A differential equation model for multi-class, multi-server queue - - PowerPoint PPT Presentation

SANUM Conference March 2016

A differential equation model for multi-class, multi-server queue networks with time dependent parameters.

Emma Gibson Prof SE Visagie

Operations Research Division, Department of Logistics, Stellenbosch University

Table of contents

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1

Introduction

2

Model

3

Queueing theory

4

DE model

5

Results

6

Conclusion

Introduction Model Queueing theory DE model Results Conclusion References

Zithulele Hospital

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Introduction Model Queueing theory DE model Results Conclusion References

Zithulele Hospital

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Introduction Model Queueing theory DE model Results Conclusion References

Hospital services

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Zithulele Hospital

Clinics

Doctors Nutrition Dental Education Optometry

Zithulele Hospital

In-patients

Maternity Surgical Paediatrics TB

Clinics

Doctors Nutrition Dental Education Optometry

Zithulele Hospital

Casualty

Fractures Burns Minor surgery

In-patients

Maternity Surgical Paediatrics TB

Clinics

Doctors Nutrition Dental Education Optometry

Zithulele Hospital

Out- patients

Doctors Dispensary Therapy Blood tests X-rays

Casualty

Fractures Burns Minor surgery

In-patients

Maternity Surgical Paediatrics TB

Clinics

Doctors Nutrition Dental Education Optometry

Introduction Model Queueing theory DE model Results Conclusion References

Aims

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Minimise patient waiting times

Introduction Model Queueing theory DE model Results Conclusion References

Aims

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Minimise patient waiting times

• 1. Understand the queueing process:

Detailed model Data

Introduction Model Queueing theory DE model Results Conclusion References

Aims

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Minimise patient waiting times

• 1. Understand the queueing process:

Detailed model Data

• 2. Make practical decisions:

Ongoing feedback Accessible with matric maths

Introduction Model Queueing theory DE model Results Conclusion References

General model

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Patient profiles

Respiratory infections Maternity Certificates Fractures Burns MVA ARV TB Diabetes

Processes

Clerk Vitals Triage Blood tests Xrays Doctor Pharmacy Dentist Therapy

Interaction

Introduction Model Queueing theory DE model Results Conclusion References

Queueing theory

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Vitals Clerk Triage Blood X-rays Doctor Pharmacy

Introduction Model Queueing theory DE model Results Conclusion References

Queueing theory

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qi: number of patients in the ith queue R: routing matrix

2 1 3 4 5 6 7

R1,2 R2,3 R3,4 R2,4 R5,6 R3,5 R3,6 R6,7 R4,6

Introduction Model Queueing theory DE model Results Conclusion References

Queueing theory

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λi: arrival rate µi: service rate

qi

λi µi

Introduction Model Queueing theory DE model Results Conclusion References

Patient profiles

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λp

i : arrival rate for profile p

µp

i : service rate for profile p

q1

i

. . . qm

i λ1

i

µ1

i

. . . . . . λm

i

µm

i

Introduction Model Queueing theory DE model Results Conclusion References

Problems

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Multi-class queues (Kelly network) Non-stationary arrival rates λp

i (t)

Time-dependent servers si(t) ∈ N0 Transient queues - variation in traffic intensity Large state space

Introduction Model Queueing theory DE model Results Conclusion References

Fluid approximations

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Deterministic model for expected queue length First-order DE’s Approximate discrete queues with continuous functions Represent arrivals/exits with continuous (mean) flows

Introduction Model Queueing theory DE model Results Conclusion References

Equations

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qi(t) =

m

• p=1

qp

i (t)

dqp

i

dt = λp

i (t) − µp i (t)

Introduction Model Queueing theory DE model Results Conclusion References

Arrival rate

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λp

i (t) = αp i (t) + n

• j=1

Rp

j,iµp j (t)

αp

i : external arrival rate for profile p

Rp

j,i: probability of moving from process j → i

qp

i

λp

i

µp

i

Rp

1,iµp 1

. . .

Rp

j,pµp j

. . .

