The Fluid View, or Flow Models

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1 Service Engineering November 30, 2005 The Fluid View, or Flow Models Introduction: Legitimate models: Simple, General, Useful Approximations (strong) Tools Scenario analysis vs. Simulation, Averaging, Steady-State Typical scenario, or very atypical (eg. catastrophy ) Predictable Variability Averaging scenarios, with small CV A puzzle (the human factor state dependent parameters) Sample size needed increases with CV Predictable variability could also turn unpredictable Hall: Chapter 2 (discrete events); 4 Pictures: Cummulants Rates ( Peak Load) Queues ( Congestion) Outflows ( end of rush-hour) Scales (Transportation, Telephone (1976, 1993, 1999)) Simple Important Models: EOQ, Aggregate Planning Skorohod s Deterministic Fluid Model (of a service station): teaching note Phases of Congestion: under-, over- and critical-loading. Rush Hour Analysis: onset, end Mathematical Framework in approximations Queues with Abandonment and Retrials (=Call Centers; Time- and State-dependent Q s). Bottleneck analysis in a (feed-forward) Fluid Network, via National Cranberry Fluid Networks (Generalizing Skorohod): The Traffic Equations Addendum 1

4 Predictable Queues Fluid Models Service Engineering Queueing Science Eurandom September 8, 2003 e.mail : avim@tx.technion.ac.il Website: 1

5 3. Supporting Material (Downloadable) Gans, Koole, and M.: Telephone Call Centers: Tutorial, Review and Research Prospects. MSOM. Brown, Gans, M., Sakov, Shen, Zeltyn, Zhao: "Statistical Analysis of a Telephone Call Center: A Queueing-Science Perspective." Submitted. Jennings, M., Massey, Whitt: "Server Staffing to Meet Time- Varying Demand." Management Science, PRACTICAL 0. M., Massey, Reiman: "Strong Approximations for Markovian Service Networks." QUESTA, M., Massey, Reiman, Rider: "Time Varying Multi-server Queues with Abandonment and Retrials", ITC-16, M., Massey, Reiman, Rider and Stolyar: "Waiting Time Asymptotics for Time Varying Multiserver Queues with Abandonment and Retrials", Allerton Conference, M., Massey, Reiman, Rider and Stolyar: "Queue Lengths and Waiting Times for Multiserver Queues with Abandonment and Retrials", Fifth INFORMS Telecommunications Conference,

6 Labor-Day Queueing in Niagara Falls Three-station Tandem Network: Elevators, Coats, Boats Total wait of 15 minutes from upper-right corner to boat How? Deterministic constant motion

7 Shouldice Hospital: Flow Chart of Patients Experience Day 1: Surgeons Admit Waiting Room Exam Room (6) Acctg. Office Nurses Station Patient s Room 1:00-3:00 PM min 10 min 5-10 min 1-2 hours Orient n Room Dining Room Rec Lounge Patient s Room 5:00-5:30 PM 5:30-6:00 PM 7:00-9:00 PM 9:30 PM- 5:30 AM Day 2: Pre Op Room Operating Room Post Op Room Patient s Room Dining Room 5:30-7:30 AM to 3:00 PM 45 min min 9:00 PM Day 3: Remove Clips Patient s Room Dining Room Clinic Room? Rec Room Grounds Dining Room 6:00 AM 7:45-8:15 AM 9:00 PM Day 4: Remove Rem. Clips Dining Room Clinic External types of abdominal hernias. 82% 1 st -time repair. 7:45-8:50 AM Stay Longer Go Home 18% recurrences operations in Recurrence rate: 0.8% vs. 10% Industry Std.

8 Matching Supply and Demand (Wharton) 6

9 Staffing Matters (on Fridays, 7:00 am) 7

14 Bank Queue Queue Time of Day Catastrophic Heavy Load Regular

19 Arrival Rate Q-Science May 1959! Time 24 hrs (Lee A.M., Applied Q-Th) % Arrivals Dec 1995! Time 24 hrs (Help Desk Institute)

20 Time-Varying Queues: Predictable Variability (with Jennings, Massey, Whitt) Arrivals Queues Waiting 45

38 From Data to Models: (Predictable vs. Stochastic Queues) Fix a day of given category (say Monday = M, as distinguished from Sat.) Consider data of many M s. What do we see? Unusual M s, that are outliers. Examples: Transportation : storms,... Hospital: : military operation, season,...) Such M s are accommodated by emergency procedures: redirect drivers, outlaw driving; recruit help. Support via scenario analysis, but carefully. Usual M s, that are average. In such M s, queues can be classified into: Predictable: queues form systematically at nearly the same time of most M s + avg. queue similar over days + wiggles around avg. are small relative to queue size. e.g., rush-hour (overloaded / oversaturated) Model: hypothetical avg. arrival process served by an avg. server Fluid approx / Deterministic queue :macroscopic Diffusion approx = refinements :mesoscopic Unpredictable: queues of moderate size, from possibly at all times, due to (unpredictable) mismatch between demand/supply Stochastic models :microscopic Newell says, and I agree: Most Queueing theory devoted to unpredictable queues, but most (significant) queues can be classified as predictable. 3

