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5
G. Ayyappan, S. Sankeetha
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RT&A, No 1 (77)
Volume 19, March 2024
STREAMLINING PRODUCT DEPLOYMENT:
ENHANCING EFFICIENCY THROUGH KITTING
PROCESSES
G. Ayyappan, S. Sankeetha
•
Department of Mathematics, Puducherry Technological University, India.
Department of Mathematics, Saradha Gangadharan College, India.
ayyappanpec@hotmail.com sangeetha.sivarajp@gmail.com
Abstract
Considering a single server with two queues that is prone to unreliability. The server offers a kitting
process and performs necessary checks and rectifications when required. The arrival of items follows
a Markovian arrival process, while the service is distributed based on a phase type distribution. The
incoming products may exhibit issues such as poor quality or defects. If either of the queues is empty,
the server is unable to provide the requested service and remains inactive. Furthermore, if all queues
are empty, the server goes into a vacation mode. Breakdowns, repairs, instances of customers leaving
without service (reneging), and vacation periods are all modeled using an exponential distribution. To
gain insights into the performance of the queueing model, various performance metrics are analyzed and
represented through 2D and 3D graphs.
Keywords: Markovian Arrival Process, PH distribution, Vacation, Optional service, Breakdown
and Repair.
1. Introduction
The Markov arrival process (MAP) is a widely employed modeling approach that captures the
dynamic Markov structure underlying point processes. It offers adaptability and versatility,
making it suitable for probabilistic models that employ matrix analysis techniques. Neuts [15]
made significant contributions by proposing and extensively investigating the flexible nature of
Markov point processes. MAP shares similarities with other point processes, including Markov-
modulated Poisson processes, phase-like updating processes, and semi-Markov point processes.
It enables the simulation of both updating and non-updating models, making it a valuable tool
for studying arrival patterns. Chakravarthy [7] has provided in-depth insights and extensive
discussions on MAP, specifically focusing on its m-dimensional parameter matrix (D0, D1, D2),
where D0 governs transitions associated with no arrivals and D1 and D2 controls alternations
related to arrival events. This parameterization allows for effective control and analysis of arrival
dynamics in various systems.
Wang et al. [28] presented a framework for optimizing the kitting process in manufacturing. It
addresses the challenges of efficiently organizing and sequencing materials required for assembly
operations. The authors propose a mathematical model to minimize the overall kitting time,
reduce material handling, and improve productivity in manufacturing settings. Yadav et al.
[26] focused on optimizing the kitting process in an automotive assembly line. It investigates
the challenges associated with kitting and proposes a mathematical model for optimizing the
allocation of parts to kits. A hybrid optimization approach is applied that combines genetic
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algorithms and simulated annealing to minimize the total distance traveled by workers during the
kitting process. The study provides insights into improving the efficiency of the kitting process
in automotive manufacturing. Ayyappan and Nithya [6] studied a retrial feature that allows
customers who experience service unavailability to reattempt service after a certain period. The
model considers priority services, where one type of customer is given priority over the other in
terms of service. Breakdowns and repairs are differentiated, meaning that the server may require
different amounts of time to recover from different types of failures. Synchronized reneging
is taken into account, which means that customers may abandon the queue simultaneously if
their waiting time exceeds a specific threshold. Additionally, the model incorporates an optional
vacation, allowing the server to take breaks during certain periods.
Zhang and Fang [29] introduce a novel optimization algorithm designed to enhance the
efficiency of bulk service systems. These systems are frequently encountered in various industries,
including manufacturing and transportation, where multiple units of work or customers are
processed simultaneously. The primary objective of the proposed algorithm is to minimize
service time and decrease waiting times for customers within bulk service systems. To achieve
this, the algorithm combines two powerful optimization techniques: stochastic optimization and
reinforcement learning. The algorithm works in iterations, continuously refining its policies based
on feedback from the system. It collects data on customer arrival patterns, service times, and
queue lengths, which are then used to update the stochastic optimization models and reinforce
the learned policies. This iterative process allows the algorithm to adapt to dynamic changes
in the system and continuously optimize its performance. Li and Li [13] focused on optimizing
bulk service systems that involve parallel servers. It addresses the challenges associated with
efficiently allocating and coordinating multiple servers to improve system performance. The
authors propose novel optimization algorithms and strategies to minimize service time and
reduce waiting times for customers.
