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Chapter 10 Sample path sensitivity analysis Learning objectives : • • • Understand the mathematics behind discrete event simulation Learn how to derive gradient information during a simulation Able to verify the correctness of sample gradients Textbook : C. Cassandras and S. Lafortune, Introduction to Discrete Event Systems, Springer, 2007 1 Plan • • • • • • • • • Introduction Perturbation analysis by example GSMP - Generalised Semi-Markovian Process Some mathematical basis of discrete event simulation Perturbation analysis of GSMP Correctness of gradient estimators Applications to manufacturing systems Sensitivity estimation revisited Extensions of perturbation analysis 2 Introduction 3 Sensitivity measures Two types of sensitivity measures: • Derivatives : d J(q)/dq • Finite difference : J(q+D) - J(q) where J(q) is the performance funtion of a system at parameter q Sensitivity measures are for • determination of the direction of change to take • optimization of a system design • performing "what-if" analysis 4 Sensitivity measures Difficulties of sensitivity analysis: • Unknown performance function J(q) • Estimation by simulation in most cases Main ideas of perturbation analysis • Prediction of the behavior of perturbed system q+D along the simulation of the system q Main steps of perturbation analysis • Perturbation generation: changes in quantities directly linked to q • Perturbation propagation: changes in other quantities generated by perturbation generation 5 Stochastic approximation Stochastic approximation (gradient based minimization) : x(n+1) = x(n) - a(n)g(n) where • x(n) : value of x at iteration n • a(n) > 0 : stepsize • g(n) : gradient estimation of a performance function f(x) Round-Robin Theorem: Assume that f(x) is continuously differentiable and there exists a single x* such that df(x*)/dx = 0. If E[g(n)] = df(x)/dx and limn a(n) = 0, Sn a(n) = ∞ then x(n) converges with probability 1 to x*. 6 Perturbation analysis by example 7 An inspection machine • • • • • • • Products arrive randomly to an inspection machine Each product requires a random number of tests The inspection of a production requires a setup of time g Each test takes q time units Products are inspected in FIFO order The buffers are of unlimited capacity The system is empty initially • Performance measure = T(q,g), average time at the inspection station. Input Buffer Inspection Station Ouput Buffer Arrival 8 A G/G/1 queueing model Inter-arrival time A Service time X = g + Lq 9 Sample path Buffer level A1 P1 A2 P2 P3 A3 P4 P5 A4 X1 P6 A5 X2 X3 X4 0 P1 P2 P3 P4 Time Definition : A Sample Path is the sequence (event, state, time) of a simulation or experimentation. Example : (Start, 0, 0), (arrival, 1, A1), (arrival, 2, A1+A2), (arrival, 3, A1+A2+A3), (departure, 2, A1+X1), … Property : The sample path contains all information of the simulation. 10 An inspection machine Each simulation run or a sample path is characterized by : • Li : number of tests of product i • Xi = g + Liq : service time of product i • Ai : inter-arrival time between product i and product i-1 • ti : time at the inspection station of product i (also called system time) Attention ti ≠ Xi Input Buffer Inspection Station Ouput Buffer Arrival 11 An inspection machine The best estimator of the average time at the inspection station: N 1 Tˆ q, g t i N i 1 Under some regularity conditions lim Tˆ q, g T q, g , w .p .1 N Goal of perturbation analysis Evaluate ∂T(q, g)/∂ g and ∂T(q, g)/∂ q Input Buffer Inspection Station Ouput Buffer Arrival 12 Event-based system dynamics {e1, e2, ..., en, ...