Queuing system 'Expections' their affects and non-queue state : a new stochastic approach.
UNIVERSITI TEKNIKAL MALAYSIA MELAKA
Faculty of Electronics and Computer Engineering
QUEUING SYSTEM 'EXCEPTIONS', TIIEIR AFFECTS AND NONQUEUE STATE: A NEW STOCHASTIC APPROACH
Harpreet Singh s/o Bhagat Singh Amd
Master of Science in Electronic Engineering
2014
© Universiti Teknikal Malaysia Melaka
QUEUING SYSTEM 'EXCEPTIONS', THEffi AFFECTS AND NON-QUEUE
STATE: A NEW STOCHASTIC APPROACH
HARPREET SINGH S/O BHAGAT SINGH AZAD
A thesis submitted
in fulfillment of the requirement for the degree of Master of Science
in Electronic Engineering
Faculty of Electronics and Computer Engineering
UNIVERSITI TEKNIKAL MALAYSIA MELAKA
2014
© Universiti Teknikal Malaysia Melaka
APPROVAL
I hereby declare that I have read this thesis and in my opinion this thesis is sufficient in
terms of scope and quality for the award of Master of Science in Electronic Engineering.
Signature:
Supervisor Name: Professor Dr. Muhammad Ghazie Ismail
Date: 15 May 2014
. ;' ••:.· ·
© Universiti Teknikal Malaysia Melaka
DECLARATION
I declare that this thesis entitle "Queuing System 'Exceptions', Their Affects and Non-Queue
State: A New Stochastic Approach" is the result of my own research except as cited in the
references. The thesis has not been accepted for any degree and is not concurrently
submitted in candidature of any other degree.
Name: Harpreet Singh s/o Bhagat Singh Azad
Date: 15 July 2014
© Universiti Teknikal Malaysia Melaka
ABSTRACT
Most queuing models are based on organized behavior of customers served by a server or
multiple servers. However, there are many instances that the queuing model does not stand
by real world queue ethics. The consequences can go towards non-queue state caused by
overcrowding and cultural background of the customers. This research portrays the
successive outcomes that sourced non-queue state due to customer ignored behaviors as
well as the unsystematic policy of a single serving system which becomes the main
research problem. Such activities are termed as 'exceptions' and their disturbances in
queuing system. The main objective of this research is to present 'exceptions', their affects
and to resolve the non-queue state. It is important so that the waiting time distribution,
behaviour of the system during 'exceptions' and queuing system can be evaluated,
demonstrated and reinstated to a single server queuing state, respectively. A non-queue is a
state addressed when system utilization goes beyond I 00 percent. Furthermore, this would
indicate no formal queue at all as customers could be assumed at random locations. This
research will explain the state of the system during 'exception' and non-queue
transformation largely with the support of probability theory in time index values. Various
'exceptions' categories with constructed waiting time probability models and their results
have proven that the customers large waiting do actually overcrowd the queuing system.
This thesis discusses 'exceptions' affects through the Markov chain, random walk and
birth-death process. The above concepts illustrate system behaviour, as 'exception' that
disrupts services. In exceptional case, stochastic matrix considered two types such as actual
job and 'exception' to occur during state transition. Their limitation is further simplified by
random walk in convolution idea. The system parameters such as late departures, queue
loss and others are affirmed in continuous-time by birth-death process. Simulation of
change of discrete-event of queuing system such as service times shows, in long-run on
averages service time fluctuates, utilization gets large, service finishing late and customers
get increased in the system causing non-queue state. A platform of dissimilar manners used
by stochastic process in non-queue transformation analysis. One of the manner employed
are Markov conditional selection procedure is presented to recover queuing system through
selecting and ordering random states. In other manner, random state(s) proceeds to get in
order are expanded by proceeding walk and multinomial-pure death view. The interrelation
of such models is grounded through the random states and distance decays by an interval.
All such manners in stochastic process have likely to bring into being a queuing model
after non-queue transformation. This research finds that the stochastic process has
tremendous potential to model 'exception' outcomes and to restore the 'exception'
eventual affects.
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ABSTRAK
Kebanyakan model baris-gilir adalah berdasarkan kepada tingkah laku terancang para
pelanggan yang dilayan oleh satu pelayan atau pelayan berganda. Sungguhpun begitu,
terdapat banyak contoh di mana model baris-gilir tidak berdiri dengan etika giliran dunia
yang sebenar. Kesannya boleh sampai kepada keadaan bukan baris-gilir yang disebabkan
oleh kesesakan dan latar belakang budaya para pelanggan. Kajian ini menggambarkan
hasil berturut-turut yang bersumberkan keadaan bukan-bergilir kerana pengabaian
perlakuan pelanggan serta dasar tidak sistematik sistem layanan tunggal yang menjadi
masalah utama penyelidikan. Aktiviti sedemikian disebut sebagai 'pengecualian' dan
gangguannya dalam sistem baris-gilir. Objektif utama penyelidikan ini adalah untuk
membentangkan 'pengecualian-pengecualian ', kesannya dan penyelesaian kepada
keadaan tidak berbaris-gilir. la adalah penting agar taburan masa menunggu, tingkah
laku sistem semasa 'pengecualian-pengecualian' dan sistem baris-gilir masing-masing
boleh dinilai, ditunjukkan dan dinyatakan kembali kepada keadaan bergilir layanan
tunggal. Bukan-bergilir adalah keadaan yang dinyatakan apabila penggunaan sistem
melampaui 100 peratus. Tambahan pula, ini boleh menunjukkan tiada baris-gilir formal
langsung kerana para pelanggan boleh diandaikan berada di lokasi-lokasi rawak Kajian
ini akan menjelaskan keadaan sistem semasa 'pengecualian 'dan transformasi bukanbergilir dengan sokongan besar oleh teori kebarangkalian dalam nilai indeks masa.
Pelbagai kategori 'pengecualian-pengecualian' dengan pembentukan model-model
kebarangkalian masa menunggu dan keputusannya telah membuktikan bahawa penantian
besar para pelanggan sebenarnya menyesakkan sistem baris-gilir. Tesis mz
membincangkan kesan 'pengecualian-pengecualian' menerusi rantaian Markov,
perjalanan rawak dan proses kelahiran-kematian. Konsep-konsep di atas menggambarkan
sistem tingkah laku sebagai 'pengecualian' yang mengganggu perkhidmatan. Dalam kes
yang istimewa, matrik stokastik mengambil kira duajenis keadaan yakni kerja sebenar dan
'pengecualian' untuk berlaku semasa peralihan tahap. Limitasi mereka kemudiannya
dipermudahkan dengan perjalanan rawak dalam idea kekusutan. Parameter-parameter
sistem seperti keberangkatan lewat, kehilangan giliran dan lain-lain adalah ditegaskan
dalam masa-berterusan oleh proses kelahiran-kematian. Simulasi perubahan acara-diskret
sistem baris-gilir seperti pertunjukan masa perkhidmatan, purata masa perkhidmatan yang
turun-naik dalam jangka panjang, penggunaan semakin meningkat, perkhidmatan tamat
lewat dan para pelanggan semakin meningkat dalam sistem menyebabkan keadaan tidakberbaris. Satu platform adab yang berbeza digunakan oleh proses stokastik dalam analisis
transformasi tidak-berbaris. Sa/ah satu cara yang digunakan adalah prosedur pemilihan
bersyarat Markov yang dibentangkan untuk mendapatkan semula sistem beratur melalui
pemilihan dan mengarah keadaan rawak Dalam cara lain, keadaan rawak diteruskan
untuk mendapatkan aturan yang diperkembangkan oleh perjalanan berterusan dan
pandangan kematian tu/en-multinomial. Hubungkait model-model tersebut adalah
berasaskan kepada keadaan rawak dan pereputan jarak menerusi hentian sementara.