Smith and Johnson [22] investigated the influence of bulk service providers on the overall per-
formance of supply chains. Also examines how the involvement of bulk service providers affects
various aspects of supply chain operations, including efficiency, cost, and customer satisfaction.
The impact of bulk service providers on key performance indicators are analyzed such as order
fulfillment, inventory management, and lead times. Additionally, it highlights the importance of
establishing effective collaboration and coordination mechanisms between bulk service providers
and other supply chain stake holders. Also emphasize on the significance of information shar-
ing, communication, and performance monitoring to ensure optimal supply chain performance.
Wang et al. [27] presents a hybrid optimization approach specifically tailored for bulk service
systems in e-commerce warehouses. The authors combine mathematical modeling, simulation,
and metaheuristic algorithms to enhance the efficiency of warehouse operations, such as order
picking, packing, and shipping. The proposed approach aims to reduce order fulfillment time
and improve customer satisfaction in e-commerce fulfillment centers.
Arun et al.[2] analyzed a bulk service queue with server breakdowns, balking, and reneging.
It provides a detailed analysis of the system's performance measures, such as the expected
waiting time and the expected queue length, under different scenarios. Sun and Zhang [23]
focused on the development of a bulk service system specifically designed for autonomous
mobile robots, the growing demand for efficient and flexible service systems in industries where
autonomous mobile robots are utilized. These systems involve the simultaneous processing of
multiple tasks or requests, and efficient management is crucial to optimize performance and
resource utilization. A comprehensive design framework for a bulk service system is proposed
that integrates autonomous mobile robots. They outline the key components of the system,
including task allocation, robot navigation, and coordination mechanisms. The findings of the
study demonstrate the advantages of incorporating autonomous mobile robots into bulk service
systems. The proposed design framework provides a blueprint for developing efficient and
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scalable systems that can adapt to changing demands and optimize resource allocation.
Arivudainambi and Arivudainambi [1] studied a mathematical model for analyzing a bulk
service queue with multiple vacations, server breakdowns, and general service times. It provides
a detailed analysis of the system's performance measures, such as the expected waiting time
and the expected queue length. Li and Zhang [14] proposed an optimal control policies for
a bulk service queue with impatient customers and time-varying arrival rates. The proposed
policies are designed to minimize the total expected cost, including waiting costs and service
costs, under different operating conditions. Saroja and Saravanarajan [20] studied bulk service
queueing models with server vacations and feedback controls. It provides a detailed analysis
of the system's performance measures, such as the expected waiting time and the expected
queue length, under different scenarios. Ayyappan and Meena [5] examined the service rate that
gradually declines until degradation is fixed. After completing a certain number of services (K),
the degradation is addressed. During the service period, the server may experience a breakdown
at any moment, triggering an immediate repair process. Once the service is complete, the server
transitions to the close-down process. If there are no customers in the system when the server
returns from vacation, the server will wait until a customer arrives. If a customer arrives without
a starting failure, the server provides service. However, if there is a starting failure, the server
immediately goes into the repair process.