} : sequence of events with en {a, d} tn : occurrence time of en with tn = 0 qn : number of parts in the system at time tn+ with qn = 0 Dn : time in state qn En : set of active events at tn+ ren : remaining time till occurrence of event e at time tn+ 13 Event-based system dynamics en : sequence of events Time to next event :Dn = Min (ren , eEn) tn : occurrence time of en Next event : en+1 = Argmin (ren , eEn) qn : number of parts at time tn+ Dn : time in state qn New state En : set of active events at tn+ qn+1 = qn +1 if en+1 = a ren : remaining time of event e qn+1 = qn -1 if en+1 = d Update event clocks ren+1 = ren - Dn En+1 = En -{en+1} Create new event Case e = d and qn+1 > 0 Case e = a and less than N arrivals AND En+1 = En+1 +{a} Case e = a and qn = 0 ran+1 = An+1 En+1 = En+1 +{d} ran+1 = g+Ln+1q 14 Event-based system dynamics Performance evaluation : Sn = total time of all parts in the system up to event n en : sequence of events tn : occurrence time of en qn : number of parts at time tn+ Dn : time in state qn En : set of active events at tn+ ren : remaining time of event e Sn+1 = Sn + qnDn T(q,g) = SV /N where eV is the departure event of N-th part 15 Event-based system dynamics Simulation algorithm 1/ Initialization : q = 0, t = 0, E = {a}, generate r.v. A, ra = A Na = Nd = S = 0 2/ Next event : e = Argmin (re, eE), D = re Statistics : Na = Na+ 1(e=a), Nd = Nd+ 1(e=d), S = S + qD 3/ State update : q = q + 1(e = a)-1(e= d) 4/ Update event clocks : re = re - D, E = E -{e} 5/ Generate new events If e=a Na < N, E = E +{a}, generate r.v. A, ra = A If (e=a q= 1) or (e=d Nd < N), E = E +{d}, generate r.v. L, rd = g+Lq 16 Perturbation analysis Typical question (What-If) : What would be the average system time if the experimentation were performed with parameter q+Dq? Definitions : • • • • Nominal system : the system with parameter q of the simulation Perturbed system : the system with parameter q+Dq of the What-If question? Nominal sample path : sample path of the nominal system Perturbed sample path : sample path of the perturbed system (preferably under the same random condition as the nominal sample path 17 Perturbation analysis Typical perturbation analysis answer: • Perturbation generation The perturbation of parameter q generates directly a perturbation ∆Xi = Li∆q to each product service time (recall that Xi = g + Liq) • Perturbation propagation The perturbation of the service time of a product will be propagated to other products • Fundamental PA condition : the sequence of events does not change for small enough perturbation Dq 18 Perturbation analysis Nb of products P1 A1 A2 P2 P3 A3 P4 P5 A4 X1 P6 A5 X2 X3 X4 0 P1 DX1 DX2 DX3 P2 P3 P4 Time DX4 19 Event-based perturbation analysis Next event : en+1 =e*= Argmin (ren , eEn) Time to next event : Dn = re*n , Dn /q = re*n /q New state qn+1 Update event clocks ren+1 = ren - Dn , ren+1/q = ren+1 /q - Dn/q En+1 = En -{en+1} Create new event New arrival : En+1 = En+1 +{a}, ran+1 = An+1, ran+1/q = 0 New depart : En+1 = En+1 +{d}, rdn+1 = g + Ln+1q, rdn+1/q = Ln+1 Statistics : Sn+1 = Sn + qnDn, Sn+1 /q = Sn /q + qnDn/q 20 Event-based system dynamics Simulation algorithm 1/ Initialization : q = 0, t = 0, E = {a}, generate r.v. A, ra = A Na = Nd = S = 0, grad_ra = 0 2/ Next event : e = Argmin (re, eE), D = re , grad_D = grad_re Statistics : Na = Na+ 1(e=a), Nd = Nd+ 1(e=d), S = S + qD, grad_S = grad_S + q*grad_D 3/ State update : q = q + 1(e = a)-1(e= d) 4/ Update event clocks : E = E -{e}, re = re - D, grad_re = grad_re - grad_D 5/ Generate new events If e=a Na < N, E = E +{a}, generate A, ra = A, grad_ra = 0 If (e=a q= 1) or (e=d Nd < N), E = E +{d}, generate L, rd = g+Lq, grad_rd = L 6/ If Nd < N, go to 2; otherwise, T = S/N, grad_T = grad_S /N, END 21 Busy period Definition : A busy period (BP) is a period of time during which the inspection station is never idle. For a product i of the 1st busy period (BP1): • arrival date : A1+A2 + ...