Outline. Basic Components of a Queue. Queueing Notation. EEC 686/785 Modeling & Performance Evaluation of Computer Systems.

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1 EEC 686/785 Modelg & Performace Evaluato of Computer Systems Lecture 5 Departmet of Electrcal ad Computer Egeerg Clevelad State Uversty webg@eee.org (based o Dr. Raj Ja s lecture otes) Outle Homework #5 ad #6 Revew for the fal exam Fal exam: Dec, Moday 4:-6:pm Fal deadle for project report: Dec 4, mdght Records of your course work wll be posted o the course Web ste. If there s ay mstake, please cotact me as soo as possble December 5 EEC686/ Basc Compoets of a Queue Queueg Notato Kedall otato: A/S/m/B/K/SD A: arrval process S: servce tme dstrbuto m: umber of servers B: umber of buffers (system capacty) K: populato sze SD: servce dscple December 5 EEC686/785 December 5 EEC686/785

2 5 6 Commo Dstrbutos Key Varables M: Expoetal E k : Erlag wth parameter k H k : hyperexpoetal wth parameter k D: determstc > costat G: geeral > all M [x] : bulk Posso arrval or bulk servce process wth expoetal servce tmes G [x] a bulk arrval or servce process wth geeral tergroup tmes December 5 EEC686/785 December 5 EEC686/ Rules for All Queues (G/G/m) Stochastc Processes Stablty codto: < mμ (m: umber of servers) Fte-populato ad the fte-buffer systems are always stable Number system versus umber queue: q + s Number versus tme (Lttle s law): f jobs are ot lost due to suffcet buffers, r, smlarly q w Tme system versus tme queue: r w + s Process: fucto of tme Stochastc process: radom varables, whch are fuctos of tme Example : (t) umber of jobs at the CPU of a computer system Take several detcal systems ad observe (t). The umber (t) s a radom varable Ca fd the probablty dstrbuto fuctos for (t) at each possble value of t Example : w(t) watg tme a queue December 5 EEC686/785 December 5 EEC686/785

3 Types of Stochastc Processes 9 Dscrete/Cotuous State Processes Dscrete or cotuous state processes Markov processes Brth-death processes Posso processes Dscrete fte or coutable umber of values the state ca take Number of jobs a system (t),,, (t) s a dscrete state process The watg tme w(t) s a cotuous state process Stochastc cha: dscrete state stochastc process December 5 EEC686/785 December 5 EEC686/785 Markov Processes Brth-Death Processes Markov Processes: Future states are depedet of the past ad deped oly o the preset Markov cha: dscrete state Markov process Markov > It s ot ecessary to kow how log the process has bee the curret state > state tme has a memoryless (expoetal) dstrbuto M/M/m queues ca be modeled usg Markov processes The tme spet by a job such a queue s a Markov process ad the umber of jobs the queue s a Markov cha μ j j- μ μ 3 μ j j Brth-Death Processes: The dscrete space Markov processes whch the trastos are restrcted to eghborg states Process state ca chage oly to state + or - Example: the umber of jobs a queue wth a sgle server ad dvdual arrvals (ot bulk arrvals) μ j j j j+ μ j+ j+ μ j+ December 5 EEC686/785 December 5 EEC686/785 3

4 Posso Processes 3 Relatoshp Amog Stochastc Processes 4 Iterarrval tmes IID + expoetal Number of arrvals over a gve terval (t, t+x) has a Posso dstrbuto The arrval process s referred to as a Posso process or a Posso stream Markov process broader tha brth-death process broader tha Posso process The Posso process ca be modeled as a pure brth process wth costat brth rate All brth-death processes are Markov processes wth the restrcto that the trastos are restrcted to eghborg states December 5 EEC686/785 December 5 EEC686/785 Brth-Death Processes 5 Theorem: State Probablty 6 State-trasto dagram j j- μ μ μ 3 μ j I state : New arrvals take place at a rate The servce rate s μ j Both the terarrval tmes ad servce tmes are assumed expoetally dstrbuted μ j j j j+ μ j+ j+ μ j+ The steady-state probablty p of a brth-death process beg state (.e., there are jobs the system) s gve by: p p,,..., μ μ μ Here, p s the probablty of beg the zero state (o job system,.e., system s dle) December 5 EEC686/785 December 5 EEC686/785 4

