Mathematical
Modeling of Human Performance and Cognition

[Features] 
The
goal of this document is to summarize and integrate all mathematical modeling
work that quantifies the different aspects and components of human cognition
and performance. It will also serve as a learning material platform for new
users of mathematical modeling of human performance.
1. Unique Features of
Mathematical Modeling of Human Performance and Cognition
Mathematical
equations can predict, quantify and analyze human performance, workload, brain
waves, and other indexes of human behavior in a rigorous way. Compared with
computer simulation,
1) Mathematical models and equations of human
behavior clearly quantify and extract the mechanisms of human behavior by clear
quantifications of the relationships of variables including the relationship
between the input and output of each equation. Users of these mathematical
models will be much easier to understand and extract the relationships among
variables than reading computer codes.
2) Mathematical models and equations of
human behavior can be relatively easily to be edited, modified, improved, and
integrated together to develop new mathematical equations.
3) Mathematical models and equations of
human behavior and performance can be relatively easily be implemented in
different programming languages and be imbedded in different intelligent
systems to work together with system design.
4) Mathematical models and equations can
lead to analytical solutions, which are more accurate than simulation results.
5) There are mathematical models and
equations quantifying the entire human cognition system (See equations in the
entire network) which is another unique feature of the mathematical modeling
approach.
6) There are mathematical models and
equations can be proved by mathematics derivation directly with no need to be
verified by empirical data (See Equations in Wu, C., Berman, M., & Liu, Y.,
2010).
2. Usage of Mathematical Models and
Equations in this Summary Webpage
The
equations summarized here can also serve as an index page and a guideline tool
for modelers who can:
1) Use
those mathematical models to quantify and predict new phenomena and tasks in
human performance
2) Add and develop new equations and
mathematical models to quantify new components of human cognition and
performance to further grow with the framework of the Queuing NetworkModel
Human Processor (QNMHP)
3) To be used and imbedded in different
intelligent systems and tool design for human performance and behavior
predictions
1) How to
build and verify models of human performance (General descriptions) (Wu, 2016)
2) How to
build mathematical models (e.g., Page 910) (Wu & Liu, 2008a)
3) How to integrate and build new mathematical
models in human performance modeling (Coming soon)
4) To become a member (user or contributor)
of mathematical modeling group in human performance modeling, please email to changxu.wu@gmailcom (Please list your full name and
institution/company name), we will send you recent updates, new modeling work,
and new tutorials. All of them are free.
3. Human
Machine System Design Tools based on the Equations on this Page [Link]
4. Mathematical Equations in
the Queuing NetworkModel Human Processors (QNMHP) as the Framework
The
General Structure of Queuing NetworkModel Human Processor (QNMHP)
(a.) Perceptual Subnetwork 

(b.) Cognitive Subnetwork 

(c.) Motor Subnetwork 
1. Common visual processing 2. Visual recognition 3. Visual location 4. Visual recognition and location integration 5. Common auditory processing 6. Auditory recognition 7. Auditory location 8. Auditory recognition and location integration 

A. Visuospatial sketchpad B. Phonological loop C. Central executive D. Longterm procedural memory E. Performance monitor F. Complex cognitive function G. Goal initiation H. Longterm declarative & spatial memory 

V. Sensorimotor integration W. Motor program retrieval X. Feedback information collection Y. Motor program assembling and error detecting Z. Sending information to body parts 2125 etc.: Body parts: eye, mouth, left hand,
right hand, foot 
Server Information Processing Time and
Information Processing Capacities
Server Name 
Processing Time^{a}: Exponential
Distribution (Mean, Min) (ms) 
Capacity (Entities^{a}) 

Server Name 
Processing Time: Exponential Distribution
(Mean, Min) (ms) 
Capacity (Entities^{b}) 
1 
Exp
(42, 25) 
4 

5 
Exp
(42, 25) 
2 
2 
Exp (42, 25) 
4 

6 
Exp (42, 25) 
1 
3 
Exp (42, 25) 
4 

7 
Exp (42, 25) 
1 
4 
Exp (42, 25) 
5 

8 
Exp (42, 25) 
1 
A 
Exp (18, 6) 
4 

E 
Exp (18, 6) 
Infinite^{c} 
B 
Exp (18, 6) 
4 

F 
Exp (18, 6) per cycle 
1 
C 
Exp (18, 6) 
3 

G 
Exp (18, 6) 
Infinite^{c} 
D 
Exp (18, 6)^{e} 
Infinite 

H 
Exp (18, 6)^{e} 
Infinite 
V 
Exp (24, 10) 
Infinite^{c} 

X 
Exp (24, 10) 
Infinite^{c} 
W 
Exp (24, 10) 
1^{c} 

21 (Eye Motor) 
Saccade and Fixation Time^{d} 
1 
Y 
Exp (24, 10) 
2 

22
(Mouth) 
As a function of number of syllables (Voice key closure time: 100 ms, Wu & Liu, 2008a) 
1 
Z 
Exp (24, 10) 
2 

