1 Queueing Analysis of Production Systems (Factory Physics)

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Queueing Analysis of Production Systems

(Factory Physics)

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Reading Material

Chapter 8 from textbook

Handout: Single Server Queueing Model by Wallace Hopp (available for download from class website)

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Queueing analysis is a tool for

evaluating operational performance Utilization

Time-in-system (flow time, leadtime)

Throughput rate (production rate, output

rate)

Waiting time (queueing time)

Work-in-process (number of parts or batches

in the systems)

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A Single Stage System

Raw material

Processing unit

Finished parts

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The Queueing Perspective

Server (production facility)

Queue (logical or physical) of

jobs

Arrival (release) of

jobs

Departure (completion) of

jobs

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E[A]: average inter-arrival time between

consecutive jobs : arrival rate (average number of jobs that arrive per unit

time), = 1/E[A] E[S]: average processing time : processing rate (maximum average number of jobs that can be processed per unit time), = 1/E[S]

: average utilization, = E[S]/E[A] = /

System Parameters

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E[W]: average time a job spends in the

system

E[Wq]: average time a job spends in the

queue

E[N]: average number of job in the system (average WIP in the system)

E[Nq]: average number of jobs in the queue (average WIP in the queue)

TH: throughput rate (average number of jobs produced per unit time)

Performance Measures

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E[W] = E[Wq] + E[S]

E[N] = E[Nq] +

Performance Measures (Continued…)

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Little’s Law

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E[N] = E[W]E[Nq] = E[Wq]

= E[S]

Little’s Law

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Example 1

Jobs arrive at regular & constant intervals

Processing times are constant Arrival rate < processing rate ( < )

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Example 1

Jobs arrive at regular & constant intervals Processing times are constant Arrival rate < processing rate ( < )

E[Wq] = 0

E[W] = E[Wq] + E[S] = E[S] = / E[N] = E[W] = E[S] = E[Nq] = E[Wq] = 0 TH =

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Case 2

Jobs arrive at regular & constant intervals

Processing times are constant Arrival rate > processing rate ( > )

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Case 2

Jobs arrive at regular & constant intervals Processing times are constant Arrival rate > processing rate ( > )

E[Wq] = E[W] = E[Wq] + E[S] = = 1E[N] = E[W] = E[Na] = E[Wa] = TH =

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Case 3

Job arrivals are subject to variability Processing times are subject to variability Arrival rate < processing rate ( < )

Example:

Average processing time = 6 min Inter-arrival time = 8 min

0.25y probabilitmin with 2

0.50y probabilitmin with 6

0.25y probabilitmin with 10

timeProcessing

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Case 3 (Continued…)

= 6/8 = 0.75 TH = 1/8 job/min = 7.5 job/hour E[Wq] > 0 E[W] > E[S] E[Nq] > 0 E[N] >

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In the presence of variability, jobs may wait for processing and a queue in front of the processing unit may build up.

Jobs should not be released to the system at a faster rate than the system processing rate.

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Sources of Variability

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Sources of Variability

Sources of variability include: Demand variability Processing time variability Batching Setup times Failures and breakdowns Material shortages Rework

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Measuring Variability

Var[ ] Variance of processing time

Var[ ] Variance of inter-arrival time

Var[ ]Coefficient of variation (CV) in processing time

[ ]

Var[A]Coefficient of variation (CV) in inter-arrival time

[ ]

S

A

S

A

SC

E S

CE A

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Variability Classes

0.75

High variability(HV)

Moderate variability(MV)

Low variability(LV)

0 1.33CV

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Illustrating Processing Time VariabilityTrial Machine 1 Machine 2 Machine 3

1 22 5 5 2 25 6 6 3 23 5 5 4 26 35 35 5 24 7 7 6 28 45 45 7 21 6 6 8 30 6 6 9 24 5 5