5 Results for M/M/ Queues j- j j+ μ μ μ μ μ μ μ Brth-death processes wth,,,..., μ μ,,..., Probablty of jobs the system p p,,..., μ ρ /μ: traffc testy 7 Results for M/M/ Queues Utlzato of the server: Mea umber of jobs the system: Varace of the umber of jobs: Mea respose tme: U p ρ ρ E [ ] ρ ρ Var[ ] / μ ( ρ) Er [] ρ Cumulatve dstrbuto fucto of the respose rμ ( ρ ) tme: F( r) e 8 December 5 EEC686/785 December 5 EEC686/785 Results for M/M/ Queues 9 Results for M/M/ Queues Cumulatve dstrbuto fucto of the watg tme F( w) ρe wμ ( ρ ) Ths s a trucated expoetal dstrbuto. Its q-percetle s gve by ρ w q l μ( ρ) q The above formula apples oly f q s greater tha (-ρ). All lower percetles are E[ w] ρ w q max, l ρ q Mea umber of jobs the queue E[ q ] ( ) p ( )( ρ) ρ ρ ρ Busy perod: the tme terval betwee two successve dle tervals December 5 EEC686/785 December 5 EEC686/785 5

6 M/M/m Queue Aalyss of M/M/m Queue The state of the system s represeted by the umber of jobs the system μ μ 3μ m- (m )μ mμ m mμ m+ mμ Number of jobs the system s a brth-death process:,,,..., μ,,..., m μ mμ m, m +,..., Probablty of jobs the system: ( mρ) p,,..., m! p m ρ m p, +,..., m m m! December 5 EEC686/785 December 5 EEC686/785 Aalyss of M/M/m Queue Probablty of zero jobs the system: p + + m!( ρ)! m m ( mρ) ( mρ) Probablty that a arrvg job has to wat the queue: Ths s kow as Erlag s C formula. Notce that for m, ϑ ρ m ( mρ) ϑ P( m jobs) p m!( ρ) 3 Aalyss of M/M/m Queue Mea umber of jobs the queue E[ q ]: ρϑ E [ q] ( m) p m+ ρ Expected umber of jobs servce: m E[ s] p + mp mρ m Utlzato of each server: Busy tme per server ( T / μ) / m U Total tme T mμ 4 December 5 EEC686/785 December 5 EEC686/785 6

7 Aalyss of M/M/m Queue 5 Aalyss of M/M/m/B Queue 6 Mea respose tme: ϑ Er [] + μ m( ρ) Mea watg tme: ϑ Ew [ ] m( ρ) q-percetle of watg tme: ϑ w q max, l mμ( ρ) q μ μ Brth-death process: m- m m+ (m )μ μ mμ,,,..., B μ,,..., m μ mμ m, m +,..., B mμ mμ B December 5 EEC686/785 December 5 EEC686/785 Aalyss of M/M/m/B Queue Probablty of jobs the system: ( mρ) p,,..., m! p m ρ m p m, m +,..., B m! Probablty of zero jobs the system: p + + m!( ρ)! B m+ m m ( ρ )( mρ) ( mρ) 7 Aalyss of M/M/m/B Queue Mea umber of jobs the system: E[ ] Mea umber of jobs the queue: E[ ] All arrvals occurrg whe the system s the state B are lost. Rate of the jobs actually eterg the system, called effectve arrval rate, s: B B ' p p ( p The dfferece p B represets the loss rate B ) B B p ( m) 8 q p m+ December 5 EEC686/785 December 5 EEC686/785 7