Rp

n,iµp n

αp

i

Introduction Model Queueing theory DE model Results Conclusion References

Traffic intensity

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ρi(t) = m

p=1 λp i (t)τ p i

si(t) τ p

i : minutes to treat patient type p at process i

si(t): staff on duty at process i Case 1: no backlog ρi(t) ≤ 1 and m

p=1 qp i (t) = 0

Case 2: backlog ρi(t) > 1

• r

m

p=1 qp i (t) > 0

Introduction Model Queueing theory DE model Results Conclusion References

State function

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φi(t) =

• 1

m

p=1 λp i (t)τ p i ≤ si(t)

&& qi(t) = 0

• therwise

Introduction Model Queueing theory DE model Results Conclusion References

Service rate: no backlog

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Patients are treated on arrival: µp

i (t) = λp i (t)

Introduction Model Queueing theory DE model Results Conclusion References

Service rate: backlog

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New arrivals join the queue. Staff must divide their time between different patient profiles: µp

i (t) = βp i (t)si(t)

τ p

i

Introduction Model Queueing theory DE model Results Conclusion References

Service rate: backlog

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New arrivals join the queue. Staff must divide their time between different patient profiles: µp

i (t) = βp i (t)si(t)

τ p

i

Weights βp

i are proportional to the number of patients and their

treatment needs: βp

i (t) = ci(t)qp i (t)τ p i

Introduction Model Queueing theory DE model Results Conclusion References

Service rate: backlog

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New arrivals join the queue. Staff must divide their time between different patient profiles: µp

i (t) = βp i (t)si(t)

τ p

i

Weights βp

i are proportional to the number of patients and their

treatment needs: βp

i (t) = ci(t)qp i (t)τ p i

ci(t) = 1 m

p=1 qp i (t)τ p i

Introduction Model Queueing theory DE model Results Conclusion References

Service rate

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µp

i (t) = φiλp i (t) + (1 − φi)

si(t)qp

i (t)

m

k=1 qk i (t)τ k i + φi

φi(t) =

• 1

m

p=1 λp i (t)τ p i ≤ si(t)

&& qi(t) = 0

• therwise.

Introduction Model Queueing theory DE model Results Conclusion References

Initial DE model

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dqp

i

dt = λp

i (t) − µp i (t)

λp

i (t)

= αp

i (t) + n j=1 Rp j,iµp j (t)

µp

i (t)

= φiλp

i (t) + (1 − φi)

si(t)qp

i (t)

m

k=1 qk i (t)τ k i + φi

φi(t) =

• 1

m

p=1 λp i (t)τ p i ≤ si(t)

&& qi(t) = 0

• therwise.

Introduction Model Queueing theory DE model Results Conclusion References

Initial DE model

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Unknown functions: qp

i , λp i , µp i , φi

Function Units Equation qp

i

Patients Differential λp

i

Patients/time Algebraic µp

i

Patients/time Algebraic φi Binary Algebraic

Introduction Model Queueing theory DE model Results Conclusion References

Substitutions

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λp

i (t) =dΛp i

dt Λp

i (t) =

t λp

i (x)dx

µp

i (t) =dUp i

dt Up

i (t) =

t µp

i (x)dx

Introduction Model Queueing theory DE model Results Conclusion References

Substitutions

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dqp

i

dt = dΛp

i

dt − dUp

i

dt dΛp

i

dt = αp

i (t) + n j=1 Rp j,i

dUp

i

dt dUp

i

dt = φi

• αp

i (t) + n j=1 Rp j,i

dUp

j

dt

• + (1 − φi)

si(t)qp

i (t)

m

k=1 qk i (t)τ k i + φi

φi(t) =    1 m

k=p

dΛp

i

dt τ p

i ≤ si(t)

&& qi(t) = 0

• therwise.

Introduction Model Queueing theory DE model Results Conclusion References

Substitutions

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qp

i (t) =

Λp

i (t) − Up i (t)

Λp

i (t) =

Ap

i (t) + n j=1 Rp j,iUp i (t)

Introduction Model Queueing theory DE model Results Conclusion References

Substitutions

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dUp

i

dt = φi

• αp

i (t) + n j=1 Rp j,i

dUp

j

dt

• + (1 − φi)

si(t)qp

i (t)

m

k=1 qk i (t)τ k i + φi

φi(t) =    1 m

p=1

dΛp

i

dt τi(t) ≤ si(t) && qi(t) = 0

• therwise.