+Ai • departure time: A1+ X1+X2 + ...+Xi • system time : ti = Xi + (X1+X2 + ...+Xi-1) - (A2 + ...+Ai) Total system time of products of BP1 4 i X j A j i 1 j 1 j 2 i 22 Busy period • Each busy period BPm is initiated by the arrival of a product that finds an empty system • Let km +1 be the product that initiates busy period BPm • System time of the i-th product of BPm i ti X j 1 i km + j Ak m + j j2 • Total system time of products of BPm nm i 1 i i X k m + j Ak m + j j2 j 1 where nm is the number of products inspected in BPm 23 Busy period • Total system time of products of BPm T q , g 1 N M nm m 1 i 1 i i X k m + j Ak m + j j2 j 1 where M is the total number of busy periods observed during the simulation • M is a random variable that depends on the simulation realization. 24 Busy period-based Perturbation analysis • Busy periods remain the same dnder the fundamental condition. • System time in the perturbed system ti q + D q i X j 1 km + j q i + D q Ak m + j t i q + j2 i DX j 1 km + j • As a result, ti q Tˆ q i j 1 1 N X km + j q N L j 1 ti q i 1 i 1 N km + j M nm m 1 i 1 i Lkm + j j 1 25 Busy period-based Perturbation analysis • To summarize, for Xi = g + Liq, Tˆ q Tˆ g 1 N 1 N N ti q , w ith i 1 N ti g , i 1 w ith ti q ti q i Lj j km +1 i 1 j km +1 26 Busy period-based Perturbation analysis Perturbation analysis algorithm during the simulation 0) Initialization : gradXq = 0, gradtq = 0, gradTq = 0 gradXg = 0, gradtg = 0, gradTg = 0 1) During the simulation : If the end of service of product with L tests, 1.1) Perturbation generation : gradXq L and gradXg 1 1.2) Perturbation propagation (system time) gradtq = gradtq + gradXq (Total system time) gradTq = gradTq + gradtq gradtq = gradtg + gradXg gradTg = gradTg + gradtg 1.3) If the system is empty, then gradtq = 0 and gradtg = 0 2) At the end of the simulation, ∂T/∂q = gradTq/N and ∂T/∂q = gradTq/N 27 Validation of the results Unbiasedness E T q, g , N q E T q, g , N g T q , g , N E q ? T q , g , N E g ? Consistency lim lim T q , g , N N q T q , g , N N g ? ? T q , g , q T q , g , g 28 GSMP - Generalised SemiMarkovian Process A general framework for representation of Discrete event systems 29 Basic concepts States : a state is a possible system configuration Ex : number of products to inspect Events : the system state changes only upon the occurrence of events Ex : arrival and end of inspection of a product Transition probabilities : which rule the change of the state when an event occurs Ex : routing probabilities among multiple alternatives Clock : which indicate the remaining time to occurrence of events. • A clock is set according to some random variable when it is activated • It then decrements as long as it is active. • An event occurs when its clock reaches zero. Attention : Remaining processing times of products are associated with clocks of related events and not in the definition of the state 30 GSMP A GSMP is defined by : • S : set of states • E : set of events • E(s) : set of events that are possible at state s • p(s'; s, e) : probability of jumping to state s' when event e occurs at state s • Fe : probability distribution of a new clock of event e Example (Station d'inspection) : • S = {0, 1, …} : number of products waiting for inspection • E = {a, d} with a = product arrival, d = product departure • E(s) = {a, d} if s > 0 and E(0) = {a} • p(s+1; s, a) = 1, p(s-1; s, d) d 1 • Fa = distribution of inter-arrival time, Fd = distribution of service time 31 GSMP Non-Interruption (NI) condition : s, s' Œ S and e Œ E(s), p(s'; s, e) > 0 E(s') E(s) - {e} The NI condition implies that, once activated, an event remain active till its occurrence. 