Simulation of Discrete Event Systems

Transcription

1 Siulation of Discrete Event Systes Unit 9 Queueing Models Fall Winter 207/208 Prof. Dr.-Ing. Dipl.-Wirt.-Ing. Sven Tackenberg Benedikt Andrew Latos M.Sc.RWTH Chair and Institute of Industrial Engineering and Ergonoics RWTH Aachen University Bergdriesch Aachen phone: eail: des@iaw.rwth-aachen.de

2 Contents. Introduction and odel specification 2. Perforance and dynaics 3. Siple arkovian queueing systes 9-2

3 Focus of lecture and exercise odel static dynaic tie-varying tie-invariant linear nonlinear continuous states tie-driven discrete states Focus of lecture and exercise event-driven deterinistic stochastic discrete-tie continuous-tie 9-3

4 . Introduction and Model Specification. Introduction and Model Specification 9-4

5 Definition of a queueing syste The ter queueing syste refers to a set of objects or subjects which are waiting for treatent by one or several achines. In queueing theory, a queueing odel is constructed so that queue lengths and waiting tie can be predicted. With a queueing syste you can answer the following questions: Average occupancy of the syste Average length of the queue Average waiting tie of objects or subjects Source:

6 Queueing syste production syste Production process (e.g. gear) consits of several production steps (cut to length, drilling, turning, curing ) Often the production stations are logistically linked and decoupled by a buffer Due to breakdowns or anoalies of the achine, set up ties of achines and personal allowance unforeseen fluctuations of the processing tie can occur In an extree case the fluctuations spread over the whole syste Quelle: The queueing theory offers an analytical approach to design a buffer and to prevent bottlenecks 9-6

7 Functional diagra Specification of queueing odels three aspects. Stochastic paraeters: Arrival and service process A(t) and B(t) 2. Structural paraeters: e.g., the storage capacity of the queue, nuber of servers 3. Specification of the operating policies: e.g., conditions under which arriving objects are accepted, preferential treatent (prioritization) of specific objects Scheatic representation of analytical queueing odels: Arrival of Object or Subject (Event a: arrival) A(t) X(t) B(t) Operator (Server) Object oder Subject dispatched (Event d: departure) Autoaton odel: E { a, d} va va,, va,2,... : interarrival ties X {0,, 2,...} vd vd,, vd,2,... : service ties ( x) { a, d} for all x 0 and (0) { a} x if e a f ( x, e) x if e d and x 0 9-7

8 Arrival and service processes The k th arrival event a k is characterized: The single probability distribution A(t) = P[Y t] describes the interarrival tie sequence, k th object occurs event a k generated P y (t) 0,3 0,2 = 0,4 Arrival events a are considered as a stochastic sequence {Y, Y 2, } where rando variable Y k is the k th interarrival tie between the (k ) th and k th arrival, k =, 2,... The objects occur with an average arrival rate of = / E(Y). E(Y) describes the assued interarrival tie: E Y = λ 0, 0,0 X(t) Length of the queue y Object occurs Object 2 occurs a a 2 y 2 A(t) 0 t a k- k t a2 t 9-8

9 Waiting Queue Characteristics of a waiting queue: Pre-defined capacity of the queue: K =, 2... Miniu capacity: Nuber of operators (processing stations) Waiting tie W(t) is statistically described by the waiting period, starting with the tie of occurrence in the queue and processing by the operator Length of queue X(t) describes the objects which are non processed and the objects which are currently in progress: X(t) = N a (t) N d (t) N a = Nuber of arrivals N d = Nuber of objects which have been fully processed Definition Nr. Interarrival tie: A(t) = P(Y t) Expected (ean) arrival rate: = / E(Y) X(t) Length of queue a a 2 Service tie: B(t) = P(Z t) Expected service rate: = / E(Z) Waiting tie: W(t) = P(W t) 0 t a t a2 t 9-9