Semua kaedah ini dalam proses stokastik mempunyai kemungkinan membawa kepada
penciptaan model baris-gilir selepas transformasi bukan-bergilir. Kajian ini mendapati
bahawa proses stokastik mempunyai potensi besar untuk membentuk hasil model
'pengecualian ' dan akhirnya untuk memulihkan kesan 'pengecualian '.
11
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ACKNOWLEDGEMENTS
As it pleases You, You assign tasks to one and all.
All things are Your Doing; we can do nothing ourselves.
(Guru Granth Sahib: 103)
I would like to express my sincerest gratitude to my supervisor, Professor Dr.
Muhammad Ghazie Ismail for introducing me to the queuing theory, giving parallel
coordination on thesis contents and inspired me to publish articles in international journals.
Likewise, the most important is the arriving at the rest of thesis through his 'higher degree
of independence' morals. I am grateful for the constructive comments and suggestions
given by the external examiner, Professor Dr. Kamaruzzaman Seman for the advancement
of this thesis.
I must also appreciate the current existence of electronic media that had helped me in
understanding the conceptual framework of the queuing theory. Especially the e-learning
video tutorials, which was uploaded by the volunteers/universities. I sincerely thanks to
Dr. Charanjit Kaur for supporting my research finances, providing required literatures,
translating thesis abstract in Bahasa Melayu, editing assistance, dealings with university
administrating staff and outside the university in local language throughout my research
duration. I attribute the level of my Masters degree to their encouragements and efforts.
Without them, this thesis would not have been integrated, improved and completed.
I would like to thanks my parents Sardar Bhagat Singh Azad and Sardarni Balbir Kaur, my
wife Dr. Charanjit Kaur, my siblings Satbir Kaur, Rupinder Singh and rest of other family
members for their endless love and spiritual support during the course of my studies. Last
but not least, a very acknowledgment to the new-born daughter Eykam Parteet Kaur during
my study period. Her cheerful presence boosts the enthusiasm of our everyday life.
lll
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TABLE OF CONTENTS
PAGE
DECLARATION
ABSTRACT
ABSTRAK
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OFTABLES
LIST OF FIGURES
LIST OF APPENDICES
LIST OF SYMBOLS
LIST OF PUBLICATIONS
ii
iii
iv
viii
ix
xii
xiii
xvii
CHAPTER
1.
INTRODUCTION
1.1
1.2
1.3
1.4
1.5
1.6
1. 7
1.8
I .9
I .I0
I .I I
2.
Background
Research Questions
Research Objectives
Research Methodology
Problem Statement
Research Scope
Research Originality
Research Contribution
Chapters Organization
Thesis Presumptions
Thesis Pictorial Outline
1
5
5
6
6
7
8
9
9
11
13
LITERATURE REVIEW
2.1
2.2
2.3
3.
1
14
Introduction
Queuing System
Queuing Among 'Exception'
I4
15
17
MATERIALS AND METHODS
24
3. I
3.2
3.3
3.4
24
3.5
3.6
3.7
Introduction
Research Design
Research Design: Conceptual Approach
Probability Theory
3.4.1 Probability models
3.4.2 Stochastic process concepts
Data Collection
Research Design: Simulation Approach
Research Design Route
iv
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26
26
27
27
30
33
37
4.
'EXCEPTIONS' AND THEIR SEQUENTIAL AFFECT
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
5.
Introduction
'Exceptions' in Queuing System
Categorization and Advanced Classification of ' Exceptions'
Queuing System with Server Disreputable Approach
4.4.1 Bypass serving system
4.4.2 Delay commencing of services
4.4.3 Tailored waiting
Queuing System with Unsystematic and Fluctuate Policy
4.5.l Devices unconcern system
4.5.2 Less informative system
4.5.3 Policy fluctuate system
4.5.4 Broadcasted appointments
Queuing System with Changes in Arrival
4.6.1 Inconsiderate underpinning
4.6.2 Brazenness feeding
4.6.3 Unorganized customer
4.6.4 Untimely arrival
Untimely Arrivals: Waiting Time Distribution Modeling
4.7.1 First Scenario: Unknown opening time
4.7.2 Earlier queuing up: Waiting time distribution
4.7.3 Later queuing up: Waiting time distribution
4.7.4 Result and discussion
4.7.5 Second scenario: Known opening time
4.7.6 Result and discussion
Waiting Time Distribution Evaluation: Brazenness
4.8.1 Result and discussion
Service Variations: Waiting Time Distribution
4.9.1 Result and discussions
Relationship Among Waiting Models
BEHAVIOUR OF THE QUEUING SYSTEM IN 'EXCEPTION'
5.1
5.2
5.3
5.4
5.5
Introduction
Queuing System Declination: Markov Chain
5.2.1 Stochastic concept approach
5.2.2 Overcrowding probability
5.2.3 Result and discussion
' Exception' Through Random Walk
5.3.1 Convolution of two types occurrences
5.3.2 Result and discussion:
System Declination: Regular Birth Single Death Chain
5.4.1 Birth-Death 'exception' chain
5.4.2 Result and discussion: 1
5.4.3 Result and discussion:2
Continuous-Time Deviation Process
5.5.1 Assumption 1: Queue fading
5.5.2 Result and discussion
5.5.3 Assumption 2: Accumulated dual-state
5.5.4 Result and discussion: 1
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39
39
40
41
41
42
42
43
43
44
44
44
45
45
46
46
47
48
49
49
51
54
58
60
64
69
71
74
77
81
81
83
88
89
90
92
93
95
97
99
101
102
105
106
109
110
113
5.6
5.7
5.8
5.9
6.
114
116
117
120
120
122
123
125
128
129
130
NON-QUEUE STATE TRANSFORMATION: A STOCHASTIC APPROACH
I
132
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
7.
5.5.5 Result and discussion: 2
5.5.6 Assumption 3:Pure birth and pure death
5.5.7 Result and discussion
Discrete-Event System Simulation
5.6.1 Different time sets and their affects
Behaviour Verification and Discussion
5.7.1 Comparison of HTS and NHTS
5. 7.2 Utilization (p)
5.7.3 System behaviour comparison in discrete-event system
System Behaviour Comparison in Time Index Parameter
Law of Large Numbers Function Comparisons
Transformation Approach
General Idea of Transformation
Transformation of Non-Queue State
6.3.1 Markov conditional selection procedure
6.3.2 Discrete-time continuous-state MCSP
Simulation ofMCSP Random State Transitions
6.4.1 Simulation of the non-queue state segment
6.4.2 Result and discussion: MCSP simulation
Result and Discussion: State Transitions and Queue Straightening
6.5.1 Fix-modes transition Markov chain procedure
Random Walk to Proceeding Walk
Relationship Between Random Walk and Proceeding Walk
Introduction to Proceeding Walk
Proceeding Walk: Random Digit Signs
Proceeding Walk: Non-Queue Transformation
6.10.1 Effective positions range ( < r >)
6.10.2 Result and discussion: Limitless proceeding walk
6.10.3 Result and discussion: Limited proceeding walk
6.10.4 Result and discussion: Expected time distribution
6.10.5 Partial proceeding walk: Coordinate system
6. 10.6 Result and discussion
132
133
135
135
136
140
140
141
143
148
151
152
153
155
157
160
163
165
166
168
169
NON-QUEUE STATE TRANSFORMATION A STOCHASTIC APPROACH
II
172
7.1
7.2
7.3
7.4
7.5
Introduction
Proceeding Walk Counting Approach
7 .2.1 Coordinate base counting method: Combinatorial analyses
Result and Discussion: Number of Ways Probabilities
Effective Positions General Probability Interpretation
7.4.1 Result and discussion: For P(uP) = P(fp)
172
172
174
175
189
180
7.4.2 Result and discussion: For P(uP) > P(fp)
7.4.3 General probabilities comparison
Dual Multinomial Distribution
7.5.1 Distribution formulation
181
182
184
185
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7 .6
7.7
7.8
7.9
7 .10
7 .11
8.