23 (Right
Hand & Right Arm) 
Arm and hand movement time, see Fitts's Law; Finger movement
time, see (Wu & Liu, 2008b) 
1 (If one movement per time) 
25 (Right Foot) 
Foot movement time, see (Zhang, Wu, & Wan, 2016;
Zhao & Wu, 2013; Zhao, Wu, & Qiao, 2013) 
1 (If one movement per time) 

24
(Left Hand & Left Arm) 
Arm and hand movement time, see Fitts's Law; Finger movement
time, see (Wu & Liu, 2008b) 
1 (If one movement per time) 
26 (Left Foot) 
Foot movement time, see (Zhang et al., 2016; Zhao &
Wu, 2013; Zhao et al., 2013) 
1 (If one movement per time) 

27 (Head), 28 (Body),
etc. 
Head, body movement time etc.^{c} 
1 (If one movement per time) 
a. Processing speed and capacities were set
based on Model Human Processor (Card, et al., 1983), Wu et al (20082017), and
Jacobson (1999).
b. Entity is defined as the smallest
information processing unit in a given task. For example, in a typing task, one
letter is an entity. In a speech warning responding task, each short word can
be regarded as one entity. For long words in speech warning, each syllable can
be represented as one entity.
c. Needs further modeling work and
investigation.
d. See Model Human Processor (Card, et al.,
1983).
e. Also depends on level of information
retrieval (e.g., familiarity and number of time of retrieval).
Equation
Set EN1: Mental workload modeling measured by NASATLX: Equation (1012) (Wu &
Liu, 2007)
Variables 

PD 
Physical Demand 
How much physical activity was required? Was
the task easy or demanding, slack or strenuous? 
TD 
Temporal Demand 
How much time pressure did you feel due to
the pace at which the tasks or task elements occurred? Was the pace slow or
rapid? 
EF 
Effort 
How hard did you have to work (mentally and
physically) to accomplish your level of performance? 
PE 
Performance 
How successful were you in performing the
task? How satisfied were you with your performance? 
FR 
Frustration 
How irritated, stressed, and annoyed versus
content, relaxed, and complacent did you feel during the task? 
MD 
Mental Demand 
How much mental and perceptual activity was
required? Was the task easy or demanding, simple or complex? 
A 
A factor of aging (A ≥ 1) 
The value of A is directly
proportional to age, set based on literature 

Arrival rate 
The arrival rate of the
subnetwork i 

Original processing
speed 
The original processing
speed of server j for the young
adults in QNMHP 

Number of servers 
The total number of
servers in the subnetwork m 
T 
Total time of a trial 
The
total task time of each trial 
a 
Constant 
The constants in
representing the direct proportional relation between the averaged
utilizations and the subjective responses (a > 0), see the published work 
b 
Constant 
Same above 
Equation
Set EN2: Mental workload modeling measured by P300 amplitude and latency:
Equation (1011) (Wu, Liu,
& QuinnWalsh, 2008)
Variables 



Amplitude of the ERP
potential P300 

L_{i} 
Latency of the P300 

k 
Constant 
A constant in this
relationship I = kN. 
b 
Constant 
A constant in this
inverse relationship 
NE 
Amount of NE 
Modeling NE (norepinephrine) in Synaptic Transmission 

Number 
Number of information
entities 

Number 
Number of information
entities of other tasks concurrently processed in server j 

Number 
Number of processing cycles
for each of those entities at server j 

A random factor 
Normally distributed
random factor with mean being equal to zero 
r 
Distance 
Distance from the
electrical field point (the location where NE is released) to locations of
the electrodes on the scalp 

Processing times 
Processing times of task
i at the perceptual subnetwork, at
Server A or B, and at Servers C and E, respectively 
Equation
Set EN3: Bold signal in fMRI modeling: Equation (27) (Wu &
Liu, 2008a)
Variables 


