Modeling Service Networks with Time-Varying Demand

Transcription

1 Modeling Service with Time-Varying Demand - Performance Approximations and Staffing Controls (with Beixiang He, Liam Huang, Korhan Aras, Ward Whitt) Department of Industrial and Systems Engineering NC State University A survey on recent results of time-varying queueing models

2 Education B.E. in Electrical Engineering, Tsinghua Univ., Beijing, 2002 M.S. & Ph.D. in Oper. Res., Columbia Univ., NY, 2007, 2011 Research Interests Method: stochastic modeling, applied prob., queueing theory Application: call centers, health care, manufacturing systems When I m not teaching or writing papers...

3 Current Ph.D. students at NCSU Beixiang He (Graduating in August 2014) Korhan Aras (Graduating in Spring 2015) Liam Huang, Yao Yu (Ongoing) Former Ph.D. advisor Prof. Ward Whitt (Columbia University)

4 Nonstationary Systems: Time-Varying Arrivals service call center emergency room Green et al. (2007) Yom-Tov and Mandelbaum (2011)

5 Realistic Models Features: General Distributions Non-exponential service and abandonment service service and abandonment abandonment Brown et al. (2005)

6 The Base Queueing Model G t /GI/s t + GI Time-varying arrival rate λ(t) (the G t ) (e.g., non-homogeneous Poisson, M t ) I.I.D. service times G(x) P(S x) (the first GI) Time-varying staffing level s(t) (the s t ) I.I.D. abandonment times F (x) P(A x) (the +GI) First-Come First-Served (FCFS) Unlimited waiting capacity

7 The Base Queueing Model G t /GI/s t + GI Time-varying arrival rate λ(t) (the G t ) (e.g., non-homogeneous Poisson, M t ) I.I.D. service times G(x) P(S x) (the first GI) Time-varying staffing level s(t) (the s t ) I.I.D. abandonment times F (x) P(A x) (the +GI) First-Come First-Served (FCFS) Unlimited waiting capacity Textbook models: M/M/1 No longer useful!!

8 The Base Queueing Model Performance measures Q(t) and B(t): number waiting in queue and in service at t X (t) Q(t) + B(t): total number in system W (t) and V (t): head-of-line and potential waiting time at t Exact analysis HARD! Time-varying arrival rate and staffing function Customer abandonment Large scale (for large λ(t) and s(t)) Non-Markovian probability structure: non-poisson, non-exponential Complicated network structure

9 Many-Server Heavy-Traffic Approximations A sequence of queues indexed by n arrival rate: λ n (t) = n λ(t) number of servers: s n (t) = n s(t) system size grows with n customer individual behavior unscaled: service cdf G and patience cdf F held fixed independent of n

10 Many-Server Heavy-Traffic Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n

11 Many-Server Heavy-Traffic Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n

12 Many-Server Heavy-Traffic Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n Approximations Q n (t) = n Q(t) + n ˆQ(t) + o( ) n) d N (n Q(t), n σ 2ˆQ (t) ( ) W n (t) = W (t) + Ŵ (t) n + o( 1 n ) d N W (t), 1n σ2ŵ (t)

13 MSHT Fluid Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n Approximations Q n (t) = n Q(t) + n ˆQ(t) + o( ) n) d N (n Q(t), n σ 2ˆQ (t) ( ) W n (t) = W (t) + Ŵ (t) n + o( 1 n ) d N W (t), 1n σ2ŵ (t)

14 What Are s?

15 MSHT Fluid Limit (FWLLN) random deterministic; discrete continuous; finite infinitely divisible.

16 Realistic Models Features: Network Structure Markovian routing Once finishing at one station, flip a coin to decide next station Pros: capture randomness in routing; tractable due to Markov property Cons: completely ignore service history; too crude to approximate real systems Prescribed paths Each customer follows a prescribed itineraries/paths Customers with same itineraries belong to same class Pros: characterizes different classes, capture service history Cons: difficult to analyze (no Markov property)

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18 with

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20 : A Dynamical System Key functions: fluid densities Q(t, y) : quantity of fluid in queue for up to y at t Q(t) : y q(t, x)dx 0 B(t, y) : quantity of fluid in service for up to y at t B(t) : y b(t, x)dx 0

21 : A Dynamical System Key functions: fluid densities Q(t, y) : quantity of fluid in queue for up to y at t Q(t) : y q(t, x)dx 0 B(t, y) : quantity of fluid in service for up to y at t B(t) : y b(t, x)dx 0 Fluid densities: transport PDE q(t,x) t + q(t,x) b(t,x) t x = h F (x) q(t, x), 0 x w(t), t 0. + b(t,x) t = h G (x) b(t, x), w(t)x 0, t 0. Fluid waiting time: ODE head-of-line (HOL) and potential waiting times at t w (t) = 1 b(t,0) q(t,w(t)) and v (t) = q(t+v(t),v(t)) b(t+v(t),0) 1 Rate into service: fixed-point equation (FPE) b(t, 0) = S (t) + t b(t x, 0)g(x)dx, 0 x t. 0