Thottan and DeVeciana [24] presented a vacation model that incorporates autonomous server
vacations and customer impatience. The research focuses on analyzing the performance of queue-
ing systems under such conditions and investigates the impact of autonomous server vacations
and customer impatience on system efficiency. Huang and Li [8] investigated on optimization of
vacation queues that involve multiple vacation periods and general service times. The authors
investigate the problem of determining optimal control policies for allocating vacation time and
managing service rates in order to optimize various performance measures. They consider system
characteristics such as queue length, waiting time, and system utilization. By analyzing the
impact of different control policies on the system's performance, the authors provide insights
into the efficient management of vacation queues. Their research contributes to the development
of strategies for optimizing service allocation and improving the overall efficiency of queueing
systems with multiple vacation periods and general service times. Anis et al. [4] explored the
analysis of a finite-buffer queue that incorporates server vacations and customer impatience. It
investigates the performance measures of the queueing system, including queue length, waiting
time, and server utilization. The study provides understanding the enhancement of buffer size,
vacation policies, and customer impatience management.
Srinivasan and Sriram [21] analyzed on studying vacation queues where the server is subject to
breakdowns and repair. The authors analyze the impact of server breakdowns on the performance
of the queueing system. They investigate various performance measures such as queue lengths,
waiting times, and server utilization during both normal operation and breakdown periods. The
study provides insights into the optimization of repair policies to minimize system downtime and
improve overall system performance. By considering the combined effect of vacations and server
breakdowns, the authors contribute to the understanding of real-world queueing systems where
service interruptions due to breakdowns are common. Kim et al.[10] researched on vacation
models that consider customer abandonments. It investigates the impact of customer abandon-
ment behavior on queueing systems during vacation periods. The study provides perception on
optimization of vacation policies and customer abandonment management.
Rakesh Kumar et al. [19] examined a single-server Markovian queuing model that incorpo-
rated customer impatience, including balking and reneging, alongside a threshold mechanism
and customer retention. They employed probability generating functions to analyze the model's
transient behavior. Kalyanaraman and Janani [9] addresses a finite population Poisson queue em-
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ploying a fixed batch service rule. Following each service, the server goes on vacation, regardless
of queue size, providing service at a reduced rate during this period. The research calculates
system size probabilities, derives performance metrics, and also explores an infinite population
model with limited waiting room capacity as a secondary model. Krishnamurthy et al. [11]
centers on the examination of a queuing system characterized by its multi-stage bulk service
approach and the availability of service in batches. Within this system, incoming customers are
initially grouped into batches before undergoing bulk servicing. The research extensively presents
mathematical derivations pertaining to performance metrics, including system size, mean waiting
time, and mean service time.
Raina Rajand Selvamuthu Dharmaraja [17] introduces an architectural framework that priori-
tizes energy efficiency within the SAT network, with a particular focus on HAPs. Furthermore, a
stochastic model is proposed to account for three distinct states of energy conservation for HAPs,
including modes of power conservation, standby, and rest, where energy consumption is minimal
or negligible. Upon the arrival of a data packet, HAPs promptly transition to active service
mode, ensuring the entire system operates in an active state. Anilkumar and Jose [3] examines a
discrete-time inventory model (s, S) is investigated, featuring Bernoulli process customer arrivals
and geometrically distributed service and replenishment times. When inventory drops to zero
due to customer service or lack of replenishment, the system can accommodate a maximum of k
customers, with any excess customers considered lost until replenishment occurs. Rakesh Kumar
et al. [18] conducted a comprehensive study examining the utilization of queuing theory in the
analysis of cloud computing systems. Their research specifically delved into the phenomenon of
task reneging, where requests are dropped from the queue due to user impatience, deadlines,
security protocols, or active queue management strategies.
2. Motivation
In a software development company, a team is working on creating a new application that
consists of multiple modules and features. Rather than developing and delivering each module
individually, they adopt a kitting process to streamline the deployment process and improve
efficiency. In this kitting process, each module or feature is treated as a separate item and is
placed in a dedicated queue. The server, which represents the deployment team, retrieves the
modules from the queues and starts assembling the software kit. They integrate the modules,
perform necessary configurations, and ensure compatibility between different components.
