IS 709/809: Computational Methods in IS Research. Queueing Theory Introduction

Transcription

1 IS 709/809: Compuaional Mehods in IS Research Queueing Theory Inroducion Nirmalya Roy Deparmen of Informaion Sysems Universiy of Maryland Balimore Couny

2 Inroducion: Saisics of hings Waiing in Lines Wai in line in our cars in raffic jam or a oll boohs Wai on hold for an operaor o pick up elephone calls Wai in line a supermarkes o check ou Wai in line a fas food resaurans Wai in line a bank and pos offices.. More demand for service han here is faciliy for service available Shorage of available servers; economically infeasible; space limi

3 SERVER Inroducion Limiaions can be removed parially Need o know how much service should hen be made available How long mus a cusomer wai? And how many people will form in he line? Queueing heory aemp o answer hose quesions Deailed mahemaical analysis PACKET QUEUE ROUTER Probabiliy models o characerize empirical processes mahemaical analysis o calculae performance measures probabiliy disribuions o model rouer s inerarrival and service imes help o deermine average queueing delay/queue lengh

4 Characerisics of Queueing Process Arrival paerns of cusomers Service paerns of servers Queue disciplines Sysem capaciy Number of service channel Number of service sages

5 Arrival paerns of cusomers Process of arrivals is sochasic Necessary o know he probabiliy disribuion describing he imes beween successive cusomer arrivals Inerarrival imes Bach or Bulk arrivals Reacion of a cusomer upon enering he sysem Irrelevan o he waiing ime Decides no o ener he queue upon arrival --- balked Ener he queue, bu afer some ime loose paience and decide o leave --- reneged

6 Arrival Paerns Arrival paern changes or does no change wih ime An arrival paern ha does no change wih ime Probabiliy disribuion describing he inpu process is imeindependen A saionary arrival paern No ime-independen A non-saionary arrival paern

7 Service Paerns A probabiliy disribuion is needed o describe he sequence of cusomer service imes Service may be single or bach One cusomer being served a a ime by a given server Cusomer may be served simulaneously by he same server A compuer wih parallel processing Sighseers on a guided our People boarding a rain

8 Queue Discipline The manner in which cusomers are seleced for service Firs come firs served (FCFS) Las come firs served (LCFS) Sored iems in a warehouse, easier o reach he neares iems A variey of prioriy schemes Higher prioriies will be seleced for service ahead of hose wih lower prioriies Preempive Higher prioriy is allowed o ener service immediaely Non-Preempive Highes prioriy cusomer goes o he head of he queue bu canno ge ino service unil he cusomer presenly in service is compleed

9 Sysem Capaciy Physical limiaion o he amoun of waiing room Line reaches a cerain lengh, no furher cusomers are allowed o ener unil space becomes available Finie queueing siuaions Limied waiing room can be viewed as one wih force balking

10 Number of Service Channels Design muli-server queueing sysems o be fed by a single line Specify he number of service channels Typically refer o he number of parallel service saions

11 Single server queue Single server (arrivals when server is full is queued) Cusomer Arrivals Buffer Server Deparures Example: Small Grocery Sore

12 Muliple single-server queue Muliple single-server queues (A queue for each channel) Example: Large supermarke wih muliple cashiers

13 Muliple single-server queue Muliple server (single queue feeds ino muliple servers) Example: Bank Teller. Example: Call Cener

14 Sages of Service Single sage of service Hair syling salon Mulisage of service Physical examinaion procedure Recycling or feedback may occur Manufacuring process, qualiy conrol Telecommunicaion nework may process messages hrough a randomly seleced sequence of nodes Some messages may require rerouing on occasion hrough he same sage

15 Hisory of Queueing Theory How many runk lines are required o provide service o a own? Agner Erlang Faher of he field of queueing heory and eleraffic engineering

16 Telecommunicaion Nework Design (TND) Imagine a small village having a populaion of 100 elephone users; how many runk lines are needed o connec his village s elephone exchange o a long-disance elephone exchange? Local exchange Single runk line long wais and blocked calls A runk line for every user economically inefficien Trunk line: a circui connecing elephone swichboards or oher swiching equipmen.

17 Soluion I will be highly unlikely ha all 100 users will wan o use heir service all he ime Wha is he average demand like? Assume ha on average here are 5 calls/ hr in he busies hour of he day and he average call lass for an hour each.