22 Algorithm for the (G t /GI/s t + GI) m /M t Fluid Network

23 A Non-Markovian Example M t /H 2 /s t + E 2 fluid model λ(t) = sin(t) S = 1 (note: not a single-server queue) H 2 service: p = 0.11, µ 1 = 0.23, µ 2 = 1.77 (Cs 2 = 4) E 2 abandonment: A = X 1 + X 2, where X i i.i.d. exp(1) System initially empty λ(t) and S will be scaled by n!

24 Fluid Algorithm: Alternating between OL and UL

25 Fluid Algorithm: Alternating between OL and UL

26 Fluid Algorithm: Alternating between OL and UL

27 Fluid Algorithm: Alternating between OL and UL

28 Fluid Algorithm: Alternating between OL and UL

29 Fluid Algorithm: Alternating between OL and UL

30 Simulation Comparisons M t /H 2 /s t + E 2 queueing model n = 20, 100, 2000 λ n (t) = n λ(t) = n n sin(t) S n (t) = n S(t) = n Want to see When n is large: ( Qn(t) n, Bn(t) n When n is small: ( E[Qn(t)] n, E[Bn(t)] n, Xn(t) n, W n (t), E[Xn(t)] n, E[W n (t)] ) (Q(t), B(t), X (t), w(t)) ) (Q(t), B(t), X (t), w(t))

31 Simulation Comparisons: M t /H 2 /s t + E 2 n = 100 and 3 sample paths

32 Simulation Comparisons: M t /H 2 /s t + E 2 n = 2000 and a single sample path

33 Simulation Comparisons: M t /H 2 /s t + E 2 n = 100 and a average of 100 sample paths

34 Example: A Network of Two Queues (M t /LN/s t + E 2 ) 2 /M t fluid network Sinusoidal arrival: λ (0) 1 (t) = n( sin(t)), Sinusoidal arrival: λ (0) 2 (t) = n( sin(t 3)) Constant staffing: s 1 (t) = n, s 2 (t) = 2n Lognormal service: Ḡ 1 LN(1, 2), Ḡ2 LN(2, 8) Erlang abandonment: F 1 E 2 (0.5), F 2 E 2 (0.3) [ ] Routing probability: P(t) System initially empty

35 Example: (M t /LN/s t + E 2 ) 2 /M t Simulation comparison: n = 2000, one path

36 Example: (M t /LN/s t + E 2 ) 2 /M t Simulation comparison: n = 50, 2000 paths

37 Example: (M t /LN/s t + E 2 ) 2 /M t Algorithm convergence ɛ: error tolerance (10 9 to 0.1) I(ɛ): # of iterations T (ɛ): computation time (seconds) in MatLab

38 Example: An (M t /M/s t + M) 10 /M t Network For 1 i, j 10, Sinusoidal arrival: λ (0) i (t) = a i + b i sin(c i t + φ i ), a i = 0.5, b i = i 10 a i, φ i = π ( 1.5 i Constant staffing: s i (t) = 1 Exponential service: Ḡ i (x) = e µ i x, µ i = 1 10), Exponential abandonment: F i (x) = e θ i x, θ i = 0.5 Routing probability: P i,j (t) 1 20 System initially empty

39 Example: Fluid Paths of (M t /M/s t + M) 10 /M t

40 Large Scale Network: Running Time Complexity

41 MSHT (FCLT) Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n Approximations Q n (t) = n Q(t) + n ˆQ(t) + o( ) n) d N (n Q(t), n σ 2ˆQ (t) ) W n (t) = W (t) + Ŵ (t) n + o( 1 n ) d N (W (t), 1n σ2ŵ (t)

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43 Characterizing the Diffusion Process Partition {1,..., m} = O U O {1, 2,..., m }, U {m + 1, m + 2,..., m}. For OL queues 1,..., m A Stochastic Differential Equation (SDE): dŵ(t) = H(t)Ŵ(t)dt + J s(t)db s(t) + J a(t)db a(t) + J λ (t)db λ (t) dŵ (t) = H(t)Ŵ(t)dt + J(t)dB (t) Bλ : m-dim BM (arrival process) Ba: m -dim BM (abandonment times) Bs: (m + 1) 2 -dim BM (service times) H, Js, J a, J λ and J : analytic matrices of λ, s, F, µ, Cλ, 2 P and fluid functions Var(Ŵ(t)) = t 0 Ĵs(u)ĴT s (u) + Ĵa(u)ĴT a (u) + Ĵλ(u)ĴT λ (u) du ˆQ(t): Brownian integrals w.r.t. B λ, B a and B s For UL queues m + 1,..., m d ˆB(t) = H (t)ˆb(t)dt + J s (t)db s (t) + J λ (t)db λ(t)