Once the kit is assembled, the server performs thorough testing and quality assurance checks
to verify the functionality and stability of the software. If any issues are identified, such as bugs or
compatibility conflicts, the server rectifies them before proceeding. Once the kit passes the testing
phase, it is packaged for release to the end-users or clients. The server ensures that all required
documentation, user guides, and support materials are included in the kit before delivering it. By
employing the kitting process in software development, the company streamlines the deployment
process, reduces errors, and ensures that the end-users receive a comprehensive and well-tested
software package.
3. Mathematical Formulation
This model considers two types of arrivals within a system. The first type follows a Markovian
arrival process and has infinite capacity, while the second type has a finite capacity of K. The
server is responsible for the packing service, which follows a phase type distribution denoted as
(«1, T1). The equation T0 + T1 e = 0 holds true, where T0 represents a column vector. Once the
packing is completed using the kitting process, the server proceeds to verify the checklist for the
packed product.
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If the checklist is satisfied, the product is deemed ready for the outlet. Otherwise, the server
initiates the rechecking and rectification process. This rechecking process follows a phase type
distribution denoted as (a2,T2). The equation T0 + T2e = 0 holds true, where T0 represents a
column vector. If either of the queues becomes empty, the server remains idle. However, when
both queues are empty, the server goes on vacation, with the vacation parameter ц following an
exponential distribution.
Additionally, the server is subject to breakdown during both the packing service and recheck-
ing, with a breakdown parameter £ following an exponential distribution. When the server
experiences a breakdown while serving, it completes the ongoing service and then enters a repair
process with a parameter 7 following an exponential distribution. Moreover, the products in both
queues are susceptible to reneging, indicated by parameters 81 and 32, respectively, following an
exponential distribution. Reneging can occur due to factors such as lack of quality or defects.
Figure 1: Schematic Representation of Our Model
In pursuit of a matrix-geometric solution, the model is explored within the framework of a
QBD (Quasi-Birth-Death) process. For a comprehensive exploration of Matrix Analytic Methods,
refer to the works of Neuts [16] and Latouche and Ramaswami [12]. The QBD model's state space
is formally defined, and an examination of the infinitesimal generator's structure is carried out,
leveraging the subsequent notational conventions.
Let
• Ij is the identity matrix of dimension j.
• e1 is the column vector of dimension M1 m2 [(П1 + n2)(1 + K) + (4 + 3K) with its entries 1.
• N1 (t) indicates the total number of items in the type I queue.
• N2 (t) indicates the total number of items in the type II queue.
• S(t) indicates the position of the server.
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if server is idle
if the server is engaged with packing
if the server is engaged with rework
if the server faces breakdown while packing
if the server faces breakdown while rework
if server is on vacation
indicates the service phase when the server is engaged with packing.
• J2(t) indicates it the service phase when the server is engaged with packing.
• M1 (t) indicates the phase of the Markovian Arrival Process for type I queue.
• M2(t) indicates the phase of the Markovian Arrival Process for type II queue.
Let {(N1 (t), N2(t), S(t), J1 (t), J2(t),M1 (t),M2(t)); t > 0} represent the continuous time Markov
chain for the QBD process with the state space.
n = l(0) u i(i)
where,
/(0) = {(0,j,0,s1,s2) : 0 < j < K, 1 < s1 < m1, 1 < s2 < m2}
For i > 0,
l(i) = U{(0,j,1,r1,s1,s2) : 1 < j < K,1 < r1 < n1,1 < s1 < m1,1 < s2 < m2}
U {(0,j,2,r2,s1,s2) : 1 < j < K,1 < r2 < n2,1 < s1 < m1,1 < s2 < m2}
U {(0, j, l, si, s2) : 0 < j < K,3 < l < 5,1 < Si < mi, 1 < s2 < m2}
For i > 1, l(i) = {(i,0,0,s1,s2) : 1 < s1 < m1,1 < s2 < m2}
The infinitesimal matrix generation of the QBD process is given by
where
S(t) =
0,
1,
2,
3,
4,
5,
q
B00 B01 0 0 0 0
B10 A1 A0 0 0 0
0 A2 A1 A0 0 0
0 0 A2 A1 A0 0
where each of its block matrix are as follows,
b10
rB11 b00 B12 b00 0 0 ■ ■ ■ 0 0
B21 b00 bgg b03 0 ■ ■ ■ 0 0
B00 = 0 b30 b02 b23 ••• 0 0
0 0 0 0 ■ ■ ■ B32 b00 BM+1M+1
B011 a1qT^ ® I m1 m2 ^ Im1 m2 0 B111
0 B211 0 ^ IM1 m2 B211
Y Im1 m2 0 B011 b33 0 0
1 Im1 m2
0
B011
0
0 Im2 ® D0 ® Im1
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В??? = Im1 0 (T? ® (D0 - £lrn2 ))
B°)41 = em2 0 a?pT?0 0 Im?