7.5.2 Result and discussion
Chances of Contravene
7 .6.1 Result and discussion
Multinomial-Pure Death View (MPD)
7. 7 .1 Result and discussion
7.7.2 Result and discussion: Modes set probabilities
Multinomial-Pure Death Distribution Relationship
MCSP Relationship and Further Presentment
7.9.1 MCSP significance in communication
Transformation Concepts Relationship
Expected Outcome
187
188
191
195
200
202
204
206
207
211
212
CONCLUSION AND RECOMMENDTIONS FOR FUTURE RESEARCH 214
8.1
8.2
8.3
8.4
8.5
Summary
Limitation of Study
Attainment of Research Objective
Significance of Research Outcomes
Suggestion for Future Work
214
220
221
222
223
REFERENCES
225
APPENDIX A
233
APPENDIXB
273
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LIST OFTABLES
TITLE
TABLE
PAGE
2.1
Study comparison
23
4.1
Statistical norms for waiting time distribution
54
4.2
2; value descriptions \;/ r
55
4.3
Statistical description for
4.4
Characteristic relationship
77
5.1
Two type transition matrix
87
5.2
Regular births irregular single death sample
103
5.3
Absorption matrix
103
5.4
Parameters notation
121
5.5
Discrete-event system performance measure models
122
5.6
Various times comparisons
128
5.7
Time index parameter comparison
129
5.8
Long-run functions comparison
130
6.1
MCSP based on state tag
139
6.2
MCSP random state transition matrix
143
6.3
State transition table
149
6.4
Different walk types relationship table
152
7.1
Probabilities for absolute distance
174
7.2
Probabilities for absolute distance
175
7.3
Number of ways with Nj
176
7.4
Probability for number of ways with Nj
178
7.5
P(uP) > P(fp)
182
7.6
Individual average for individual step in < r >
183
X n q Ai
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62
LIST OF FIGURES
FIGURE
TITLE
PAGE
1.1
Thesis outline
13
2.1
Literature review concerns
17
3.1
Research formation
37
4.1
A typical queuing system with 'exception'
41
4.2
Comparison of R(r);,, R(r)qA.i and R(r)A,nqA.i
55
4.3
Comparison of F(r);,, F(r)A.i and F(r)A,nqA.i
56
4.4
Combined plots of
4.5
X(r)continuous-time early arrival waiting
61
4.6
4.7
Density functions for successive w
Waiting cumulative distribution by Markovian service
68
69
4 .8
Erlangian waiting distribution function for different phases (states)
71
4.9
Service rate variation layout
72
4.10
Waiting inside the queuing system during ' exception'
75
4.11
w; of L
76
4.12
Expected service time for each customer
76
5.1
Stochastic matrix prototype
90
5.2
Stochastic matrix of two types
91
5.3
N x M matrix random walk for 11 steps
93
5.4
Convolution of probabilities in N
5.5
Birth-death process in random walk illustration
5.6
Regular births single death finite chain
5.7
Transition matrix for regular births single death
5.8
Long term state probability by linear equation method
102
5.9
Long term state probability by limiting probability method
104
5.10
State transition rate diagram for regular births single death
105
5.11
Probability distribution of queue disappear
109
5.12
Dual-state of accumulated arrival rates, and then a departure
110
5.13
Next state probabilities during kin the system
113
5.14
MEk + l andµ during overcrowding
115
5.15
Joint view of pure birth then single death process
116
5.16
Joint probability of pure birth and single death
118
X,
XnR(r)q,t2 and F(r)xnqJq
= 51
IX
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60
95
99
99
99
6.11
Average IAT vs Avg. Service finish time
Long term service times during HTS and NHTS
Utilization comparison during both situations
Utilization when no 'exception' occurs
Utilization (p) when 'exception' occurs
System operation times during HTS and NHTS
Non-queue state
General idea of transformation
Non-queue segment (with tag and current position digits)
MCSP (based on least distance with earlier position number)
Comparison between MCSP and Headless selection
Random state transition diagram
Long-term state probability vector (3-D)
Pictorial view of transformation of non-queue segment
Straightening out queue process
Simple symmetric random walk
Illustration of proceeding walk, j p
158
6.12
One-dimensional proceeding walk on xy -plane
159
6.13
6.14
6.15
6.16
Limited positions
Ranged proceeding walk
Limitless end-queue proceeding walk
Limited end-queue proceeding walk
161
161
164
165
6.17
Expected time from
6.18
Probability of landing at various levels for n = 13
170
6.19
Logarithmic probability plot of I for n = 13
170
7.1
A sample for counting number of ways for
7.2
Samples of proceeding walk for end-queue positions
174
7.3
Logarithmic plot for probabilities based on Nj
178
7.4
For P(uP) = P(fp)
181
7.5
For P(uP) > P(fp)
182
7.6
7.7
Average comparison
Cartesian coordinate proceeding walk model
182
186
7.8
For P(uP) = P(fp)
188
7.9
For l =20 levels of proceeding walks reduction
192
7.10
Stack probabilities of customer (A, B)
193
7.11
Sample diagram of multinomial-pure death
197
5.17
5.18
5.19
5.20
5.21
5.22
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
I-YI distance
123
125
126
127
127
128
134
134
140
142
142
144
144
146
147
157
167
Jp
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173
7.12
7.13
7.14
7.15
7.16
7.17
7.18
Probability at each distance
Modes set flicking probabilities in various intervals
Pure death single mode
Recurrence from acknowledged to previous state
Recurrence form responded state to origin
Interrelation among concepts
Single server with multiple queues
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200
202
205
208
208
211
212
LIST OF APPENDICES
APPENDIX
A
B
TITILE
Additional substantial of analysation of 'exception' and non-queue
transformation and Pseudo-codes of some mathematical models.
Basic definition of the concept used in thesis.
xii
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PAGE
239
279
LIST OF SYMBOLS
Symbol
p
Characteristics
Rho
Explanation
Reciprocal service rate
Mean service time of a single customer over
mean interval between arrivals of successive
individual customers.
Mean service time.
Reciprocal arrival rate
Mean interval between arrivals.
Markov chain process
Set of random variable at step n.
Next state random
variable
Variable that will take some state value at n
step.
Random variables set
Set of series of n random variable
Implies
It implies that both
+1
R• .
( *) and (.) have same
meaning.
Ｊｾﾷ＠
Random variable
Random variable for exponential distribution
that exceed define time.
7b(•)
Before waiting time
X-Exp(.-1)
Distribution function
parameters
Waiting time before the outlet opens for
customer to arrive at Ti.
It is a short hand notation which implies X is an
exponential random variable with parameter .-1 .
Waiting time set after the queuing system begins.
After waiting time
Distribution function
R(r)
Survivor function
R(r).
Arrival time of k
Median
System opening time
*
R7b(i)
Long-run average of
Customer's waiting
time
Cumulative distribution function at time T.
Cumulative distribution function at time T of ( *)
form.
Survivor function of exponential distribution or
probability of waiting more than r.
Survivor function of ( *) form.
k number of customer arrived at time Ti .
The median is used to determine the location of
the reasonable time when the distribution
is skewed.
Time Tn, at which the queuing system will open.
The accent mark on any character is used to
symbolize the average or expectation, whichever
related to relative section.
Long-run stationary value in discrete time.
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Distribution function
Cumulative distribution for sum of N random
variable ZN .
x'h
Customer number
Random variable for
do displacement act.
R(T)s1a1e•
R(T).
Survivor function
Survivor function of state*·
Survivor function of any state.
Service rate
Service rate before ith customer.
Service rate after ith customer.
Traffic intensity during
service variations
where (. ) represents the state of the system and
( *) for before or after service rate.
Probability that the queuing system is empty.
Length of interval i.
Length of interval j.
I
Po
I;
State probability
Time length
11
ith
affected customer due
(i, j)v ,
State transitions pair
State transition from i to j at interval label v.
Aj
Actual job
Probability at interval
The actual job stand for the service supposed to
perform for a customer in queue in a specified
interval.
Behaviour that can brings the queuing system to
non-queue state.
Probability of type(*) at interval label v.
Survival probability
The probability of actual job done so far.