Introduction Model Queueing theory DE model Results Conclusion References

Substitutions

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dUp

i

dt = φi

• αp

i (t) + n j=1 Rp j,i

dUp

j

dt

• + (1 − φi)

si(t)qp

i (t)

m

k=1 qk i (t)τ k i + φi

φi(t) =    1 m

p=1

dΛp

i

dt τi(t) ≤ si(t) && qi(t) = 0

• therwise.

qp

i (t)

→ Ap

i (t) + n j=1 Rp j,iUp i (t) − Up i (t)

dΛp

i

dt → αp

i (t) + n j=1 Rp j,i

dUp

i

dt

Introduction Model Queueing theory DE model Results Conclusion References

Final DE model

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dUp

i

dt =φi  αp

i (t) + n

• j=1

Rp

j,i

dUp

j

dt   + (1 − φi)   si(t)

• Ap

i (t) + n j=1 Rp j,iUp j − Up i

• m

k=1 τ k i

• Ak

i (t) + n j=1 Rk j,iUk j − Uk i

• + φi

  φi(t) =              1 m

p=1(αp i (t) + n j=1 Rp j,i

dUp

i

dt )τ p

i ≤ si(t)

&& Ap

i (t) + n j=1 Rp j,iUp j − Up i = 0

• therwise.

Introduction Model Queueing theory DE model Results Conclusion References

Numerical solution

28/33 1 Solve DE’s for Up

i and φi:

Initialise t = 0, Up

i (0) = 0 and φi(0) = 0.

Introduction Model Queueing theory DE model Results Conclusion References

Numerical solution

28/33 1 Solve DE’s for Up

i and φi:

Initialise t = 0, Up

i (0) = 0 and φi(0) = 0.

1

Calculate Up

i (t + ∆t)

2

Update φi(t + ∆t)

Introduction Model Queueing theory DE model Results Conclusion References

Numerical solution

28/33 1 Solve DE’s for Up

i and φi:

Initialise t = 0, Up

i (0) = 0 and φi(0) = 0.

1

Calculate Up

i (t + ∆t)

2

Update φi(t + ∆t)

3

If φi(t + ∆t) = φi(t):

t = t + ∆t

Introduction Model Queueing theory DE model Results Conclusion References

Numerical solution

28/33 1 Solve DE’s for Up

i and φi:

Initialise t = 0, Up

i (0) = 0 and φi(0) = 0.

1

Calculate Up

i (t + ∆t)

2

Update φi(t + ∆t)

3

If φi(t + ∆t) = φi(t):

t = t + ∆t

4

If φi(t + ∆t) = φi(t):

Locate discontinuity at t + δ Calculate Up

i (t + δ)

Calculate φi(t + δ+) t = t + δ

Introduction Model Queueing theory DE model Results Conclusion References

Results: Queue length

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Introduction Model Queueing theory DE model Results Conclusion References

Results: Queue length

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Introduction Model Queueing theory DE model Results Conclusion References

Results: Rate of change

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Introduction Model Queueing theory DE model Results Conclusion References

Results: Rate of change

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Introduction Model Queueing theory DE model Results Conclusion References

Results: Accuracy

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Introduction Model Queueing theory DE model Results Conclusion References

Results: Accuracy

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Introduction Model Queueing theory DE model Results Conclusion References

Conclusion

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DE results give less information than simulations Fairly accurate for long queues/high traffic intensity Can usually predict queue growth

Introduction Model Queueing theory DE model Results Conclusion References

References

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Springer Science & Business Media, 2013. [3] C. Duguay and F. Chetouane. Modeling and improving emergency department systems using discrete event simulation. Simulation, 83(4):311–320, 2007. [4] L. Moreno, R. M. Aguilar, C. Martin,

• J. D. Pi˜

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• Geneva. Technical Program,

Conference Record, IEEE International Conference on, volume 1, pages 352–358. IEEE, 1993. [6] J. A. White. Analysis of queueing systems. Elsevier, 2012.