the erlang-r queue: time-varying qed queues with...

29
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

Upload: doankiet

Post on 28-Apr-2018

222 views

Category:

Documents


5 download

TRANSCRIPT

Page 1: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

Motivation The Erlang-R Queue Results

The Erlang-R Queue:Time-Varying QED Queues with Reentrant

Customersin Support of Healthcare Staffing

Galit Yom-Tov Avishai Mandelbaum

Industrial Engineering and ManagementTechnion

MSOM Conference, June 2010

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

Page 2: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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, andtests.=> Deterioration in medical state.Patients leave ED without being seen or abandon duringtheir stay.=> Patient return in severe state.

We use Service Engineering approach to reduce theseeffects.

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

Page 3: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

Motivation The Erlang-R Queue Results The Problem Studied

The Problem Studied ) 08/01/2009(

7

Figure 4 - Activities-Resources (Flow) Chart

Physician NurseImagineLabOther

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 inbalance between service quality and efficiency?

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

Page 4: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

Motivation The Erlang-R Queue Results The Problem Studied

The Problem Studied

Standard assumption in service models: service time iscontinuous.But we find systems in which: service is dis-continuous andcustomers re-enter service again and again.

M/M/s/N with cycles

1exp( ), ( ) ( )Q s Qμ μ μ=

Service (Needy) Content

1-p

p

s

2exp( ), ( )Q Q

∞ Blocked

( )( )

PoissQ

λλ λ=

δ δ δ=

N

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 forstaffing?

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

Page 5: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

Motivation The Erlang-R Queue Results The Problem Studied

The Problem Studied

Standard assumption in service models: service time iscontinuous.But we find systems in which: service is dis-continuous andcustomers re-enter service again and again.

M/M/s/N with cycles

1exp( ), ( ) ( )Q s Qμ μ μ=

Service (Needy) Content

1-p

p

s

2exp( ), ( )Q Q

∞ Blocked

( )( )

PoissQ

λλ λ=

δ δ δ=

N

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 forstaffing?

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

Page 6: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 haveCyclic 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

Page 7: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

Model Definition

The (Time-Varying) Erlang-R Queue:

Erlang-R:

Needy (s-servers)

Content (Delay)

1-p

p

1

2

Arrivals Patient discharge

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

Page 8: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 hou

r

8

10

nts

per

4

6

Patie

n

0

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 230 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

Page 9: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 planningintervals, calculate average arrival rate and steady-stateoffered-load for each interval, then staff according tosteady-state recommendation (i.e., R(t) ≈ λ̄(t)E [S]).MOL/IS - assuming no constraints on number of servers,calculate the time-varying offered-load. For example, in asingle service system:R(t) = E [

∫ tt−S λ(u)du] = E [λ(t − Se)]E [S].

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

Page 10: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 planningintervals, calculate average arrival rate and steady-stateoffered-load for each interval, then staff according tosteady-state recommendation (i.e., R(t) ≈ λ̄(t)E [S]).MOL/IS - assuming no constraints on number of servers,calculate the time-varying offered-load. For example, in asingle service system:R(t) = E [

∫ tt−S λ(u)du] = E [λ(t − Se)]E [S].

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

Page 11: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 planningintervals, calculate average arrival rate and steady-stateoffered-load for each interval, then staff according tosteady-state recommendation (i.e., R(t) ≈ λ̄(t)E [S]).MOL/IS - assuming no constraints on number of servers,calculate the time-varying offered-load. For example, in asingle service system:R(t) = E [

∫ tt−S λ(u)du] = E [λ(t − Se)]E [S].

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

Page 12: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 thenumber 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:

ddt

R1(t) = λt + δR2(t)− µR1(t),

ddt

R2(t) = pµR1(t)− δR2(t).

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

Page 13: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

Case Study: Sinusoidal Arrival Rate

Periodic arrival rate: λt = λ̄+ λ̄κsin(ωt).λ̄ is the average arrival rate, κ is the relative amplitude, and ω isthe frequency.

External / Internal arrivals rate, Offered-load, and Staffing120

100

60

80

g level

40

60

Staffin

g

20

40λ(t)

λ+(t)

0

20R1(t)

s(t)

0 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

Page 14: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

0.1

0.2

0 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

Page 15: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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-WhittEmpirical

0

0.1

0.2

0 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

Page 16: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

on

0 8

0.85

Util

izat

io

0.75

0.8U

0.70 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

Page 17: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140 160 180 200 220 240

P(W>0)

TimeErlang‐R Erlang‐C PSA

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

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

Page 18: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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[

11− p

S1

]Erlang-R offered-load:

R1(t) = E

[ ∞∑i=1

piλ(

t − S∗i1 − S∗i2 − S1,e

)]E [S1]

Needy

Content

1-p

p

1

2

Arrivals Patient discharge

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

Page 19: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

Ad

28

30

32

85

90

Arrival Rred Load

26

28

75

80

RateOffer

22

24

65

70 Erlang‐CErlang‐Rλ(t)

2060

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

Page 20: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 ω1

0 98

0.99

io

0.97

0.98

ude Ra

ti

0.96Amplitu

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

Page 21: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 ω1

0 98

0.99

io

0.97

0.98ud

e Ra

ti

0.96Amplitu

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

Page 22: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 ω2.3

2.5

1 9

2.1

1.7

1.9

e Ra

tio

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

Page 23: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 onthe period’s length.

36105λ=30, µ=1, δ=0.5, p=2/3, cycles per day=1

34100

30

32

90

95 Arrivad 

Load

28

30

85

90

al Rate

Offered

2680 Erlang‐CErlang‐R

2475

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9

λ(t)

Time

3696λ=30, µ=1, δ=0.5, p=2/3, cycles per day=4

3494

30

32

90

92 Arrivad 

Load

28

30

88

90

al Rate

Offered

2686 Erlang‐CErlang‐R

2484

9.5 9.6 9.7 9.8 9.9 10.0

λ(t)

Time

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

Page 24: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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 5Constrains:

Staffing resolution: 1 hourMinimal staffing: 1 doctor per typeInteger 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

Page 25: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

0.9P(W>0)

0

0.1

0.2

0 1000 2000 3000 4000 5000 6000 7000

TimeBeta=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

Page 26: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

Conclusions

In time-varying systems where patients return for multipleservices:

1 Using the MOL (IS) algorithm for staffing stabilizesperformance.

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 arrivalsTransient behavior even with constant parameters

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

Page 27: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

What next?

Fluid and diffusion approximations for mass-casualtyeventsQED - MOL approximations for the processes:

Number of customers in systemVirtual waiting time

Extension: upper limit on the number of customers withinthe system

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

Page 28: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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

What next?

The semi-open Erlang-R queue

Erlang-R: Closed Erlang-R: IW model closed network:

Needy (s-servers)

Content (Delay)

1-p

p

1

2

Arrivals Patient discharge

1-p

p

3

1

exp( ), Sμ

exp( ),δ ∞

exp( ),1λ

2

4

exp( ),γ ∞

Arrivals: Needy:

Content:

Cleaning:

N

Patient is Content

1-p

p

1

2

Arrivals

Blocked patients

Patient is Needy

N beds

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

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

Page 29: The Erlang-R Queue: Time-Varying QED Queues with …ie.technion.ac.il/serveng/References/MSOM_2010/pres_MSOM_2010.pdf · MotivationThe Erlang-R QueueResults The Erlang-R Queue: Time-Varying

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