xi −ρi s

1−ρ,

with s representing total mean switch-over time in cycle and xi expectedamount of time guaranteed per visit to Qi

• ki -limited: xi = kiE{Bi}

• τi -limited: xi = τi

• Bernoulli(pi ): xi =E{Bi }1−pi

Typical constraints are of the form xi >ρi s

1−ρ , i = 1, . . . , N , and∑N

i=1 γi xi ≤

G, imposing upper bound on (weighted) sum of expected visit periods, e.g.,ensuring maximum expected cycle time



Optimization problem takes the form

minN∑

i=1

ciλiai + bi xi

xi −ρi s

1−ρ

subject to xi >ρi s

1−ρ , i = 1, . . . , N , and

N∑i=1

γi xi ≤ G



Optimal solution is

x∗i =ρis

1− ρ+

(G −

N∑i=1

γiρis1− ρ

)κi∑N

j=1 κ j,

where

κi =

√ciλi(ai +

biρis1− ρ

)/γi

• increasing in ci , λi , ai , E{Bi}

• decreasing in bi , γi

Interpretation

• Qi needs to receive at least ρi s1−ρ service capacity per cycle

• residual service capacity is divided in proportion to κi ’s



Determining optimal service shares exactly is extremely difficult, even withzero switch-over times

• Resembles problem of selecting suitable weights in differentiated sched-ulers, such as

– Weighted Round Robin (WRR)

– Weighted Fair Queueing (WFQ)

– Generalized Processor Sharing (GPS)

– Discriminatory Processor Sharing (DPS)

• While such schedulers are widely believed to be useful, setting suitableweights for given target performance metrics remains huge challenge


Optimization of routing policy / visit frequencyand order

Now suppose service policy at each of queues is given, e.g., exhaustive,gated, or 1-limited

Aim to determine optimal parameters for routing policy, specifying orderand frequency of visits to various queues

• random polling: probability vector (p1, . . . , pN )

• routing table: deterministic sequence (q1, q2, . . . , qM) =

(i1, i2, . . . , iM) ∈ {1, . . . , N }M

Denote by yi relative visit frequency of Qi

• random polling: yi = pi

• routing table: yi =miM , with mi =

∑Mj=1 I{{q j=i}}


Optimization of routing policy / visit frequency and order (cont’d)

Starting point is (approximate) expression for mean waiting time as functionof (y1, . . . , yN ):

E{Wi} ≈ Gi(y1, . . . , yN ),

assuming visits to each individual queue to be evenly spaced

Function Gi(·) further depends on traffic-related parameters, such as arrivalrate λi , service time moments, and switch-over time moments


i=1 ciλi Gi(y1, . . . , yN )

subject to constraint∑N

i=1 yi = 1

Depending on specific form of Gi(y1, . . . , yN ), optimization problem mayallow closed-form solution or require numerical solution procedure


Optimization of routing policy / visit frequencyand order (cont’d)

For exhaustive and gated policies, typical approximation is of the form

Gi(y1, . . . , yN ) =ai

yi

N∑j=1

y js j

For 1-limited policies, typical approximation is of the form

Gi(y1, . . . , yN ) =bi

yi −λi

1−ρ∑N

j=1 y js j

N∑j=1

y js j

and typical constraint is of the form yi >λi

1−ρ∑N

j=1 y js j , i = 1, . . . , N

Note that both problems are zero-degree homogeneous in (y1, . . . , yN )



For gated and exhaustive policies, optimization problem takes the form

minN∑

i=1

ciλiai

yi

N∑j=1

y js j

subject to∑N

i=1 yi = 1

Optimal solution is of the form y∗i =κi∑N

j=1 κ j,

where κi =√

ciλiai/si

• increasing in ci , λi , ai

• decreasing in si



For 1-limited policy, optimization problem takes the form

minN∑

i=1

ciλibi

yi −λi

1−ρ∑N

j=1 y js j

N∑j=1

y js j

subject to∑N

i=1 yi = 1 and yi >λi s

1−ρ∑N

j=1 y js j , i = 1, . . . , N



Optimal solution is of the form

y∗i ∼ λi +

1− ρ −N∑

j=1

λ js j

κi

si∑N

j=1 κ j,

where κi =√

ciλibi/si

• increasing in ci , λi , bi

• decreasing in si

Interpretation

• Qi needs to obtain at least λi visits (per time unit)