CB(t) 
The integrated BOLD
signal 
Modeling of BOLD signal and
its percentage of change: The integrated BOLD (blood oxygen level dependent)
signal 
s 
Latency scale 

M 
Magnitude scale 

k,a,b 
Parameters 
k, a, and b come from
the equations of Cohen (1997) and Anderson et al. (2003), determined by the
properties of the brain regions with certain fMRI measurement techniques 
t 
The duration of each
trial 
Modeling of BOLD Signal
and Its Percentage of Change 

The length of time being
occupied at a server 
In queuing networks can
be quantified by Equation 28 (Gross & Harris, 1998): 
Equation
Set EN4: Entity route selections and skill acquisitions based on reinforcement
learning algorithms: Equation (9.79.8) (Wu, Berman,
& Liu, 2010)
Variables 


Processing speed of server i 

The minimal of processing time of server i after intensive practice 

The change of expected value of processing
time of server i from the beginning
to the end of the practice 

Learning rate of server i 

Number of entities processed by server i 
Variables 



Online Q value 
is the online Q value if entity routes
from server i to server j in t+1th transition 

Maximum Q value 
Maximum Q value routing from server j to
next k server(s) 

Processing speed 
is the reward and is the processing speed
of the server j if entity enters it at
tth transition 

Discount parameter 
The discount parameter of routing to the
next server() 

Learning rate 
The learning rate of Q online learning() 
Equation
Set EN5: Information processing speed and its variability changes in learning
process: Equation (6) (Wu &
Liu, 2008b)
Variables 


X 
Summation of processing
time of servers (Y) 

Y_{i} 
Processing time of
server i 

k 
Number of servers in the
route 


Arrival rates of
entities/information 
Equation
Set EN6: Modeling the expected utilization as the mental workload under the
time stress. Equation (2) (Cao &
Liu, 2015)
Variables 


a,b 
Parameter 
Parameters a and b are the constants in
representing the direct proportional relation between the averaged
utilizations and the subjective responses (a > 0) 

The average utilization of motor
subnetwork 
The score of PD reflects workload at the
motor component, and therefore, it is in direct proportion to the average
utilization of motor subnetwork 
Equation Set EN7: Modeling the response
time of speech warnings. Equation (11, 12, 13) (Zhang, Wu & Wan, 2016)
Variables 
Description 
Tk 
Notation of processing time of the
stimulus at Server k (k =18, A,B,C,F,H, W−Z) 
T_{6(0)}and T_{8(0)} 
The initial entity
processing time in Server 6 and Server 8, respectively 
U_{L} 
The perceived urgency as
a function of warning loudness 
U_{S} 
The perceived urgency as
a function of signal world choice 
p_{i} 
Notation of probability of a warning
stimulus traveling through a route i
(i=I or II) 
Perceptual Subnetwork
－Visual perceptual subnetwork:
Server
1. Equation Set VP1: Eye movement modeling in textual information perception:
Equation (12) (Wu &
Liu, 2008b)
Variables 
Sources 

E(FC) 
The expected position of the first
character in each chunk 
Calculation
of the Expected Position of the First Character i 
E(FP) 
The expected position of the fixation
point 


The halfrange of each chunk under
extensive practice condition 
Server
1. Equation Set VP2: Eye movement modeling in picture information perception:
Equation (3) (Lim &
Liu, 2009)
Variables 



Importance index of function k 
The relatively important function can be
given a value 1, and a value 0 is given to the other. The importance index of
function k can be calculated. 

The importance value for function k 
The importance value for function k
obtained from each pairwise comparisons, either 1 or 0. 
Sever
3. Equation Set VP3: Visual optical flow perception and speed perception:
Equation (1) (Zhao &
Wu, 2013)
Variables 


Perceived speed 
V 
Actual speed 

The current texture density 

The texture density in the last driving
scenario 

The eye height in the last driving
scenario 

The current eye height 

two constant parameters 
Sever
4. Equation Set VP4: Visual detection modeling with detection distance and
image matrix: Equation (12) (Bi,
Tsimhoni, & Liu, 2009)
Variables 


RPOT 
Square root of the number of pixels on a
target 

f 
Focal length 

S 
Size of the area of a target object. 

