Choosing Between Several Queuing Policies pierre.douillet@ensait.fr - PowerPoint PPT Presentation

✬ ✩ ISKE 2009 Choosing Between Several Queuing Policies pierre.douillet@ensait.fr École Nationale Supérieure des Arts et Industries Textiles Roubaix, France ✫ ✪

✬ ✩ Douillet ISKE 2009 ⇒ • How to manage a G/GI/n system ? . . . . . 3 averaging process criteria assumptions batch mean method • manager’s point of view . . . . . . . . . . . . 7 • customer’s point of view . . . . . . . . . . . . 11 • scaling and pooling . . . . . . . . . . . . . . . 14 • conclusion . . . . . . . . . . . . . . . . . . . . 17 ✫ ✪ Ensait - Roubaix 2

✬ ✩ Douillet ISKE 2009 How to manage a G/GI/n system ? averaging process • customer : "one man, one vote" E c ( X ) • manager : "one clock tick, one vote" E t ( X ) criteria • manager : exhaustivity and number of waiting customers • customer : mean and variance of sojourn time ✫ ✪ • fairness, and perceived fairness, are important Ensait - Roubaix 3

✬ ✩ Douillet ISKE 2009 assumptions • n identical servers, any service is independent from anything else • total capacity of service µ = n/ E B ( t ) • independent (...) arrivals, flow λ = E A ( t ) , ρ = λ/µ < 1 • distributions : anything except from M/M • Here : B is Gamma (svc = 0 . 4 ), A is Gamma (svc= 1 . 25 ), ρ = 0 . 93 or 0 . 97 ✫ ✪ Ensait - Roubaix 4

✬ ✩ Douillet ISKE 2009 batch mean method • each result has been obtained with K = 400 batches of N = 50000 events • containing rounding errors, allowing parallelization (with suitable random generator) • estimation of the sd of the estimators (and checking for independence) ✫ ✪ Ensait - Roubaix 5

✬ ✩ Douillet ISKE 2009 √ • How to manage a G/GI/n system ? . . . . . 3 ⇒ • manager’s point of view . . . . . . . . . . . . 7 ordinary policies number of busy servers jockeying • customer’s point of view . . . . . . . . . . . . 11 • scaling and pooling . . . . . . . . . . . . . . . 14 • conclusion . . . . . . . . . . . . . . . . . . . . 17 ✫ ✪ Ensait - Roubaix 6

✬ ✩ Douillet ISKE 2009 manager’s point of view ordinary policies B5s, N= 25000, K= 400, T= 10000040 1.0 fast • rand, robn , load 0.9 size rtwo robn 0.8 rand • rtwo (distributed) 0.7 0.6 • size (shortest queue) 0.5 0.4 • load (how can we ... ?) 0.3 0.2 • fast (single, µ , scv) 0.1 ✫ ✪ 0.0 0 5 10 15 20 25 30 Ensait - Roubaix 7

✬ ✩ Douillet ISKE 2009 number of busy servers • load is pooling, size is not the optimal B5s N= 25000 K= 400 T= 10000040 1.0 • Little : E t ( n b ) = nρ , fast load 0.9 size rtwo robn doesn’t depend on policy 0.8 rand 0.7 • probability ρ ∗ of full use 0.6 0.5 of the capacity of service 0.4 • fast is ρ ∗ = ρ 0.3 0.2 • load ensures exhaustivity 0.1 ✫ ✪ 0.0 -1 0 1 2 3 4 5 6 Ensait - Roubaix 8

✬ ✩ Douillet ISKE 2009 jockeying • jsiz = size then jockeying, jran = rand then jockeying • same distribution of queue length and servers business as load : jockeying solves (mostly) the manager’s problem. B5s, N= 25000, K= 400, T= 10000016 B5s N= 25000 K= 400 T= 10000016 1.0 1.0 fast load load jsiz 0.9 0.9 jsiz jran jran 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 ✫ ✪ 0.0 0.0 0 5 10 15 20 25 30 35 40 -1 0 1 2 3 4 5 6 Ensait - Roubaix 9

