# On the impact of heterogeneity and back-end scheduling in load balancing designs

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5 TABLE I SUMMARY OF RESULTS FOR PARETO JOB SIZES THE RESULTS FOR PS WERE FIRST DERIVED IN [3] [36] BUT ALSO FOLLOW IMMEDIATELY FROM THE RESULTS FOR SRPT Back-ed scheduler SRPT PS Job sze dstrbuto Pareto wth α < 2 Pareto wth α > 2 Geral C λ log m µ λ µ λ where m = α 2 α Optmal assgmet µ µ µ m/m+ µ m/m+ µ λ µ µ Lemma 8 Lemma 9 Lemma 9 Nash assgmet λ Prce of aarchy Satsfy λ = λ µ λ st = µ µ m /m µ m /m µ /µ µ wth µ λ /µ µ Lemma 0 Lemma Lemma µ k k m m Theorem 4 Theorem 5 Proposto whe k s small However whe k s large PS ca actually outperform SRPT I the remader of ths secto we prove the prce of aarchy results Table IV-B To prove these results we frst ed to characterze the λ ad These results are summarzed Table IV-B but we defer the dervatos to the appedx We start wth the case of fte-varace Pareto ob szes ad the move to the case of fte-varace Theorem 4 Whe C λ = log s k µ λ the prce of aarchy Proof: We beg by provg a upper boud o the prce of aarchy ad the we llustrate that t s asymptotcally tght Sce the form of λ s mplct we caot smply drectly compare E[T ] ad E[T ] Istead we wll wrte each terms of the remag servce capacty at each queue e the gaps γ = µ λ Def c to be the average respose tme for the -th queue E[T ] otce that every queue has the same average respose tme uder Nash equlbrum The we have E[T ] = c µ Note that the remag capacty at each server Q at the Nash assgmet s γ = µ e cµ Next we wll calculate E[T ] terms of c ad γ Sce the total gap s dstrbuted equally uder the mal allocato see Table IV-B we have = = µ e c Thus recallg that µ = k µ we have that E[T ] s as follows E[T µ ] = log = = log = = log + k µ e cµ k e ck µ logk + log k e ck µ Notg that k e ckµ s decreasg for large eough cµ we ca boud the thrd term above as follows: log log e cµ / whch gves E[T ] k e ck µ = cµ log logk + cµ Now we ca boud the prce of aarchy by c µ logk + cµ = k logk cµ + k k Next we wll show that ths boud o the prce of aarchy s asymptotcally tght Cosder the specfc example where µ = kµ ad µ 2 = = µ = µ The we ca aga calculate E[T ] as E[T ] = c µ = cµk +

6 ad E[T ] as E[T ] = = µ log = log µ e cµ µ µ e cµ = logµ + logk + log kµ + log µ e cµ µ e cµ Sce µ e c for large eough c c ca be chose arbtrarly large heavy-traffc we ca boud the last term above by log log e cµ = cµ µ e cµ Whe cosderg the heavy-traffc regme e c ths yelds the followg lower boud o the prce of aarchy: cµk + logµ + logk + cµ = k + logµ+logk cµ + For costat ths gves Ωk as desred k/ as c Theorem 5 Whe C λ = µ λ m the prce of aarchy s k m m Proof: Sce λ ad are so smlar see Table IV-B we ca compare the mea respose tme uder these polces drectly I partcular straghtforward calculato yelds E[T ; λ k] = E[T ; λ λ k] = m m µ m/m+ = = µ = m+ µ m m whch gves that the prce of aarchy s the soluto to the followg mzato: m We ca wrte the above more succctly usg orms as follows: max k x /m k x /m+ st k k k m x /p where y p = = yp We ca ow boud the soluto to ths mzato: k x /m k m k x /m+ k x /m+ k k /m+ m k m m k m The frst step follows from upper boudg x The secod step follows from lower boudg x k m The thrd step follows from observg that k /m+ k To see that ths boud o the prce of aarchy s asymptotcally tght let us cosder the stuato wth queues where k = k ad k 2 = = k = I ths case = k = k m m m = k m m+ m+ = m + k + k m m + k m m+ m+ m+ + k m + k m/m+ m+ + k = m+ + km/m+ Now suppose k wth k m/m+ the we have + k + km/m+ m+ k k m m+ = m k m max = k m = km /m m+ = km/m+ st k k A upper boud o the soluto to ths mzato s the followg reformulato: max = k m = x/m = k x /m+ m+ st k k k m x VI THE CASE OF ARBITRARY SCHEDULING POLICIES AND GENERAL JOB SIZES I the prevous secto we derved bouds o the prce of aarchy whe the back-ed scheduler performs SRPT ad ob szes follow a Pareto dstrbuto I ths secto we dscuss geralzatos of those results to both geral ob sze dstrbutos ad to arbtrary polces Though geralzg the schedulg polcy ad geralzg the ob sze dstrbuto seem dfferet from o aother they both have the same effect they chage the form of E[T ] For example Basal [4] recetly showed that the heavy-traffc growth rate of SRPT uder Expotal ob szes s E[T ] = θ µ λ logµ /µ λ 2

9 Notce the tuto to the above lemma: the total excess servce capacty µ Λ s dvded evely amog the servers I ths xt lemma the total excess servce capacty µ Λ s ot dvded evely amog the servers aymore stead t s dvded proporto to µ m/m+ m Lemma 9 Cosder C λ = µ λ The mal dspatcher uses m/m+ µ = µ µ Λ µm/m+ Proof: We ca aga fd the socally mal arrval rates by solvg 4 Ths gves m+ m µ µ µ = m µ from whch we obta = So we ca wrte = whch gves = µ µ µ m/m+ µ m/m+ m/m+ µ µm/m+ m+ 6 Equvaletly we have m/m+ µ = µ µ Λ µm/m+ Note that whe m = we get the mal arrval rates for PS APPENDIX B CALCULATING THE NASH ASSIGNMENT I ths secto we wll calculate the Nash assgmet the case of SRPT schedulg ad Pareto ob szes We kow that heavy-traffc all queues are used ad thus the arrval rates must satsfy E[T ] = E[T ] { } From ths codto t s possble to derve the arrval rates explctly ad we atta the results summarzed Table IV-B Lemma 0 Cosder C λ = log satsfes µ/µ λ = λ µ µ µ λ where λ s the soluto to /µ µ Λ = µ λ µ The λ Proof: At a Nash assgmet all obs must have the same expected respose tme e f λ λ represet the arrval rates at a Nash assgmet the for all { } µ log µ µ λ = µ log µ µ λ 7 From here we ca calculate λ explctly From the equato above t follows that /µ / µ γ = γ 8 Summg both sdes gves γ / = from whch t follows that γ /µ = µ / µ /µ µ/ Combg 8 ad 9 gves γ = γ = µ µ γ µ µ γ /µ µ µ/µ γ / 9 /µ whch s equvalet to the equato the statemet of the lemma Note that though the form of λ the above lemma s mplct t ca be solved easly may specal cases For stace whe µ = µ for all the Nash assgmet s the same as the mal assgmet The λ the xt lemma are explct I fact ths case λ has arly the same form as Lemma 9 Lemma Cosder C λ = m µ λ The m /m µ λ = µ µ Λ µm /m µ Proof: The Nash codto gves us that m = m µ { } 0 µ λ µ µ λ Notg that ths s parallel to 6 except that m + s ow chaged to m we ca mmedately wrte m /m µ γ = µm /m whch s equvalet to the statemet of the lemma

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