10 28 4 4 11 27 7 7 12 25 50 500 13 24 6 6 14 23 6 6 15 22 5 5

Mean 25.1 13.2 43.2 Std dev 2.5 15.9 127.0

CV 0.1 1.2 2.9 CV 0.01 1.4 8.6

Class LV MV HV

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Illustrating Arrival Variability

t

Low variability arrivals

t

High variability arrivals

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The G/G/1 Queue

If (1) < , (2) the distributions of job processing and inter-arrival times are independent and identically distributed (iid), and (3) jobs are processed on a first come, first served (FCFS) basis, then average waiting time in the queue can be approximated by the “VUT” formula:

2 2

E[W ]

[ ]2 1

q

A S

V U t

C CE S

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Example

CA = CS = 1 E[S] = 1 Case 1: = 0.50 E[W] = 2, E[N] = 1 Case 2: = 0.66 E[W] = 3, E[N] = 1.98 Case 2: = 0.75 E[W] = 4, E[N] = 3 Case 1: = 0.80 E[W] = 5, E[N] = 4 Case 1: = 0.90 E[W] = 10, E[N] = 9 Case 1: = 0.95 E[W] = 20, E[N] = 19 Case 1: = 0.99 E[W] = 100, E[N] = 99

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Example

CA = 1 E[S] = 1 = 0.8 Case 1: CS = 0 E[W] = 3, E[N] = 2.4 Case 2: CS = 0.5 E[W] = 4, E[N] = 3.2 Case 1: CS = 1 E[W] = 5, E[N] = 4 Case 1: CS = 1.5 E[W] = 6, E[N] = 4.8 Case 1: CS = 2 E[W] = 7, E[N] = 5.6

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Facilities should not be operated near full capacity.

To reduce time in system and WIP, we should allow for excess capacity or reduce variability (or both).

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A Single Stage System with Parallel facilities

Servers (production facilities)

Queue (logical or physical) of

jobs

Arrival (release) of

jobs

Departure (completion) of

jobs

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The G/G/m Queue

If (1) < m, (2) the distributions of job processing and inter-arrival times are independent and identically distributed (iid), and (3) jobs are processed on a first come, first served (FCFS) basis, then average waiting time in the queue can be approximated by the “VUT” formula:

2( 1) 12 2 ( / )[ ]

2

mA S

q

C C mE W

m

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Increasing Capacity

Capacity can be increased by either increasing the production rate (decreasing processing times) or increasing the number of production facilities

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Increasing Capacity

Capacity can be increased by either increasing the production rate (decreasing processing times) or increasing the number of production facilities

In a system with multiple parallel production facilities, maximum throughput equals the sum of the production rates

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Dedicated versus Pooled Capacity

Dedicated system: m production facilities, each with a single processor with production rate and arrival rate

Pooled system: A single production facility with m parallel processors, with production rate per processor, and arrival rate m

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Dedicated versus Pooled Capacity

Dedicated system:

2 2

[W ]2

A S

q

C CE

2( 1) 12 2 ( / )[ ]

2 ( )

mA S

q

C C mE W

m

Pooled system:

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Pooling reduces expected waiting time by more than a factor of m

Pooling makes better use of existing capacity by continuously balancing the load among different processors

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The M/M/1 Queue

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GX/GY/k/N

A Common Notation

G: distribution of inter-arrival

times

X: distribution of arrival batch

(group) size

G: distribution of service times

Y: distribution of service batch

size

k: number of servers

N: maximum number of customers

allowed

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Common examples

M/M/1M/G/1 M/M/k M/M/1/NMX/M/1GI/M/1M/M/k/k

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Notation in the Book versus Notation in the Lecture

Notes CT (cycle time): E(W)

CTq (cycle time in the queue): E(Wq)

WIP: E(N)

WIPq (WIP in the queue): E(Nq)

u: U (=/)

ra:

te: E(S); ts: E(X);

ca: cA

ce: cS

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A single server queue The distribution of inter-arrival times

is exponential (Markovian arrivals) The distribution of processing times is

exponential (Markovian processing times)

Assumptions

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Distribution of Inter-arrival Times

0 0

( ) : density function for the time interval between any two

successive arrivals

( ) , 0

1[ ] ( )

Pr( ) ( )

t

t

t T

T T

f t t

f t e t

E A tf t dt t e dt

t T f t dt e dt e

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The Memoryless Property

( )Pr( )Pr( | ) Pr( )

Pr( )

T hh

T

t T h et T h t T e t h

t T e

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The Taylor Series Expansion

2 3( ) ( )Pr(no arrivals in any interval of length ) 1 ...