✬ ✩ Douillet ISKE 2009 √ • How to manage a G/GI/n system ? . . . . . 3 √ • manager’s point of view . . . . . . . . . . . . 7 ⇒ • customer’s point of view . . . . . . . . . . . . 11 sojourn time some results • scaling and pooling . . . . . . . . . . . . . . . 14 • conclusion . . . . . . . . . . . . . . . . . . . . 17 ✫ ✪ Ensait - Roubaix 10

✬ ✩ Douillet ISKE 2009 customer’s point of view sojourn time B5s N= 25000 K= 400 T= 10000040 1.0 0.9 • mean sojourn time 0.8 0.7 • MM1 : parameter µ − λ 0.6 0.5 0.4 0.3 • variance and fairness fast 0.2 load size rtwo 0.1 robn ✫ ✪ rand 0.0 0 500 1000 1500 2000 2500 3000 3500 4000 Ensait - Roubaix 11

✬ ✩ Douillet ISKE 2009 some results ± 2 σ ± 2 σ τ 0 . 1% τ/µ mean ratio sd ratio fast 470.53 7.00 1.00 445.61 10.01 1.00 3808 8.1 load 586.72 7.36 1.25 463.88 10.60 1.04 4189 7.2 jsiz 606.51 7.89 1.29 496.94 11.40 1.12 4435 7.4 jran 675.53 7.09 1.44 670.32 11.94 1.50 7479 11.2 size 621.48 7.69 1.32 500.77 10.48 1.12 4423 7.1 rtwo 710.71 7.36 1.51 527.89 10.37 1.18 4817 6.8 robn 997.21 13.15 2.12 904.20 18.61 2.03 8793 8.9 rand 2102.66 31.03 4.47 1999.41 45.64 4.49 17560 8.4 ✫ ✪ τ is the last 1 / 1000 fractile Ensait - Roubaix 12

✬ ✩ Douillet ISKE 2009 √ • How to manage a G/GI/n system ? . . . . . 3 √ • manager’s point of view . . . . . . . . . . . . 7 √ • customer’s point of view . . . . . . . . . . . . 11 ⇒ • scaling and pooling . . . . . . . . . . . . . . . 14 how to model scaling ? pooling factor pooling reshapes towards Poisson • conclusion . . . . . . . . . . . . . . . . . . . . 17 ✫ ✪ Ensait - Roubaix 13

✬ ✩ Douillet ISKE 2009 scaling and pooling how to model scaling ? • shape • independence (short range) • independence (long range) ??? • distributed, with coupling ? ✫ ✪ Ensait - Roubaix 14

✬ ✩ Douillet ISKE 2009 pooling factor • when customer flow increases and the number of servers increases accordingly, the mean sojourn time decreases exhaustive non exhaustive n fast load jsiz jran size rtwo robn rand 7 339 458 481 550 497 627 921 2070 5 470 586 606 675 621 710 997 2102 3 794 881 898 965 911 948 1242 2141 1 2378 2378 2378 2378 2378 2378 2378 2378 • pooling can also result in reduced staff... ✫ ✪ Ensait - Roubaix 15

✬ ✩ Douillet ISKE 2009 pooling reshapes towards Poisson • S 1 ( z ) , S n ( z ) , S r ( z ) are the mgf of A , n scaled A , and the resulting rand -arrivals in a single queue “ z S n ( z ) Z ” S 1 ( z )= a ( t ) exp ( t z ) d t, S n ( z ) = S 1 , S r ( z ) = n n − ( n − 1) S n ( z ) • Expanding in series : λ + z 2 λ − z + z 2 S 1 ( z )=1+ z λ ( svc − 1) „ 1 « ( λ − z ) 2 + z 3 O 2 λ 2 (1 + scv )+ · · · , S r ( z )= n 2 2 n svc ( a r ) = 1 + ( svc ( a ) − 1) /n ✫ ✪ Ensait - Roubaix 16

✬ ✩ Douillet ISKE 2009 conclusion • manager/customer perceptions are not based on the same averaging process • moreover manager’s focus is exhaustivity • customer’s focus is variance and fairness • centralized/distributed systems are different • aggregation changes drastically the results ✫ ✪ Ensait - Roubaix 17