Dual classification of a
variable
Sequence of random
variable
It defines k is an element of natural number and
Long-run random variable of being in state k.
D..
Long run random
variable
Long run random
variable
Discrete-time
C'
Continuous-time
'Exception'
State probability
Arrival rate
also not equals to zero.
Sequence of N random variable from 1 to oo.
Sequence of i steps random variables.
Random variable at
ith
step of being in state k.
Discrete-time used during Markov chain.
Continuous-time used during birth and death
process.
Probability of being in state of k > o at (t + ll!) ,
where 8 is default queue size and k is total of
default number of customers and overcrowded
customers.
Arithmetic collection of back to back
occurrences of arrival rates.
xiv
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State probability
Probability of accumulated states from 0 until
EK.
ljlN
Conditional probability
Expected value
Combination
Probability that single event will occur given N
births.
Expected value of long-run function of(.).
It signifies (.) choose ( *), as same as n choose
r.
Coordinate pair
GLB
Greatest lower bound
LUB
Least upper bound
Proceeding walk with
up and flat proceeds
The location of dissipated customer with its
previous value P, prior to non-queue state.
Greatest lower bound will be the next nearest
value.
Least upper bound will be the rarest customer.
p with u and f only indicates the to proceed up
and flat.
Probability of up step
Probability of flat step
Probability of steps during up proceeding.
Probability of steps during flat proceeding.
Proceeding walk with
random digits
Coordinate pair
It screens out the directions of proceedings in up
Effective position
nth effective position from (xp,- Yq).
Ranged of effective positions at the end-queue.
I-Yi
Ranged
Absolute distance
X+x axis
Location at abscissa
Distance from (xp,- Yq)to
position.
Location at x of +x axis.
Location at x of - y axis.
Location at 0 of +x axis.
Y-y axis
Location at ordinate
Location at y of -y axis.
Infinity of
Positive infinity of {x,y} or negative infinity of
{x,y}.
Level proceeding
Random digit sign (0 or 1) added up to I.
Number of steps
Number of j steps that lead to effective positions.
Random variable
The process that performs counting number of
ways collected by random variable, which is
used to reach the destination, where l represent
ordinate level.
Probability of effective
position at s
Probability of kth effective position is s distance
far from (xp,- Yq).
jp
:Jp
P(lEPk)s
and flat, positively.
Position of dissipated customer in 4th quadrant
xv
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first
effective
Wx(x,- y)
Probability function
Mutually overlapping
(s-l)x(s)
(s-2)x(s)
cells
(s+l)x(s-2)
(l)x(s-2)
Probability function when (xp,- Yq)is x
proceeding far.
Numbers of column and row arrangement during
overlapping of ranged of two different locations
when effective position is s distance far.
D(t)
D(ta
Random process
Counting process of decreasing distance in time
period t or ti .
d {a .P.rl
Distance through modes
d distance covered or declined through
r modes.
ith
distance
a,/3 and
Covered ith distance through the combination of
a,/) and r .
xvi
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LIST OF PUBLICATIONS
Singh, Harpreet and Ismail, M. Ghazie, 2014. 'Exceptions' in Queuing Theory.
International Journal of Computer and Information Technology, 3 (1), pp. 110-119.
Singh, Harpreet and Ismail, M. Ghazie, 2014. 'Exception' in Queuing Theory through
Markov Chain. International Journal of Electrical and Electronics Engineering, 1 (1 ), pp.
14-29.
xvii
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CHAPTERl
INTRODUCTION
1.1
Background
In a system of line up there are various factors that can develop the chain of
complications before breaking the queuing system into unlike situation. Any system that
constitutes individual giving services to seekers either mechanically or electronically or
manually or else and a seeker that gets services from either authorized or experienced
individual is known as serving system. In the queuing literatures, it is recognized as
queuing system. Generally the host(s) provides services to the chronological arrivals of
seeker that waits for their turn patiently. The study that mathematically analyzes the
number of customers in the system, queue length, system size, arrival discipline, service
time, service discipline, expected number of waiting, time in queue, load in the system and
many other interested terms, can know from the queuing theory (Hlynka, 2007). Danish
engineer, Agner Krarup Erlang considered as a discoverer of such theory when he
published first paper in 1909 on queuing theory in telephonic networks at Copenhagen.
The theory results in the form of queuing models with many applications m
different fields of study. This theory extensively used in telecommunications, hospitals,
traffic, computers, retails, airports, banks and many others. The arrival and service can be
individual or group, completely random or constant. A server can be single or multiple.
Queue size can be both finite and infinite. The queue controlling can be chronological,
random, or unordered. Customers permitted in the system are finite or infinite. The makeup of all these choices can collectively define in Kendall notation given as
(P/Q/R):(X/Y/Z), where P/Q is the arrival and a server probabilistic approach, R is server
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quantity, X is queuing controlling, Y is number of customers permitted and Z is size of
customer population.
It is routinely believed that the mutual contribution of socially acceptable behaviour
by the seekers and the host(s) supports queuing system performances. This can only be
possible if they are following ethics that are organized. If the queuing system considers
human involvements then most probably the only term relevant is the waiting. Waiting
requires patience and patience requires practice. Besides this, due to their achievement rate
the system in services does not desirable to keep customers in the waiting. However, the
reason that put into concern that why the waiting begins and gets large. The reason is
simple, when the undesirable exception comes in during the current task and its regularity
cause the beginning of waiting more than expected.
Large waiting completely influenced by the large accumulation of small-scale of
hindrances. It possibly will be an unnecessary exception that caused either the service
provider gets undesirably slow or service seekers are undesirably arriving faster than
normal. If there were no hindrances then the servicing rate is infinite and queue size is
zero, i.e., sojourn time in such state is nothing. Otherwise, it causes growing of queue
besides overcrowding prior to the queue less state. For e.g., in a queue network, service
rate variation within a larger time frame gives sudden input to subsequent server or specific
time of arrival of requests in a system in a short period of time.
The services of the customer may be either constant or stochastic, and the
organization of queuing system varies globally. However, in some area of the world the
queue routines are not following properly. From place to place the cultural practice of
queuing includes awful customers, dishonor and for few the existence of a queue is almost
alien to them. Such practices cause pressure on the serving system. However, the proper
restricted policy can control customer behavior inside the serving system to avoid from
2
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delay in services. Gross et al. (2008) suggested that if such behaviours are not controlled in
time the chances are that the queues get bigger and bigger unless at some point the
customers are not allowed to enter the system.
Ever since Erlang, most of the researches have been focused on designing wellorganized system by assuming static conditions to perform without any disruptions. In
general, queue analysis is based on the stationary assumption in customer's variability, but
in reality there are other variables that can direct to a different state of the system. Many
literatures analyzed the results using a queuing model on the basic assumption that p < 1,
otherwise later p
ｾ＠
1, the queuing system goes imbalance or unimagined situation. But the
series of outcomes with an undesired event and the queuing system later p > 1 exclusively
not at all exhibited. As Gross et al. (2008) highlighted, in queuing theory the queuing
models that predict the performance of serving system with the assumption that demands
arising randomly, but not unnaturally, which can be either service or arrival.