44 Special Case: One-Queue Model dŵ (t) = H(t)Ŵ (t)dt + J s(t)db s (t) + J a (t)db a (t) + J λ (t)db λ (t) dŵ (t) = H(t)Ŵ (t)dt + J (t)db (t) σ 2 (t) Var(Ŵ (t)) = ) t (Ĵ2 Ŵ 0 s (t, u) + Ĵa 2 (t, u) + Ĵλ 2(t, u) du ( ) H(t) = (1 w (t)) λ (t w(t)) λ(t w(t)) + h F (w(t)) J s (t) = J a (t) = b(t,0) s (t) λ(t w(t)) F (w(t)) F (w(t))b(t,0) λ(t w(t)) F (w(t)) J λ (t) = C λ J (t) = Want to see ) Var (Ŵn (t) ( ) Var ˆQn (t) F (w(t))b(t,0) λ(t w(t)) F (w(t)) b(t,0) s (t)+(f (w(t))+cλ 2 F (w(t))) b(t,0) λ(t w(t)) F (w(t)) ( ) σ 2 (t), Var ˆVn (t) σ 2ˆV (t), Ŵ σ (t) 2ˆQ

45 Example: M t /M/s t + H 2 in Both UL and OL Intervals λ(t) = n( sin(t)), s(t) = n, µ = 1, θ = 0.5 n = 2000 and 500 sample path

46 Engineering Refinement for Smaller n λ(t) = n( sin(t)), s(t) = n, µ = 1, θ = 0.5 n = 100 and 2000 sample path

47 Engineering Refinement for Smaller n λ(t) = n( sin(t)), s(t) = n, µ = 1, θ = 0.5 n = 20 and 5000 sample path

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49 Prescribed Routing Paths

50 Multiclass Queueing Network with (G t /GI + GI) n /s m t /PRP n customer classes and m service pools Class i (1 i n) characterized by a deterministic path πi with π i (k) denotes the destination at step k (the PRP) an external arrival rate λi (the G t) non-exponential service dist. Gi,k (the first GI) non-exponential abandonment dist. Fi,k (the +GI) Time-varying staffing level s j (t) at queue j, 1 j m (the s t ) First-Come First-Served (FCFS) at each queue j Unlimited waiting capacity

51 Multiclass Queueing Network with A two-class two-queue example Class 1: π 1 = [1, 2, 1] Class 2: π 2 = [2].

52 MSHT Fluid Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n Approximations Q n (t) = n Q(t) + n ˆQ(t) + o( ) n) d N (n Q(t), n σ 2ˆQ (t) ( ) W n (t) = W (t) + Ŵ (t) n + o( 1 n ) d N W (t), 1n σ2ŵ (t)

53 Algorithm for the (G t /GI + GI) n /s m t /PRP Fluid Network

54 Multiclass Queueing Network with A two-class two-queue example

55 Example: An (M t /LN + E 2 ) 2 /s 2 t /PRP Network Simulation comparison

56 Achieving Time-Stable Performance Shi et al. (2014) Avoid large delay spikes Smooth performance level across time System becomes stationary

57 Staffing to Achieve Time-Stable Performance Design staffing and shifts Meet service level agreements P(waiting < 2 mins)>0.8 E(wait) 3 mins = 0.05 hr P(Abandonment)<0.02

58 Mean Delay At each station j, design staffing s j (t) to stabilize delay at w j Approximate other performance functions (e.g., queue length) Treating both high QoS (small w j ) and low QoS (big w j ) Class dependent service levels calls: to reply in minutes (small wj ) s: to reply in hours (big wj )

59 Delayed Infinite-Server (DIS) Approx. for M t /GI /s t + GI DIS staffing: mean # of busy servers s(t) = E[B(t)] F (w) (t w) + 0 λ(t w x)ḡ(x)dx Approximations for other performance measures t E[Q(t)] = λ(t x) F (x)dx (t w) + P(A < W (t)) = α F (w) Asymptotic stability: when scale becomes large sup 0<t T E[W (t)] w 0, sup 0<t T E[P(A < W (t)) α] 0

