Modeling and Performance Analysis with Discrete-Event Simulation

Transcription

1 Simulatio Modelig ad Performae Aalysis with Disrete-Evet Simulatio

2 Chapter 8 Queueig Models

3 Cotets Charateristis of Queueig Systems Queueig Notatio Kedall Notatio Log-ru Measures of Performae of Queueig Systems Steady-state Behavior of Ifiite-Populatio Markovia Models Steady-state Behavior of Fiite-Populatio Models Networks of Queues Chapter 8. Queueig Models 3

4 Purpose Simulatio is ofte used i the aalysis of queueig models. A simple but typial queueig model Callig populatio Waitig lie Server Queueig models provide the aalyst with a powerful tool for desigig ad evaluatig the performae of queueig systems. Typial measures of system performae Server utilizatio, legth of waitig lies, ad delays of ustomers For relatively simple systems, ompute mathematially For realisti models of omplex systems, simulatio is usually required Chapter 8. Queueig Models 4

5 Outlie Disuss some well-kow models Not developmet of queueig theory, for this see other lass! We will deal with Geeral harateristis of queues Meaigs ad relatioships of importat performae measures Estimatio of mea measures of performae Effet of varyig iput parameters Mathematial solutios of some basi queueig models Chapter 8. Queueig Models 5

6 Charateristis of Queueig Systems Chapter 8. Queueig Models 6

7 Charateristis of Queueig Systems Key elemets of queueig systems Customer: refers to aythig that arrives at a faility ad requires servie, e.g., people, mahies, truks, s. Server: refers to ay resoure that t provides the requested servie, e.g., repairpersos, retrieval mahies, ruways at airport. System Customers Server Reeptio desk People Reeptioist Hospital Patiets Nurses Airport Airplaes Ruway Produtio lie Cases Case-paker Road etwork Cars Traffi light Groery Shoppers Chekout statio Computer Jobs CPU, disk, CD Network Pakets Router Chapter 8. Queueig Models 7

8 Callig Populatio Callig populatio: the populatio of potetial ustomers, may be assumed to be fiite or ifiite. Fiite populatio model: if arrival rate depeds o the umber of ustomers beig served ad waitig, e.g., model of oe orporate jet, if it is beig repaired, the repair arrival rate beomes zero. - Ifiite populatio model: if arrival rate is ot affeted by the umber of ustomers beig served ad waitig, e.g., systems with large populatio of potetial ustomers. Chapter 8. Queueig Models 8

9 System Capaity System Capaity: a limit o the umber of ustomers that may be i the waitig lie or system. Limited apaity, e.g., a automati ar wash oly has room for ars to wait i lie to eter the mehaism. Waitig lie Server Ulimited apaity, e.g., oert tiket sales with o limit o the umber of people allowed to wait to purhase tikets. Waitig lie Server Chapter 8. Queueig Models 9

10 Arrival Proess For ifiite-populatio models: I terms of iterarrival times of suessive ustomers. Radom arrivals: iterarrival times usually haraterized by a probability distributio. - Most importat model: Poisso arrival proess (with rate ), where A represets the iterarrival time betwee ustomer - ad ustomer, ad is expoetially distributed (with mea /). Sheduled arrivals: iterarrival times a be ostat or ostat plus or mius a small radom amout to represet early or late arrivals. - Example: patiets to a physiia or sheduled airlie flight arrivals to a airport At least oe ustomer is assumed to always be preset, so the server is ever idle, e.g., suffiiet raw material for a mahie. Chapter 8. Queueig Models

11 Arrival Proess For fiite-populatio models: Customer is pedig whe the ustomer is outside the queueig system, e.g., mahie-repair problem: a mahie is pedig whe it is operatig, it beomes ot pedig the istat it demads servie from the repairma. Rutime of a ustomer is the legth of time from departure from the queueig system util that ustomer s s ext arrival to the queue, e.g., mahie-repair problem, mahies are ustomers ad a rutime is time to failure (TTF). Let A (i), A (i), be the suessive rutimes of ustomer i, ad S (i), S (i) be the orrespodig suessive system times: Chapter 8. Queueig Models

12 Queue Behavior ad Queue Disiplie Queue behavior: the atios of ustomers while i a queue waitig for servie to begi, for example: Balk: leave whe they see that the lie is too log Reege: leave after beig i the lie whe its movig too slowly Jokey: move from oe lie to a shorter lie Queue disiplie: the logial orderig of ustomers i a queue that determies whih ustomer is hose for servie whe a server beomes free, for example: First-i-first-out (FIFO) Last-i-first-out (LIFO) Servie i radom order (SIRO) Shortest proessig time first (SPT) Servie aordig to priority (PR) Chapter 8. Queueig Models