32 Simulation algorithm of an GSMP 1. • • • • Initialization : Number of events : n = 0 Set initial state : S0 Set clocks of activated events : C0(e) = X(e, 1), e Œ E(S0) Set event counter : N(e, 0) = 0, e Œ E 2. Determine the next event : e n +1 arg min C n e e E s 3. Sojourn time at Sn : tn+1 = Cn(en+1) 4. Update the event counters : N e , n + 1, if e e n + 1 N e, n + 1 N e , n , otherw ise 33 Simulation algorithm of an GSMP 5. Update the system state : Sn+1 = F(Sn , en+1 , U(en+1, N(en+1, n+1))) where • F is a function such that P{F(s, e, U) = s'} = p(s'; s, e) • U is a random variable with uniform distribution on [0,1]. 6. Update existing clocks : Cn+1(e) = Cn+1(e) - tn+1, e E(sn) -{en+1} 7. Generate new clocks : Cn+1(e) = X(e, N(e, n) +1), e E(sn+1) - (E(sn) -{en+1}) 8. If the end of simulation, stop. Otherwise, n:= n+1 and go to step 2 34 Simulation algorithm of an GSMP Routing mechanisms • Each active event e is associated a uniform random variable U(e), called routing indicator Step 5 of the GSMP simulation algorithm is decompose into two steps: • 5.1. Generation of the routing indicator : U(en+1,N(en+1, n+1)) • 5.2. Determination of the new state : Sn+1 = F(Sn , en+1 , U(en+1,N(en+1, n+1)) ) where F is such that P{F(s, e, U) = s'} = p(s'; s, e). Example of F : Let S = { s1, s2, … , sm }. Sn+1 := si such that : i1 p s j j 1 ; S n , e n +1 U i p s j j 1 ; S n , e n +1 35 Simulation algorithm of an GSMP Performance measures Let Zt : be the state of the system at time t and f(Zt) : the cost associated with state Zt . The following performance measures are considered • LT T 0 f Z t dt where T is a given constant • LN tN 0 f Z t dt where N is a given integer and tN is occurrence time of N-th event, i.e. tN = t1 + t2 + ... + tN ; • L a,k T a ,k 0 f Z t dt where T(a,k) is the time of k-th occurrence of event a . 36 Some mathematical basis of discrete event simulation 37 Random Number Generation The linear congruential technique Goal: • Generate samples of U[0, 1] distribution The linear congruential technique : Xk+1 = (aXk + c) mod m, k = 0, 1, ... where a = multiplier, c = increment, m = modulus X0 = seed Quality strongly depends on the selection of parameters a, c, m. The "pseudo" random samples have cycle of length at most m. m = 2n , as large as possible with 231 for 32-bit computer, is often used to maximize the cycle and to ease to the modula arithmetic 38 Random variate generation The inverse transform technique Goal: • Generate samples of probability distribution F(x) The inverse tranform technique : • Generate a sample U of distribution U[0, 1] • Transform U into X with X = F-1(U) Online proof Examples : • EXP(l) with F = 1 - exp(-lx) • Weibull with F = 1 - exp (-(l(x-g))b) Drawback: • The computation of the inverse transform can be time consuming. 39 Random variate generation Acceptance-rejection technique Majorizing function: • g(x) such that g(x) ≥ f(x) Density function related to g(x) hx g x g y dy Acceptance-rejection technique : 1. Select a majorizing function g(x) and its related density function h(x) 2. Generate a random variate Y with pdf h(x) 3. Generate a random number U[0,1] independently from Y 4. If U ≤ f(Y)/g(Y), then set X = Y. Otherwise, repeat step 2. Criteria for selection of majorizing function: • Ease of generation of distribution h(x) • Small rejection points, i.e. small surface between g(x) and f(x) 40 Terminating simulation output analysis Point estimation Observations: • X1, X2, ..., Xn : simulation observations of true but unknown performance q Sample mean 1 qˆn n n Xi i 1 Unbiasedness: E qˆn q For iid observations X1, X2, ..., Xn, 2 E qˆn E X i q , V ar qˆn , lim qˆn q n n strong law of large number Sample variance: S 2 n 1 n 1 n i 1 X i qˆn 2 41 Terminating simulation output