• residual visits 1− ρ −∑N

j=1 λ js j (per time unit) in proportion to κi/si



It remains to construct routing table based on derived visit frequencies

• Determine length of table, e.g., M =∑N

i=1 mi , where M̂ yi ≈ mi ∈ N forsome M̂

• Determine order of visits so that visits to each individual queue areevenly spaced

Even spacing is not always feasible: (m1,m2,m3) = (1, 2, 3)

• (1, 2, 3, 3, 2, 3): visits to Q2 evenly spaced, but not to Q3

• (1, 3, 2, 3, 2, 3): visits to Q3 evenly spaced, but not to Q2



Approaches for spacing visits ‘as evenly as possible’

• Golden Ratio rule (Hofri & Rosberg, Panwar)

• balanced sequences (Altman, Gaujal, Hordijk, Van der Laan)

• interleaving of optimal splitting sequences (Arian & Y. Levy, B.)

• quadratic programming techniques (B. & Ramakrishnan)


Optimization of service policy / visit lengthIn special case of random polling with mixture of exhaustive (i ∈ e) andgated (i ∈ g) service policies at various queues and ci ≡ E{Bi},pseudo-conservation law provides exact expression for objective function:

N∑i=1

ρiE{Wi } = ρ

∑Ni=1 λiE{B

(2)i }

2(1− ρ)−

s1− ρ

∑i∈e

ρ2i

pi+

s1− ρ

N∑i=1

ρi

pi−

N∑i=1

ρi si +ρ

2s

N∑i=1

pi s(2)i

In case si ≡ s1 and s(2)i ≡ s(2)1 , we obtain

p∗i =√ρi(1− ρi)∑

j∈e√ρ j(1− ρ j)+

∑j∈g√ρ j

i ∈ e,

and

p∗i =√ρi∑

j∈e√ρ j(1− ρ j)+

∑j∈g√ρ j

i ∈ g


Joint optimization of visit frequency / order and length

Now suppose we wish to optimize visit lengths (x1, . . . , xN ) as well as visitorder and frequencies (y1, . . . , yN )

Assume that condition is imposed of the form

xi ≥bi∑N

j=1 y j x j

yi+ di ,

reflecting for example that visit must be sufficiently long to clear all workthat arrives during intervisit period with high probability


Joint optimization of visit frequency/order andlength (cont’d)

As before, starting point is (approximate) expression for mean waiting timeas function of (x1, . . . , xN ) and (y1, . . . , yN )

E{Wi} ≈ Hi(x1, . . . , xN ; y1, . . . , yN )

Function Hi(·) further depends on traffic-related parameters, such as arrivalrate λi , service time moments, and switch-over time moments


i=1 ciλi Hi(x1, . . . , xN ; y1, . . . , yN )

subject to possible constraints on (x1, . . . , xN ) and xi ≥bi∑N

j=1 y j x jyi

+ di ,i = 1, . . . , N

Depending on form of Hi(x1, . . . , xN ; y1, . . . , yN ), optimization problemmay allow closed-form solution or require numerical solution procedure



Typical approximation is of the form

Hi(x1, . . . , xN ; y1, . . . , yN ) ≈ai

yi

N∑j=1

y j x j

Optimization problem then takes the form

minN∑

i=1

ciλiai

yi

N∑j=1

y j x j

subject to xi ≥bi∑N

j=1 y j x jyi

+ di , i = 1, . . . , N



Optimality requires latter constraint to be satisfied with equality, yielding

N∑j=1

y j x j =

∑Nj=1 d j y j

1−∑N

j=1 b j,

and hence optimization problem reduces to

minN∑

i=1

ciλiai

yi

N∑j=1

d j y j



Optimal solution is of the form y∗i =κi∑N

j=1 κ j,

where κi =√

ciλiai/di

Same as optimal visit frequency for gated and exhaustive service policies


Dynamic optimization

We have focused on ‘static’ optimization of parametrized/structured policies

Dynamic policies have received relatively limited attention

• optimization of visit order from cycle to cycle (Browne, Weiss, Yechiali)

• stochastic optimality results for total workload and queue length (Liu,Nain & Towsley)

• dominance relationships for total workload (H. Levy, Sidi & Boxma)

• stochastic monotonicity results for queue lengths at visit epochs (Alt-man, Konstantopoulos & Liu)


Dynamic optimization (cont’d)

Two-class queues with set-up times / costs

• in symmetric scenario, optimal service policy is exhaustive, with thresh-old rule for switching from empty to non-empty queue (Hofri & Ross)

• in asymmetric scenarios, dynamic programming yields threshold rulefor switching from ‘cheap’ to ‘expensive’ queue (Koole)

• asymptotically optimal policies in heavy-traffic regime (Reiman & Wein)

• heavy-traffic analysis of dynamic cyclic policies (Markowitz, Reiman &Wein)

Spatial settings

• vehicle routing strategies

• message gathering strategies in wireless networks


Mean value analysis

Approach for determining mean queue lengths and waiting times based on

• PASTA property: Poisson Arrivals See Time Averages

• Little’s law: E{L} = λE{W}

N queues 1, 2, . . . , N , served by single server in fixed cyclic order

• Exhaustive service discipline and FCFS within each queue

• Customers arrive at queue i as Poisson process of rate λi

• Customers at queue i have generally distributed service requirements Bi

• Mean residual service requirement E{RBi } =E{B2

i }

2E{Bi }

• Traffic intensity at queue i is ρi = λiE{Bi}

• Total traffic intensity is ρ =∑N

i=1 ρi < 1


Mean value analysis (cont’d)

• Generally distributed switch-over time Si from queue i to queue(imodN )+ 1

• Mean total switch-over time in cycle E{S} =∑N

i=1 E{Si}

• Cycle time of queue i is time between two successive arrivals of server atthis queue; mean cycle time is

E{C} = E{S}1− ρ

• Visit time Ti of queue i is service period of queue i plus precedingswitchover time

E{Ti} = E{Si−1} + ρiE{C}



Let Li denote length of queue i (excluding customer possibly in service),and let Wi denote waiting time of customer at queue i

Objective: Determine E{Li} and E{Wi}

Derive (i) arrival relation for mean waiting time and use (ii) Little’s law

First ordinary M/G/1 queueArrival relation

E{W} = E{L}E{B} + ρE{RB}

in combination with Little’s law

E{L} = λE{W}

yields

E{W} = ρ

1− ρE{RB}



Now two queuesArrival relation

E{W1} = E{L1}E{B1} + ρ1E{RB1}

+E{S2}

E{C} E{RS2} +E{T2}

E{C} (E{RT2} + E{S2})

in combination with Little’s law

E{L1} = λ1E{W1}

yields

E{W1} =1

1− ρ1

[ρ1E{RB1} +

E{S2}

E{C} E{RS2} +E{T2}

E{C} (E{RT2} + E{S2})

]But what is E{RT2}??



Let Li,n denote length of queue i at arbitrary point in time during visit timeof queue n

Then, since each of L2,2 customers initiates busy period,

E{RT2} = E{L2,2}E{B2}

1− ρ2+ρ2E{C}E{T2}

E{RB2}

1− ρ2+

E{S1}

E{T2}

E{RS1}

1− ρ2

Further

E{L2} =E{T1}

E{C} E{L2,1} +E{T2}

E{C} E{L2,2}

Finally L2,1 is equal to number of arrivals at queue 2 during age of T1, andage has same distribution as residual lifetime of T1, so