13 Servie Times ad Servie Mehaism Servie times of suessive arrivals are deoted by S,S,S 3. May be ostat or radom. {S, S, S 3, } is usually haraterized as a sequee of idepedet ad idetially distributed radom variables, e.g., expoetial, Weibull, gamma, logormal, ad truated ormal distributio. A queueig system osists of a umber of servie eters ad iteroeted queues. Eah servie eter osists of some umber of servers,, workig i parallel, upo gettig to the head of the lie, a ustomer takes the st available server. Chapter 8. Queueig Models 3

14 Servie Times ad Servie Mehaism Example: osider a disout warehouse where ustomers may: Serve themselves before payig at the ashier Chapter 8. Queueig Models 4

15 Servie Times ad Servie Mehaism Wait for oe of the three lerks: Bath servie (a server servig several ustomers simultaeously), or ustomer requires several servers simultaeously. Chapter 8. Queueig Models 5

16 Servie Times ad Servie Mehaism Chapter 8. Queueig Models 6

17 Example Cady produtio lie Three mahies separated by buffers Buffers have apaity of adies Assumptio:Allways suffiiet supply of raw material. Chapter 8. Queueig Models 7

18 Queueig Notatio Kedall Notatio Chapter 8. Queueig Models 8

19 Queueig Notatio Kedall Notatio A otatio system for parallel server queues: A/B//N/K A represets the iterarrival-time distributio B represets the servie-time distributio represets the umber of parallel servers N represets the system apaity K represets the size of the allig populatio N, K are usually dropped, if they are ifiity Commo symbols for A ad B M Markov, expoetial distributio D Costat, determiisti E k El Erlag distributio tib ti of order k H Hyperexpoetial distributio G Geeral, arbitrary Examples M/M// / same as M/M/: Sigle-server with ulimited apaity ad allpopulatio. Iterarrival ad servie times are expoetially distributed G/G//5/5: Sigle-server with apaity 5 ad all-populatio 5. Chapter 8. Queueig Models 9

20 Queueig Notatio Geeral performae measures of queueig systems: P steady-state probability of havig ustomers i system P (t) probability of ustomers i system at time t arrival rate e effetive arrival rate servie rate of oe server ρ server utilizatio A iterarrival i time betwee ustomers - ad S servie time of the -th arrivig ustomer W total time spet i system by the -th arrivig ustomer W Q total time spet i the waitig lie by ustomer L(t) the umber of ustomers i system at time t L Q (t) the umber of ustomers i queue at time t L log-ru time-average umber of ustomers i system L Q log-ru time-average umber of ustomers i queue w log-ru average time spet i system per ustomer w Q log-ru average time spet i queue per ustomer w Q Chapter 8. Queueig Models

21 Log-ru Measures of Performae of Queueig Systems Chapter 8. Queueig Models

22 Log-ru Measures of Performae of Queueig Systems Primary log-ru measures of performae are L log-ru time-average umber of ustomers i system L Q log-ru time-average umber of ustomers i queue w log-ru average time spet i system per ustomer w Q log-ru average time spet i queue per ustomer ρ server utilizatio Other measures of iterest are Log-ru proportio of ustomers who are delayed loger tha t time uits Log-ru proportio of ustomers tured away beause of apaity ostraits Log-ru proportio of time the waitig lie otais more tha k ustomers Chapter 8. Queueig Models

23 Log-ru Measures of Performae of Queueig Systems Goal of this setio Major measures of performae for a geeral G/G//N/K queueig system How these measures a be estimated from simulatio rus Two types of estimators Sample average Time-itegrated sample average Chapter 8. Queueig Models 3

24 Time-Average Number i System L Cosider a queueig system over a period of time T Let T i deote the total time durig [,T ] i whih the system otaied exatly i ustomers, the time-weighted-average umber i a system is defied by: Lˆ T iti i i Ti i T Cosider the total area uder the futio is L(t), the, Lˆ iti T T i T L( t) dt The log-ru time-average umber of ustomers i system, with probability : Lˆ T L ( t ) dt L T T Chapter 8. Queueig Models 4

25 Time-Average Number i System L Number of ustomers i the system Time Chapter 8. Queueig Models 5

26 Time-Average Number i System L The time-weighted-average umber i queue is: Lˆ Q T Q iti LQ ( t) dt T T i T L G/G//N/K example: osider the results from the queueig system (N 4, K 3). L ˆ [(3) + () + (4) + 3()]/ 3/.55 ustomers L Q Q, if L(t) ( t) L ( t ), if L(t) ˆ (5) + (4) + () L Q.3 ustomers Chapter 8. Queueig Models 6

27 Average Time Spet i System Per Customer w Computer Siee, Iformatik 4 The average time spet i system per ustomer, alled the average system time, is: wˆ N N W i where W, W,, W N are the idividual times that eah of the N ustomers sped i the system durig [,T]. For stable systems: w ˆ w as N If the system uder osideratio is the queue aloe: i wˆ Q N N i W Q i w N Q Chapter 8. Queueig Models 7

28 Average Time Spet i System Per Customer w Computer Siee, Iformatik 4 G/G//N/K example (ot.): The average system time is W... (8 3)... ( 6) ˆ + W + + W w 5 5 The average queuig time is 4.6 time uits ˆ. time uits 5 w Q Chapter 8. Queueig Models 8

29 The Coservatio Equatio Little s Law Coservatio equatio (a.k.a. Little s law) Average # i system Lˆ ˆ wˆ Average System time Arrival rate L w as T ad N Holds for almost all queueig systems or subsystems (regardless of the umber of servers, the queue disiplie, or other speial irumstaes). G/G//N/K example (ot.): O average, oe arrival every 4 time uits ad eah arrival speds 4.6 time uits i the system. Hee, at a arbitrary poit i time, there are (/4)(4.6).5 ustomers preset o average. Chapter 8. Queueig Models 9

30 Server Utilizatio Defiitio: the proportio of time that a server is busy. Observed server utilizatio, ρˆ, is defied over a speified time iterval [,T ]. Log-ru server utilizatio is ρ. ρ For systems with log-ru stability: ˆ ρ ρ as T Chapter 8. Queueig Models 3

31 Server Utilizatio For G/G// / queues: Ay sigle-server queueig system with average arrival rate ustomers per time uit, where average servie time E(S)/ time uits, ifiite it queue apaity ad allig populatio. Coservatio equatio, L w, a be applied. For a stable system, the average arrival rate to the server, s, must be idetial to. The average umber of ustomers i the server is: Lˆ s T ( ) T T L( t) LQ ( t) dt T T Chapter 8. Queueig Models 3

32 Server Utilizatio I geeral, for a sigle-server server queue: Lˆ s ad ˆ ρ L ρ T ρ E( s) s - For a sigle-server stable queue: ρ < - For a ustable queue ( > ), log-ru server utilizatio is. Chapter 8. Queueig Models 3

33 Server Utilizatio For G/G// / queues: A system with idetial servers i parallel. If a arrivig ustomer fids more tha oe server idle, the ustomer hooses a server without favorig ay partiular server. For systems i statistial equilibrium, the average umber of busy servers, L s,is: Clearly L S L S E(S) The log-ru average server utilizatio is: ρ L s, where < for stablesystems Chapter 8. Queueig Models 33

34 Server Utilizatio ad System Performae System performae varies widely for a give utilizatio ρ. For example, a D/D/ queue where E(A) / ad E(S) /, where: L ρ /, w E(S) /, L Q W Q - By varyig ad, server utilizatio a assume ay value betwee ad. - Yet there is ever ay lie. I geeral, variability of iterarrival ad servie times auses lies to flutuate i legth. Chapter 8. Queueig Models 34

35 Server Utilizatio ad System Performae Example: A physiia who Cosider the system is simulated shedules patiets every with servie times: S 9, S, miutes ad speds S i miutes S 3 9, S 4 9, S 5 9,. with the i-th patiet: The system beomes: S i 9 miutes with probability.9 miutes with probability. Arrivals are determiisti, A A -. Servies are stohasti ti - E(S i ) 9.3 mi - V(S ).8 mi - σ.9 mi O average, the physiia's utilizatio ρ /.93 <. The ourree of a relatively log servie time (S ) auses a waitig lie to form temporarily. Chapter 8. Queueig Models 35

36 Costs i Queueig Problems Costs a be assoiated with various aspets of the waitig lie or servers: System iurs a ost for eah ustomer i the queue, say at a rate of $ per hour per ustomer. W Q jq is the time - The average ost per ustomer is: N j $ W N Q j $ wˆ Q ustomer j speds i queue - If ˆ ustomers per hour arrive (o average), the average ost per hour is: $ ˆ ˆ ustomer wq hour ustomer $ ˆ wˆ Q $ Lˆ hour Server may also impose osts o the system, if a group of parallel servers ( ) have utilizatio r, eah server imposes a ost of $5 per hour while busy. - The total server ost is: $5 ρ Q Chapter 8. Queueig Models 36

37 Steady-state Behavior of Ifiite-Populatio Markovia Models Chapter 8. Queueig Models 37

38 Steady-State Behavior of Markovia Models Markovia models: Expoetial-distributed arrival proess (mea arrival rate /). Servie times may be expoetially (M) or arbitrary (G) distributed. Queue disiplie i is FIFO. A queueig system is i statistial equilibrium if the probability that the system is i a give state is ot time depedet: P ( L( t) ) P ( t) P Mathematial models i this hapter a be used to obtai approximate results eve whe the model assumptios do ot stritly hold, as a rough guide. Simulatio a be used for more refied aalysis, more faithful represetatio for omplex systems. Chapter 8. Queueig Models 38

39 Steady-State Behavior of Markovia Models Properties of proesses with statistial equilibrium The state of statistial equilibrium is reahed from ay startig state. The proess remai i statistial equilibrium oe it has reahed it. Chapter 8. Queueig Models 39

40 Steady-State Behavior of Markovia Models For the simple model studied i this hapter, the steady-state state parameter, L, the time-average umber of ustomers i the system is: L P Apply Little s equatio, L w, to the whole system ad to the queue aloe: w L, w Q w, L Q w G/G// / example: to have a statistial equilibrium, a eessary ad suffiiet i oditio is: ρ < Q Chapter 8. Queueig Models 4

41 M/G/ Queues Sigle-server ge e queues with Poisso arrivals a ad ulimited apaity. Suppose servie times have mea / ad variae σ ad ρ / <, the steady-state parameters of M/G/ queue: ρ P ρ L ρ + ρ ( + σ ) ( ρ) (/ + σ ) w + ( ρ) L Q w Q ρ ( + σ ) ( ρ) (/ + σ ) ( ρ) The partiular distributio is ot kow! Chapter 8. Queueig Models 4

42 M/G/ Queues There are o simple expressios for the steady-state state probabilities P, P, P, L L Q ρ is the time-average umber of ustomers beig served. Average legth of queue, L Q, a be rewritte as: L Q ρ ( ρ) + σ ( ρ) If ad are held ostat, L Q depeds o the variability, σ, of the servie times. Chapter 8. Queueig Models 4

43 M/G/ Queues Example: Two workers ompetig for a job, Able laims to be faster tha Baker o average, but Baker laims to be more osistet, Poisso arrivals at rate per hour (/3 per miute). Able: / 4 miutes ad σ 4 miutes : L Q (/ 3) [4 + 4] ( 4 / 5).7ustomers - The proportio of arrivals who fid Able idle ad thus experiee o delay is P -ρ /5 %. Baker: / 5 miutes ad σ 4 miutes : L Q (/ 3) [5 + 4] ( 5/ 6).97 ustomers - The proportio of arrivals who fid Baker idle ad thus experiee o delay is P -ρ /6 6.7%. Although workig faster o average, Able s greater servie variability results i a average queue legth about 3% greater tha Baker s. Chapter 8. Queueig Models 43

44 M/M/ Queues Suppose the servie times i a M/G/ queue are expoetially distributed with mea /, the the variae is σ /. M/M/ queue is a useful approximate model whe servie times have stadard d deviatio approximately equal to their meas. The steady-state parameters ρ ( ρ) ρ ρ L ρ w ( ρ ) P L Q w Q ( ) ρ ρ ρ ( ) ( ρ) P ρ Chapter 8. Queueig Models 44

45 M/M/ Queues M/M/ Queues Sigle-hair uisex hair-stylig shop Sigle hair uisex hair stylig shop Iterarrival ad servie times are expoetially distributed ustomers/hour ad µ 3 ustomers/hour 3 ρ Customers 3 L 3 P P ρ h hour w L P P Customers 3 4 ) 3(3 4 ) ( hour 3 3 Q Q L w w P P Customers ) 3(3 ) ( + + Q Q L L Chapter 8. Queueig Models 45