analysis Interval estimation Central limit theorem Zn qˆn E qˆn V a r qˆn tends to a standard normal distribution as n goes to infinity. (1-a)-percent confidence interval (why): P qˆn z a / 2 2 / n q qˆn + z a / 2 2 / n 1a where • za/2 is a defined on a standard normal distribution F(z). • 2 is evaluated by sample variance Student's t distribution is also used to avoid checking how large n is for normal distribution approximation 42 Nonterminating simulation output analysis Problem setting Observations: • X1, X2, ... : simulation observations of a nonterminating simulation Unknown steady state performance : q lim E X k k Time average: lim n 1 n n Xi i 1 Under ergodicity condition lim n 1 n n Xi q i 1 43 Nonterminating simulation output analysis Replications with deletions I. • Simulation for a time horizon of m+r steps • Ignore data of the first r steps • Estimation with data of the remaining m steps qˆm , r 1 mr m Xi i r +1 II. • Perform K replications of I • Use point estimation methods to evaluate the performance Key questions : • How long should the warmup interval r be? 44 Nonterminating simulation output analysis Batch means • Based on a single but long simulation run • Observed data X1, X2, ... are grouped into Batches Assume batches of equal length m with a warmup period of length r • Batch mean Bj is estimated Bj • 1 m r + jm Xi i r + ( j 1) m + 1 Batch means are then used in point estimation 1 ˆ qn n n B j j 1 45 Nonterminating simulation output analysis Regenerative simulation • • • Assumption: There exists some regenerative state (regeneration point) after which the system is independent of past history Regeneration cycles R1, R2, ... : regeneration times [Rj, Rj+1) : regenerative cycle or period Cycle total (Lj) and cycle length (Nj) of cycle j R j 1 Lj X i , N j R j R j 1 i R j 1 • Steady state mean 1 q E L E N lim n n 1 n n L n j 1 lim n N j 1 L j n j j j 1 n N j j 1 46 Perturbation analysis of GSMP 47 Functional GSMP System paramets : • Assume that the probability distribution of new clocks depend on a continuous parameter q. Let : • Fe(x; q) : distribution of new clocks of event e • Xq(e, n) : n-th clock of event e Hypothesis (H1) : • e and q, Fe(x; q) is continuous in x and Fe(x; q) = 0. Hypothesis (H2) : • e and q, Xq(e, n) is continuous differentiable in q. Remark : • Under H1, the probability that two clocks reach zero at the same time is null 48 Perturbation generation Remark : U = Fe(Xq(e, n); q) est une v.a. uniforme dans [0, 1] The inverse transform technique for generation of event clocks X : 1. Generate a random variable U uniformly distributed on [0, 1] 2. Xq(e, n) = Fe-1(U, q) Property : When the inverse tranform technique is used for generation of event clocks, the following infinitesimal perturbations are generated : dX q e, n q F X q e , n ; q x F X q e , n ; q dq 49 Perturbation generation Special cases : • scale parameters : Fe(x; q) = Ge(x/q) dXq(e, n) / dq = Xq(e, n) / q Ex : a exponentially distributed random variable with mean q • location parameters : Fe(x; q) = Ge(x - q) dXq(e, n) / dq = 1 Ex : X = g + L q where g is a location parameter 50 Perturbation propagation Theorem: Under hypothese NI, H1 and H2, with probability 1, (i) the events en and the states sn remain unchanged within a small enough neighorhood of q; (ii) the sojourn time tn and event clocks Cn(e) are continuously differentiable at q. Further, d t n +1 dq dC n e n + 1 dq dC n e dC n e n + 1 , e E S n en +1 dC n + 1 e d q dq dq dX e , N e , n + 1 , e E S n +1 E S n e n +1 dq 51 Sensitivity of performance measure Theorem : The performance measures LT, LN et La, k are all differentiable in q with dL T dq dL N dq dL a , k dq NT n 1 N n 1 dt n dq dt n dq f S n 1 f S N T n 1 n 1 dt n dq f S n 1 