D 
Distance of the image
forming 
－Auditory perceptual subnetwork:
Sever
6. Equation Set AP1: Modeling the effect of loudness on speech
warning perception: Equation
(1, 11) (Zhang,
Wu, & Wan, 2016)
Variables 



The perceived urgency 
Modeling the relationship between loudness
and perceived urgency 
, 
Constants 
The relationship between intensity and
perceived urgency was quantified: = 1.33, = −0.64, 

Random factors 
distributed random factors following
distribution [0, 0.7] 
L 
Loudness level 
Variables 


The effect of loudness on reaction time 

The initial entity processing time in
Server 6 

The effect of loudness on perceived
urgency 
Sever
8. Equation Set AP2: Modeling the effect of signal word on speech
warning perception: Equation
(12) (Zhang, Wu, & Wan, 2016)
Variables 


The effect of signal word choice on
reaction time 

The entity processing time in Server 8 

The urgency level expressed by the initial
words 

The number of words in the ith speech warning 
Cognitive Subnetwork
－Server
B:
Equation Set C1: Modeling of optimal
chunking of textual information: Equation (22) (Wu & Liu, 2008b)
Variables 
Description 
Z 
Objective
function of task completion time 

Overall
duration of processing each chunk at servers after server B 
N 
Total
number of entities processed 
x 
Chunk
size 

Rate
of retrieval failure at server B 
R 
Average duration to correct an error caused by a wrongly processed
entity or character 
Equation
Set C2: Modeling of memory decay of speech information: Equation (7) ( Zhang, Wu, & Wan, 2016)
Variables 
Description 

The
probability of memory decay 

Lead
time of a speech warning 
Equation
Set C3: Modeling the probability of route choice in reinforcement learning of
the speech warnings: Equation (45) ( Zhang, Wu, & Wan, 2016)
Variables 


The
route choice error rate 

The
error rate when a speech warning travels via route i 

The
probability of a speech warning entity processed via route i 
Variables 


The
error rate of route choice 
L 
Loudness
level in dB 

Perceived
urgency level with different signal word choice 
, 
Parameters
to quantify the power law of perceived urgency and loudness 
, 
Parameters
to quantify the power law of perceived annoyance and loudness 
p_{I}, p_{II} 
Probabilities
of choosing route I (the shorter route) and route II (the longer route) 
－Server C:
Equation Set C4: Inhibiting incompatible responses
modeling: Equation (46) (Wu &
Liu, 2008a)
Variables 

T2,Ccomp and T2,Fcomp, 
Processing
times of Server C and F in the compatible conditions 
T2,Cincomp and T2,Fincomp 
Processing times of
Server C and F in the incompatible conditions 
SOA (stimulus onset
asynchrony) 
The
delay between the presentation of the stimuli of T1 and T2 
T_{k} 
Processing time at server k (k=AP,
VP, A, B, C, F, W, Y, Z, X) 
Equation Set C5: Dual task interference
modeling: Equation (89) (Lin &
Wu, 2012)
Variables 
Description 
Sources 


DL_{i} 
Delay time 


T_{i,C} 
The entity processing time needed at
Server C 


PT_{i1} 
Time lapse for the previous key to be
pressed 


I_{v} 
Inter stimulus interval 


I_{v} + TAP+ TB 
Time lapse for the entity of the ongoing
stimulus to leave Server B 


PT_{i1}(I_{v} +TAP+TB) 
The least duration that the current
stimulus needs to wait at Server C 


TC 
Cycle time at Server C 


－Server
E:
Equation
Set C6: Background noise in motor control: Equation (15) (Lin &
Wu, 2012)
Variables 



The extent of SDN added with muscle
activation level u; 
Modelling baseline errors in numerical
typing 
, 
Experimental constants 
Modelling baseline errors in numerical
typing 
u 
Muscle activation level 
Modelling baseline errors in numerical
typing 
c 
The extent of temporal noise 
c is the extent of TN which accumulates as
movement time increases 
I 
Interference index 
I was an interference index accounting for
the relative extent of the dualtask interference in background noise (CN). 
－Server F
Equation Set C7: Choice reaction modeling
in multiple tasks: Equation (B16) (Wu & Liu, 2008a)
Variables 


E(RT2) 
Expected reaction time 

SOA 
Stimulusonset asynchrony 
The time difference between the onset of
the two stimuli from two tasks 
Ti 
Processing time of servers see (Wu & Liu, 2008a) 

T_{Fst} 

Equation Set C8: Modeling the effects of
response complexity (using a single finger or multiple fingers at the same
time): Equation (113) (Lin &
Wu, 2012)
Variables 