8 Aalyss of M/M/m/B Queue 9 Queueg Networks 3 Mea respose tme (usg Lttle s law): E[ ] E[ ] E[ r] ' ( Utlzato of each server: Busy tme per server ( ' T / μ) / m ' U ρ( Total tme T mμ Probablty of the system full capacty p B p B ) p B ) Network model whch jobs departg from oe queue arrve at aother queue (or possbly the same queue Ope queueg etworks Exteral arrvals departures Closed queueg etworks No exteral arrvals or departures Mxed queueg etworks ope for some workloads ad closed for others December 5 EEC686/785 December 5 EEC686/785 Product-Form Network 3 No-Markova Product Form Networks 3 Ay queueg etwork whch: P( M,, 3,..., M ) f ( ) G( N) Whe f ( ) s some fucto of the umber of jobs at the th faclty, G(N) s a ormalzg costat ad s a fucto of the total umber of jobs the system. Job flow balace: for each class, the umber of arrvals to a devce must equal the umber of departures from the devce. Oe step behavor: o bulk arrvals ad o smultaeous job-moves 3. Devce homogeety: a devce s servce rate for a partcular class does ot deped o the state of the system ay way except for the total devce queue legth ad the desgated class s queue legth December 5 EEC686/785 December 5 EEC686/785 8

9 No-Markova Product Form Networks The assumpto 3 mples the followg: Sgle resource possesso No blockg Idepedet job behavor Local formato Far servce Routg homogeety 33 Mache Reparma Model Oe of the earlest model of computer systems Orgally developed for modelg mache repar shops A umber of workg maches A repar faclty wth oe or more servers (reparme) Wheever a mache breaks dow, t s put the queue for repar ad servced as soo as a reparma s avalable Termals System 34 December 5 EEC686/785 December 5 EEC686/785 Cetral Server Model Itroduced by Buze (973) The CPU s the cetral server that schedules vsts to other devces After servce at the I/O devces the jobs retur to the CPU Termals CPU Dsk A Dsk B 35 Types of Servce Ceters Fxed-capacty servce ceters: servce tme does ot deped upo the umber of jobs the devce Delay ceters or IS (fte server): o queueg. Jobs sped the same amout of tme the devce regardless of the umber of jobs t Load-depedet servce ceters: servce rates may deped upo the load or the umber of jobs the devce 36 December 5 EEC686/785 December 5 EEC686/785 9

10 Operatoal Laws 37 Operatoal Laws 38 Relatoshps that do ot requre ay assumptos about the dstrbuto of servce tmes or terarrval tmes Operatoally testable assumptos > assumptos that ca be verfed by measuremets For example, whether umber of arrvals s equal to the umber of completos (.e., job flow balace) s operatoally testable A set of observed servce tmes s ot operatoally testable Operatoal quattes: quattes that ca be drectly measured durg a fte observato perod T observato terval A umber of arrvals C umber of completos B busy tme Operatoal quattes are varables that ca chage from oe observato perod to the ext December 5 EEC686/785 December 5 EEC686/785 Operatoal Laws Relatoshps that hold every observato perod are called operatoal laws Operatoal quattes Utlzato Law Number of arrvals A Arrval Rate Tme T Number of completos C Throughput X Tme T Busy Tme B Utlzato U Total Tme T Total tme served B Mea servce tme S Number served C U or U T T C December 5 EEC686/785 B C B X S 39 Vst Rato M Jobs V vsts per job Each job makes V requests for th devce the system: C C CV or V C V s called vst rato December 5 EEC686/785 4