B0?1 = Im2 ® (T2 ® (Do - £Im? ))
B04 = Em2 0 ^2To ® Im?
B01 = IM2 0 (D0 7Вм? )
0 1п?м? 0 D2 0 0 0 0
B?2 = B00 = 0 0 Вп2м? 0 D2 0 0 0
0 0 0 Bm? 0 D2 0 0
0 0 0 0 Bm? 0 D2 0
0 0 0 0 0 Bm? 0 D2
0 0 0 0 ^2 BM? М2
^2 In?m? mo 0 0 0 0
Boo = о о ^2 Вп2 м? м2 0 0 ^2 Bm? м2 О о о о
0 0 0 ^2 Вм? М2 0
0 0 0 0 ^2 ВМ? М2_
B?0?22 0 0 0 0 0
a?pT?0 0 Im B20222 a^T?1 0 Im £ Bmn 0 0
a2 T2? 0 Im 0 B°0322 0 £ Bmn 0
0 7 Bmn 0 B022 B44 0 0
0 0 7 Bmn 0 B022 B44 0
0 0 0 0 0 B?0?22
B022 = B?? = IM2 0 (D0- ^2 Bm? )
B022 = B22 = Bm? 0 (T? ® (D0- (£ + ^2) BM2 ))
B022 = B33 = BM2 0 (T2 ® (D0- (£ + ^2) Bm? ))
B022 = B44 = BM2 0 (D0- (7 + ^2 ) Bm? )
Bm? 0 D2 0 0 0 0 0 "
0 Вп? м? 0 D2 0 0 0 0
0 0 ВП2М? 0 D2 0 0 0
0 0 0 Bm? 0 D2 0 0
0 0 0 0 0 D2 0
0 0 0 0 0 Bm? 0 D2_
M?M2 0 0 0 0 0
0 ^2 Вп? м? М2 0 0 0 0
0 0 ^2 ВП2 m? м2 0 0 0
0 0 0 ^2 Bm? м2 0 0
0 0 0 0 ^2 BM? М2 0
0 0 0 0 0 ^2 BM?M2
L w2*- M?M2J
■в0м+lм+l 0 0 0 0 0
?? B0?? B0M+?m+? B0M+?m+? £ Вп?м? М2 0 0
bm+?m+? = b0?? 0 b0M+?m+? 0 £ BM? М2 0
00 = 0 7 BM? М2 0 B0M+IM+! 0 0
0 0 7 BM? М2 0 B0M+?m+? 0
0 0 0 0 0 B0M+?m+? ^55
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B0M+1M+1 = 0 (Dq + D2 - $2Imi)
BGM+IM+1 = Imi 0 (Ti ® (Dq + D2 - (Z + $2) Im2 ))
BqM+1M+1 = Em2 0 aiqT0 0 Im1
В0М+1м+1 = lmi 0 (T2 ® (Dq + D2 - (Z + $2)Imi))
B0M+1M+1 = Im2 0 (Dq + D2 - (7 + $2)lmi)
BQM+1M+1 = lm2 0 Dq + D2 - $2 lmi
Boi
rBii b01 o o ■ ■ o "
o b22 o ■ ■ o
_ o o o ■ ■ B22 _
Bii
b01
B22
b01
■0 lNiM2 0 D1 o o o o
o o lN2М2 0 D 1 o o o
o o o Im2 0 D 1 o o
o o o o lM2 0 D1 0
0 o o o o lM2 0 D
lM2 0 D1 o o o o '
INiM2 0 D1 o o o o
o lN2М2 0 D1 o o o
o o lm2 0 Do o o
o o o Im2 0 Do 0
o o o o lM2 0 D1-
[Bio o o o o '
b20 B22 Bio o o o
Bio = o B32 Bio B22 Bio o o
. o o o B32 Bio B22 BioJ
o o o o $1 lMi М2
$1 l«i mi m2 o o o o
11 o $1 lN2mi М2 o o o
10 o o $1 lmi m2 o o
o o o $1 lmi m2 o
o o o o $1 lMi М2-
B21
b10
em2 0 aipT0 0 lmo o o o o'
eM2 0 a2 T;g 0 lmi o o o o
o o o o o
o o o o o
o o o o o
B22
b10
"0 $1 l«iMi М2 o o o o "
o o $11«2Mi М2 o o o
o o o $1 lMiM2 o o
o o o o $1 lmi m2 o
_o o o o o $1 lmi M2_
i