2! 3!

Pr(no arrivals in any interval of length ) 1 (when is small)

Pr(a single arrival in any interval of length )

hh h

h e h

h h h

h h

(when is small)h

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Exponential Inter-arrival Times and the Poisson Process

( )Pr( arrivals in an interval of length ) , =0,1,2...

!

n TT en T n

n

Poisson distribution

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Distribution of Processing Times

0 0

( ) : density function for the time to process any job

( ) , 0

1[ ] ( )

t

t

g t t

g t e t

E S tg t dt t e dt

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Similarly, when h is small,

Pr(processing time is not completed in interval of length ) 1

Pr(processing time completes in interval of length )

h h

h h

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The Distribution of the Number of Jobs in the System

( ) Pr( ( ) ), probability that customers are in

the system at time nP T N T n n

T

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The Distribution of the Number of Jobs in the System

1

1

( ) Pr( ( ) ), probability that customers are in

the system at time

P ( ) ( )(1 ) ( ) (1 )(1 ) ( )

( )( ) ( ) (1 )( ) ( )

n

n n n

n n

P T N t n n

t

T h h h P T h h P T

h h P T h h P T

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0 1 1

00 1

P ( ) ( ) Plim ( ) ( ) ( ) ( ), for >0

P( ) ( ), for =0

n n nh n n n

T h P T dP T P T P T n

h dT

dP T P T n

dT

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1 1

0 1

If 1, we can show that lim P ( ) , where is a constant

0

0 ( ) , for 1,2,3,...

0 , for =0

T n n n

n

n n n

T P P

dP

dT

P P P n

P P n

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10 2

3

0 1

1 2 0

State rate out of state rate into state

0

1 ( )

2

j j

P P

P P P

2 3 1

1 1

( )

1 ( ) n n n

P P P

n P P P

The Birth-Death Model

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1 0 0 0

00 00 0

( )

1 =1- 1

(1 )

nn

nnn n

nn

P P P P P

PP P P

P

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0

0

20

20

2

: number of customers in the system (in the long run), a random variable

( ) Pr( )

(1 )

1

Var( ) Pr( )

(1 )

(1 )

n

nn

n

nn

N

E N n N n

n

N n N n

n

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1

0

2

Pr( ) Pr( )

1 Pr( )

( ) ( )

1

1

n s

s

n

s

q

N s N n

N n

E N E N

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( )( )

(1 )

1

E NE W

2

( )( )

(1 )

( )(1 ) 1

qq

E NE W

E S

Applying Little’s law

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The M/G/1 Queue

A single server queue The distribution of inter-arrival times

is exponential (Markovian arrivals) The distribution of processing times is

general

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The M/G/1 Queue (Continued…)

2

2 2

1[ ] [ ]

2 1

1[ ]

2 1

[ ] [ ] [ ]

[ ] [ ]

Sq

Sq

q

q

CE W E S

CE N

E W E W E S

E N E N

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The G/G/1 Queue Revisited

2 2 2 2

2 2 2

[ ] [ ]2 1 2

[ ]2 1

[ ] [ ] [ ]

[ ] [ ]

A S A Sq

A Sq

q q

q

C C C CE W E S

C cE N

E W E W E S

E N E N

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The M/M/m Queue

A queue with m servers The distribution of inter-arrival times

is exponential (Markovian arrivals) The distribution of processing times is

exponential (Markovian arrivals)

