the strict priority scheduler - intranet...

Quality of Service in IP networks Paolo GiacomazziCopyright © 2009 paolo Giacomazzi. All rights reserved.1

Course of Multimedia Internet (Sub-course”Reti Internet Multimediali”), AA 2010-2011 Prof. Paolo GiacomazziPolitecnico di Milano, Dipartimento di Elettronica e Informazione, Via Ponzio, 34/5, 20133 MILANO, ITALY

16. SP and GPS schedulers Pag. 1

The strict priority schedulerThe strict priority scheduler is very simple and efficientIn the following, we will carry out the analysis of this scheduler todetermine the formulas for performance analysis, resource allocationand admission controlThe service envelope of service priority i, in the strict priorityscheduler, is

Where C is the line capacityIn fact, service class i is interfered only by service classes with a better service priority, that is, classes from 1 to i-1

( ) ( )1

1max 0,

i

i jj

S t Ct X t−

=

⎛ ⎞= −⎜ ⎟

⎝ ⎠∑




The strict priority schedulerThus

The specific form of αi(t) depends on the average values and variances of traffic flows from 1 to i-1

( )( ) ( ) ( )

( ) ( ) ( )

1

1

1

1

max 0,

var max 0,

i

i i j ij

ii

i j ij

E X t C t d X t dt

X t C t d X t d

α

−

=

−

=

⎛ ⎞⎛ ⎞− + − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= −

⎛ ⎞⎛ ⎞− + − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑

∑




The strict priority schedulerWhen all traffic flows admit a linear variance envelope, a closed-formanalysis is possibleIn this case, the average value of Si(t) is approximated as

And the variance of Si(t) is approximated as

Thus,

( )( )1

1

i

i j jj

E S t Ct N r t−

=

= −∑

( )( )1

1

vari

i j j jj

S t N r b t−

=

= ∑

( )( ) ( )

( )

1

1

1

1

i

i i i j j ij

i i

i i i j j j ij

N rt C t d N r t dt

N rb t N r b t dα

−

=

−

=

− + + − +=

+ +

∑

∑




The strict priority schedulerThe absolute minimum of αi(t) is

where

From which we can calculate the closed-form formulas forDelay distributionCapacity planningAdmission control

( ) ( ) ( )( )1 1,min 22 i i i i i

i ii

C A C A B C A Bd

Bα − −− − − −

=

1 1

i i

i j j i j j jj j

A N r B N r b= =

= =∑ ∑




The strict priority scheduler: delay distribution with linear-bounded variance traffic

Delay distribution

Average delay

( ) ( ) ( )( )1 12Pr exp 2 ii i i i i i i

i

C AD d C A B C A B dB − −

⎛ ⎞−> ≈ − − − −⎜ ⎟

⎝ ⎠

( ) ( ) ( ) ( )( )2

1 12i

ii i i i i

BE DC A C A B C A B− −

≈− − − −




The strict priority scheduler: capacity planning with linear-bounded variance traffic

Given a strict priority scheduler with n service classes, where the traffic of class i, Xi(t) has average value E(Xi(t))=rit and varianceupper bounded by var(Xi(t))≤ribitGiven the statistical delay SLA of class i: (di, pi)The line capacity C needed to satisfy concurrently all SLAs isobtained by inverting the delay distribution formula

Then:

( ) ( ) ( )( )

2 2 2 2 2 2 2 2 2 31 1 1 1 1

1

2 2 2 ln 2 ln2

i i i i i i i i i i i i i i i i i i i i i i i ii

i i i

A B A B A B d A B d A A B d A B d B B d p B d pC

B B d− − − − −

−

− − − − + + −≥

−

( )1sup i

i nC C

≤ ≤=




The strict priority scheduler: capacity planning with linear-bounded variance traffic

The required capacity depends on the priority level assigned to eachservice classGiven n service classes, there are n! possible permutations for thisassignmentThe problem of finding the optimal permutation is NP-completeFrequently, the best permutation is that assigning the traffic with the largest average value to priority 1, the second-largest average valueto priority 2 and so onHowever, this is not necessarily true in the general case, as alsovariance has a significant effect




The strict priority scheduler: admission region with linear-bounded variance traffic

Given a strict priority scheduler with n service classes, where the traffic of class i, Xi(t) has average value E(Xi(t))=rit and varianceupper bounded by var(Xi(t))≤ribitGiven the statistical delay SLA of class i: (di, pi)Given the line capacity CThe admission region is the set of tuples (N1, N2, ..., Nn) for which the SLAs are concurrently fulfilledThe admission region is calculated by inverting the delay curveAlgebraic calculations are very complex, but it is possible to find a closed-form expression of the admission region




The strict priority scheduler: an example of calculation of delaycurves

For example, let us consider the case of two service classes, with the following parameters

r1 = 200 kbit/s, b1 = 9,600 bitr2 = 200 kbit/s, b1 = 4,600 bitN1 = 30 sourcesN2 = 10 sourcesC=1x107 bit/s

The delay curves is this case are calculated with the proposedformula and they are plotted in the next slide




The strict priority scheduler : an example of calculation of delay curves

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Delay, d

Pr(D

>d)

Service class 1

Service class 2




The strict priority scheduler : an example of calculation of capacity planning

Continuing with the previous example, let us assume that capacity isunknown and let us add statistical delay SLAs

r1 = 200 kbit/s, b1 = 9,600 bitr2 = 200 kbit/s, b1 = 4,600 bitd1 = 50 ms, p1 = 1x10-3

d2 = 70 ms, p2 = 2x10-3

N1 = 30 sourcesN2 = 10 sources

The resulting capacity requested to fulfill concurrently all SLAs is:C1 = 6.6 x106 bit/sC2 = 9.35 x106 bit/s

ThusC = max(6.6 x106 bit/s, 9.35 x106 bit/s) = 9.35 x106 bit/s




The strict priority scheduler : an example of calculation of capacity planning

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Delay, d

Pr(D

>d)

Service class 1

Service class 2

In the scenario of the previous example, the delay curve is that plotted in the figure; the SLA of service class 2 is provided exactly, while the dimensioningfor class 1 is large




The strict priority scheduler: admission region with linear-bounded variance traffic

where

( )( )

2 221 1 1

1 1 1 1 1

*,max

11

*

ln ln2 2 4 4 2

0 min , , 1,2,...,ln4 2

0

i i i i ii i i i i

i i i i i

i i k i k i ki i i

i ii i

i i k i k

B B p B pC A B C A C A Bb b d b d

N for N N N and i k nB b pr C Ab d

N for N N

− − −− − − − −

− − −

−−

− −

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟− − + − − − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠≤ ≤ ≤ > =

⎛ ⎞⎛ ⎞− − + −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠= ≤ ≤ ,max, 1,2,...,i kN and i k n−

⎧⎪⎪⎪⎨⎪⎪⎪ > =⎩

( ) ( )( )21 1 1*

2

,max

ln ln ln 2 2

22 ln

i i k i i i k i i k i i i k i k i i ki k

i k i

i ki k

i k i k i k i k i k

Cd b p d A p b p d b C A d BN

r d

d CNd r C r b p

− − − − − − − − −

−−

−−

− − − − −

− − − − − + − +=

=−




Admission control curves for the SP schedulerFor example, with the selected parameters, the admission control regionis plotted in the figureIt is interesting to note thatfor more than 34 sourcesof class 1, no sources of class 2 can be admittedThis depends on the specific values of the TCA and SLA parametersWith other parameters, this phenomenon does notnecessarily occur

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50

N1

N2

class 1 class 2r1 2.00E+05 bit/s r2 2.00E+05 bit/sb1 9.60E+03 bit b2 4.60E+03 bit/sr1b1 1.92E+09 bit2/s r2b2 9.20E+08 bit2/sd1 0.05 d2 0.07p1 1.00E-03 p2 2.00E-03

C 1.00E+07




Admission control curves for the SP schedulerThis is another example, providing a qualitativelydifferent admission region

class 1 class 2r1 2.00E+05 bit/s r2 2.00E+05 bit/sb1 9.60E+03 bit b2 4.60E+03 bit/sr1b1 1.92E+09 bit2/s r2b2 9.20E+08 bit2/sd1 0.05 d2 0.8p1 1.00E-03 p2 1.00E-01