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b32
b10
Ai =
0 Em2 © aipT.° © Im4 0 0 0 0
0 EM2 © a 2 t2 © Im4 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
ral1 al2 0 0 ■ ■ ■ 0 0
a^1 a^2 al3 0 ■ ■ ■ 0 0
0 b32 al2 al3 ... 0 0
0 0 0 0 ■ ■ ■ «32 flf+1M+1.
rail 0 0 0 0 0 ■
b044 a22 B0M+1M+1 & IMlM2 0 0
al4 — B011 b24 0 23 aii a33 0 & Imi m2 0
0 Y IMi M2 0 a44 0 0
0 0 Y IMl M2 0 a44 0
0 0 0 0 0 a4i a66
a
12
1
a11 — ^От2 © (°0 5i ^mi )
a22 — 1m1 © (T4 ® D0 — (& + 5i)^m2)
a33 — !m2 © (T2 ® D0 — (& + 5i) 1m1 )
a44 — Im2 © D0 (7 + ^1) ^mi
a66 — 1m2 © D0 - 5i Im4
Imi © D2 0 0 0 0
Inimi © D2 0 0 0 0
0 I«2Mi © D2 0 0 0
0 0 ^mi © D2 0 0
0 0 0 Imi © D2 0
0 0 0 0 Imi © D2_
a
21
1
0 52 I«l Ml M2 0 0 0 0
0 0 52 ^2Ml M2 0 0 0
0 0 0 52 Imi m2 0 0
0 0 0 0 52 Imi m2 0
0 0 0 0 0 52 Imi M2_
a
22
1
Г ai22 aii Em2 © ai pTi © ^Ml &^«4 Ml M2 0 0
0 a422 0 &^N4 Mi m2 0
Y Imi M2 0 a422 a33 0 0
0 Y ^Ml M2 0 a422 и44 0
_n Imi m2 0 0 0 a422 И44 J
a442 — Im4 © (Ti © D0 — (& + 5i + 52))^m2
a222 — ^m2 © (T2 © D0 — (& + 5i + 52)) 1m1
a322 — Im4 © D0 — (7 + 51 + 52)^m2
a4!2 — Im4 © D - (n + 5l + &) Im2
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InjMi 0 D2 0 0 0 0
aJ3 = 0 I«2Mi 0 D2 0 0 0
0 0 Jmi 0 D2 0 0
0 0 0 JM! 0 D2 0
0 0 0 0 JM! 0 D2
Я
M+1M+1
1
1M+1M+1
11
0
Y Jmi m2
0
n Jmi m2
EM2 0 «1 pT0 0 ^Mj
Я
1M+1M+1
22
0
Y JMi M2
0
£ 1щМ1 M2
0
1M+1M+1
33
0
0
Я
0
£ I«2M1 М2
0
1M+1M+1
Я
33
Я
0
0
0
0
1M+1M+1
44
0
a1M+1M+1 — Jm! ® (T1 ® Do + D2 - (£ + *1 + $2))Im2
a1M+1M+1 — Jm2 ® (T2 ® Do + D2 - (£ + *1 + *2))Imj
aJM+1M+1 = Im! ® Do + D2 - (y + *1 + *2)IM2
aJM+1M+1 = Imj 0 Do + D2 - (n + *1 + *2)Im2
Ao
яО1 0 0
0 яО2 0
0
0
0 0 0 ■ ■ ■ яО2
JM2 0 D1 0 0 0 0 0 "
0 Jni m2 0 D1 0 0 0 0
Я11 — Я0 — 0 0 JN2M2 0 D1 0 0 0
0 0 0 JM2 0 D1 0 0
0 0 0 0 JM2 0 D1 0
0 0 0 0 0 JM2 0 D1_
JniM2 0 D1 0 0 0 0
Я22 — Я0 — 0 JN2M2 0 D1 0 0 0
0 0 JM2 0 D1 0 0
0 0 0 JM2 0 D1 0
0 0 0 0 JM2 0 D
я121 0 0 ■ ■ 0 0
A2 — я2221 я222 0 ■ ■ 0 0
_ 0 0 0 ■ ■ я2! я222
*1 JMi М2 0 0 0 0 0 "
0 *1 Jni Mj М2 0 0 0 0
я121 — 0 0 *1 J«2 Ml М2 0 0 0
0 0 0 *1 IM1 M2 0 0
0 0 0 0 *1 IM1 M2 0