Response time to i th stimulus with a finger
strategy under an urgency condition 


Finger strategy 

i 
Response order 

T 
Processing time 

Equation Set C9: Complex decision making
with value matrix: Equation (5) (Zhao &
Wu, 2013)
Variables 

P(t) 
Speed
choice at time t 
V(t) 
Momentary
valence 
M(n) 
Human
subjective attribute matrix 
W(t) 
Attention weight matrix 
S 
Feedback matrix 
Equation Set C10: Perceived risk modeling:
Equation (5) (Zhuang &
Wu, 2013)
Variables 

PRv 
Human perceived risk increases with higher
risk from vehicles 
PRl 
Human perceived risk increases with higher
risk from localdefined risk 
a_{g} 
A coefficient adjusting effect of group
size of human 
N_{group} 
Group size of human 
Equation Set C11: Decision making in
lateral control: Equation (1, 2) (Bi, Gan,
Shang, & Liu, 2012)
Variables 


Increment of steering angle 
k_{p}, k_{d} 
The coefficients of proportional
derivative controller 
a'_{y} 
The first derivative of acceleration 
E 
Error
between the desired lateral position gained with the predefined desired path
and predictive lateral position computed with the internal vehicle dynamics
model 
v 
Current
lateral velocity 
t_{p} 
Preview time 
Equation Set C12: Modeling hazard
evaluation accuracy: Equation (8, 16, 21) (
Zhang, Wu, & Wan, 2016)
Variables 



The effect of hazard evaluation accuracy
on error rate 


Perceived value of hazard 


Actual value of hazard 


Estimated distance 


Threshold of perceived distance 


Actual distance between the current
position of warning receiving vehicle 

v(t) 
Instant speed 
The instant speed (v) and acceleration
(at) at time t is modeled in [23] as follows: 

Global optic flow rate of the textured
ground surface 
φ is the global optic flow rate of the
textured ground surface, a proportion of speed as long as eye height is
constant 
k 
Parameter 
The parameter k is quantified by the
annual mileage divided by a maximum value of annual mileage in general 

Perceived timetocollision 
The perceived timetocollision (TTCp)
will be affected by the existence of the lead vehicle. TTC is the actual time
to collision that the vehicle will be able to avoid a collision without
exceeding the assumed maximum deceleration 
LV 
Lead vehicle status 
LV is a dichotomous variable of the lead
vehicle in order to model the effect of the lead vehicle on TTCp (0 =
without lead vehicle; 1 = with lead vehicle) 

Lead time of speech warning 
－Server G:
Equation
Set C13: Urgency and Motivation Modeling: Equation (12) (Lin &
Wu, 2012)
Variables 



Response time to i th stimulus with a
finger strategy under an urgency condition 

RT 
Reaction time 

DL 
Delay time caused by dualtask
interference 

MT 
Movement time 


Keyclosure Time 


Finger strategy 
notation of finger
strategy. =0 →Single finger typing; =1 →Multifinger typing 

Urgency 
notation of urgency. =1→nonurgent condition; =0→urgent condition 
i 
Response order 
notation of response
order. i=1→first
response in 9digit number, and so on. 
Motor Subnetwork
－Server W:
Equation Set M1: Motor program retrieval
modeling in the learning process: Equation (2) (Wu & Liu, 2008b)
Variables 



Processing
time in each server 
Reduction
of Server Processing Time. 

Expected
minimal processing time (Ti) at server i after intensive practice 
Feyen
(2002) 

Change
in the expected processing time from the beginning to
the end of practice 
Reduction
of Server Processing Time. 

Learning
rate of server i 
Heathcote
et al. (2000) 

Number
of entities processed by server i 
Reduction
of Server Processing Time. 
－Server X:
Equation Set M2: Error correction modeling
in closeloop motor control: Equation (2432) (Lin & Wu, 2012)
Variables 



The
uncorrected portion of endpoint variability 
Endpoint
variability in different conditions 
MT 
Movement
time 
Modelling
response time of numerical typing: the general equation of response time 
DL 
Delay
time caused by dualtask interference 
Modelling
response time of numerical typing: the general equation of response time 
Err% 
Estimations
of error rates 
Estimations
of error rates (Err%) in jth typing
conditions 
P(), P() 
Parameters 
The probability of errors in Xdirection
and Ydirection during jth experimental
condition 
－Hand Servers (Server 23, 24):
Equation Set M3: Hand and finger movement time and errors in QWERTY keyboard typing: Equation (19)