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Motivation The Erlang-R Queue Results

The Erlang-R Queue: Time-Varying QED Queues with Reentrant Customers in Support of Healthcare Staffing

Galit Yom-Tov Avishai Mandelbaum

Industrial Engineering and Management Technion

MSOM Conference, June 2010

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results The Problem Studied

The Problem Studied

Problems in Emergency Departments: Hospitals do not manage patients flow. Long waiting times in the ED for physicians, nurses, and tests. => Deterioration in medical state. Patients leave ED without being seen or abandon during their stay. => Patient return in severe state. We use Service Engineering approach to reduce these effects.

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results The Problem Studied

The Problem Studied

Physician Nurse Imagine Lab Other Follow-up Instruction prior discharge Awaiting evacuation Administrative release First Examination Decision Labs Treatment Administrative reception Vital signs & Anamnesis Imagine: X-Ray, CT, Ultrasound Treatment Consultation First Examination Alternative Operation

• Recourse Queue - Synchronization Queue –

Ending point of alternative operation - C A A A A B B B C C C

Can we determine the number of physicians (and nurses) needed to improve patients flow, and control the system in balance between service quality and efficiency?

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results The Problem Studied

The Problem Studied

Standard assumption in service models: service time is continuous. But we find systems in which: service is dis-continuous and customers re-enter service again and again. μ μ μ = Service (Needy) Content ∧ ∞ λ λ λ = δ δ δ = What is the appropriate staffing procedure? What is the significance of the re-entering customers? What is the implication of using simple Erlang-C models for staffing?

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results The Problem Studied

The Problem Studied

Standard assumption in service models: service time is continuous. But we find systems in which: service is dis-continuous and customers re-enter service again and again. μ μ μ = Service (Needy) Content ∧ ∞ λ λ λ = δ δ δ = What is the appropriate staffing procedure? What is the significance of the re-entering customers? What is the implication of using simple Erlang-C models for staffing?

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results The Problem Studied

Related Work

Mandelbaum A., Massey W.A., Reiman M. Strong Approximations for Markovian Service Networks. 1998. Massey W.A., Whitt W. Networks of Infinite-Server Queues with Nonstationary Poisson Input. 1993. Green L., Kolesar P .J., Soares J. Improving the SIPP Approach for Staffing Service Systems that have Cyclic Demands. 2001. Jennings O.B., Mandelbaum A., Massey W.A., Whitt W. Server Staffing to Meet Time-Varying Demand. 1996. Feldman Z., Mandelbaum A., Massey W.A., Whitt W. Staffing of Time-Varying Queues to Achieve Time-Stable Performance. 2007.

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue

Model Definition

The (Time-Varying) Erlang-R Queue:

Needy (st-servers) rate- μ Content (Delay) rate - δ 1-p p 1 2 Arrivals Poiss(λt) Patient discharge

λt - Arrival rate of a time-varying Poisson arrival process. µ - Service rate. δ - Delay rate (1/δ is the delay time between services). p - Probability of return to service. st - Number of servers at time t.

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue

Patients Arrivals to an Emergency Department

2004-2008

16 18 12 14

r hour

8 10

nts per

4 6

Patien

2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Hour of day

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue

Staffing: Determine st, t ≥ 0

Based on the QED-staffing formula: s = R + β √ R, where R = λE[S] In time-varying environments: s(t) = R(t) + β

• R(t),

where β is chosen according to the steady-state QED. Two approaches to calculate the time-varying offered load (R(t)):

PSA / SIPP (lag-SIPP) - divide the time-horizon to planning intervals, calculate average arrival rate and steady-state

• ffered-load for each interval, then staff according to

steady-state recommendation (i.e., R(t) ≈ ¯ λ(t)E[S]). MOL/IS - assuming no constraints on number of servers, calculate the time-varying offered-load. For example, in a single service system: R(t) = E[ t

t−S λ(u)du] = E[λ(t − Se)]E[S].

