The strict priority scheduler

Transcription

1 6. SP and GPS schedulers Pag. The strct prorty scheduler The strct prorty scheduler s very smple and effcent In the followng, we wll carry out the analyss of ths scheduler to determne the formulas for performance analyss, resource allocaton and admsson control The servce envelope of servce prorty, n the strct prorty scheduler, s Where C s the lne capacty S t Ct X j t j= () = max 0, () In fact, servce class s nterfered only by servce classes wth a better servce prorty, that s, classes from to -

2 6. SP and GPS schedulers Pag. The strct prorty scheduler Thus α () t = E X t C t d X t d () max 0, ( + ) ( + ) j j= var X t max 0, C t d X t d () ( + ) ( + ) j j= The specfc form of α (t) depends on the average values and varances of traffc flows from to -

3 6. SP and GPS schedulers Pag. 3 The strct prorty scheduler When all traffc flows admt a lnear varance envelope, a closed-form analyss s possble In ths case, the average value of S (t) s approxmated as ( ()) And the varance of S (t) s approxmated as E S t Ct N rt = j j j= Thus, α () t = var ( ()) S t = N rb t j j j j= ( ) ( ) Nrt+ C t+ d N r t+ d j j j= ( ) Nrbt+ N rb t+ d j j j j= 3

4 6. SP and GPS schedulers Pag. 4 The strct prorty scheduler The absolute mnmum of α (t) s α ( ) ( ) ( ) ( ) C A C A B C A B,mn = B d where j j j j j j= j= A = Nr B= Nrb From whch we can calculate the closed-form formulas for Delay dstrbuton Capacty plannng Admsson control 4

5 6. SP and GPS schedulers Pag. 5 The strct prorty scheduler: delay dstrbuton wth lnearbounded varance traffc Delay dstrbuton C A Pr ( D > d ) exp (( C A ) B ( C A) B ) d B Average delay ( ) E D ( ) ( ) ( ) ( ) B C A C A B C A B 5

6 6. SP and GPS schedulers Pag. 6 The strct prorty scheduler: capacty plannng wth lnearbounded varance traffc Gven a strct prorty scheduler wth n servce classes, where the traffc of class, X (t) has average value E(X (t))=r t and varance upper bounded by var(x (t)) r b t Gven the statstcal delay SLA of class : (d, p ) The lne capacty C needed to satsfy concurrently all SLAs s obtaned by nvertng the delay dstrbuton formula C 3 ( AB A B AB ) d A B d A AB d + A B d + B B dln( p) Bdln( p) ( B B ) d Then: C = sup n ( C ) 6

7 6. SP and GPS schedulers Pag. 7 The strct prorty scheduler: capacty plannng wth lnearbounded varance traffc The requred capacty depends on the prorty level assgned to each servce class Gven n servce classes, there are n! possble permutatons for ths assgnment The problem of fndng the optmal permutaton s NP-complete Frequently, the best permutaton s that assgnng the traffc wth the largest average value to prorty, the second-largest average value to prorty and so on However, ths s not necessarly true n the general case, as also varance has a sgnfcant effect 7

8 6. SP and GPS schedulers Pag. 8 The strct prorty scheduler: admsson regon wth lnearbounded varance traffc Gven a strct prorty scheduler wth n servce classes, where the traffc of class, X (t) has average value E(X (t))=r t and varance upper bounded by var(x (t)) r b t Gven the statstcal delay SLA of class : (d, p ) Gven the lne capacty C The admsson regon s the set of tuples (N, N,..., N n ) for whch the SLAs are concurrently fulflled The admsson regon s calculated by nvertng the delay curve Algebrac calculatons are very complex, but t s possble to fnd a closed-form expresson of the admsson regon 8

9 6. SP and GPS schedulers Pag. 9 The strct prorty scheduler: an example of calculaton of delay curves For example, let us consder the case of two servce classes, wth the followng parameters r = 00 kbt/s, b = 9,600 bt r = 00 kbt/s, b = 4,600 bt N = 30 sources N = 0 sources C=x0 7 bt/s The delay curves s ths case are calculated wth the proposed formula and they are plotted n the next slde 9

10 6. SP and GPS schedulers Pag. 0 The strct prorty scheduler : an example of calculaton of delay curves.0e+00.0e-0.0e-0 Pr(D>d).0E-03.0E-04.0E-05 Servce class.0e-06 Servce class.0e-07.0e-08.0e Delay, d 0

11 6. SP and GPS schedulers Pag. The strct prorty scheduler : an example of calculaton of capacty plannng Contnung wth the prevous example, let us assume that capacty s unknown and let us add statstcal delay SLAs r = 00 kbt/s, b = 9,600 bt r = 00 kbt/s, b = 4,600 bt d = 50 ms, p = x0-3 d = 70 ms, p = x0-3 N = 30 sources N = 0 sources The resultng capacty requested to fulfll concurrently all SLAs s: C = 6.6 x0 6 bt/s C = 9.35 x0 6 bt/s Thus C = max(6.6 x0 6 bt/s, 9.35 x0 6 bt/s) = 9.35 x0 6 bt/s

