EP2200 Queueing theory and teletraffic systems

Transcription

1 EP2200 Queueing heory and eleraffic sysems Vikoria Fodor Laboraory of Communicaion Neworks School of Elecrical Engineering Lecure 1 If you wan o model neworks Or a comple daa flow A queue's he key o help you see All he hings you need o know. (Leonard Kleinrock, Ode o a Queue from IETF RFC 1121) 1

2 Wha is queuing heory? Wha are eleraffic sysems? Queuing heory Mahemaical ool o describe resource sharing sysems, e.g., elecommunicaion neworks, compuer sysems Requess arrive dynamically Reques may form a queue o wai for service Applied probabiliy heory Teleraffic sysems Sysems wih elecommunicaion raffic (daa neworks, elephone neworks) Are designed and evaluaed using queuing heory Why do we need a whole heory for ha? 2

3 Why do we need eleraffic heory? Waiing delays a oupu buffers in nework rouers How will he delay change if he number of packes arriving wihin a second doubles? 3

4 Why do we need eleraffic heory? Throughpu (useful ransmissions) in a wireless nework wih random access If ransmissions may collide, how would he hroughpu change if he number of packes o be sen doubles? Wha is he effec of packe collisions? 4

5 Blocking probabiliy Throughpu Why do we need eleraffic heory? Call blocking probabiliy in a elephone nework vs. load TCP hroughpu vs. packe loss lines 10 lines RTT= RTT= Load Eperienced packe loss Teleraffic sysems are non-linear, and mahemaical ools are needed o predic heir performance 5

6 Course objecives Basic heory undersand he heoreical background of queuing sysems, apply he heory for sysems no considered in class Applicaions find appropriae queuing models of simple problems, derive performance merics Basis for modeling more comple problems advanced courses on performance evaluaion maser hesis projec indusry (elecommunicaion engineer) Prerequisies mahemaics, saisics, probabiliy heory, sochasic sysems communicaion neworks, compuer sysems 6

7 Course organizaion Course responsible Vikoria Fodor Lecures Vikoria Fodor Reciaions Ioannis (John) Glarapoulos Liping Wang Course web page KTH Social EP2200 (hps:// Home assignmens, projec, messages, updaed schedule and course informaion Your responsibiliy o say up o dae! Useful resources: Erlang and Engse calculaors, Java Aples Useful links: on-line books Links o probabiliy heory basics 7

8 Course maerial Course binder Lecure noes by Jorma Viramo, HUT, and Philippe Nain, INRIA Used wih heir permission Ecerps from L. Kleinrock, Queueing Sysems Problem se wih oulines of soluions Old eam problems wih oulines of soluions Erlang ables (ge more from course web, if needed) Formula shee, Laplace ranforms For sale a STEX, Q2 building. Coss 100 SEK. No e book needed! If you would like a book, hen you can ge one on your own Ng Chee Hock, Queueing Modeling Fundamenals, Wiley, (simple) L. Kleinrock, Queueing Sysems, Volume 1: Theory, Wiley, 1975 (well known, engineers) D. Gross, C. M. Harris, Fundamenals of Queueing Theory, Wiley, 1998 (difficul) Beware, he noaions migh differ 8

9 Course organizaion 12 lecures cover he heoreical par 12 reciaions applicaions of queuing models Home assignmens and projec (1.5 ECTS, compulsory, pass/fail) Home assignmen problems and (laer) soluions on he web individual submission, only handwrien version you need 75% saisfacory soluion o pass his momen submission on Nov. 20, submi a he STEX office Small projec compuer eercise deails laer submission deadline: Jan 4 +5 poins for ousanding projecs 9

10 Eam There is a wrien eam o pass he course, 5 hours Consiss of five problems of 10 poins each Passing grade usually 20 3 poins Allowed aid is he Bea mahemaical handbook (or similar) and simple calculaor. Probabiliy heory and queuing heory books are no allowed! The shee of queuing heory formulas will be provided, also Erlang ables and Laplace ransforms, if needed (same as in he course binder and on he web) Possibiliy o complemenary oral eam if you miss E by 2-3 poins (F) Complemen o E Regisraion is mandaory for all he eams A leas wo weeks prior o he eam Sudens from previous years: conac STEX (se@ee.kh.se) if you are no sure wha o do 10

11 Lecure 1 Queuing sysems - inroducion Teleraffic eamples and he performance riangle The queuing model Sysem parameers Performance measures Sochasic processes recall 11

12 Eample Packe ransmission a a large IP rouer Inpu pors Oupu pors Swiching engine Rouing processor We simplify modeling ypically he swiching engine is very fas he ransmission a he oupu buffers limis he packe forwarding performance we do no model he swiching engine, only he oupu buffers 12

13 Eample Packe ransmission a he oupu link of a large IP rouer packes arrive randomly and wai for free oupu link Performance: Depends on: 13

14 Eample Packe ransmission a he oupu link of a large IP rouer - packes wai for free oupu link Performance: Depends on: 14

15 Eample Voice calls in a GSM cell calls arrive randomly and occupy a channel. Call blocked if all channels busy. Performance Depends on: 15