Such unnatural randomness may perhaps be due to undesired behaviour occurrences
of customers and server. Cohen (1982) describes the behavior of customers with two
characteristic features, viz. The arrival process and the waiting process, and thus the
continuation of unnoticeable behaviour may challenge the service seekers patience. Cox
and Smith (1961) and Conolly (1975) mentioned that congestion in queuing system occurs
due to an abnormality, i.e. , the sudden changes in behaviours of either customer(s) or
server will result in an unbalanced system. Wang et al. (2010) highlighted that numerous
queuing literatures discuss two types of behaviours concerned with customer and servers
which are impatience and vacation period, respectively. In customer behaviours case
reneging, preemptive, baking and jockeying is covered largely by other researchers. Kok
and Tijms (1984), Shin, Choo and Chakravarthy (2009), Sudesh, Pere! and Yechiali (2010)
3
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Faculty of Electronics and Computer Engineering
QUEUING SYSTEM 'EXCEPTIONS', TIIEIR AFFECTS AND NONQUEUE STATE: A NEW STOCHASTIC APPROACH
Harpreet Singh s/o Bhagat Singh Amd
Master of Science in Electronic Engineering
2014
© Universiti Teknikal Malaysia Melaka
QUEUING SYSTEM 'EXCEPTIONS', THEffi AFFECTS AND NON-QUEUE
STATE: A NEW STOCHASTIC APPROACH
HARPREET SINGH S/O BHAGAT SINGH AZAD
A thesis submitted
in fulfillment of the requirement for the degree of Master of Science
in Electronic Engineering
Faculty of Electronics and Computer Engineering
UNIVERSITI TEKNIKAL MALAYSIA MELAKA
2014
© Universiti Teknikal Malaysia Melaka
APPROVAL
I hereby declare that I have read this thesis and in my opinion this thesis is sufficient in
terms of scope and quality for the award of Master of Science in Electronic Engineering.
Signature:
Supervisor Name: Professor Dr. Muhammad Ghazie Ismail
Date: 15 May 2014
. ;' ••:.· ·
© Universiti Teknikal Malaysia Melaka
DECLARATION
I declare that this thesis entitle "Queuing System 'Exceptions', Their Affects and Non-Queue
State: A New Stochastic Approach" is the result of my own research except as cited in the
references. The thesis has not been accepted for any degree and is not concurrently
submitted in candidature of any other degree.
Name: Harpreet Singh s/o Bhagat Singh Azad
Date: 15 July 2014
© Universiti Teknikal Malaysia Melaka
ABSTRACT
Most queuing models are based on organized behavior of customers served by a server or
multiple servers. However, there are many instances that the queuing model does not stand
by real world queue ethics. The consequences can go towards non-queue state caused by
overcrowding and cultural background of the customers. This research portrays the
successive outcomes that sourced non-queue state due to customer ignored behaviors as
well as the unsystematic policy of a single serving system which becomes the main
research problem. Such activities are termed as 'exceptions' and their disturbances in
queuing system. The main objective of this research is to present 'exceptions', their affects
and to resolve the non-queue state. It is important so that the waiting time distribution,
behaviour of the system during 'exceptions' and queuing system can be evaluated,
demonstrated and reinstated to a single server queuing state, respectively. A non-queue is a
state addressed when system utilization goes beyond I 00 percent. Furthermore, this would
indicate no formal queue at all as customers could be assumed at random locations. This
research will explain the state of the system during 'exception' and non-queue
transformation largely with the support of probability theory in time index values. Various
'exceptions' categories with constructed waiting time probability models and their results
have proven that the customers large waiting do actually overcrowd the queuing system.
This thesis discusses 'exceptions' affects through the Markov chain, random walk and
birth-death process. The above concepts illustrate system behaviour, as 'exception' that
disrupts services. In exceptional case, stochastic matrix considered two types such as actual
job and 'exception' to occur during state transition. Their limitation is further simplified by
random walk in convolution idea. The system parameters such as late departures, queue
loss and others are affirmed in continuous-time by birth-death process. Simulation of
change of discrete-event of queuing system such as service times shows, in long-run on
averages service time fluctuates, utilization gets large, service finishing late and customers
get increased in the system causing non-queue state. A platform of dissimilar manners used
by stochastic process in non-queue transformation analysis. One of the manner employed
are Markov conditional selection procedure is presented to recover queuing system through
selecting and ordering random states. In other manner, random state(s) proceeds to get in
order are expanded by proceeding walk and multinomial-pure death view. The interrelation
of such models is grounded through the random states and distance decays by an interval.
All such manners in stochastic process have likely to bring into being a queuing model
after non-queue transformation. This research finds that the stochastic process has
tremendous potential to model 'exception' outcomes and to restore the 'exception'
eventual affects.
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ABSTRAK
Kebanyakan model baris-gilir adalah berdasarkan kepada tingkah laku terancang para
pelanggan yang dilayan oleh satu pelayan atau pelayan berganda. Sungguhpun begitu,
terdapat banyak contoh di mana model baris-gilir tidak berdiri dengan etika giliran dunia
yang sebenar. Kesannya boleh sampai kepada keadaan bukan baris-gilir yang disebabkan
oleh kesesakan dan latar belakang budaya para pelanggan. Kajian ini menggambarkan
hasil berturut-turut yang bersumberkan keadaan bukan-bergilir kerana pengabaian
perlakuan pelanggan serta dasar tidak sistematik sistem layanan tunggal yang menjadi
masalah utama penyelidikan. Aktiviti sedemikian disebut sebagai 'pengecualian' dan
gangguannya dalam sistem baris-gilir. Objektif utama penyelidikan ini adalah untuk
membentangkan 'pengecualian-pengecualian ', kesannya dan penyelesaian kepada
keadaan tidak berbaris-gilir. la adalah penting agar taburan masa menunggu, tingkah
laku sistem semasa 'pengecualian-pengecualian' dan sistem baris-gilir masing-masing
boleh dinilai, ditunjukkan dan dinyatakan kembali kepada keadaan bergilir layanan
tunggal. Bukan-bergilir adalah keadaan yang dinyatakan apabila penggunaan sistem
melampaui 100 peratus. Tambahan pula, ini boleh menunjukkan tiada baris-gilir formal
langsung kerana para pelanggan boleh diandaikan berada di lokasi-lokasi rawak Kajian
ini akan menjelaskan keadaan sistem semasa 'pengecualian 'dan transformasi bukanbergilir dengan sokongan besar oleh teori kebarangkalian dalam nilai indeks masa.
Pelbagai kategori 'pengecualian-pengecualian' dengan pembentukan model-model
kebarangkalian masa menunggu dan keputusannya telah membuktikan bahawa penantian
besar para pelanggan sebenarnya menyesakkan sistem baris-gilir. Tesis mz
membincangkan kesan 'pengecualian-pengecualian' menerusi rantaian Markov,
perjalanan rawak dan proses kelahiran-kematian. Konsep-konsep di atas menggambarkan
sistem tingkah laku sebagai 'pengecualian' yang mengganggu perkhidmatan. Dalam kes
yang istimewa, matrik stokastik mengambil kira duajenis keadaan yakni kerja sebenar dan
'pengecualian' untuk berlaku semasa peralihan tahap. Limitasi mereka kemudiannya
dipermudahkan dengan perjalanan rawak dalam idea kekusutan. Parameter-parameter
sistem seperti keberangkatan lewat, kehilangan giliran dan lain-lain adalah ditegaskan
dalam masa-berterusan oleh proses kelahiran-kematian. Simulasi perubahan acara-diskret
sistem baris-gilir seperti pertunjukan masa perkhidmatan, purata masa perkhidmatan yang
turun-naik dalam jangka panjang, penggunaan semakin meningkat, perkhidmatan tamat
lewat dan para pelanggan semakin meningkat dalam sistem menyebabkan keadaan tidakberbaris. Satu platform adab yang berbeza digunakan oleh proses stokastik dalam analisis
transformasi tidak-berbaris. Sa/ah satu cara yang digunakan adalah prosedur pemilihan
bersyarat Markov yang dibentangkan untuk mendapatkan semula sistem beratur melalui
pemilihan dan mengarah keadaan rawak Dalam cara lain, keadaan rawak diteruskan
untuk mendapatkan aturan yang diperkembangkan oleh perjalanan berterusan dan
pandangan kematian tu/en-multinomial. Hubungkait model-model tersebut adalah
berasaskan kepada keadaan rawak dan pereputan jarak menerusi hentian sementara.
Semua kaedah ini dalam proses stokastik mempunyai kemungkinan membawa kepada
penciptaan model baris-gilir selepas transformasi bukan-bergilir. Kajian ini mendapati
bahawa proses stokastik mempunyai potensi besar untuk membentuk hasil model
'pengecualian ' dan akhirnya untuk memulihkan kesan 'pengecualian '.