18 Soluion Now, ha we know here are 5 calls/hr on average in he busy-hour-ime, we should jus pu 5 runk lines. RIGHT? WRONG! Calls bunch up! The number of calls in a hour is a random variable and follows a Poisson disribuion wih expeced value of 5 calls/hour

19 Soluion Now, ha we know here are 5 calls/ hr on average in he busy-hour-ime, we should jus pu 5 runk lines. RIGHT? WRONG! Calls bunch up! The number of calls in a hour is a random variable and follows a Poisson disribuion wih expeced value of 5 calls/ hour

20 11 Blocking rae is 0.01 To carry 5 erlangs of raffic (5 calls/ hr wih average call duraion of 1 hour) wih blocking probabiliy of 0.01 only, 11 runk lines are needed

21 A Simple Deerminisic Queue Inerarrival ime is exacly 1 minue and Service imes is also exacly 1 minue. Cusomers in sysem Arrival Deparure Time (min)

22 A Sochasic Queue Inerarrival ime is.5 minue or 1.5 minue wih equal probabiliy expeced inerarrival ime = 1 minue Service ime is.5 minue or 1.5 minue wih equal probabiliy expeced service ime = 1 minue Cusomers in sysem Arrival Deparure Time (min) This queue is unsable since he inerarrival ime and service ime are equal (Due o sochasic naure of arrivals, arrivals would bunch up)

23 Sabiliy of Queue For a sable queue, he average service rae (μ) mus be more han he arrival rae (λ); he raffic inensiy ρ (= λ/ μ) < 1 ρ close o 1

24 Lighly loaded queue Poisson arrival process (rae λ = 0.1 arrivals/ minue ); Exponenial service (rae μ = 1 deparure/ minue) ρ= λ/μ= 0.1

25 Moderaely loaded queue Poisson arrival process (rae =λ = 0.5 arrivals/ minue); Exponenial service (rae= μ = 1 deparure/ minue) ρ= λ/μ= 0.5

26 Heavily loaded queue Poisson arrival process (rae =λ = 0.99 arrivals/ minue); Exponenial service (rae= μ = 1 deparure/ minue) ρ= λ/μ= 0.99

27 Noaion of a Queueing Sysem A/S/m/B/K/SD (Kendall s noaion) 5. Cusomer Populaion (finie or infinie) 1. Arrival process disribuion (M,D,G) 6. Service discipline (e.g., FIFO) 2. Service ime disribuion (M,D,G) 4. Buffers available (finie or infinie) 3. Number of servers (1, m, ) Inerarrival ime and service ime is assumed o be IID, herefore, only he family of disribuions needs o be specified

28 Kendall s noaion M/G/1: Poisson arrivals; General service disribuions; 1 server (Infinie buffer, populaion; FCFS) G/G/1: General arrival and service disribuions; 1 server (Infinie buffer, populaion; FCFS) M/D/2/ /FCFS: Poisson arrivals; deerminisic service ime; wo parallel servers; no resricions on he maximum # allowed in he sysem; and FCFS queue disciplines

29 Measuring Sysem Performance Effeciveness of a queueing sysem Generally 3 ypes of sysem responses of ineres Waiing ime Cusomer accumulaion manner A measure of he idle ime of he server Queueing sysems have sochasic elemens Measures are ofen random variables and heir probabiliy disribuions

30 Measuring Sysem Performance Two ypes of cusomer waiing imes Time a cusomer spends in he queue amusemen park Toal ime a cusomer spends in he sysem (queue + service) Machines ha need repairs Cusomer accumulaion measures Number of cusomers in he queue Toal number of cusomers in he sysem Idle-service measures of a server Time he enire sysem is devoid of cusomers

31 Lile s Theorem One of he mos powerful relaionship in queueing heory Developed by John D. C. Lile in early 1960s Relaed he seady-sae mean sysem sizes o he seady sae average cusomer waiing imes T q = ime a cusomer spends waiing in he queue T = oal ime a cusomer spends in he sysem= response ime T = T q + S; where S is he service ime and T, T q, and S all are random variables Two ofen used measures of sysem performance Mean waiing ime in queue; W q = E [T q ] Mean waiing ime in he sysem; W = E [T]

32 Lile s Theorem Mean number of cusomers (backlog) Number of cusomers in queue: N q Number of cusomers obaining service: N s Number of cusomers in sysem N = (N q + N s ) Average waiing ime of cusomers (delay) T q T Waiing ime for cusomers in queue: T q Service ime for cusomers: S Toal ime (response ime) of cusomers in sysem T = (T q + S) All he above merics are RVs, and herefore we will calculae heir expecaions

33 Lile s Law Lile s Law relaed he wo primary performance measures of any queue Mean number of cusomers Average waiing ime of cusomers L S E[S] L q L W q W