46 M/M/ Queues Example: M/M/ queue with servie rate ustomers per hour. Cosider how L ad w irease as arrival rate,, ireases from 5 to 8.64 by iremets of % ρ L w L w If /, waitig lies ted to 6 otiually grow i legth 4 Irease i average system 8 time (w) ad average umber 6 i system (L) is highly 4 oliear as a futio of ρ. Num mber of Customers rho Chapter 8. Queueig Models 46

47 Effet of Utilizatio ad Servie Variability For almost all queues, if lies are too log, they a be redued by dereasig server utilizatio (ρ) or by dereasig the servie time variability (σ ). A measure of the variability of a distributio, ib ti oeffiiet of variatio (v): ( v ) V (X X ) [ E( X )] The larger v is, the more variable is the distributio relative to its expeted value For expoetial servie times with rate µ - E(X) /µ - V(X) ( ) /µ µ v Chapter 8. Queueig Models 47

48 Effet of Utilizatio ad Servie Variability Cosider L Q for ay M/G/ queue: L Q ρ ( + σ ) ( ρ) ρ + ( v) ρ L Q for M/M/ queue Correts the M/M/ formula to aout for a o-expoetial servie time dist Chapter 8. Queueig Models 48

49 Multiserver Queue M/M/ M/M// / queue: servers operatig i parallel Arrival proess is poisso with rate Eah server has a idepedet ad idetial expoetial servie-time distributio, with mea /. To ahieve statistial equilibrium, the offered load (/) must satisfy / <, where /(m) ρ is the server utilizatio. Callig populatio Waitig lie Chapter 8. Queueig Models 49

50 Multiserver Queue M/M/ Multiserver Queue M/M/ The steady-state parameters for M/M/ The steady state parameters for M/M/ ρ P +!! ) / ( ( ) ( ) ρ ρ ρ ρ L P P P L P + ) ( ) ( )!( ) ( ) ( Probability that ll ( ) ρ ρ ρ ρ ρ ρ L w L P P L + + ) ( )!)( ( ) ( all servers are busy ( ) ρ ρ L P L Q ) ( Chapter 8. Queueig Models 5 ρ L L Q

51 Multiserver Queue M/M/ Probability of empty system Number of ustomers i system Chapter 8. Queueig Models 5

52 Multiserver Queue Commo Models Other ommo multiserver queueig models L Q ρ + ( v ) ρ L Q for M/M/ queue Correts the M/M/ formula M/G// : geeral servie times ad parallel server. The parameters a be approximated from those of the M/M// / model. M/G/ : geeral servie times ad ifiite umber of servers. M/M//N/ : servie times are expoetially distributed at rate ad servers where the total system apaity is N ustomer. Whe a arrival ours ad the system is full, that arrival is tured away. Chapter 8. Queueig Models 5

53 Multiserver Queue M/G/ M/G/ : geeral servie times ad ifiite umber of servers - ustomer is its ow server - servie apaity far exeeds servie demad - whe we wat to kow how may servers are required so that ustomers are rarely delayed ( ) P e,,, K! P w e w Q L L Q Chapter 8. Queueig Models 53

54 Multiserver Queue M/G/ How may users a be logged i simultaeously i a omputer system Customers log o with rate 5 per hour Stay oeted i average for /µ µ 8 miutes 3 hours For plaig purposes it is preteded that the simultaeous logged i users is ifiite Expeted umber of simultaeous users L L To esure providig adequate apaity 95% of the time, the umber of parallel users has to be restrited P( L( ) ) P e 5 (5)!.95 The apaity 564 simultaeous users satisfies this requiremet Chapter 8. Queueig Models 54

55 Multiserver Queue with Limited Capaity Multiserver Queue with Limited Capaity M/M//N/ : servie times are expoetially distributed at rate ad p y servers where the total system apaity is N ustomer Whe a arrival ours ad the system is full, that arrival is tured away Effetive arrival rate is defied as the mea umber of arrivals per Effetive arrival rate e is defied as the mea umber of arrivals per time uit who eter ad remai i the system a a P N ρ + + a P a P N N N ρ +!!! ρ ( ) P N P a L N e N N Q ρ ρ ρ ρ ρ ) ( ) ( ) ( )!( w w L w e Q Q + ( - P N ) probability that a ustomer will fid a spae ad be able to Chapter 8. Queueig Models 55 w L w w e Q + spae ad be able to eter the system