11 Forced Flow Law Forced Flow Law relates the system throughput to dvdual devce throughputs. If observato perod T s such that A C > Devce satsfes the assumpto of job flow balace Forced Flow Law System throughput: Jobs completed C System throughput X Total tme T Throughput of th devce: Devce throughput X C C C or X XV T C T 4 Forced Flow Law Combg the forced flow law ad the utlzato law, we get: Utlzato of th devce U XS XVS or U XD Here D V S s the total servce demad o the devce for all vsts of a job The devce wth the hghest D has the hghest utlzato ad s the bottleeck devce 4 December 5 EEC686/785 December 5 EEC686/785 Trasto Probabltes p j probablty a job movg to jth queue after servce completo at th queue Vst ratos ad trasto probabltes are equvalet the sese that gve oe we ca always fd the other I a system wth job flow balace, the umber of completos at jth queue: M Cj Cpj Subscrpt s used to deote vsts to the outsde lk p probablty of a job extg from the system after completo of servce at th devce 43 Vst Rato Equatos Dvdg both sdes by C we get: M Vj Vpj Sce each vst to the outsde lk s defed as the completo of the job, we have: V The above two equatos are called vst rato equatos 44 December 5 EEC686/785 December 5 EEC686/785

12 Vst Rato Equatos for Cetral Server Models I cetral server models, after completo of servce at every queue, the jobs always move back to the CPU queue: p pj, j Vp V + V + V3+ + VM p j Vj Vpj j,3,..., M p Thus, we ca fd the vst ratos by dvdg the probablty p j of movg to jth queue from CPU by the ext probablty p 45 Lttle s Law Mea umber the devce arrval rate mea tme the devce Q R If the job flow s balaced, the arrval rate s equal to the throughput ad we ca wrte: Q X R 46 December 5 EEC686/785 December 5 EEC686/785 Geeral Respose Tme Law 47 Iteractve Respose Tme Law 48 There s oe termal per user ad the rest of the system s shared by all users Geeral respose tme law: M R RV Ths law holds eve f the job flow s ot balaced Termals Cetral Subsystem If Z thk-tme, R respose tme, the total cycle tme of requests s R + Z Each user geerates about T/(R + Z) requests T If there are N users, the system throughput X total umber of requests / total tme N[T/(R+Z)]/T N/(R+Z) or R (N/X) Z Ths s the teractve respose tme law December 5 EEC686/785 December 5 EEC686/785

13 Bottleeck Aalyss 49 Asymptotc Bouds 5 From forced flow law: U D the devce wth the hghest total servce demad D has the hghest utlzato ad s called the bottleeck devce Note: delay ceters ca have utlzatos more tha oe wthout ay stablty problems. Therefore, delay ceters caot be a bottleeck devce Oly queueg ceters used computg D max The bottleeck devce s the key lmtg factor achevg hgher throughput Throughput ad respose tmes of the system are boud as follows: N X( N) m, Dmax D+ Z RN ( ) max { DND, max Z} Here, D D s the sum of total servce demads o all devces except termals These are kow as asymptotc bouds December 5 EEC686/785 December 5 EEC686/785 Asymptotc Bouds The asymptotc bouds are based o the followg observatos: The utlzato of ay devce caot exceed oe. Ths puts a lmt o the maxmum obtaable throughput The respose tme of the system wth N users caot be less tha a system wth just oe user. Ths puts a lmt o the mmum respose tme The teractve respose tme formula ca be used to covert the boud o throughput to that o respose tme ad vce versa December 5 EEC686/785 5 Typcal Asymptotc Bouds Both throughput ad respose tme bouds cosst of two straght les The pot of tersecto of the two les s called the kee For both respose tme ad throughput, the kee occurs at the same value of umber of users Respose Tme D December 5 EEC686/785 -Z Kee 5 SlopeD max Bouds N* Number of Users /D max Slope/(D+Z) Throughput Bouds Number of Users 3

14 Typcal Asymptotc Bouds 53 The umber of jobs N* at the kee s gve by: * D NDmax Z * D + Z N D If the umber of jobs s more tha N*, the we ca say wth certaty that there s queueg somewhere the system The asymptotc bouds ca be easly explaed to people who do ot have ay backgroud queueg theory or performace aalyss max December 5 EEC686/785 4