60 DIS Approx. for (G t /GI + GI) n /s m t /PRP DIS staffing formulas construct DIS approx. for each class i, step k m i,k (t) F (t wj ) + (w j ) λ i,k (t w j x)ḡi,k(x)dx 0 staffing at queue j: sum of all the required staffing for each type of customers that visits queue j s j (t) = N m i m i,k (t) 1 {πi (k)=j} i=1 k=1 Approximations for other performance measures t E[Qi,k (t)] λ i,k (t x) F i,k (x)dx (t w j ) + N m i E[Qj (t)] E[Q i,k (t)] 1 {πi (k)=j} αi,k (t) i=1 k=1 n F i,k (w j ) 1 {πi (k)=j} j=1

61 DIS Approx. for (G t /GI + GI) n /s m t /PRP Asymptotic stability: when scale becomes large sup E[W j (t)] w j 0 0<t T sup Pi,k ab (t) α j] 0 0<t T

62 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Class i 1 2 Arrival Rate λ (0) i (t) sin t cos(t + 0.5) Path π i π 1 = [1, 2, 1], N 1 = 3 π 2 = [2], N 2 = 1 Stage k, 1 k N i Service distribution LN(0.5, 0.75) c1,1 s =1.5 LN(1.5, 3) 1,2 =2 LN(1, 3) 1,3 =3 LN(2, 5) 2,1 =2.5 Abandonment distribution H 2 (0.15, 0.3) H 2 (1.15, 2.5) H 2 (1, 3) H 2 (0.25, 0.6) c1,1 a =2 ca 1,2 =2.17 ca 1,3 =3 ca 2,1 =2.4 LN(µ, σ 2 ) g(x) = 1 x (ln x µ) 2 2πσ e 2σ 2 H 2 (µ, σ 2 ) f (x) = pλ 1 e λ1x + (1 p)λ 2 e λ2x where p = µ2 +σ 2 + σ 4 µ 4 2(σ 2 +µ 2 ), λ 1 = 2p/µ, λ 2 = 2(1 p)/µ

63 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Arrival rates

64 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Low QoS: w 1 = 0.1, w 2 = 0.3

65 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Low QoS: w 1 = 0.1, w 2 = 0.3

66 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Low QoS: w 1 = 0.1, w 2 = 0.3

67 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP High QoS: w 1 = 0.005, w 2 = 0.02

68 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP High QoS: w 1 = 0.005, w 2 = 0.02

69 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Mixed: w 1 = (high QoS), w 2 = 0.1 (low QoS)

70 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Mixed: w 1 = (high QoS), w 2 = 0.1 (low QoS)

71 Summary of Fluid Summary Markovian routing: (G t /GI/s t + GI) m /M t network Deterministic paths: (G t /GI + GI) m /s n t /PRP network Fluid (FWLLN) and diffusion (FCLT) approximations Staffing to achieve time-stable performance Simulations comparisons to verify effectiveness Future work Fit into realistic applications (e.g., health care) Incorporate both uncertainty and history

72 References [1] He & Liu, the Tail Probability of Delay in Service Systems with Time-Varying Demand. Submitted to Operations Research (2014) [2] Aras, Liu & Whitt, Heavy-Traffic Limit for the Initial Content Process. Submitted to Mathematics of Operations Research (2014) [3] Liu & Whitt, Stabilizing Performance in Many-Server Queues with Time-Varying Arrivals and Customer Feedback. Submitted to Operations Research (2014) [4] Liu & Whitt, Many-Server Heavy-Traffic Limits for Queues with Time-Varying Parameters. Annals of Applied Probability 24(1), (2014) [5] Liu & Whitt, Algorithms for Time-Varying of Many-Server Fluid Queues. INFORMS Journal on Computing 26(1), (2014) [6] Liu & Whitt, The G t/gi /s t + GI Many-Server Fluid Queue. Queueing Systems 71(4), (2012) [7] Liu & Whitt, Stabilizing Customer Abandonment in Many-Server Queues with Time-Varying Arrivals. Operations Research 60(6) (2012) [8] Liu & Whitt, A Network of Time-Varying Many-Server Fluid Queues with Customer Abandonment. Operations Research 59(4) (2011) Available at

73 THANK YOU!

74 The G t /GI /s t + GI Queueing Model Constructing the G t arrivals Deterministic variability: time-varying λ(t) (avg. behavior) Stochastic variability: constant c λ (magnitude of variance) Composition methods Step 1: generate rate-1 (equilibrium) renewal process N 0 (t), with variance of interrenewal time Var(X ) = cλ 2 Step 2: set N(t) N 0 (Λ(t)), with Λ(t) t 0 λ(u)du Properties: Mean value: E[N(t)] = Λ(t), Variance-to-mean ratio: Var(N(t))/E[N(t)] = Var(N(t))/Λ(t) c λ (Liu et al. 2014)