40 Scales (Fig. 2.1 in Newell s book: Transportation) Horizon Max. count/queue Phenom (a) 5 min 100 cars/5 10 (stochastic) instantaneous queues (b) 1 hr 1000 cars/200 rush-hour queues (c) 1 day = 24 hr 10,000 /? identify rush hours (d) 1 week 60,000 / daily variation (add histogram) (e) 1 year seasonal variation (f) 1 decade trend Scales in Tele-service Horizon Decision e.g. year strategic add centers / permanent workforce month tactical temporary workforce day operational staffing (Q-theory) hour regulatory shop-floor decisions 4

42 Arrival Process Yearly Monthly Daily Hourly

43 Arrival Process, in 1976 (E. S. Buffa, M. J. Cosgrove, and B. J. Luce, An Integrated Work Shift Scheduling System ) Yearly Monthly Daily Hourly

Custom Inspections at an Airport Number of Checks

Checks Week in Year Day in Month Average Number of

45 Custom Inspections at an Airport Number of Checks Made During 1993: Predictable? Strike Holiday Number of Checks Made in November 1993: Weekend Weekend Weekend Weekend # Checks # Checks Week in Year Day in Month Average Number of Checks During the Day: # Checks Hour Source: Ben-Gurion Airport Custom Inspectors Division

59 Predictable Queues Fluid Models and Diffusion Approximations for Time-Varying Queues with Abandonment and Retrials with Bill Massey Marty Reiman Brian Rider Sasha Stolyar 1

60 Sudden Rush Hour n = 50 servers; µ = 1 λ t = 110 for 9 t 11, λ t = 10 otherwise Lambda(t) = 110 (on 9 <= t <= 11), 110 (otherwise). n = 50, mu1 = 1.0, mu2 = 0.1, beta = 2.0, P(retrial) = Q1 ode Q2 ode Q1 sim Q2 sim variance envelopes time 2

61 Call Center: A Multiserver Queue with Abandonment and Retrials 1 2 µ t Q 2 (t) λ t 2 Q 1 (t)... 1 µ t (Q 1 (t) n t ) n t β t ψ t ( Q 1 (t) n t ) + β t (1 ψ t ) ( Q 1 (t) n t ) + Q 2 (t)

62 Primitives (Time-Varying Predictably) λ t exogenous arrival rate e.g., continuously changing, sudden peak µ 1 t service rate e.g., change in nature of work or fatigue n t β t ψ t number of servers e.g., in response to predictably varying workload abandonment rate while waiting e.g., in response to IVR discouragement at predictable overloading probability of no retrial 1/µ 2 t average time to retry Large system: η scaling parameter. Now define Q η ( ) via λ t ηλ t n t ηn t What do we get, as η? 4

63 Fluid Model Replacing random processes by their rates yields Q (0) (t) = (Q (0) 1 (t), Q(0) 2 (t)) Solution to nonlinear differential balance equations d dt Q(0) 1 (t) = λ t µ 1 t (Q (0) 1 (t) n t) +µ 2 t Q(0) 2 (t) β t (Q (0) 1 (t) n t) + d dt Q(0) 2 (t) = β 1(1 ψ t )(Q (0) 1 (t) n t) + µ 2 t Q(0) 2 (t) Justification: Functional Strong Law of Large Numbers, with λ t ηλ t, n t ηn t. As η, 1 η Qη (t) Q (0) (t), uniformly on compacts, a.s. given convergence at t = 0 5

64 Diffusion Refinement Q η (t) d = η Q (0) (t) + η Q (1) (t) + o ( η ) Justification: Functional Central Limit Theorem [ ] 1 η η Qη (t) Q (0) d (t) Q (1) (t), in D[0, ), given convergence at t = 0. Q (1) solution to stochastic differential equation. If the set of critical times {t 0 : Q (0) 1 (t) = n t} has Lebesque measure zero, then Q (1) is a Gaussian process. In this case, one can deduce ordinary differential equations for EQ (1) i (t), Var Q (1) i (t) : confidence envelopes These ode s are easily solved numerically (in a spreadsheet, via forward differences). 6

65 What if P r {Retrial} increases to 0.75 from 0.25? Lambda(t) = 110 (on 9 <= t <= 11), 10 (otherwise). n = 50, mu1 = 1.0, mu2 = 0.1, beta = 2.0, P(retrial) = Q1 ode Q2 ode Q1 sim Q2 sim variance envelopes time Lambda(t) = 110 (on 9 <= t <= 11), 110 (otherwise). n = 50, mu1 = 1.0, mu2 = 0.1, beta = 2.0, P(retrial) = Q1 ode Q2 ode Q1 sim Q2 sim variance envelopes time 7