10 2. Step in generalization: Structural paraeters Definition Nr. 2 The storage capacity of the queue is denoted by the variable K =, 2,... and specifies, how any objects can be queued at axiu. An infinite capacity is denoted by K =. Definition Nr. 3 The nuber of servers =, 2,... denotes, how any parallel instances of servers are available for processing of the objects in the queue. Exaple:. For the siple odel we have K = and = 2. Multi-processor syste with K = and servers: Server Arrival of object (event a) Queue... Object departure (event d) Server 9-0

11 3. Step in generalization: Operating policies Nuber of custoer classes Scheduling Queueing disciplines Adission policies Single-class syste Multiple-class syste Single-class syste Single-class syste Sae service requireents for all objects Sae service tie distribution for all objects. Multiple-class syste Objects are distinguished according to their service requireents Prioritization of specific objects Server ust decide which object to process next. Order in which the server selects objects to be processed Multiple-class syste Order in which the server selects objects to be processed For finite and infinite storage capacity, it ay be desirable to deny adission to soe arriving objects. Multiple-class syste Higher priority objects ay always be aditted, but lower priority objects ay only be aditted if the queue is epty. 9 -

12 Processing sequence of a waiting queue Processing sequence FIFO (First-In-First-Out) Relative priority Absolute priority SIRO (Selection in Rando Order) LIFO (Last-In-First-Out) RR (Round Robin) Operation only for one fixed interval Anrufbeantworter 9-2

13 Design of queueing systes (Selection) Bus driver Server, Waiting queue Callcenter 2 Server, Waiting queue Superarket checkout 2 Server, 2 Waiting queues Server which can serve several objects/subjects siultaneously, Waiting queue Traffic light syste 9-3

14 A/B//K Notation In queuing theory the notation for a queue is as follows: A/B//K A is the interarrival tie distribution function B is the service tie distribution function M is the nuber of service present, =, 2, K is the storage capacity of the queue, K =, 2, Note: If the storage capacity of the queue is infinite: K = The paraeter is dropped 9-4

15 Distribution function of interarrival tie and service tie Coon notation for distribution A and B Characteristic of distribution function for the interarrival tie A(t) and the service tie B(t) A/B//K G General distribution when nothing is known about the statistics of the process D Deterinistic, where the interarrival/service ties are constant M Markovian, where the interarrival/service ties are exponentially distributed: A( t ) Exp( ) F Z( t ) Exp( ) F Y Z ( t ) e ( t ) e t t respectively with, >0 Specification of probability of a new arrival or service in the period 0 to t 9-5

16 Rando variables with probability distribution Exaple of a (cuulative) distribution function F(t) for the interarrival tie Y Exponential distribution with different arrival rates : F A ( t ) e t ~0,77 [Tie units] Lets assue, that is set to: = 0,25 With a probability of approxiate 77%, a axiu of 6 tie units occur between two arrivals 9-6

17 Exaple of the notation M / M / Subject or object occur (Event a) Subject or object is part of the set which wait for processing (Queue) Service (Server) Subject or object is fully processed (Event d) Characteristics of the syste: Interarrival tie and service tie are described by an exponential distribution One server processes the jobs K =, infinite capacity of the queue All arriving objects/subjects are accepted and will be served The processing sequence is: FIFO (First-In-First-Out) Characteristics of a odified syste: M / M / / 3: Siilar to M / M /, but: K = 3, liited capacity of the queue 9-7

18 Characteristics of M / G / 2 Subject or object occur (Event a) Subject or object is part of the set which wait for processing (Queue) M / G / 2 Service no. (Server) Service no. 2 (Server) Subject or object no. is fully processed (Event d ) Subject or object no. 2 is fully processed (Event d 2 ) Characteristics of the syste: Interarrival tie is described by an exponential distribution Service tie is described by an general distribution Two server process the jobs K =, infinite capacity of the queue All arriving objects/subjects are accepted and will be served The processing sequence is: FIFO (First-In-First-Out) 9-8

