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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
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
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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