Vaalty, Randomness and Lttle s Law Geoge Leopoulos Lttle s Law Assumptons Any system (poducton system) n whch enttes (pats) ave, spend some tme (pocessng tme + watng) and eventually depat Defntons = (long-un expected) aveage thoughput of the system (pats/unt tme) IP = (long-un expected) aveage wok n pocess (nume of pats n the system) (pats) = (long-un expected) aveage cycle tme (tme spent n the system) (tme unts) IP Lttle s Law: IP IP ΤΗ IP 2
Bottleneck ate, aw pocess tme, ctcal IP Poducton lne (wokstatons (S) n sees) S S 2 S 3 S t aveage pocess tme n S s nume of paallel machnes n S s : aveage (maxmum) pocess ate of S t mn{ }: ottleneck ate aveage pocess ate of slowest S maxmum of the l ne T t : aveage aw pocess tme n the lne mnmum T : ctcal IP IP fo whch a lne wth no vaalty acheves maxmum ( ) wth mnmun ( T ) 3 Bottleneck ate, aw pocess tme, ctcal IP Example : (Balanced lne) t = 2 hs s = = ½ pats/h t 2 = 2 hs s 2 = 2 = ½ pats/h t 3 = 2 hs s 3 = 3 = ½ pats/h t = 2 hs s = = ½ pats/h mn { } mn{.5,.5,.5,.5}.5 pats/hs,, T t 2 2 2 2 8 hs T (.5)(8) pats ( no. of machnes) 2
Bottleneck ate, aw pocess tme, ctcal IP Example 2: (Unalanced lne) t = 2 s = = ½ =.5 t 2 = 5 s 2 = 2 2 = 2/5 =. t 3 = s 3 = 6 3 = 6/ =.6 t = 3 s = 2 = 2/3 =.67 mn{ } mn{.5,.,.6,.67}. pats/h,, T t 2 5 3 2 hs T (.)(2) 8 pats ( no. of machnes) 5 Pefomance as a functon of IP Example : (Balanced lne) Best case pefomance (no vaalty: constant pocess tmes) IP (w) %T = IP/ % 8 /8 =.25 25 2 8 2/8 =.25 5 3 8 3/8 =.375 75 8 /8 =.5 5 25 5/ =.5 6 2 5 6/2 =.5 7 75 7/ =.5 8 6 2 8/6 =.5 6 3
Pefomance as a functon of IP Example : (Balanced lne) Best case pefomance 3 T 2 2 6 8 est T, f w w,f w 2 6 8 2 IP (w) 7 Pefomance as a functon of IP Example : (Balanced lne) Best case pefomance,6,5,,3,2, T est w,f w T, f w 2 6 8 2 IP (w) 8
Pefomance as a functon of IP Example : (Balanced lne) ose case pefomance (hghest vaalty: hghly vaale pocess tmes) Assumpton: Fo IP = w, thee ae 2 types of pats: slow pat (wth pocess tme t s = wt at S ) followed y w fast pats (wth zeo (neglgle) pocess tmes t f = ) w wt Aveage pocess tme n S t w Example: w = : t s s = tt = 2 = 8h hs = 8 + 8 + 8 + 8 = 32 hs! (= T ); = /32 = /8 9 Pefomance as a functon of IP Example : (Balanced lne) ose case pefomance IP (w) t s %T = IP/ % 2 2 = 8 /8 =.25 25 2 = 6 2 2/6 =.25 25 3 6 6 = 2 3 3/2 =.25 25 8 8 = 32 /32 =.25 25 5 = 5 5/ =.25 25 6 2 2 = 8 6 6/8 =.25 25 7 = 56 7 7/56 =.25 25 8 6 6 = 6 8 8/6 =.25 25 5
Pefomance as a functon of IP Example : (Balanced lne) ost case pefomance 3 T 2 2 6 8 T wt wost 2 6 8 2 IP (w) Pefomance as a functon of IP Example : (Balanced lne) Best case pefomance T,6,5,,3,2, wost T 2 6 8 2 IP (w) 2 6
Pefomance as a functon of IP Example : (Balanced lne) Pactcal wose case pefomance (hghest andomness medum vaalty) Fo IP = w, all possle states ae equally lkely Example: w = 3 (2 states) N N2 N3 N 3 3 3 3 2 2 3 Pefomance as a functon of IP Example : (Balanced lne) Pactcal wose case pefomance Assumptons. Lne s alanced 2. All statons have a sngle machne 3. Pocess tmes ae exponentally dstuted Defntons N = nume of sngle-machne wokstatons t = aveage pocessng tme at each S Aveage tme spent at a staton watng tme pocess tme = w tt N w w Nt t Nt ( w) t T N IP w w w T w 7
Pefomance as a functon of IP Example : (Unalanced lne example) Pactcal wost case pefomance T 6 5 3 2 wost PC / Unal. lne 2 est w T PC 8 2 6 2 2 IP 5 Pefomance as a functon of IP Example : (Balanced lne) Pactcal wost case pefomance,5,,3,2, 8 2 6 2 2 IP est Unal. lne 2 wost PC PC w w 6 8