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue

Staffing: Determine st, t ≥ 0

Based on the QED-staffing formula: s = R + β √ R, where R = λE[S] In time-varying environments: s(t) = R(t) + β

• R(t),

where β is chosen according to the steady-state QED. Two approaches to calculate the time-varying offered load (R(t)):

PSA / SIPP (lag-SIPP) - divide the time-horizon to planning intervals, calculate average arrival rate and steady-state

• ffered-load for each interval, then staff according to

steady-state recommendation (i.e., R(t) ≈ ¯ λ(t)E[S]). MOL/IS - assuming no constraints on number of servers, calculate the time-varying offered-load. For example, in a single service system: R(t) = E[ t

t−S λ(u)du] = E[λ(t − Se)]E[S].

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue

Staffing: Determine st, t ≥ 0

Based on the QED-staffing formula: s = R + β √ R, where R = λE[S] In time-varying environments: s(t) = R(t) + β

• R(t),

where β is chosen according to the steady-state QED. Two approaches to calculate the time-varying offered load (R(t)):

PSA / SIPP (lag-SIPP) - divide the time-horizon to planning intervals, calculate average arrival rate and steady-state

• ffered-load for each interval, then staff according to

steady-state recommendation (i.e., R(t) ≈ ¯ λ(t)E[S]). MOL/IS - assuming no constraints on number of servers, calculate the time-varying offered-load. For example, in a single service system: R(t) = E[ t

t−S λ(u)du] = E[λ(t − Se)]E[S].

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue

The Offered-Load

Offered-Load in Erlang-R = The number of busy servers (or the number of customers) in a corresponding (Mt/M/∞)2 network.

Theorem: (Massey and Whitt 1993) R(t) = (R1(t), R2(t)) is determined by the following expression: Ri(t) = E[λ+

i (t − Si,e)]E[Si]

where, λ+

1 (t) = λ(t) + E[λ+ 2 (t − S2)]

λ+

2 (t) = pE[λ+ 1 (t − S1)]

Theorem: If service times are exponential, R(t) is the solution of the following Fluid ODE: d dt R1(t) = λt + δR2(t) − µR1(t), d dt R2(t) = pµR1(t) − δR2(t).

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Case Study: Sinusoidal Arrival Rate

Periodic arrival rate: λt = ¯ λ + ¯ λκsin(ωt). ¯ λ is the average arrival rate, κ is the relative amplitude, and ω is the frequency.

External / Internal arrivals rate, Offered-load, and Staffing

120 100 60 80

g level

40 60

Staffing

20 40 λ(t) λ+(t) 20 R1(t) s(t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Case Study: Sinusoidal Arrival Rate

Simulation of P(Wait) for various β (0.1 ≤ β ≤ 1.5)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P(W>0)

0.1 0.2 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100105110115

Time [Hour]

beta=0.1 beta=0.3 beta=0.5 beta=0.7 beta=1 beta=1.5

Performance measure is stable! (0.15 ≤ P(Wait) ≤ 0.85)

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Case Study: Sinusoidal Arrival Rate

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P(W>0)

Halfin-Whitt Empirical 0.1 0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

β

Relation between P(wait) and β fits steady-state theory!

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Case Study: Sinusoidal Arrival Rate

Simulation results of servers’ utilization for various β

0 95 1 0.9 0.95

• n

0 8 0.85

Utilizatio

0.75 0.8

U

0.7 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225

Time

beta 0.1 beta 0.3 beta 0.5 beta 0.7 beta 1.0 beta 1.5

Performance measure is stable! (0.85 ≤ Util ≤ 0.98)

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Can We Use Erlang-C?

Simulation results of P(wait): Erlang-R vs. Erlang-C and PSA

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 20 40 60 80 100 120 140 160 180 200 220 240

P(W>0) Time

Erlang‐R Erlang‐C PSA

Using Erlang-C’s R(t), does not stabilize systems’ performance.

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Why Erlang-C Does Not Fit Re-entrant Systems?

Compare R(t) of Erlang-C and Erlang-R: Erlang-C offered-load (with concatenated services) : R(t) = E

• λ
• t −

1 1 − pS1,e

• E
• 1

1 − pS1

• Erlang-R offered-load:

R1(t) = E ∞

• i=1

piλ

• t − S∗i

1 − S∗i 2 − S1,e

• E[S1]

Needy Content

1-p p 1 2

Arrivals Patient discharge

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Comparison between Erlang-C and Erlang-R

Erlang-C under- or over-estimates the Erlang-R offered-load.