C 1.00E+07

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50

N1

N2




The GPS schedulerIn a GPS with n service classes the statistical service of the class-itraffic is given by

Where the j index represents non-backlogged classes while the kindex refers to backlogged classesTherefore, the average value of the statistical service envelopeavailable for service class i can be evaluated as

( ) ( )( ) 1,...,max 0, ,

1,..., backlogged classesi

i i j jj i k

k j

j nwS t w Ct w Ct X tk nw≠

≠

=⎧= + − ⎨ =⎩

∑∑

( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

max 0,

var max 0,

ii i i j i j i

j i kk j

i

ii i j i j i

j i kk j

wE X t wC t d w C t d X t dw

t

wX t wC t d w C t d X t dw

α≠

≠

≠≠

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟− + + + − +⎜ ⎟⎜ ⎟

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= −⎛ ⎞⎜ ⎟− + + + − +⎜ ⎟⎜ ⎟⎝ ⎠

∑∑

∑∑




The GPS schedulerWith linear-variance bounded traffic,

thus

( )( ) ( )

( )( )

2

var

ii i j j j

j i kk j

ii j j j

j i kk j

wE S t wCt w Ct N r tw

wS t N r b tw

≠≠

≠≠

≈ + ⋅ −

⎛ ⎞⎜ ⎟≈ ⎜ ⎟⎜ ⎟⎝ ⎠

∑∑

∑ ∑

( )

( ) ( )

( )

2

ii i i j j j i

j i kk j

i

ii i i j j j i

j i kk j

wN rt w C w C N r t dw

t

wN rb t N r b t dw

α≠

≠

≠≠

⎛ ⎞⎜ ⎟− + + ⋅ − +⎜ ⎟⎜ ⎟⎝ ⎠=

⎛ ⎞⎜ ⎟+ +⎜ ⎟⎜ ⎟⎝ ⎠

∑∑

∑ ∑




The GPS schedulerThe absolute minimum of ai(t) is

where

( ) ( )( ),min 2

4 i i i i i i i i i i i ii

i i i i

N rd w bC b A B w C A N rN rb B

α+ + ⋅ + −

=+

( )2

max 0, ; .i ii j j j i j j j

j i j ik kk j k j

w wA w C N r B N r bw w≠ ≠

≠ ≠

⎛ ⎞⎜ ⎟= ⋅ − = ⎜ ⎟⎜ ⎟⎝ ⎠

∑ ∑∑ ∑




The GPS scheduler: probability of violation of delay boundThus,

( ) ( ) ( )( )2

2Pr exp .i i i i i i i i i i i

i i ii i i i

N r wbC b A B wC A N rD d d

N rb B

⎛ ⎞+ + ⋅ + −> ≈ −⎜ ⎟

⎜ ⎟+⎝ ⎠




The GPS scheduler: admission control ruleFor the admission control rule, we define

Then the following condition is checked

( )( )

2

,min2

.2 ln

i ii

i i i i i

d wCN

r d wC b p=

−( ) ( ) ( ) ( )( )

( )( )

2 2

,max 2

ln 2 ln.

2 lni i i i i i i i i i i i i i i i i i i i i i

ii i i i i i i i i

d wC A wbC b A B bB p d wbC b A B d wC A B pN

r d wbC b A B b p

+ + + + + + + + +=

+ + −

( )( ) ( )2,min ,max

,min

2 ln 0 max ,i i i i i i i i

i i

If d wC A B p then N N N

else N N

+ + > ≤

≤




SP versus GPS: an exampleThe figure shows a comparison between the admission curves of a Strict Priority and a GPS schedulerThe GPS schedulerguarantees a minimum bandwidth for all classesThus, service class 2 is nottruncated as in the case of the SP scheduler

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70 80

N1

N2 N2 SP

N2 gps

r1 600000b1 40000r2 800000b2 100000d1 0.01d2 0.05p1 0.000001p2 0.000001C 45000000w1 0.8w2 0.2

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