19 2. Perforance and Dynaics 2. Perforance and Dynaics 9-9

20 Perforance easures (I) Definition Nr. 4 In order to analyze the queueing syste perforance, the following rando variables are defined: Arrival tie of k th custoer: Interarrival tie Service tie Waiting tie of k th custoer: W k = ax(0, W k- + Z k- Y k ) (fro arrival instant until beginning of service) Y k Z k Syste tie of k th custoer: S k = W k + Z k or S k = D k - A k (fro arrival instant until beginning of service) Departure tie of k th custoer: D k = A k + W k + Z k (fro arrival instant until beginning of service) Initial state: The queue is epty {W 0 = 0, Z 0 = 0, Y 0 = 0, S 0 = 0, D 0 = 0, A 0 = 0} 9-20

21 Perforance easures (II) We also define the rando variables: X(t) Queue length at tie t, X(t) 0,, 2, U(t) Workload at tie t, U(t) 0,, 2, Aount of tie required to epty the syste at t Stochastic behavior of the waiting tie sequence {W k } provides iportant inforation regarding the syste s perforance. Probability distribution function of {W k }, P[W k t] depends on k Steady state: k existence of a stationary distribution, P[W t], independent of k: li k P W k t = P W t A sufficiently large nuber of objects is processed Every new custoer experiences a stochastically identical waiting process: P[W t]. Notation: π n, n = 0,,, to denote the stationary queue length probability: π n = P X = n, n = 0,, 9-2

22 Perforance easures (III) Objective: Designing a queueing syste which let a typical custoer at steady state waits as little as possible. Trade-off proble: Server highly utilized: Tolerate soe potentially long waiting ties Server long idle period: To provide low waiting ties Assue: Steady state can be reached Definition Nr. 5 The characteristic perforance indicators of a queueing syste are: E[W]: Expected (average) waiting tie of object E[S]: Expected syste tie of objects E[X]: Expected queue length Syste utilization : The fraction of tie that the server is busy Throughput : The rate at which objects leave the syste after service 9-22

23 Syste dynaics The dynaics of G/G/ queueing systes is specified by the following recursive state equations: Waiting Tie: W k = ax(0, W k- + Z k- - Y k ) Syste Tie: S k = ax(0, S k- - Y k ) + Z k Departure Tie: D k = ax(a k, D k- ) + Z k Usually, it is assued that the queue is initially epty and W 0 = 0, Z 0 = 0, S 0 = 0 and D 0 = 0 Exaple: Arrival of Object or Subject (Event a: arrival) A(t) X(t) B(t) Operator (Server) Object oder Subject dispatched (Event d: departure) 9-23

24 Dynaic of the syste (I) Arrival of the k th object Case : Syste is epty. Waiting tie, k th custoer: W k = 0 Departure tie, k th custoer: D k- < A k Arrival tie, k th custoer: Service tie, k th custoer: A k Z k = D k - A k Case 2: Syste is not epty. Waiting tie, k th custoer: W k = D k- A k > 0 Departure tie, k th custoer: Service tie, k th custoer: A k Z k = D k A k W k Interarrival tie, k th custoer: Y k = A k A k- Interarrival tie, k th custoer: Y k = A k A k- Waiting tie: W k+ = ax(0, W k + Z k Y k+ ) W k+ = ax(0, 0 + D k A k A k+ + A k ) W k+ = ax(0, 0 + D k -A k+ ) = 0, due to: D k A k+ < 0 according to the initial assuption Service tie, k th custoer: Departure tie, k th custoer: S k = Z k D k = A k + Z k Waiting tie: W k+ = ax(0, W k + Z k Y k+ ) W k+ = ax(0, W k + D k A k W k A k+ + A k ) W k+ = ax(0, D k A k+ ) = D k A k+, due to: D k A k+ > 0 according to the initial assuption Service tie, k th custoer: Departure tie, k th custoer: S k = S k- Y k + Z k D k = D k- + Z k 9-24

25 Variables in a waiting queue Waiting queue Turning Waiting queue Milling Production process under consideration p a S Queue s P e B Feeding I Idle (available for next production step) Start Process End Finished 9-25

26 Scheatic representation of the chronology of events Turning (I) First raw aterial ready for processing (a ) 2 x(t) a t p a S Queue s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

27 Scheatic representation of the chronology of events Turning (II) Start of processing the first raw aterial without waiting (w =0) 2 x(t) a w = 0 t p a S Queue s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