34 Lile s Theorem Lengh = Arrival-rae x Wai-ime Lile s law saes ha ime-average of queue lengh is equal o he produc of he arrival rae and he cusomer-average waiing ime (response ime) JDC Lile E[ N ]. E[ T ] L. W E[T]

35 Lile s Theorem Lile s formulas are: L = λw L q = λw q L q = mean number of cusomer in he queue L = mean number of cusomer in he sysem λ = arrival rae E[T] = E[T q ] + E[S] => W = W q + 1/μ

36 Lile s Theorem Proof

37 Lile s Theorem Proof Number of cusomers (say N) arrive over he ime period (0,T) is 4 Find ou he value of L and W from he graph T T T T T L ) / ( under curve) / (area )]/ 1( ) 2( ) 3( ) 2( ) 1( ) 2( ) [1( N T T W under curve) / (area 4 ) / ( 4 )]/ ( ) ( ) ( ) [(

38 Lile s Theorem Proof W L W T N L WN LT where fracion (N/T) is he number of cusomers arriving over he ime T and which is for his period, he arrival rae λ ) 1 ( ( q ) q W W L L ] [ ] [ ] [ ] [ s q q q N E N N E N E N E L L So expeced no. of cusomers in service in he seady sae is λ/μ, also denoed by r. For a single server sysem r=ρ

39 Summary of General Resuls for G/G/c queues ) (1 L L G/G/1empy sysem probabiliy 1 L L offered work load rae r Busy probabiliy for an arbirary server p value argumen Expeced - 1 W W Lile's formula L Lile's formula W L offered work load rae o a server Traffic inensiy; 0 q 0 q b q q p p r c W c q

40 Example: Lile s Law Assume ha you receive 50 s every day and ha you archive your messages afer responding o your . The number of un-responded s in your inbox varies beween ~100 o ~200 and is average value is 150. How long do you ake o answer your s? L=150 s λ=50 s/ day W=L/λ = 3 days

41 Example: Lile s Law Suppose ha 10,800 HTTP requess arrive o a web server over he course of he busies hour of he day. If we wan o limi he mean waiing ime for service o be under 6 seconds, wha should be he larges permissible queue lengh? λ=10,800 requess/ hour λ= 3 requess/ second W=6 seconds Since L q = λw q => L q = 18

42 Poisson Process & Exponenial Disribuion Sochasic queueing model Assume inerarrival imes and service imes obey he Exponenial disribuion Equivalenly arrival rae and service rae follow a Poisson disribuion Firs derive he Poisson disribuion See whieboard Show ha assuming no. of occurrences in some ime inerval o be a Poisson random variable is equivalen o assuming ime beween successive occurrences o be an exponenially disribued random variable

43 Derivaion of Poisson Disribuion Consider an arrival couning process {N(), 0} N() denoes he oal no. of arrivals up o ime Assumpion: Probabiliy ha an arrival occurs beween ime and ime ( + ) = λ + o( ) Pr{an arrival occurs beween and + } = λ + o( ) where λ is a consan independen of N() is an incremenal elemen o( ) denoes a negligible quaniy when as 0; Pr{more han one arrival beween and + } = o( ) Number of arrivals in non-overlapping inervals are saisically independen o( ) lim 0 0

44 Derivaion of Poisson Disribuion Calculae p n () = probabiliy of n arrivals in a ime inerval of lengh, where n 0 Develop differenial difference equaions for he arrival process p n ( + ) = Pr{n arrivals in and none in } + Pr{n-1 arrivals in and 1 in } + Pr{n-2 arrivals in and 2 in } Pr{no arrivals in and n in }. p n ( + ) = p n ()[1 -λ - o( )] + p n-1 ()[λ + o( )] + o( ) where he las erm o( ) represens he erms Pr{n -j arrivals in and j in ; 2 j n }

45 Derivaion of Poisson Disribuion p n ( + ) = p n ()[1 -λ - o( )] + p n-1 ()[λ + o( )] +o( ) For n = 0; p 0 ( + ) = p 0 ()[1 - λ - o( )] Rewriing; p 0 ( + ) - p 0 () = - λ p 0 () + o( ) and p n ( + ) - p n () = - λ p n () + λ p n-1 () + o( ) for (n 1) Divide by and ake he limi as 0 o obain he differenial-difference equaions ] ) ( ) ( ) ( ) ( [ lim o p p p 1) (n ] ) ( ) ( ) ( ) ( ) ( [ lim 1 n 0 o p p p p n n n