0 0 0 0 0 *1 IM1 M2
EM2 0 «1 pT1 0 JM1 0 0 0 0"
eM2 0 «1T2 0 Jmj 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0.
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^1 fn1 M1 m2 0 0 0 0
я22 = 0 ^1 in 2 M1 M2 0 0 0
0 0 ^1 ^M1 M2 0 0
0 0 0 ^1 Im1 M2 0
0 0 0 0 ^1IM1 M2_
4. Analysis of the Stability Condition
Determining the stability of a system is crucial to ensure its smooth operation and efficient
handling of incoming arrivals. The concept of traffic intensity serves as a key metric in assessing
system stability. By comparing the average arrival rate with the average service rate over the
long run, we can gauge whether the system is capable of managing the workload effectively. For
stability, it is desirable that the traffic intensity remains below 1, indicating that the system can
handle the incoming arrivals without becoming overwhelmed.
Analyzing the stability of a Markovian arrival process (MAP) presents unique challenges
compared to simpler arrival processes like the Poisson process. This is due to the diverse
inter arrival time distributions that MAPs can exhibit. To explore stability conditions in MAPs,
researchers employ matrix-analytic methods and simulation-based methods. These approaches
involve analyzing matrices and eigenvalues to ascertain the system's stability. Simulation-based
methods, in particular, prove valuable when dealing with complex systems that lack analytical
solutions, enabling researchers to simulate and study system behavior under varying conditions.
Let A be an irreducible infinitesimal generator matrix of order m1 m2[(n1 + n2)(1 + K) + (4 +
3K)]. We can decompose A as A = A0 + A1 + A2. The vector p = (p0, P1, p2,..., Pk+1 ) represents
an invariant probability vector. It satisfies the conditions pA = 0 and pe = 1, where pe denotes
the dot product between p and the vector e.
P0K1 + < + Й21 ] + p![fl21 + Й21 ] = 0.
p0[«21 ] + p^2 + я!2 + Я22 ] + p2 [В32 + flf ] = 0.
Й-1И13] + pi [я22 + я12 + я32] + Й+#32 + Я21 ] = 0, for i = 1 to К - 1.
pK [я23 ] + pk+1[a02 + яК+1,К+1 + я22 ] = 0.