11
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ACKNOWLEDGEMENTS
As it pleases You, You assign tasks to one and all.
All things are Your Doing; we can do nothing ourselves.
(Guru Granth Sahib: 103)
I would like to express my sincerest gratitude to my supervisor, Professor Dr.
Muhammad Ghazie Ismail for introducing me to the queuing theory, giving parallel
coordination on thesis contents and inspired me to publish articles in international journals.
Likewise, the most important is the arriving at the rest of thesis through his 'higher degree
of independence' morals. I am grateful for the constructive comments and suggestions
given by the external examiner, Professor Dr. Kamaruzzaman Seman for the advancement
of this thesis.
I must also appreciate the current existence of electronic media that had helped me in
understanding the conceptual framework of the queuing theory. Especially the e-learning
video tutorials, which was uploaded by the volunteers/universities. I sincerely thanks to
Dr. Charanjit Kaur for supporting my research finances, providing required literatures,
translating thesis abstract in Bahasa Melayu, editing assistance, dealings with university
administrating staff and outside the university in local language throughout my research
duration. I attribute the level of my Masters degree to their encouragements and efforts.
Without them, this thesis would not have been integrated, improved and completed.
I would like to thanks my parents Sardar Bhagat Singh Azad and Sardarni Balbir Kaur, my
wife Dr. Charanjit Kaur, my siblings Satbir Kaur, Rupinder Singh and rest of other family
members for their endless love and spiritual support during the course of my studies. Last
but not least, a very acknowledgment to the new-born daughter Eykam Parteet Kaur during
my study period. Her cheerful presence boosts the enthusiasm of our everyday life.
lll
© Universiti Teknikal Malaysia Melaka
TABLE OF CONTENTS
PAGE
DECLARATION
ABSTRACT
ABSTRAK
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OFTABLES
LIST OF FIGURES
LIST OF APPENDICES
LIST OF SYMBOLS
LIST OF PUBLICATIONS
ii
iii
iv
viii
ix
xii
xiii
xvii
CHAPTER
1.
INTRODUCTION
1.1
1.2
1.3
1.4
1.5
1.6
1. 7
1.8
I .9
I .I0
I .I I
2.
Background
Research Questions
Research Objectives
Research Methodology
Problem Statement
Research Scope
Research Originality
Research Contribution
Chapters Organization
Thesis Presumptions
Thesis Pictorial Outline
1
5
5
6
6
7
8
9
9
11
13
LITERATURE REVIEW
2.1
2.2
2.3
3.
1
14
Introduction
Queuing System
Queuing Among 'Exception'
I4
15
17
MATERIALS AND METHODS
24
3. I
3.2
3.3
3.4
24
3.5
3.6
3.7
Introduction
Research Design
Research Design: Conceptual Approach
Probability Theory
3.4.1 Probability models
3.4.2 Stochastic process concepts
Data Collection
Research Design: Simulation Approach
Research Design Route
iv
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25
26
26
27
27
30
33
37
4.
'EXCEPTIONS' AND THEIR SEQUENTIAL AFFECT
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
5.
Introduction
'Exceptions' in Queuing System
Categorization and Advanced Classification of ' Exceptions'
Queuing System with Server Disreputable Approach
4.4.1 Bypass serving system
4.4.2 Delay commencing of services
4.4.3 Tailored waiting
Queuing System with Unsystematic and Fluctuate Policy
4.5.l Devices unconcern system
4.5.2 Less informative system
4.5.3 Policy fluctuate system
4.5.4 Broadcasted appointments
Queuing System with Changes in Arrival
4.6.1 Inconsiderate underpinning
4.6.2 Brazenness feeding
4.6.3 Unorganized customer
4.6.4 Untimely arrival
Untimely Arrivals: Waiting Time Distribution Modeling
4.7.1 First Scenario: Unknown opening time
4.7.2 Earlier queuing up: Waiting time distribution
4.7.3 Later queuing up: Waiting time distribution
4.7.4 Result and discussion
4.7.5 Second scenario: Known opening time
4.7.6 Result and discussion
Waiting Time Distribution Evaluation: Brazenness
4.8.1 Result and discussion
Service Variations: Waiting Time Distribution
4.9.1 Result and discussions
Relationship Among Waiting Models
BEHAVIOUR OF THE QUEUING SYSTEM IN 'EXCEPTION'
5.1
5.2
5.3
5.4
5.5
Introduction
Queuing System Declination: Markov Chain
5.2.1 Stochastic concept approach
5.2.2 Overcrowding probability
5.2.3 Result and discussion
' Exception' Through Random Walk
5.3.1 Convolution of two types occurrences
5.3.2 Result and discussion:
System Declination: Regular Birth Single Death Chain
5.4.1 Birth-Death 'exception' chain
5.4.2 Result and discussion: 1
5.4.3 Result and discussion:2
Continuous-Time Deviation Process
5.5.1 Assumption 1: Queue fading
5.5.2 Result and discussion
5.5.3 Assumption 2: Accumulated dual-state
5.5.4 Result and discussion: 1
v
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39
39
39
40
41
41
42
42
43
43
44
44
44
45
45
46
46
47
48
49
49
51
54
58
60
64
69
71
74
77
81
81
83
88
89
90
92
93
95
97
99
101
102
105
106
109
110
113
5.6
5.7
5.8
5.9
6.
114
116
117
120
120
122
123
125
128
129
130
NON-QUEUE STATE TRANSFORMATION: A STOCHASTIC APPROACH
I
132
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
7.
5.5.5 Result and discussion: 2
5.5.6 Assumption 3:Pure birth and pure death
5.5.7 Result and discussion
Discrete-Event System Simulation
5.6.1 Different time sets and their affects
Behaviour Verification and Discussion
5.7.1 Comparison of HTS and NHTS
5. 7.2 Utilization (p)
5.7.3 System behaviour comparison in discrete-event system
System Behaviour Comparison in Time Index Parameter
Law of Large Numbers Function Comparisons
Transformation Approach
General Idea of Transformation
Transformation of Non-Queue State
6.3.1 Markov conditional selection procedure
6.3.2 Discrete-time continuous-state MCSP
Simulation ofMCSP Random State Transitions
6.4.1 Simulation of the non-queue state segment
6.4.2 Result and discussion: MCSP simulation
Result and Discussion: State Transitions and Queue Straightening
6.5.1 Fix-modes transition Markov chain procedure
Random Walk to Proceeding Walk
Relationship Between Random Walk and Proceeding Walk
Introduction to Proceeding Walk
Proceeding Walk: Random Digit Signs
Proceeding Walk: Non-Queue Transformation
6.10.1 Effective positions range ( < r >)
6.10.2 Result and discussion: Limitless proceeding walk
6.10.3 Result and discussion: Limited proceeding walk
6.10.4 Result and discussion: Expected time distribution
6.10.5 Partial proceeding walk: Coordinate system
6. 10.6 Result and discussion
132
133
135
135
136
140
140
141
143
148
151
152
153
155
157
160
163
165
166
168
169
NON-QUEUE STATE TRANSFORMATION A STOCHASTIC APPROACH
II
172
7.1
7.2
7.3
7.4
7.5
Introduction
Proceeding Walk Counting Approach
7 .2.1 Coordinate base counting method: Combinatorial analyses
Result and Discussion: Number of Ways Probabilities
Effective Positions General Probability Interpretation
7.4.1 Result and discussion: For P(uP) = P(fp)
172
172
174
175
189
180
7.4.2 Result and discussion: For P(uP) > P(fp)
7.4.3 General probabilities comparison
Dual Multinomial Distribution
7.5.1 Distribution formulation
181
182
184
185
vi
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7 .6
7.7
7.8
7.9
7 .10
7 .11
8.