66 Starting Empty and Approaching Stationarity 100 Lambda(t) = 110, n = 50, mu1 = 1.0, mu2 = 0.2, beta = 2.0, P(retrial) = Q1 ode Q2 ode Q1 sim Q2 sim variance envelopes time 700 Lambda(t) = 110, n = 50, mu1 = 1.0, mu2 = 0.2, beta = 2.0, P(retrial) = Q1 ode Q2 ode Q1 sim Q2 sim variance envelopes time 8

68 Sample Mean vs. Fluid Approximation Queue Lengths (λ t = 20 or 100) 80 n=50, mu1=1, mu2=.2, beta=2, P(retrial)=.5, lambda = 20 (t in [0,2), [4,6), [8,10) etc) else queue length means q1 ode q1 sim q2 ode q2 sim time 70 n=50, mu1=1, mu2=.2, beta=2, P(retrial)=.5, lambda = 40 (t in [0,2), [4,6), [8,10) etc) else queue length means q1 ode q1 sim q2 ode q2 sim time 9

69 Variances and Covariances Queue Lengths 140 n=50, mu1=1, mu2=.2, beta=2, P(retrial)=.5, lambda = 20 (t in [0,2), [4,6), [8,10) etc) else 100 queue length covariance matrix entries q1 variance ode q1 variance sim q2 variance ode q2 variance sim covariance ode covariance sim time queue length covariance matrix entries n=50, mu1=1, mu2=.2, beta=2, P(retrial)=.5, lambda = 40 (t in [0,2), [4,6), [8,10) etc) else 80 q1 variance ode q1 variance sim q2 variance ode q2 variance sim covariance ode covariance sim time 13

70 Sample Density vs. Gaussian Approximation Multi-Server Queue n=50,mu1=1,mu2=2,beta=.2,p(retrial)=.5,lambda = 20 (t in [0,2),[4,6),[8,10) etc) else t=6 t=5 "x " = queue length empirical law " " = queue length limit law 0.06 t=7 q 1 queue length density n=50,mu1=1,mu2=2,beta=.2,p(retrial)=.5,lambda = 40 (t in [0,2),[4,6),[8,10) etc) else "x " = queue length empirical law " " = queue length limit law t=6 t=5 q 1 queue length density t=

72 Sample Mean vs. Fluid Approximation Virtual Waiting Time n=50,mu1=1,mu2=2,beta=.2,p(retrial)=.5,lambda = 20 (t in [0,2),[4,6),[8,10) etc) else 100 waiting time mean ode waiting time mean sim 0.35 virtual waiting time mean n=50,mu1=1,mu2=2,beta=.2,p(retrial)=.5,lambda = 40 (t in [0,2),[4,6),[8,10) etc) else waiting time ode waiting time sim 0.25 virtual waiting time mean

73 Back to the Multiserver Queue with Abandonment and Retrials 1 2 µ t Q 2 (t) λ t 2 Q 1 (t)... 1 µ t (Q 1 (t) n t ) n t β t ψ t ( Q 1 (t) n t ) + β t (1 ψ t ) ( Q 1 (t) n t ) + Q 2 (t)

74 Sample Path Construction of a Multiserver Queue with Abandonment and Retrials ( t ) Q 1 (t) = Q 1 (0) + A a λ s ds 0 ( t ) ( t ) + A c 21 Q 2 (s)µ 2 s ds A c (Q 1 (s) n s )µ 1 s ds 0 ( t ) A b 12 (Q 1 (s) n s ) + β s (1 ψ s )ds 0 ( t ) A b (Q 1 (s) n s ) + β s ψ s ds 0 and Q 2 (t) = ( t ) Q 2 (0) + A b 12 (Q 1 (s) n s ) + β s (1 ψ s )ds ( t ) A c 21 Q 2 (s)µ 2 s ds A d = Poisson(1), independent. 2

75 Fluid Limit for the Multiserver Queue with Abandonment and Retrials (2 O.D.E. s) d dt Q(0) 1 (t) = λ t + µ 2 t Q (0) 2 (t) µ1 t ( ) + β t Q (0) 1 (t) n t ( ) Q (0) 1 (t) n t and d ( ) + dt Q(0) 2 (t) = β t(1 ψ t ) Q (0) 1 (t) n t µ 2 t Q (0) 2 (t). Can be solved numerically (forward Euler) in a spreadsheet. 3