36 38 105 110 32 34 95 100

A d

28 30 32 85 90

Arrival R red Load

26 28 75 80

Rate Offer

22 24 65 70 Erlang‐C Erlang‐R λ(t) 20 60 7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 λ(t)

Time

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Comparison between Erlang-C and Erlang-R

Theorem: The ratio of amplitudes between Erlang-R and Erlang-C is given by

• (δ2 + ω2)(((1 − p)µ)2 + ω2)

((µ − iω)(δ − iω) − pµδ)((µ + iω)(δ + iω) − pµδ) Plot of amplitudes ratio as a function of ω

1 0 98 0.99

io

0.97 0.98

ude Rati

0.96

Amplitu

0.95 0.94 0.00 0.26 0.53 0.79 1.06 1.32 1.58 1.85 2.11 2.38 2.64 2.90

Omega

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Comparison between Erlang-C and Erlang-R

Plot of amplitudes ratio as a function of ω

1 0 98 0.99

io

0.97 0.98

ude Rati

0.96

Amplitu

0.95 0.94 0.00 0.26 0.53 0.79 1.06 1.32 1.58 1.85 2.11 2.38 2.64 2.90

Omega

Erlang-C over-estimate the amplitude of the offered-load. The re-entrant patients stabilize the system. Minimum ratio achieved when: ω =

• δµ(1 − p) (for example

ED).

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Comparison between Erlang-C and Erlang-R

Plot of the ratio of phases as a function of ω

2.3 2.5 1 9 2.1 1.7 1.9

e Ratio

1.3 1.5

Phase

0.9 1.1 0.7 0.9 0.0 0.5 1.1 1.6 2.1 2.6 3.2 3.7 4.2 4.8 5.3 5.8

Omega

Erlang-C under- or over-estimates the time-lag.

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Comparison between Erlang-C and Erlang-R

Erlang-C under- or over-estimates this time-lag depending on the period’s length.

36 105 λ=30, µ=1, δ=0.5, p=2/3, cycles per day=1 34 100 30 32 90 95

Arriva d Load

28 30 85 90

al Rate Offered

26 80 Erlang‐C Erlang‐R 24 75 7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 λ(t)

Time

36 96 λ=30, µ=1, δ=0.5, p=2/3, cycles per day=4 34 94 30 32 90 92

Arriva d Load

28 30 88 90

al Rate Offered

26 86 Erlang‐C Erlang‐R 24 84 9.5 9.6 9.7 9.8 9.9 10.0 λ(t)

Time

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Small systems - Hospitals

Small systems: No of doctors range from 1 to 5 Constrains: Staffing resolution: 1 hour Minimal staffing: 1 doctor per type Integer values: s(t) = [R1(t) + β

• R1(t)]

Example: R = 2.75

β range s P(W > 0) [0, 0.474] 3 82.4% (0.474, 1.055] 4 34.0% (1.055, 1.658] 5 11.4% (1.658, 2.261] 6 3.0% 1.658 and up 7 0%

=> Can not achieve all performance levels!

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Small systems - Hospitals

0.3 0.4 0.5 0.6 0.7 0.8 0.9

P(W>0)

0.1 0.2 1000 2000 3000 4000 5000 6000 7000

Time

Beta=0.1 Beta=0.5 Beta=1 Beta=1.5

P(Wait) is stable and separable!

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Conclusions

In time-varying systems where patients return for multiple services:

1

Using the MOL (IS) algorithm for staffing stabilizes performance.

2

Re-entrant patients stabilize the system.

3

Using single-service models, such as Erlang-C, is problematic in the re-entrant ED environment:

Time-varying arrivals Transient behavior even with constant parameters

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

What next?

Fluid and diffusion approximations for mass-casualty events QED - MOL approximations for the processes:

Number of customers in system Virtual waiting time

Extension: upper limit on the number of customers within the system

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

What next?

The semi-open Erlang-R queue

μ δ ∞ λ γ ∞ Patient is Content 1-p p 1 2 Arrivals Blocked patients Patient is Needy N beds

Does MOL approximation works? yes, stabilizing performance is achieved. Is it close to M/M/s/n model? no.

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue

Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function

Thank You

Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue