M/G/1/MLPS Queue Mean Delay Analysis

1aalto.ppt EURO 2006, Reykjavik, Iceland, 2.5.7.2006

M/G/1/MLPS QueueMean Delay Analysis

Samuli Aalto (TKK)in cooperation with

Urtzi Ayesta (LAAS-CNRS) and Eeva Nyberg-Oksanen

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Outline• Introduction• DHR service times• IMRL service times• Ongoing work

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Application: bandwidth sharing in IP networks

• Consider a bottleneck link in an IP network– loaded with elastic flows, such as file transfers using TCP– if RTTs are of the same magnitude, then approximately

fair bandwidth sharing among the flows

• Intuition says that – favouring short flows reduces the total number of flows,

and, thus, also the mean delay at flow level (that is, the average file transfer time)

• How to service flows and how to analyse?– Guo and Matta (2002), Feng and Misra (2003),

Avrachenkov et al. (2004), Aalto et al. (2004,2005,2006)

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Queueing model• Assume that

– flows arrive according to a Poisson process with rate

– flow size distribution is ”heavier” than exponential, such as hyperexponential or Pareto

• So, we have a M/G/1 queue at flow level– customers in this queue are flows (and not packets)– service time = the total number of packets to be

sent– attained service = the number of packets already

sent– remaining service = the number of packets still left

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Optimality results for M/G/1• Schrage (1968)

– If the remaining service times are known, then• SRPT is optimal minimizing the mean delay

• Yashkov (1987)– If only the attained service times are known, then

• DHR implies that FB is optimal minimizing the mean delay

• Remark: We consider work-conserving (WC) and non-anticipating (NA) service disciplines such as FB, MLPS, and PS (but not SRPT) for which only the attained service times are known

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Service disciplines at flow level• FB = Foreground-Background = LAS = Least Attained

Service– Choose a packet from the flow with least packets sent

• MLPS = Multi Level Processor Sharing– Choose a packet from a flow with less packets sent

than a given threshold

• PS = Processor Sharing– Without any specific scheduling policy at packet level,

the elastic flows are assumed to divide the bottleneck link bandwidth evenly

• Reference model: M/G/1/PS

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MLPS disciplines• Definition: MLPS discipline, Kleinrock, vol. 2 (1976)

– based on the attained service times Xi(t)

– N1 levels defined by N thresholds a1 … aN – between levels, a strict priority is applied– within a level, an internal discipline (FB, PS, or FCFS)

is applied

a

FCFS+PS(a)Xi(t)

t

PSFCFS

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Our objective• We compare MLPS disciplines in terms of the mean

delay– MLPS vs PS– MLPS vs MLPS

• We assume that service times are – DHR– IMRL

• Note: – DHR IMRL

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References• K. Avrachenkov, U. Ayesta, P. Brown and E. Nyberg (2004)

– IEEE INFOCOM• S. Aalto, U. Ayesta and E. Nyberg-Oksanen (2004)

– ACM SIGMETRICS – PERFORMANCE• S. Aalto, U. Ayesta and E. Nyberg-Oksanen (2005)

– Operations Research Letters 33• S. Aalto and U. Ayesta (2006a)

– IEEE INFOCOM• S. Aalto and U. Ayesta (2006b)

– to appear in Journal of Applied Probability• S. Aalto (2006)

– to appear in Mathematical Methods of Operations Research

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DHR service times• Service time distribution:

• Density function:

• Hazard rate:

• Definition: – Service time distribution belongs to class DHR

(Decreasing Hazard Rate) if h(x) is decreasing• Examples:

– Pareto (taking values from 0 on) and hyperexponential

)(1)( },{)( xFxFxSPxF

}{)( dxSPxf

x dyyf

xfxFxfxh

)(

)()()()(

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Unfinished truncated work Ux

• Customers with attained service Xi(t) less than x:

• Unfinished truncated work with truncation threshold x:

• Unfinished work:

}},min{)(|)({)( xStXtAitN iix

)( ))(},(min{)( tNi iix xtXxStU

)( ))(()()( tNi ii tXStUtU

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Optimality of FB• Aalto et al. (2004):

– FB minimizes the unfinished truncated work for any x and t in each sample path

s

FCFSXi(t)

t

x

FB

t

Ux(t)

sx

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Idea of the mean delay comparison• Kleinrock (1976):

– For all WC & NA service disciplines

– so that (by applying integration by parts)

• Thus, 0

'1' )]([)( xhdUUTT xx

'' & DHR TTxUU xx

0

1 ][ )(

xUdxhT

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MLPS vs PS• Aalto et al. (2004):

– Two levels with FB and PS allowed as internal disciplines

• Aalto et al. (2005): – Any number of levels with FB and PS allowed as

internal disciplines

PSPSPSPSFBFB DHR TTTT

PSMLPSFB DHR TTT

FB/PSFB/PSFB/PS

PSFB

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Mean unfinished truncated work

bounded Pareto service time distribution

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MLPS vs MLPS: changing internal disciplines

• Aalto and Ayesta (2006a):– Any number of levels with all internal disciplines

allowed– MLPS derived from MLPS’ by changing an internal

discipline from PS to FB (or from FCFS to PS)MLPS'MLPS DHR TT

FB/PS PS/FCFS

MLPS’MLPS

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MLPS vs MLPS: splitting FCFS levels• Aalto and Ayesta (2006a):

– Any number of levels with all internal disciplines allowed

– MLPS derived from MLPS’ by splitting any FCFS level and copying the internal disciplineMLPS'MLPS DHR TT

FCFS FCFS

MLPS’MLPS

FCFS

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MLPS vs MLPS: splitting PS levels• Aalto and Ayesta (2006a):

– Any number of levels with all internal disciplines allowed

– The internal discipline of the lowest level is PS– MLPS derived from MLPS’ by splitting the lowest level

and copying the internal discipline

• Splitting any higher PS level is still an open problem (contrary to what we thought in an earlier phase)!

MLPS'MLPS DHR TT

PS PS

MLPS’MLPS

PS

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IMRL service times• Service time distribution:

• H-function:

• Mean residual lifetime:

• Definition: – Service time distribution belongs to class IMRL

(Increasing Mean Residual Lifetime) if H(x) is decreasing• Examples:

– all DHR service time distributions, Exp+Pareto

)(1)( },{)( xFxFxSPxF

)(1

)()(

]|[ xHxFdyyFxxSxSE

x dyyF

xFxH)(

)()(

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Level-x workload• Customers with attained service less than x:

• Unfinished truncated work with truncation threshold x:

• Level-x workload:

• Workload = unfinished work:

}},min{)(|)({)( xStXtAitN iix

)( ))(},(min{)( tNi iix xtXxStU

)())(()()( )( tUtXStVtV tNi ii

)( ))(()( tNi iix xtXStV

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Non-optimality of FB• Aalto and Ayesta (2006b):

– FB does not minimize the level-x workload

s

FCFSXi(t)

t

x

FB

t

Vx(t)

sx

FB notoptimal

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Idea of the mean delay comparison• Righter et al. (1990):

– For all WC & NA service disciplines

– so that (by applying integration by parts)

• Thus, 0

'1' )]([)( xHdVVTT xx

'' & IMRL TTxVV xx

0

1 ][ )(

xVdxHT

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MLPS vs PS• Aalto (2006):

– Any number of levels with FB and PS allowed as internal disciplines

• Aalto and Ayesta (2006b): – Any number of levels with FB and PS allowed as

internal disciplines

PSMLPS IMRL TT

MLPSFB IMRL TT

FB/PSFB/PSFB/PS

PS

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Mean level-x workload

bounded Pareto service time distribution

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Non-optimality of FB• Aalto and Ayesta (2006b):

– FB does not necessarily minimize the mean delay for IMRL service times

• Counter-example:– Exp+Pareto belongs to IMRL but not DHR (for 1 c

e):

– There is 0 such that

,0 ,)(

cxxcxcxF c

x

FB)(FBFCFS TT c

FBFCFS FB

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Gittins index• Gittins (1989):

– J-function

– Gittins index for a customer with attained service a:

– Gittins discipline serves the customer with highest index

• Gittins discipline minimizes the mean delay in M/G/1 (among NA disciplines):– If DHR, then FB optimal– If NBUE, then FCFS optimal– If CHR+DHR, then FB, FCFS, or FCFS+FB optimal

)(),( ),()0,( ,),()(

)(aHaJahaJaJ a

a

aa

dyyF

dyyf

),(sup)( 0 aJaG

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Bandwidth sharing networks• Bandwidth sharing network

– multiple links shared by elastic flows with different routes

– necessary stability conditions: for all links l,

• Verloop et al. (2005):– global SRPT instable in the network case, that is

necessary stability conditions are not sufficient

• However, local SRPT (applied to each route separately) is stable and reduces the mean delay in an optimal way

llRr r C )(

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THE END

M/G/1/MLPS Queue Mean Delay Analysis

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