76 Diffusion Moments for the Multiserver Queue with Abandonment and Retrials Let E 1 (t) = E Assume the set Then [ [ Q (1) 1 ], (t) E 2 (t) = E Q (1) 2 ]. (t) { } t Q (0) 1 (t) = n t has Lebesque measure zero. d ( ) dt E 1(t) = µ 1 t 1 {Q (0) (t) n 1 t} + β t1 {Q (0) (t)>n E 1 t} 1 (t) + µ 2 t E 2(t) and d dt E 2(t) = β t (1 ψ t )E 1 (t)1 {Q (0) (t) n 1 t} µ2 t E 2 (t). 4

77 Let and V 1 (t) = Var More Diffusion Moments (A Grand Total of 7 O.D.E. s) [ C(t) = Cov [ Q (1) 1 ], (t) V 2 (t) = Var Q (1) 1 (t), Q(1) 1 (t) ]. Then [ Q (1) 2 (t) ], d ( ) dt V 1(t) = 2 β t 1 {Q (0) (t)>n 1 t} + µ1 t 1 {Q (0) (t) n V 1 t} 1 (t) + λ t + β t ( Q (0) 1 (t) n t + µ 2 t Q (0) 2 (t), d dt V 2(t) = 2µ 2 t V 2 (t) + β t (1 ψ t ) and ) + + µ 1 t ( Q (0) 1 (t) n t ( ) + Q (0) 1 (t) n t + µ 2 t Q (0) 2 (t) + 2β t(1 ψ t )C(t)1 {Q (0) 1 (t) n t}, d ( dt C(t) = β t 1 {Q (0) (t) n 1 t} + µ1 t 1 {Q (0) (t)<n 1 t} + µ 2 t (V 2 (t) C(t)) β t (1 ψ t ) µ 2 t Q(0) 2 (t). ) C(t) ) ( ) Q (0) 1 (t) n t 5

78 Example: Spiked Arrival Rate: λ(t) = 110, if 9 t 11 otherwise λ(t) = 10, µ 1 = 1.0, µ 2 = 0.1, β = 2.0, n = 50, ψ = 0.25 Lambda(t) = 110 (on 9 <= t <= 11), 110 (otherwise). n = 50, mu1 = 1.0, mu2 = 0.1, beta = 2.0, P(retrial) = Q1 ode Q2 ode Q1 sim Q2 sim variance envelopes time 6

79 Theory Generalizes to Jackson Networks with Abandonment i ij µ t φ t (Q i (t) n t ) i 1 λ t j Q j (t) 2... j j µ t (Q j (t) n t ) j n t j j β t (Q j (t) n t ) + k kj k β t ψ t (Q k (t) n t ) + Further generalizations: Pre-Emptive Priorities 7

80 Bottleneck Analysis Inventory Build-up Diagrams, based on National Cranberry (Recall EOQ,...) (Recall Burger-King) (in Reading Packet: Fluid Models) A peak day: 18,000 bbl s (barrels of 100 lbs. each) 70% wet harvested (requires drying) Trucks arrive from 7:00 a.m., over 12 hours Processing starts at 11:00 a.m. Processing bottleneck: drying, at 600 bbl s per hour (Capacity = max. sustainable processing rate) Bin capacity for wet: 3200 bbl s 75 bbl s per truck (avg.) - Draw inventory build-up diagrams of berries, arriving to RP1. - Identify berries in bins; where are the rest? analyze it! Q: Average wait of a truck? - Process (bottleneck) analysis: What if buy more bins? buy an additional dryer? What if start processing at 7:00 a.m.? Service analogy: front-office + back-office (banks, telephones) service production hospitals (operating rooms, recovery rooms) ports (inventory in ships; bottlenecks = unloading crews,router) More? 6

86 Types of Queues Perpetual Queues: every customers waits. Examples: public services (courts), field-services, operating rooms,... How to cope: reduce arrival (rates), increase service capacity, reservations (if feasible),... Models: fluid models. Predictable Queues: arrival rate exceeds service capacity during predictable time-periods. Examples: Traffic jams, restaurants during peak hours, accountants at year s end, popular concerts, airports (security checks, check-in, customs)... How to cope: capacity (staffing) allocation, overlapping shifts during peak hours, flexible working hours,... Models: fluid models, stochastic models. Stochastic Queues: number-arrivals exceeds servers capacity during stochastic (random) periods. Examples: supermarkets, telephone services, bank-branches, emergency-departments,... How to cope: dynamic staffing, information (e.g. reallocate servers), standardization (reducing std.: in arrivals, via reservations; in services, via TQM),... Models: stochastic queueing models. 3

Lecture 9: Deterministic Fluid Models and Many-Server Heavy-Traffic Limits. IEOR 4615: Service Engineering Professor Whitt February 19, 2015

Lecture 9: Deterministic Fluid Models and Many-Server Heavy-Traffic Limits IEOR 4615: Service Engineering Professor Whitt February 19, 2015 Outline Deterministic Fluid Models Directly From Data: Cumulative