N T a ,k N T dt n dq f S n 1 where N(t) is the number of events occurred in (0, t). What about the average cost? 52 Simulation algorithm and perturbation analysis of an GSMP 1. • • • Initialization : Number of events : n = 0 Set initial state : S0 Set clocks of activated events : C0(e) = X(e, 1), e Œ E(S0) Gradient of C0(e) (perturbation generation): d C n + 1 e d X e ,1 • Set event counter : N(e, 0) = 0, e Œ 2. Determine the next event : e n +1 arg min C n e dq dq e E s 3. Sojourn time at Sn : tn+1 = Cn(en+1) Gradient of tn+1 (perturbation propagation): d t n +1 dq dC n e n + 1 dq 4. Update the event counters : 53 Simulation algorithm and perturbation analysis of an GSMP 5. Update the system state : Sn+1 = F(Sn , en+1 , U(en+1, N(en+1, n+1))) 6. Update existing clocks : Cn+1(e) = Cn+1(e) - tn+1, e E(sn) -{en+1} Gradient of Cn+1(e) (perturbation propagation): dC n + 1 e dq 7. dC n e dq dC n e n + 1 dq Generate new clocks : Cn+1(e) = X(e, N(e, n) +1), e E(sn+1) - (E(sn) -{en+1}) Gradient of Cn+1(e) (perturbation generation): dC n + 1 e dq dX e , N e , n + 1 dq 8. If the end of simulation, stop. Otherwise, n:= n+1 and go to step 2 54 Correctness of gradient estimators 55 Correctness of gradient estimators We only check the unbiasedness, i.e. dE L dq dL E dq , (1) A counter-example 1, if q L 0, if q where w is a uniform random variable on [0, 1] Necessary and sufficient condition for (1) : uniform differentiability of L Uniform differentiability is in general impossible to check directly. 56 Commuting Condition (CU) CU condition: For all s1, s2, s3, a, and b such that a, b E(s1) and p(s2; s1, a) p(s3; s2, b) > 0 , there exist a state s4 such that: p(s4; s1, b) = p(s3; s2, b) and p(s3; s4, a) = p(s2; s1, a). Further, p(z1 ; s, a) = p(z2 ; s, a) > 0 z1 = z2. s2 b a s1 s3 b a s4 Example: G/G/1 (CU), G/G/1/K (no CU) 57 Commuting Condition (CU) Remarks : • CU condition ensures the continuity of the state trajectory. • The perturbation of two events only generates temporal perturbations of the state trajectory. • The last part of the CU condition is not restrictive and we usually limit ourselves to the first part. Example : CU condition holds for the inspection system example. 58 Commuting Condition (CU) Hypothis (H3) : There exists a constant B > 0 such that: dX q e, n B X q e, n + 1 dq Theorem CU : Under (NI), (H1) - (H3) and (CU). • • • If E[supq N(T)2] < ∞ , then GSMP gradient estimator of LT is unbiased, If E[supq tN] < ∞, then GSMP gradient estimator of LN is unbiased, If E[supq N(T(a, k))2] < ∞ and E[supq T(a, k)2] < ∞, then GSMP gradient estimator of La,k is unbiased, where N(t) is the number of event occurred in in [0, t] Remark : • Conditions E[supq N(T)2] < ∞ , E[supq tN] < ∞, E[supq N(T(a, k))2] and E[supq T(a, k)2] < ∞ hold for most models of production systems. • In general, we limit ourselves to the verification of other conditions. 59 Correctness conditions of Perturbation Analysis Conditions on event clocks Hypothesis (H1) : e and q, Fe(x; q) is continuous in x and Fe(x; q) = 0. Hypothesis (H2) : e and q, Xq(e, n) is continuous differentiable in q. Hypothèse (H3) : There exists a constant B > 0 such that: dX q e, n B X q e, n + 1 dq Conditions on the GSMP structure NI condition: s, s' Œ S and e E(s), p(s'; s, e) > 0 E(s') E(s) - {e} CU condition: For all s1, s2, s3, a, and b such that a, b E(s1) and p(s2; s1, a) p(s3; s2, b) > 0 , there exist a state s4 such that: p(s4; s1, b) = p(s3; s2, b) and p(s3; s4, a) = p(s2; s1, a). Further, p(z1 ; s, a) = p(z2 ; s, a) > 0 z1 = z2. 