16 Performance of queuing sysems The riangular relaionship in queuing Service demand sochasic Server capaciy Performance Works in 3 direcions sochasic Given service demand and server capaciy achievable performance Given server capaciy and required performance accepable demand Given demand and required performance required server capaciy 16

17 Lecure 1 Queuing sysems - inroducion Teleraffic eamples and he performance riangle The queuing model Sysem parameers Performance measures Sochasic processes recall 17

18 Block diagram of a queuing sysem Queuing sysem: absrac model of a resource sharing sysem buffer and server(s) Cusomers arrive, wai, ge served and leave he queuing sysem cusomers can ge blocked, service can be inerruped Arrival Reurn o sysem Blocking Inerruped service Compleed service Waiing Under service Ei sysem Buffer Server 18

19 Descripion of queuing sysems Sysem parameers Number of servers (cusomers served in parallel) Buffer capaciy Infinie: enough waiing room for all cusomers Finie: cusomers migh be blocked Order of service (FIFO, random, prioriy) Service demand (sochasic) Arrival process: How do he cusomers arrive o he sysem given by a sochasic process Service process: How long service ime does a cusomer demand given by a probabiliy disribuion Cusomer: IP packe Phone call 19

20 Eamples in deails Packe ransmission a he oupu link of a large IP rouer Number of servers: 1 Buffer capaciy: ma. number of IP packes Order of service: FIFO Arrivals: IP packe mulipleed a he oupu buffer Services: ransmission of one IP packe (service ime = ransmission ime = packe lengh / link ransmission rae) 20

21 Eamples in deails Voice calls in a GSM cell channels for parallel calls, each call occupies a channel if all channels are busy he call is blocked Number of servers: number of parallel channels Buffer capaciy: no buffer Order of service: does no apply Arrivals: call aemps in he GSM cell Service: he phone call (service ime = lengh of he phone call) 21

22 Group work Service a a bank, wih queue numbers and several clerks Draw he block diagram of he queuing sysems Arrivals: Service: Number of servers: Buffer capaciy: Order of service: 22

23 Performance measures N N q N s Number of cusomers in he sysem (N) Number of cusomers in he queue (N q ) Number of cusomers in he server (N s ) Sysem ime (T) W Waiing ime of a cusomer (W) Service ime of a cusomer () T Probabiliy of blocking (blocked cusomers / all arrivals) Uilizaion of he server (ime server occupied / all considered ime) Transien measures how will he sysem sae change in he near fuure? Saionary measures how does he sysem behave on he long run? average measures ofen considered in his course 23

24 Lecure 1 Queuing sysems - inroducion Teleraffic eamples and he performance riangle The queuing model Sysem parameers Performance measures Sochasic processes recall 24

25 Sochasic process Sochasic process A sysem ha evolves changes is sae - in ime in a random way Family of random variables Variables indeed by a ime parameer Coninuous ime: X(), a random variable for each value of Discree ime: X(n), a random variable for each sep n=0,1, Sae space: he se of possible values of r.v. X() (or X(n)) Coninuous or discree sae X() X(n) Number of packes waiing: Discree space Coninuous ime ime packe Waiing ime of consecuive packes: Discree ime Coninuous space EP2200 Queuing heory and eleraffic sysems 25

26 26 EP2200 Queuing heory and eleraffic sysems We are ineresed in quaniies, like: ime dependen (ransien) sae probabiliies (saisics over many realizaions, an ensemble of realizaions, ensemble average ): n h order saisics join disribuion over n samples limiing (or saionary) sae probabiliies (if eis) : Sochasic process - saisics } ) ( { lim ), ) ( ( lim X P F X P f ) ) ( ( ) ( ), ) ( ( ) ( X P F X P f ) ) (,, ) ( ( ),, ( 1 1 1,, 1 n n n X X P F n ensemble average

27 Sochasic process - erminology The sochasic process is: saionary, if all n h order saisics are unchanged by a shif in ime: F ( ) F ( ), F (,, n ) F (,, n), n,, 1,, n 1 1,, n 1 1, n ergodic, if he ensemble average is equal o he ime average of a single realizaion consequence: if a process ergodic, hen he saisics of he process can be deermined from a single (infiniely long) realizaion and vice versa ime average ensemble average EP2200 Queuing heory and eleraffic sysems 27

28 Sochasic process Eample on saionary versus ergodic Consider a source, ha generaes he following sequences wih he same probabiliy: ABABABAB BABABABA EEEEEEEE Is his source saionary? Is his source ergodic? ime average ensemble average EP2200 Queuing heory and eleraffic sysems 28

29 Summary Today: Queuing sysems - definiion and parameers Sochasic processes Ne lecure: Poisson processes and Markov-chains, he heoreical background o analyze queuing sysems Reciaion: Probabiliy heory and ransforms Prepare for he reciaion: read Viramo 1-3 in he course binder or download from he course web Definiion of probabiliy of evens Condiional probabiliy, law of oal probabiliy, Bayes formula, independen evens Random variables, disribuion funcions (discree and coninuous) Z and Laplace ransforms 29