46 Derivaion of Poisson Disribuion Reduces o dp0( ) p0( ) d dpn( ) d pn( ) pn 1( ) (n 1) We have an infinie se of linear, firs-order ordinary differenial equaions o solve

47 Derivaion of Poisson Disribuion p0( ) Ce where C = 1 since p 0 (0) =1 For n =1; dp1 ( ) d p ( ) p ( ) The soluion o his equaion is: 1 0 p1( ) e Ce e Using he boundary condiion p n (0) = 0 for all n > 0 yields C=0 Therefore; p1( ) p 2 ( ) e ( ) 2 2 e, p 3 ( ) ( ) 3! 3 e

48 Derivaion of Poisson Disribuion General formula is n ( ) pn( ) e n! which is a well known formula for a Poisson probabiliy disribuion wih mean λ The random variable defined as he number of arrivals o a queueing sysem by ime his random variable has a Poisson disribuion wih a mean of λ arrivals or wih a mean arrival rae of λ.

49 Markovian Propery of Exponenial Disribuion Markov process is characerized by is unique propery of memoryless he fuure saes of he process are independen of is pas hisory and depends solely on is presen sae Poisson processes consiue a special class of Markov processes even occurring paerns follow he Poisson disribuion he iner-arrival imes and service imes follow he exponenial disribuion Exponenial disribuion is he coninuous disribuion ha possesses he unique propery of memoryless-ness

50 Markovian Propery of Exponenial Disribuion Recall he condiional probabiliy law ha P( A B) P( A B) P( B) given he even B, he probabiliy of he even A is equal o he join probabiliy of A and B divided by he probabiliy of he even B Le T be he variable represening he random inerarrival ime beween wo successive arrivals a wo ime poins we have he following probabiliies for he wo muuallyexclusive evens

51 Memorylessness p n ( ) ( n ) n! e No arrival ye for a period of seconds: P( T ) Having an arrival wihin a period of seconds: P( T Prove ha P( T T ) P(0 T ) lef-hand-side represens he probabiliy of having an arrival by waiing seconds longer under he condiion ha no arrival has occurred during he pas waiing period of seconds wih 0 righ-hand-side represens he probabiliy of having an arrival if waiing for anoher seconds The equaion saes ha he probabiliy of having an arrival during he nex seconds is independen of when he las arrival occurred e ) 1 e

52 Memorylessness Prove ha P( T e e ( ) Consider here has been no arrival during he las 10 seconds. Then he probabiliy of having an arrival wihin he nex 2 seconds is independen of how long here has been no arrival so far, namely, P( T 10 2 T 10) P( T 2) Do no misakenly hink ha e T ) 1 e P[( T P( T 10 2 T 10) P( T 12) ) ( T P( T ) P(0 T ) )]

53 Sochasic Process Sochasic process is he mahemaical absracion of an empirical process governed by he probabilisic laws (such as he Poisson process) A Family of Random Variables (RV) X = {X() T} defined over a given probabiliy space, indexed by parameer ha varies over index se T, is called Sochasic Process he se T is he ime range and X() denoes he sae of he process a ime he process is classified as a discree-parameer or coninuous parameer process

54 Sochasic Process If T is counable sequence like T = {0, ±1, ±2,.} or T = {0, 1, 2,.} Sochasic process {X(), T} is said o be a discreeparameer process If T is an inerval for example T = {: - < < + } or T = {: 0 < < + } Sochasic process {X(), T} is called a coninuousparameer process

55 Markov Process A discree-parameer sochasic process {X(), = 0, 1, 2,..} or coninuous-parameer sochasic process {X(), > 0} is a Markov process when he condiional disribuion of X( n ) given he values of X( 1 ), X( 2 ), X( 3 ),., X( n-1 ) depends only on he preceding value X( n-1 ) ; mahemaically; Pr{X( n ) x n X( 1 ) = x 1,., X( n-1 ) = x n-1 } = Pr{X( n ) x n X( n-1 ) = x n-1 } More inuiively, given he presen condiion of he process, he fuure is independen of he pas The process is hus memoryless

56 Markov Process Markov processes are classified according o: The naure of he index se of he process (discree or coninuous parameer) The naure of he sae space of he process (discree or coninuous parameer) Sae Space Index se T (Type of Parameers) Discree Coninuous Discree Discree parameer Sochasic/Markov chain Coninuous parameer Sochasic/Markov chain Coninuous Discree parameer Coninuous Sochasic/ Markov process Coninuous parameer Coninuous Sochasic/ Markov process

57 Quesions?