12 6. SP and GPS schedulers Pag. The strct prorty scheduler : an example of calculaton of capacty plannng.0e+00.0e-0.0e-0.0e-03 Pr(D>d).0E-04.0E-05 Servce class.0e-06.0e-07 Servce class.0e-08.0e Delay, d In the scenaro of the prevous example, the delay curve s that plotted n the fgure; the SLA of servce class s provded exactly, whle the dmensonng for class s large

13 6. SP and GPS schedulers Pag. 3 The strct prorty scheduler: admsson regon wth lnearbounded varance traffc N for 0 N mn N, N, and > k =,,..., n B B ln p B ln p C A + 4B 4 C A + ( C A ) + B b b d b d k k k B bln p 4r C A + b d where * (,max ) * N = 0 for N k N k N,,,..., k,max and > k = n N N * k = k,max ( ) ( ) ( ) Cd b ln p d A ln p b ln p + d b C A + d B k k k k k k d kc = d r C r b ln p k k k k k r k d 3

14 6. SP and GPS schedulers Pag. 4 Admsson control curves for the SP scheduler For example, wth the selected parameters, the admsson control regon s plotted n the fgure It s nterestng to note that for more than 34 sources of class, no sources of class can be admtted Ths depends on the specfc values of the TCA and SLA parameters Wth other parameters, ths phenomenon does not necessarly occur N class class r.00e+05 bt/s r.00e+05 bt/s b 9.60E+03 bt b 4.60E+03 bt/s rb.9e+09 bt/s rb 9.0E+08 bt/s d 0.05 d 0.07 p.00e-03 p.00e-03 C.00E N 4

15 6. SP and GPS schedulers Pag. 5 Admsson control curves for the SP scheduler Ths s another example, provdng a qualtatvely dfferent admsson regon class class r.00e+05 bt/s r.00e+05 bt/s b 9.60E+03 bt b 4.60E+03 bt/s rb.9e+09 bt/s rb 9.0E+08 bt/s d 0.05 d 0.8 p.00e-03 p.00e-0 C.00E N N 5

16 6. SP and GPS schedulers Pag. 6 The GPS scheduler In a GPS wth n servce classes the statstcal servce of the class- traffc s gven by w,..., j = n S() t = wct + max ( 0, wjct X j () t ), j wk k =,..., n backlogged classes k j Where the j ndex represents non-backlogged classes whle the k ndex refers to backlogged classes Therefore, the average value of the statstcal servce envelope avalable for servce class can be evaluated as α () t w E X t wc t d wjc t d X j t d j wk k j = w var X t wc t d max 0, w C t d X t d j wk k j ( ) () ( + ) + max 0, ( + ) ( + ) ( ) () ( + ) + ( + ) ( + ) j j 6

17 6. SP and GPS schedulers Pag. 7 The GPS scheduler Wth lnear-varance bounded traffc, w E S t wct w Ct N rt w var ( () ) + ( j j j ) ( ()) j k j w S t N rb t j j j j wk k j k thus α () t = w Nrt+ wc+ wc N r t+ d ( ) ( ) j j j j w k k j w Nrbt+ N rb t+ d ( ) j j j j w k k j 7

18 6. SP and GPS schedulers Pag. 8 The GPS scheduler The absolute mnmum of a(t) s α ( + + ) ( + ) ( Nrb+ B) 4Nrd wbc ba B wc A Nr,mn = where w w A = max ( 0, wc j Njrj) ; B = Njrb j j. j w k j w k k j k j 8

19 6. SP and GPS schedulers Pag. 9 The GPS scheduler: probablty of volaton of delay bound Thus, ( + + ) ( + ) ( Nrb+ B) Nr wbc ba B wc A Nr Pr ( D > d) exp d. 9

20 6. SP and GPS schedulers Pag. 0 The GPS scheduler: admsson control rule For the admsson control rule, we defne N,mn ( ) d wc = r dwc b p ( ln ). N Then the followng condton s checked ( ) ( )( ) ( ) ( ) r ( d( wbc + ba + B) b lnp) ln ln d wc A wbc ba B bb p d wbc ba B d wc A B p =,max ( ( + ) + ln ) > 0 max (,mn,,max ) If d w C A B p then N N N else N N,mn. 0

21 6. SP and GPS schedulers Pag. SP versus GPS: an example The fgure shows a comparson between the admsson curves of a Strct Prorty and a GPS scheduler The GPS scheduler guarantees a mnmum bandwdth for all classes Thus, servce class s not truncated as n the case of the SP scheduler N r b r b d 0.0 d 0.05 p p C w 0.8 w 0. N SP N gps N