56 Multiserver Queue with Limited Capaity Sigle-hair uisex hair-stylig shop (agai!) Spae oly for 3 ustomers: oe i servie ad two waitig First omputer P P P(system is full) P N ( ) 3 P P! Average of the queue Queue time w L Q Q e System time, time i shop w w Q Expeted umber of ustomers i shop L Q.43 Effetive arrival rate e L ew.5 65 Probability of busy shop e P.585 Chapter 8. Queueig Models 56

57 Steady-state Behavior of Fiite-Populatio Models Chapter 8. Queueig Models 57

58 Steady-State Behavior of Fiite-Populatio Models Computer Siee, Iformatik 4 I pratial problems allig populatio is fiite Whe the allig populatio is small, the presee of oe or more ustomers i the system has a strog effet o the distributio of future arrivals. Cosider a fiite-allig populatio model with K ustomers (M/M//K/K) The time betwee the ed of oe servie visit ad the ext all for servie is expoetially distributed with mea /. Servie times are also expoetially distributed with mea /µ. parallel servers ad system apaity is K. K Customers Mea rutime / Waitig lie with apaity K Chapter 8. Queueig Models 58

59 Steady-State Behavior of Fiite-Populatio Models Steady-State Behavior of Fiite-Populatio Models Some of the steady-state probabilities of M/M//K/K : So e o t e steady state p obab t es o / // / K K K P K +! )! (! P K P K,,...,,! )! ( K K K P +,...,,! )! (! ρ L w P L e e K, /, K e P K ) ( to queue (or eterig/exitig servie) ustomers rate of the log ru effetive arrival is where e Chapter 8. Queueig Models 59 e ) (

60 Steady-State Behavior of Fiite-Populatio Models Computer Siee, Iformatik 4 Example: two workers who are resposible for millig mahies. Mahies ru o the average for miutes, the require a average 5-miute servie period, both times expoetially distributed: / ad /5. All of the performae measures deped o P : P! + ( )!! The, we a obtai the other P, ad a ompute the expeted umber of mahies i system: The average umber of ruig mahies: L P 3.7 mahies K L mahies Chapter 8. Queueig Models 6

61 Networks of Queues Chapter 8. Queueig Models 6

62 Networks of Queues May systems are aturally modeled as etworks of sigle queues Customers departig from oe queue may be routed to aother The followig results assume a stable system with ifiite allig populatio ad o limit o system apaity: Provided that o ustomers are reated or destroyed i the queue, the the departure rate out of a queue is the same as the arrival rate ito the queue, over the log ru. If ustomers arrive to queue i at rate i, ad a fratio p ij of them are routed to queue j upo departure, the the arrival rate from queue i to queue j is i p ij over the log ru. Chapter 8. Queueig Models 6

63 Networks of Queues The overall arrival a rate ito queue j: Arrival rate from outside the etwork j a j + i p alli ij Sum of arrival rates from other queues i etwork If queue j has j < parallel servers, eah workig at rate j, the the log-ru utilizatio of eah server is: (where ρ j < for stable queue). ρ j j j j If arrivals from outside the etwork form a Poisso proess with rate a j for eah queue j, ad if there are j idetial servers deliverig expoetially distributed servie times with mea / j, the, i steady state, queue j behaves likes a M/M/ j queue with arrival rate j a j + i pij all i Chapter 8. Queueig Models 63

64 Network of Queues Customer 8 Cust. Populatio hour.4 Disout store example: Suppose ustomers arrive at the rate 8 per hour ad 4% hoose self-servie. servie. Hee: - Arrival rate to servie eter is 8(.4) 3 per hour - Arrival rate to servie eter is 8(.6) 48 per hour. 3 lerks ad ustomers per hour. The log-ru utilizatio of the lerks is: ρ 48/(3*).8 All ustomers must see the ashier at servie eter 3, the overall rate to servie eter 3 is per hour. - If 3 9 per hour, the the utilizatio of the ashier is: ρ 3 8/ Chapter 8. Queueig Models 64

65 Summary Itrodued basi oepts ep of queueig models. Showed how simulatio, ad some times mathematial aalysis, a be used to estimate the performae measures of a system. Commoly used performae measures: L, L Q, w, w Q, ρ, ρ ad e. Whe simulatig ay system that evolves over time, aalyst must deide whether to study trasiet behavior or steady-state behavior. Simple formulas exist for the steady-state t t behavior of some queues. Simple models a be solved mathematially, ad a be useful i providig a rough estimate of a performae measure. Chapter 8. Queueig Models 65