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The Balance Equations

0 1

1 1

1 1

, for =0

( 1) =( ) for 1

=( ) for n n n

n n n

P P n

P n P n P n m

P m P m P n m

Using analysis similar to the one for the M/M/1 queue:

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The Birth-Death Model

0 0 1 1

1 1 1 2 2 0 0

State rate out of state rate into state

0

1 ( )

2

j j

P P

P P P

2 2 2 3 3 1 1

1 1 1 1

( )

( )

n n n n n n n

P P P

n m P P P

10 2 3

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The / / queue is a birth and death process with

if

if

n

n

M M m

n n m

m n m

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0

0

01

0

, for 0!

, for !

1

[ ]! !( )

n

n

n

n n m

j mm

j

P P n mn

P P n mm m

P

j m mm

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0Pr( )!(1 )

[ ] Pr( )

1[ ] Pr( )

, average utilization per server

m

q

q

N m Pm

m

E N N mm

E W N mm

Um

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The G/G/m Queue

2( 1) 12 2

[ ]2

mA S

q

C C UE W

m

For a queue with a general distribution for arrivals and processing times, average time in the queue can be approximated as

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Notation in the Book versus Notation in the Lecture

Notes CT (cycle time): E(W)

CTq (cycle time in the queue): E(Wq)

WIP: E(N)

WIPq (WIP in the queue): E(Nq)

u: U

ra:

te: E(S); ts: E(X);

ca: cA

ce: cS

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Propagation of Variability

Single server queue:

Multi-server queue:

2 2 2 2 2(1 )

/D S AC U C U C

U

22 2 2 21 (1 )( 1) ( 1)

/( )

D A S

UC U C C

mU m

CD(i) = CA(i+1)i i+1

CS(i)CA(i)

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Propagation of Variability

High Utilization Station

High Process Var

Low Flow Var High Flow Var

Low Utilization Station

High Process Var

Low Flow Var Low Flow Var

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Propagation of Variability (Continued…)

High Utilization Station

Low Process Var

High Flow Var Low Flow Var

Low Utilization Station

Low Process Var

High Flow Var High Flow Var

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Variability Relationships

, CD2 , CA

2

E(S), CS2

2 2

Queue Time

[ ] ( )2 1

A Sq

C C UE W E S

U

2 2 2 2 2

Flow Variability

(1 )

/D A SC U c U C

U

Processing

Time

( )E S

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If utilization is low, reduce arrival arrival variability; if utilization is high, reduce process variability.

Operations with the highest variability should be done as late as possible in the production process.

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A Production Line

N: number of stages in the production line Si: processing time in stage i (a random

variable), i=1,…, N U(i): utilization at stage i CA(i): coefficient of variation in inter-

arrival times to stage i

CS(i): coefficient of variation in

processing time at stage i

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Time in System for a Production Line

E[Wq(i)]= V(i) U(i) E[S(i)]

2 2( ) ( )

2A SC i C i

( )[ ( )]

1 ( )

U iE S i

U i

1 1[ ] [ ( )] [ ( )] [ ( )]

N N

qi iE W E W i E W i E S i

( ) [ ]ii

U i E S

73

Time in System for a Production Line

2 2( ) ( )

2A SC i C i

( )[ ( )]

1 ( )

U iE S i

U i

( ) [ ]ii

U i E S

2 2 2 2 2( 1) (1 ( ) ) ( ) ( ) ( )A A SC i U i C i U i C i

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Reducing Time in System (Cycle Time)

2 2( ) ( )

2A SC i C i

( )[ ( )]

1 ( )

U iE S i

U i

Reduce Variability • failures• setup times• uneven arrivals• process control• worker training

Reduce Utilization • arrival rate (yield, rework, etc.)• processing time (processing speed, availability)• capacity (number of machines)

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Expected WIP in System for a Production Line

1

1

1

[ ] [ ]

[ ( )]

[ ( )]

[ ( )] ( )

N

i

N

i

N

qi

E N E W

E W i

E N i

E N i U i