28 Scheatic representation of the chronology of events Turning (III) Second raw aterial is ready for processing (a 2 ) Passed tie between the arrival of the first and the second aterial (y 2 ) 2 x(t) a a 2 y 2 w = 0 t p a S Queue s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

29 Scheatic representation of the chronology of events Turning (IV) a a 2 The first raw aterial was processed and is leaving x(t) y 2 the achine (d ) 2 Service tie of the turning achine and the worker to produce the first raw aterial: z w = 0 t z p a S Queue d s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

30 Scheatic representation of the chronology of events Turning (V) Start of processing the second raw aterial a a 2 x(t) y 2 Waiting tie of the second raw atrial until it can be processed: 2 ( = leaving of aterial ): w 2 w = 0 w 2 t z p a S Queue d s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

31 Scheatic representation of the chronology of events Turning (VI) Third raw aterial is ready for processing (a 3 ) 2 a a 2 a 3 x(t) y 2 y 3 Passed tie between the arrival of the second and the third aterial (y 3 ) w = 0 w 2 t z d p a S Queue s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial 4 9-3

32 Scheatic representation of the chronology of events Turning (VII) The second raw aterial was processed and is leaving the achine (d 2 ) 2 a a 2 a 3 x(t) y 2 y 3 Service tie of the turning achine and the worker to produce the second raw aterial: z 2 w = 0 w 2 t z z 2 p a S Queue d d 2 s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

33 Scheatic representation of the chronology of events Turning (VIII) Start of processing the third raw aterial a a 2 a 3 x(t) y 2 y 3 2 Waiting tie of the second raw atrial until it can be processed: ( = leaving of aterial 2): w 3 w = 0 w 2 w 3 t z z 2 p a S Queue d d 2 s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

34 Scheatic representation of the chronology of events Turning (IX) The third raw aterial was processed and is leaving the achine (d 3 ) 2 a a 2 a 3 x(t) y 2 y 3 Service tie of the turning achine and the worker to produce the third raw aterial: z 3 w = 0 w 2 w 3 t z z 2 z 3 p a S Queue d d 2 d 3 s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

35 Scheatic representation of the chronology of events Turning (X) Fourth raw aterial is ready for processing (a 4 ). 2 a a 2 a 3 a 4 a 5 x(t) y 2 y 3 y 4 y 5 Passed tie between the arrival of the second and the third aterial (y 4 ) w 2 w 3 w = 0 w 4 = 0 t z z 2 z 3 p a S Queue d d 2 d 3 s P e B Feeding I Idle (available for next production step) Start Process End Finished Raw aterial Raw aterial 2 Raw aterial 3 Raw aterial

36 Scheatic representation of the chronology of events Turning (XI) First raw aterial ready for processing Passed tie between the occurrence of the first and the second raw aterial Second raw aterial ready for processing Two parts of the raw aterial are in the syste a a 2 a 3 a 4 a 5 x (t) y 2 y 3 y 4 y 5 2 Start of the waiting tie of the raw aterial in the queue (= 0, due to the fact that the first raw aterial can be iediately processed) Processing tie of the first raw aterial w 2 w 3 w 5 w = 0 w 4 = 0 z z 2 z 3 z 4 z 5 d The first raw aterial is fully processed and leaves the achine ( = Tie of leaving) d 2 d 3 d 4 d 5 Waiting tie of the second raw aterial until the processing is possible (= leaving of part one) t 9-36

37 Little s law Exaple: Arrival of Object or Subject (Event a: arrival) Operator (Server) N a (t): Event score count the nuber of arrivals in the interval (0, t] A(t) X(t) B(t) Object oder Subject dispatched (Event d: departure) N d (t): Event score count the nuber of departures in the interval (0, t] X(t): Queue length X(t) = N a (t) - N d (t) Note Rectangle One unit height = One custoer Length defined by the first arrival and first departure on the tie axis represents the aount of tie spent in the syste Saple path The shaded area consisting of such rectangles up to tie t represents the total aount of tie all custoers have spent in the syste by tie t. We denote this areaby u(t). Arrival: 3 Departure:

38 Little s law u(t) = total aount of tie all custoers have spent in the syste by tie t 0 t Dividing this area by the total nuber of objects n a (t) that have arrived in (0, t], we obtain the average syste tie per object by tie t, denoted by s(t): ҧ Average syste tie per object in (0, t]: Average queue length during (0, t]: Average custoer arrival rate in (0, t]: ut () st () n () a t ut () x t na () () t t t Total aount of tie all custoers Total nuber of onjects arrived until t 9-38

39 Exaple of Little s law (continued) Average syste tie Average queue length Average custoer arrival rate st () ut () n () a t x ut () t na () () t t t x ( t) s( t) We assue the following liits to exist: t, λ(t) and s(t) ҧ both converge to fixed values λ and sҧ respectively li ( t) t li s ( t) s t We are assuing that the arrival, syste tie, and queue length processes are all ergodic In this case, xҧ t is actually the ean queue length E[X] at steady state, and sҧ t is the ean syste tie E[S] at steady state. => x s,equivalently we have E( X ) E( S) in the liit Little s law states that the ean queue length is proportional to the ean syste tie with the ean arrival rate λ being the constant of proportionality. 9-39

40 3. Siple Markovian Queueing Systes 3. Siple Markovian Queueing Systes 9-40

41 Introduction A siple queueing syste ay be viewed as a stochastic tied autoaton Arrival of Object or Subject (Event a: arrival) Event set: E = {a, d} A(t) State space: X = {0,, 2, } Queue length: X(t) at tie t X(t) B(t) Operator (Server) Object oder Subject dispatched (Event d: departure) Generalized Sei-Markov Process (GSMP) Evaluation of its stationary state distribution: Our focus A specific stochastic clock structure is given: Probability distributions characterizing the events a and d, Task: Developing explicit analytical expressions for the stationary state probabilities π n. Due to siulating or directly observing saple paths of the syste: Estiation of the stationary state probabilities π n If the observation period is of length T and the aount of tie spent at state n is T n, then an estiate of π n is the fraction T n /T, provided T is sufficiently long. 9-4

42 Equations We assue that the interarrival and service tie distributions are exponential with paraeters λ and μ. Transition functions: Arrival: f(x, a) = x+, Departure: f(x, d) = x as long as x > 0. birth death Birth death chain 9-42

43 Birth-death chain The Markov chain of the birth-death chain can be generalized towards a continuous-tie process. The birth and death rates in steady state are considered: j E Y j Birth and death processes odeled with exponential probability distributions with distinct rates: j P( Y ) t j t e j P( Z ) t j t e 0n n 0 ( n, 2,...) Key equations pertaining the stationary state probabilities π n, n = 0,,..., 0 0n 0 n n 0 ( n, 2,...) with n n n 0 where λ n and μ n are the 0birth and n death rates respectively when the state is n. n n n 9-43 with j E Z j

44 Steady state analysis of M/M/ queueing systes (I) A single-server syste with infinite storage capacity Exponentially distributed interarrival and service ties State transition diagra for all n 0,,... n for all n, 2,... n 0 n n 0 n 0 0 n n n ( n, 2,...) with The su in the denoinator is a siple geoetric series that converges as long as λ/μ < and therefore 9-44

45 Steady state analysis of M/M/ queueing systes (II) utilization Let us set ρ = λ/μ, which is the traffic intensity defined through: Reeber: [Traffic intensity] = [Average arrival rate] [Average service rate] For a single-server syste: It ust be: due to the assuption: Using: 0 n n 0 n Stationary 0 probability distribution of the queue length of the M/M/ syste 0 n n n ( n, 2,...) with we get: 9-45

46 Steady state analysis of M/M/ queueing systes (III) p a t ~Exp(λ) S Queue s t 2 ~Exp(μ) P e B Feeding I Idle (available for next production step) Start Process End Finished Average waiting tie: Average syste tie: E(W ) ( ) E( S) ( ) Utilization: Throughput: M / M / M / M / 0 Average queue length: E(X ) with: 0 These equations are only valid for steady state: t 9-46