7.5.2 Result and discussion
Chances of Contravene
7 .6.1 Result and discussion
Multinomial-Pure Death View (MPD)
7. 7 .1 Result and discussion
7.7.2 Result and discussion: Modes set probabilities
Multinomial-Pure Death Distribution Relationship
MCSP Relationship and Further Presentment
7.9.1 MCSP significance in communication
Transformation Concepts Relationship
Expected Outcome
187
188
191
195
200
202
204
206
207
211
212
CONCLUSION AND RECOMMENDTIONS FOR FUTURE RESEARCH 214
8.1
8.2
8.3
8.4
8.5
Summary
Limitation of Study
Attainment of Research Objective
Significance of Research Outcomes
Suggestion for Future Work
214
220
221
222
223
REFERENCES
225
APPENDIX A
233
APPENDIXB
273
vii
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LIST OFTABLES
TITLE
TABLE
PAGE
2.1
Study comparison
23
4.1
Statistical norms for waiting time distribution
54
4.2
2; value descriptions \;/ r
55
4.3
Statistical description for
4.4
Characteristic relationship
77
5.1
Two type transition matrix
87
5.2
Regular births irregular single death sample
103
5.3
Absorption matrix
103
5.4
Parameters notation
121
5.5
Discrete-event system performance measure models
122
5.6
Various times comparisons
128
5.7
Time index parameter comparison
129
5.8
Long-run functions comparison
130
6.1
MCSP based on state tag
139
6.2
MCSP random state transition matrix
143
6.3
State transition table
149
6.4
Different walk types relationship table
152
7.1
Probabilities for absolute distance
174
7.2
Probabilities for absolute distance
175
7.3
Number of ways with Nj
176
7.4
Probability for number of ways with Nj
178
7.5
P(uP) > P(fp)
182
7.6
Individual average for individual step in < r >
183
X n q Ai
viii
© Universiti Teknikal Malaysia Melaka
62
LIST OF FIGURES
FIGURE
TITLE
PAGE
1.1
Thesis outline
13
2.1
Literature review concerns
17
3.1
Research formation
37
4.1
A typical queuing system with 'exception'
41
4.2
Comparison of R(r);,, R(r)qA.i and R(r)A,nqA.i
55
4.3
Comparison of F(r);,, F(r)A.i and F(r)A,nqA.i
56
4.4
Combined plots of
4.5
X(r)continuous-time early arrival waiting
61
4.6
4.7
Density functions for successive w
Waiting cumulative distribution by Markovian service
68
69
4 .8
Erlangian waiting distribution function for different phases (states)
71
4.9
Service rate variation layout
72
4.10
Waiting inside the queuing system during ' exception'
75
4.11
w; of L
76
4.12
Expected service time for each customer
76
5.1
Stochastic matrix prototype
90
5.2
Stochastic matrix of two types
91
5.3
N x M matrix random walk for 11 steps
93
5.4
Convolution of probabilities in N
5.5
Birth-death process in random walk illustration
5.6
Regular births single death finite chain
5.7
Transition matrix for regular births single death
5.8
Long term state probability by linear equation method
102
5.9
Long term state probability by limiting probability method
104
5.10
State transition rate diagram for regular births single death
105
5.11
Probability distribution of queue disappear
109
5.12
Dual-state of accumulated arrival rates, and then a departure
110
5.13
Next state probabilities during kin the system
113
5.14
MEk + l andµ during overcrowding
115
5.15
Joint view of pure birth then single death process
116
5.16
Joint probability of pure birth and single death
118
X,
XnR(r)q,t2 and F(r)xnqJq
= 51
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60
95
99
99
99
6.11
Average IAT vs Avg. Service finish time
Long term service times during HTS and NHTS
Utilization comparison during both situations
Utilization when no 'exception' occurs
Utilization (p) when 'exception' occurs
System operation times during HTS and NHTS
Non-queue state
General idea of transformation
Non-queue segment (with tag and current position digits)
MCSP (based on least distance with earlier position number)
Comparison between MCSP and Headless selection
Random state transition diagram
Long-term state probability vector (3-D)
Pictorial view of transformation of non-queue segment
Straightening out queue process
Simple symmetric random walk
Illustration of proceeding walk, j p
158
6.12
One-dimensional proceeding walk on xy -plane
159
6.13
6.14
6.15
6.16
Limited positions
Ranged proceeding walk
Limitless end-queue proceeding walk
Limited end-queue proceeding walk
161
161
164
165
6.17
Expected time from
6.18
Probability of landing at various levels for n = 13
170
6.19
Logarithmic probability plot of I for n = 13
170
7.1
A sample for counting number of ways for
7.2
Samples of proceeding walk for end-queue positions
174
7.3
Logarithmic plot for probabilities based on Nj
178
7.4
For P(uP) = P(fp)
181
7.5
For P(uP) > P(fp)
182
7.6
7.7
Average comparison
Cartesian coordinate proceeding walk model
182
186
7.8
For P(uP) = P(fp)
188
7.9
For l =20 levels of proceeding walks reduction
192
7.10
Stack probabilities of customer (A, B)
193
7.11
Sample diagram of multinomial-pure death
197
5.17
5.18
5.19
5.20
5.21
5.22
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
I-YI distance
123
125
126
127
127
128
134
134
140
142
142
144
144
146
147
157
167
Jp
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173
7.12
7.13
7.14
7.15
7.16
7.17
7.18
Probability at each distance
Modes set flicking probabilities in various intervals
Pure death single mode
Recurrence from acknowledged to previous state
Recurrence form responded state to origin
Interrelation among concepts
Single server with multiple queues
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200
202
205
208
208
211
212
LIST OF APPENDICES
APPENDIX
A
B
TITILE
Additional substantial of analysation of 'exception' and non-queue
transformation and Pseudo-codes of some mathematical models.
Basic definition of the concept used in thesis.
xii
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PAGE
239
279
LIST OF SYMBOLS
Symbol
p
Characteristics
Rho
Explanation
Reciprocal service rate
Mean service time of a single customer over
mean interval between arrivals of successive
individual customers.
Mean service time.
Reciprocal arrival rate
Mean interval between arrivals.
Markov chain process
Set of random variable at step n.
Next state random
variable
Variable that will take some state value at n
step.
Random variables set
Set of series of n random variable
Implies
It implies that both
+1
R• .
( *) and (.) have same
meaning.
Ｊｾﾷ＠
Random variable
Random variable for exponential distribution
that exceed define time.
7b(•)
Before waiting time
X-Exp(.-1)
Distribution function
parameters
Waiting time before the outlet opens for
customer to arrive at Ti.
It is a short hand notation which implies X is an
exponential random variable with parameter .-1 .
Waiting time set after the queuing system begins.
After waiting time
Distribution function
R(r)
Survivor function
R(r).
Arrival time of k
Median
System opening time
*
R7b(i)
Long-run average of
Customer's waiting
time
Cumulative distribution function at time T.
Cumulative distribution function at time T of ( *)
form.
Survivor function of exponential distribution or
probability of waiting more than r.
Survivor function of ( *) form.
k number of customer arrived at time Ti .
The median is used to determine the location of
the reasonable time when the distribution
is skewed.
Time Tn, at which the queuing system will open.
The accent mark on any character is used to
symbolize the average or expectation, whichever
related to relative section.
Long-run stationary value in discrete time.
Xlll
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Distribution function
Cumulative distribution for sum of N random
variable ZN .
x'h
Customer number
Random variable for
do displacement act.
R(T)s1a1e•
R(T).
Survivor function
Survivor function of state*·
Survivor function of any state.
Service rate
Service rate before ith customer.
Service rate after ith customer.
Traffic intensity during
service variations
where (. ) represents the state of the system and
( *) for before or after service rate.
Probability that the queuing system is empty.
Length of interval i.
Length of interval j.
I
Po
I;
State probability
Time length
11
ith
affected customer due
(i, j)v ,
State transitions pair
State transition from i to j at interval label v.
Aj
Actual job
Probability at interval
The actual job stand for the service supposed to
perform for a customer in queue in a specified
interval.
Behaviour that can brings the queuing system to
non-queue state.
Probability of type(*) at interval label v.
