Lecture 5: Fluid Models - Technionie.technion.ac.il/serveng2013S/Lectures/Lecture 5.pdf ·...
Transcript of Lecture 5: Fluid Models - Technionie.technion.ac.il/serveng2013S/Lectures/Lecture 5.pdf ·...
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Lecture 5: Fluid Models
Service Engineering
Galit B. Yom-Tov
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Content
• Predictable Variability: Call Centers, Transportation, Emergency Department.
• Four “pictures”: rates, queues, outflows, cumulative graphs.
• Phases of Congestion. • From Data to Models: Scales. (Movies) • A fluid model of one station queue. • A fluid model of call centers with abandonment and
retrials. • Bottleneck Analysis, via National Cranberry
Cooperative. • Summary of the Fluid Paradigm.
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Current waiting time in ER
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Sample Path vs. Averages
From “Predicting Emergency Department Status” Houyuan Jiang, Lam Phuong Lam, Bowie Owens, David Sier, and Mark Westcott
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Predictable vs. Unpredictable
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Time variation of the mean vs. process variability
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Predictable Variability
From “Predicting Emergency Department Status” Houyuan Jiang, Lam Phuong Lam, Bowie Owens, David Sier, and Mark Westcott
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Predictable variability
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From Empirical Models to Fluid Models
• Recall Empirical Models, cumulative arrivals and departure functions.
• For large systems (bird’s eye) the functions look smoother.
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The Bird’s Eye
• Movies:
– Road transportation
– Air transportation
– Visa add
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From Empirical Models to Fluid Models
• Derived directly from event-based (call-by-call) measurements.
• For example, an isolated service-station: – A(t) = cumulative # arrivals from time 0 to time t;
– D(t) = cumulative # departures from system during [0, t];
– L(t) = A(T) − D(t) = # customers in system at t.
Arrivals and Departures from a Bank Branch Face-to-Face Service
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Phases of Congestion via Cumulates Hall, pg 189:
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Four points of view
• Cumulates
• Rates (⇒ Peak Load)
• Queues (⇒ Congestion)
• Outflows (⇒ end of rush-hour)
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Phases of Congestion via Rates
• Time-lag
• Change Service Rate
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Aggregate Planning via “cumulative picture”
A(•): given and seasonal
What should be the service rate?
• Option a) Chase demand D=A – Costly, variable workforce
• Option b) Constant workforce with no queue – Excess capacity
• Option c) Least constant capacity that accommodate all arrivals (no queue at end)
• Option d) Add capacity during peak hours
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Queueing System as a Tub (Hall, p.188)
• A(t) – cumulative arrivals function. • D(t) – cumulative departures function. • – arrival rate. • – processing (departure) rate. • c(t) – maximal potential processing rate. • q(t) – total amount in the system.
Fluid Models: General Setup
)()( tDt
)()( tAt
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Mathematical Fluid Models
Deferential equations:
• λ(t) – arrival rate at time t ∈ [0,T].
• c(t) – maximal potential processing rate.
• δ(t) – effective processing (departure) rate.
• q(t) – total amount in the system.
Then q(t) is a solution of
].,0[,)0();()()( 0 Ttqqtttq
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Mathematical Fluid Models: Multi-server queue
• n(t) statistically-identical servers, each with service rate μ.
• c(t) = μn(t): maximal potential processing rate.
• : processing rate.
i.e.,
How to actually solve? Discrete-time approximation: Start with Then, for
].,0[,)0());(),(min()()( 0 Ttqqtqtnttq
.))(),(min()()()( 1111 ttqtntttqtq nnnnn
))(),(min()( tqtnt
.)(,0 000 qtqt :1 ttt nn
.))(),(min()()0()(00 tt
duuqunduuqtq
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Mathematical Fluid Models: Multi-server queue with abandonment
• θ – Abandonment rate of customers in queue
• Processing rate:
• The fluid model:
].,0[,)0(
;)]()([))(),(min()()(
0 Ttqq
tntqtqtnttq
)]()([))(),(min()( tntqtqtnt
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Fluid Model as Approximation
• Let Q(t) be the number of customer in a queue. (Q(t) is a random process)
• Increase the arrival rate and the capacity, such that and define as the number of customer in the “η” queue.
• Then by the Functional Strong Law of Large Numbers, as
uniformly on compacts, a.s. given convergence at t=0.
(the fluid approximation) is the solution to the differential balance equations of the system.
,, tttt nn
,
),()(1 )0( tQtQ
)(tQ
)()0( tQ
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Example: Time-varying analysis
“Fluid Models and Diffusion Approximations for Time-Varying Queues with Abandonment and Retrial” Based on a series of papers of Bill Massey, Avishai Mandelbaum, Marty Reiman, Brian Rider, and Sasha Stolyar.
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Primitives
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Confidence interval are calculated using Diffusion approximations
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National Cranberry
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Queue Build-up Diagram: National Cranberry Case Study
• A peak day: – 18,000 bbl (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 per hour (Capacity = max. sustainable processing rate)
– Bin capacity for wet: 3200 bbl’s
– 75 bbl per truck (avg.)
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Queue Build-up Diagram: National Cranberry Case Study
• Draw inventory build-up diagrams of wet 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.?
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Flow Diagram: National Cranberry
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Total inventory build-up: Wet berries (bins & trucks)
Processing capacity 600 bbl/hr; Start at 11:00; Peak day 18k*70% over 12 hours.
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Trucks inventory build-up Wet berries
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Trucks queue analysis
• Area over curve =
Divide by 75. • Truck hours waiting = 40,533/75 bbl/truck = 540
truck•hours • Ave throughput rate =
• Ave WIP = 540/162/3=32.4 trucks (a “biased”
average) • Given that a truck waits, it will wait on average
32.4/7.52 = 4.3 hours (Little’s Law)
hoursbbl 533,403
2746002
18]46001000[
2
111000
2
1
hourtrucks/52.7]753
216/[]3
21560010[
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Total inventory build-up: Wet berries
Processing capacity 600 bbl/hr; Start at 7:00; Peak day 18k*70% over 12 hours.
# trucks in queue
5
17
29
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Total inventory build-up: Wet berries
Processing capacity 800 bbl/hr (i.e., add 4th dryner); Start at 7:00; Peak day 18k*70% over 12 hours.
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Summery
• More examples: EOQ (inventory models) • Service analogy:
– Front-office + back-office (banks, telephones) – Hospitals (operating rooms, recovery rooms) – Ports (inventory in ships; bottlenecks = unloading crews,
router) – More?
• Reminder: Bottleneck operation – Add resources, use alternative resources, reduce setup
(change IT system), reduce wasted times (e.g., synchronization), improve work conditions, work overtime, subcontract, higher skilled personnel.
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Summery: 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.
– 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.
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Summery: Why Fluid models? • Predictable variability is dominant (std<<Mean) • The value of the fluid-view increases with the complexity of
the system from which it originates • Legitimate models of flow systems
– Often simple and sufficient; empirical, predictive • Capacity analysis • Inventory build-up diagrams • Mean-value analysis
• Approximations – First-order fluid approximation of stochastic systems
• Strong laws of large numbers (vs. second-order diffusion approximation, Central limits)
– Long-run • Long horizon, smooth-out variability (strategic)
• Technical tools – Lyapunov functions to establish stability (Long-run) – Building blocks for stochastic models