E{L2,1} = λ2E{RT1}

Similar equations can be obtained with roles of queues 1 and 2 interchanged



Example: λ1 = 0.6, λ2 = 0.2, B1, B2, S1, S2 all exp. distr. with unit mean

E{L1,2} = 0.6E{RT2},

E{L2,1} = 0.2E{RT1},

0.7E{L1,1} + 0.3E{L1,2} = 1.5+ 0.45E{RT2},

0.7E{L2,1} + 0.3E{L2,2} = 0.25+ 0.175E{RT1},

E{RT1} = 2.5+ 2.5E{L1,1},

E{RT2} = 1.25+ 1.25E{L2,2}

Solution

E{L1,1} =12935 , E{L1,2} =

125 , E{L2,1} =

8235, E{L2,2} =

115 ,

E{RT1} =827 , E{RT2} = 4

which leads to

E{L1} = 3.3, E{L2} = 2.3, E{W1} = 5.5, E{W2} = 11.5


Impact of service discipline within queue

So far we assumed FCFS service discipline within each queue

In several computer-communication and manufacturing applications, thatis not realistic

We now analyze impact of service discipline within each queue(for given visit frequency and length)


Impact of service discipline within queue (cont’d)

N queues, gated service discipline at Qi

visitto Qi

intervisit

Ci

pastCi

res

Consider mean waiting time E{Wi(x)} of customer at Qi with service re-quirement x :

E{Wi(x)} = E{Cri } + λiE{Cp

i }E{Ki p(x)} + λiE{Cri }E{Kir(x)}

with

• Ki p(x): contribution of work from type-i customer arriving earlier incycle

• Kir(x): contribution of work from type-i customer arriving later in cycle


Impact of service discipline within queue (cont’d)

• FCFS: Ki p(x) = Bi , Kir(x) = 0:

E{Wi} = E{Wi(x)} = (1+ ρi)E{Cri }

• LCFS: Ki p(x) = 0, Kir(x) = Bi :

E{Wi} = E{Wi(x)} = (1+ ρi)E{Cri }

• PS: Ki p(x) = min{Bi , x}, Kir(x) = min{Bi , x}:

E{Wi(x)} = E{Cri }(1+ 2λiE{min{Bi , x}};

E{Wi} = E{Cri }(1+ 2λiE{min{Bi1,Bi2}}

• SPT (optimal): Ki p(x) = BiI{Bi<x}, Kir(x) = BiI{Bi<x}:

E{Wi(x)} = E{Cri }(1+ 2λiE{BiI{Bi<x}});

E{Wi} = E{Cri }(1+ λiE{min{Bi1,Bi2}})


Impact of service discipline within queue(cont’d)

Gated:Significant gains under heavy load, but smaller than in ordinary M/G/1queue

250

200

150

100

50

0.6 0.7 0.8 0.9 1

SJF

FCFS

2 queue symmetric polling system

load

mean d

ela

y


Impact of service discipline within queue(cont’d)

Exhaustive:Big gains under heavy load, matching those in ordinary M/G/1 queue

250

200

150

100

50

0.6 0.7 0.8 0.9 1

2 queue symmetric polling system

load

mean d

ela

y


References

E. Altman, B. Gaujal, A. Hordijk (2000). Balanced sequences and optimalrouting. J. ACM 47, 752–775.E. Altman, P. Konstantopoulos, Z. Liu (1992). Stability, monotonicity andinvariant quantities in general polling systems. Queueing Systems 11, SpecialIssue on Polling Models, 35–57.Y. Arian, Y. Levy (1992). Algorithms for generalized round robin routing.Oper. Res. Lett. 12, 313–319.J.E. Baker, I. Rubin (1987). Polling with a general-service order table. IEEETrans. Commun. 35, 283–288.J.P.C. Blanc, R.D. van der Mei (1995). Optimization of polling systems withBernoulli schedules. Perf. Eval. 21, 139–158.S.C. Borst, O.J. Boxma, J.H.A. Harink, G.B. Huitema (1994). Optimizationof fixed time polling schemes. Telecommunication Systems 3, 31–59.S.C. Borst, O.J. Boxma, H. Levy (1995). The use of service limits for ef-ficient operation of multi-station single-medium communication systems.IEEE/ACM Trans. Netw. 3, 602–612.


References (cont’d)

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