Survival probability
The probability of actual job done so far.
Dual classification of a
variable
Sequence of random
variable
It defines k is an element of natural number and
Long-run random variable of being in state k.
D..
Long run random
variable
Long run random
variable
Discrete-time
C'
Continuous-time
'Exception'
State probability
Arrival rate
also not equals to zero.
Sequence of N random variable from 1 to oo.
Sequence of i steps random variables.
Random variable at
ith
step of being in state k.
Discrete-time used during Markov chain.
Continuous-time used during birth and death
process.
Probability of being in state of k > o at (t + ll!) ,
where 8 is default queue size and k is total of
default number of customers and overcrowded
customers.
Arithmetic collection of back to back
occurrences of arrival rates.
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State probability
Probability of accumulated states from 0 until
EK.
ljlN
Conditional probability
Expected value
Combination
Probability that single event will occur given N
births.
Expected value of long-run function of(.).
It signifies (.) choose ( *), as same as n choose
r.
Coordinate pair
GLB
Greatest lower bound
LUB
Least upper bound
Proceeding walk with
up and flat proceeds
The location of dissipated customer with its
previous value P, prior to non-queue state.
Greatest lower bound will be the next nearest
value.
Least upper bound will be the rarest customer.
p with u and f only indicates the to proceed up
and flat.
Probability of up step
Probability of flat step
Probability of steps during up proceeding.
Probability of steps during flat proceeding.
Proceeding walk with
random digits
Coordinate pair
It screens out the directions of proceedings in up
Effective position
nth effective position from (xp,- Yq).
Ranged of effective positions at the end-queue.
I-Yi
Ranged
Absolute distance
X+x axis
Location at abscissa
Distance from (xp,- Yq)to
position.
Location at x of +x axis.
Location at x of - y axis.
Location at 0 of +x axis.
Y-y axis
Location at ordinate
Location at y of -y axis.
Infinity of
Positive infinity of {x,y} or negative infinity of
{x,y}.
Level proceeding
Random digit sign (0 or 1) added up to I.
Number of steps
Number of j steps that lead to effective positions.
Random variable
The process that performs counting number of
ways collected by random variable, which is
used to reach the destination, where l represent
ordinate level.
Probability of effective
position at s
Probability of kth effective position is s distance
far from (xp,- Yq).
jp
:Jp
P(lEPk)s
and flat, positively.
Position of dissipated customer in 4th quadrant
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first
effective
Wx(x,- y)
Probability function
Mutually overlapping
(s-l)x(s)
(s-2)x(s)
cells
(s+l)x(s-2)
(l)x(s-2)
Probability function when (xp,- Yq)is x
proceeding far.
Numbers of column and row arrangement during
overlapping of ranged of two different locations
when effective position is s distance far.
D(t)
D(ta
Random process
Counting process of decreasing distance in time
period t or ti .
d {a .P.rl
Distance through modes
d distance covered or declined through
r modes.
ith
distance
a,/3 and
Covered ith distance through the combination of
a,/) and r .
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LIST OF PUBLICATIONS
Singh, Harpreet and Ismail, M. Ghazie, 2014. 'Exceptions' in Queuing Theory.
International Journal of Computer and Information Technology, 3 (1), pp. 110-119.
Singh, Harpreet and Ismail, M. Ghazie, 2014. 'Exception' in Queuing Theory through
Markov Chain. International Journal of Electrical and Electronics Engineering, 1 (1 ), pp.
14-29.
xvii
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CHAPTERl
INTRODUCTION
1.1
Background
In a system of line up there are various factors that can develop the chain of
complications before breaking the queuing system into unlike situation. Any system that
constitutes individual giving services to seekers either mechanically or electronically or
manually or else and a seeker that gets services from either authorized or experienced
individual is known as serving system. In the queuing literatures, it is recognized as
queuing system. Generally the host(s) provides services to the chronological arrivals of
seeker that waits for their turn patiently. The study that mathematically analyzes the
number of customers in the system, queue length, system size, arrival discipline, service
time, service discipline, expected number of waiting, time in queue, load in the system and
many other interested terms, can know from the queuing theory (Hlynka, 2007). Danish
engineer, Agner Krarup Erlang considered as a discoverer of such theory when he
published first paper in 1909 on queuing theory in telephonic networks at Copenhagen.
The theory results in the form of queuing models with many applications m
different fields of study. This theory extensively used in telecommunications, hospitals,
traffic, computers, retails, airports, banks and many others. The arrival and service can be
individual or group, completely random or constant. A server can be single or multiple.
Queue size can be both finite and infinite. The queue controlling can be chronological,
random, or unordered. Customers permitted in the system are finite or infinite. The makeup of all these choices can collectively define in Kendall notation given as
(P/Q/R):(X/Y/Z), where P/Q is the arrival and a server probabilistic approach, R is server
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quantity, X is queuing controlling, Y is number of customers permitted and Z is size of
customer population.
It is routinely believed that the mutual contribution of socially acceptable behaviour
by the seekers and the host(s) supports queuing system performances. This can only be
possible if they are following ethics that are organized. If the queuing system considers
human involvements then most probably the only term relevant is the waiting. Waiting
requires patience and patience requires practice. Besides this, due to their achievement rate
the system in services does not desirable to keep customers in the waiting. However, the
reason that put into concern that why the waiting begins and gets large. The reason is
simple, when the undesirable exception comes in during the current task and its regularity
cause the beginning of waiting more than expected.
Large waiting completely influenced by the large accumulation of small-scale of
hindrances. It possibly will be an unnecessary exception that caused either the service
provider gets undesirably slow or service seekers are undesirably arriving faster than
normal. If there were no hindrances then the servicing rate is infinite and queue size is
zero, i.e., sojourn time in such state is nothing. Otherwise, it causes growing of queue
besides overcrowding prior to the queue less state. For e.g., in a queue network, service
rate variation within a larger time frame gives sudden input to subsequent server or specific
time of arrival of requests in a system in a short period of time.
The services of the customer may be either constant or stochastic, and the
organization of queuing system varies globally. However, in some area of the world the
queue routines are not following properly. From place to place the cultural practice of
queuing includes awful customers, dishonor and for few the existence of a queue is almost
alien to them. Such practices cause pressure on the serving system. However, the proper
restricted policy can control customer behavior inside the serving system to avoid from
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delay in services. Gross et al. (2008) suggested that if such behaviours are not controlled in
time the chances are that the queues get bigger and bigger unless at some point the
customers are not allowed to enter the system.
Ever since Erlang, most of the researches have been focused on designing wellorganized system by assuming static conditions to perform without any disruptions. In
general, queue analysis is based on the stationary assumption in customer's variability, but
in reality there are other variables that can direct to a different state of the system. Many
literatures analyzed the results using a queuing model on the basic assumption that p < 1,
otherwise later p
ｾ＠
1, the queuing system goes imbalance or unimagined situation. But the
series of outcomes with an undesired event and the queuing system later p > 1 exclusively
not at all exhibited. As Gross et al. (2008) highlighted, in queuing theory the queuing
models that predict the performance of serving system with the assumption that demands
arising randomly, but not unnaturally, which can be either service or arrival.
Such unnatural randomness may perhaps be due to undesired behaviour occurrences
of customers and server. Cohen (1982) describes the behavior of customers with two
characteristic features, viz. The arrival process and the waiting process, and thus the
continuation of unnoticeable behaviour may challenge the service seekers patience. Cox
and Smith (1961) and Conolly (1975) mentioned that congestion in queuing system occurs
due to an abnormality, i.e. , the sudden changes in behaviours of either customer(s) or
server will result in an unbalanced system. Wang et al. (2010) highlighted that numerous
queuing literatures discuss two types of behaviours concerned with customer and servers
which are impatience and vacation period, respectively. In customer behaviours case
reneging, preemptive, baking and jockeying is covered largely by other researchers. Kok
and Tijms (1984), Shin, Choo and Chakravarthy (2009), Sudesh, Pere! and Yechiali (2010)
3
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