47 Exaple of expected syste tie of a M/M/ queueing systes At steady state Mean queue length E[X] Mean syste tie E[S] Mean arrival rate λ E( S) ( ) Utilization [Average arrival rate] [Average service rate] E(S) As ρ 0, E[S] /μ, represents average service tie. With a ean service tie of 0.5 the utilization is approx. 60 At very low utilizations the only delay of a custoer is the service tie. Tradeoff between high utilization of server and low average syste tie of the custoers / As ρ increases, E[S] is initially insensitive to changes (the slope is close to 0). It then suddenly becoes very sensitive (as the slope rapidly approaches ). A good operating area Utilization M / M / 9-47

48 Steady state analysis of M/M/ systes (I) s P e t 2 ~Exp(μ) a S Queue Start Process End Finish t ~Exp(λ) I Idle (available for next production step) t 3 ~Exp(μ) Feeding s 2 P 2 e 2 Start Process End I 2 Idle (available for next production step) [ ] 9-48

49 Steady state analysis of M/M/ systes (I) Infinite storage capacity identical server Upon an arrival A custoer is served by any available server Upon an arrival All server are busy, the custoer is queued until next departure Interarrival ties are exponentially distributed with rate λ. Service ties at each server are exponentially distributed with rate μ. Note: The effective service rate varies depending on the state of the syste: If n < : Then there are n servers busy and the service rate is nμ If n : The service rate is fixed at its axiu value μ Therefore, we odel the syste as a birth-death chain: for all n 0,,... n n if n n if n 9-49

50 9-50 Steady state analysis of M/M/ systes (II) State transition diagra 0 0 0!!, 2,...,!,,...! n n n n n n n n n The state probabilities of a M/M/ queueing syste in steady state are:

51 9-5 Steady state analysis of M/M/ systes (III) Average waiting tie: Average syste tie: Average queue length: Capacity: Throughput: (in a state of equilibriu) with: 2 0 ) (! ) ( ) ( W E 2 0 ) (! ) ( ) ( S E 2 0 ) (! ) ( ) ( X E M M / / M / / M 0! ) (! ) ( n n n These equations are only valid for steady state: t The perforance indicators of a M/M/ queueing syste in steady state are:

52 Exaple of expected syste tie of a M/M/ queueing syste E(S) = =2... / =0 M / M / Representation of the expected average waiting tie, subject to: the utilization for a constant service rate a variable nuber of server 9-52

53 Steady state analysis of M/M/ systes (I) M/M/ queueing syste is analyzed as a special case of the M/M/ syste. If the queue capacity is infinite, the previous case differentiation concerning the nuber of objects in relation to the nuber of servers is not necessary and we siply have: n n n for all n 0,,... for all n, 2,... State transition diagra 9-53

54 Steady state analysis of M/M/ systes (II) The state probabilities of a M/M/ queueing syste in steady state are equal to the stationary state probabilities of a Poisson process with the point density paraeter (utilization as expected nuber of busy servers): 0 n n e n n! e n n! ( Poisson distribution, n, 2,...) Nuerical distribution exaple for point density rate = 5: ( 5) n n 9-54

55 Steady state analysis of M/M/ systes (III) E(S) Expected ean syste tie depending on: the utilization for a constant service rate an infinite nuber of server M / M / Auslastung Average waiting tie: E( W) 0 Capacity: M / M / e Average syste tie: Average queue length: X represents the nuber of busy servers E( S) E(X) Throughput: with: M / M / Arrival and departure rates ust be balanced 0 e These equations are only valid for steady state: t 9-55

56 References CASSANDRAS, C.,G.; LAFORTUNE, S. (2008): Introduction to Discrete Event Systes. 2 nd edition. Boston (MA): Springer Science+Business Media. PAPOULIS, A.; PILLAI, S.U. (2002): Probability, Rando Variables and Stochastic Processes. Forth Edition. Boston (MA): Mc Graw Hill. 9-56

57 Questions? Open Questions??? 9-57