60 Correctness conditions of Perturbation Analysis A GSMP can be represented by its event diagram in which • nodes = states • arcs = possible transitions Event diagrams are particular useful for checking NI and CU conditions. s1 s3 a, P1 b, 1 s a, P2 s2 61 Applications to manufacturing systems 62 A. G/G/1 queue for departure time, throughput rate, average waiting time, ...), G/M/1, M/G/1, M/M/1 B. Optimization of mean WIP of production lines M1 B1 M2 B2 M3 C. Failure-prone continuous flow production lines (wrt failure/repair/production rates) 63 D. KANBAN systems • Each Kanban stage is associated with a given number of kanbans • each product in a kanban stage is associated with a kanban • Performance measure: average product waiting time Kanban stage 2 Kanban stage 1 I1 M1 O1 I2 M2 O2 64 E. A flexible manufacturing system • The system is composed of a set of machines (servers) and a given number of pallets (clients) for material handling • The path of a client through the system is a Markov chain with a probability qij to joining machine j after leaving i • The queues are of unlimited capacity • Performance measures: time for server 1 to serve N clients. 65 F. A flexible cell • A flexible machine produces sucessively for three dedicated machines (round robin rule) • Performance measures : mean WIP B0 M0 B1 M1 B2 M2 B3 M3 66 Sensitivity estimation revisited 67 Problem setting • Performance measure: J(q) = E[L[X1(q), X2(q), ...]] • Focus on scalar case for simpliciy, i.e. J(q) = E[L[X(q)]]. • Assume that X(q) has continuous probability distribution F(x, q). • Then, J q L x q dF x , q 68 The IPA approach • Represent X as a continuous function of q by inverse transformation X(q) = F-1(U, q) where U = UNIF(0, 1) • Then • By differentiation (provided the interchange of expectation and differentiation is allowed) J q dJ q dq • 1 0 L x u , q du 1 L x u , q 0 q du Further, L X U , q q L X U , q X U , q X q perturbation propagation perturbation generation 69 Likelihood Ratio Approach (LR approach) • J q L x f x ; q dx • L(x) is used instead of L[x(q)], and a common realization of X is used while q varies • By differentiation (provided the interchange of expectation and differentiation is allowed) dJ q dq L x f x;q q dx 70 Likelihood Ratio Approach (LR approach) • • dJ q dq Since L x ln f f q x;q q • x;q f dx x;q q 1 f x;q if f(x; q) ≠0 dJ q dq LX ln f L x ln f x ; q X ;q q q ln f X ; q f x ; q dx E L X q Likehood Ratio estimator Score Function estimator 71 IPA or LR • IPA ≠ LR • • IPA needs more detailed system dynamics analysis LR is easier to obtain • • IPA is more accurate which variance goes to zero as time horizon increase LR has large variance which increases to infinity for the time horizon increase 72 Extensions of perturbation analysis 73 When IPA does not work Ex 1 : A production line where the performance measure is the average number of parts inspected per time unit. Ex 2 : Inspection machine with an input buffer of limited capacity State-dependent routing Ex 3 : Jackson network where q is a routing probability Structural parameter Discontinuity of performance measures (Ex 1) Discontinuity of the sample path (Ex 2 and 3) 74 Solution 1 : change of the representation of the stochastic process Ex : Birth-death process Solution 2 : change of measures (case 1) For a production line, from Little's law, L=lT where Lis the average WIP, l the throughput rate, T the mean sojourn time. Then, dl dq dL dq l dT dq T and dL/dq and dT/dq can often be estimated by IPA 75 Solution 3 : Conditional perturbation analysis (SPA) • The biase of an IPA estimator, i.e. E[dL/dq] - dE[L]/dq, is essentially due to the discontinuities of the random performance function L • SPA consists in evaluating the derivative of a conditional performance function, , i.e. • d[E[L /z]]/dq The SPA estimator is unbiased if : dE L / z dE z E L / z Ez dq dq • In general, E[L /z] is better smoothed than L and SPA estimator is more likely to be unbiased. 76