Given the normalizing condition pe = 1, in a stable system, it is necessary that
pA0emn[Z(K+1)+1] < pA2emn[Z(K+1)+1].
p0Я01 + (p1 + p2 + ... + pK+1)A02 < p0Я21 + (p1 + p2 + ... + pK)я21 + (p1 + p2 + ... + pK+1)A22.
5. The Vector of Invariant Probabilities
The crucial role of capturing the system's steady-state behavior is played by the invariant
probability vector, which is symbolically represented as X. In order to obtain the vector X, it is
necessary to solve the system of equations represented as XQ = 0, while simultaneously ensuring
the normalization condition Xe = 1. Once the stability requirements are fulfilled, the remaining
components of X can be computed using an iterative approach. It is important to emphasize that X
can be partitioned into sub-vectors, including X0 and Xi for i > 1, which have specific dimensions
based on the system's characteristics. The dimension of X0 ism1 m2[(n1 + n2)(1 + K) + (3 + 4K),
while Xi for i > 1 has a dimension of m1m2[(n1 + n2)(1 + K) + (4 + 3K). Precisely calculating
the values of X0 and Xi involves considering the unique properties and parameters of the system
at hand. The expression for Xi can be represented as:
Xi = X1 Ri-1, i = 2,3,4,...,
Here, R refers to the rate matrix, which serves as the minimal non-negative solution to the matrix
quadratic equation.
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R2 A2 + RA1 + A0 — 0
The boundary states, represented as X0 and Xi, are determined by solving the following equations:
X0 B00 + Xi Bw — 0
X0 B01 + Xi (Ai + RA2) — 0
These equations are subject to the normalizing condition:
X0e + X1 (I - R)-1e — 1
It's worth noting that Latouche and Ramaswamy [12] have improved the computation of the rate
matrix R by introducing the Logarithmic Reduction Algorithm. This algorithm simplifies the
process of obtaining R, making it more efficient and straightforward.
Step 1 : H ^ (-A1)-1 A0, L ^ (-A1 )-1 A2, G — L and T — H.
Step 2 : U — HL + LH;
M — H2;
H — (I - U)-1 M;
M — L2;
L — (I - U)-1 M;
G — G + TL;
T — TH;
continue Step 1 until \\e - Ge\\ » < e.
Step 3 : R — -A0(A1 + A0G)-1.
6. Examination of Busy Period
In the context of queueing theory, an essential aspect is the analysis of the busy period. This term
refers to the duration that starts when a customer enters an empty queue and concludes when
the queue once again becomes vacant. However, when dealing with Quasi-Birth-Death (QBD)
processes, a different concept known as the "fundamental period” emerges. The fundamental
period characterizes the duration needed for the system to shift from level 1 to level 1 - 1, where
1 assumes a value of 2 or greater. It's worth noting that special considerations are needed for
boundary states, particularly when 1 takes on values of 0 or 1. Furthermore, when examining all
levels 1 greater than or equal to 2, it becomes evident that there is a total of mn[l(1 + K) + 1] states.
This expression quantifies the number of states associated with each level within the queueing
model.
Notations:
• Gvv1 (k, x) corresponds to the likelihood that the QBD process enters level и - 1 at time t — 0
after undergoing precisely к leftward transitions and arriving at state (u, v'), under the
condition that it initially commenced in state (u, v) at time t — 0.
• The transition matrix Gvv' (z, s) is defined as zk e-sxdGvv' (к, x), where the conditions
are |z| < 1 and Re(s) > 0. This matrix incorporates a combination of infinite series and
integrals to capture the intricate transitions inherent in the QBD process. •
• G(z,s) takes the form of a matrix (Gvv'(z,s)) and adheres to the equation G(z,s) — z[sI -
A1 ]-1 A2 + [sI - A1 ]-1 A0G2(z,s), representing the interplay among various elements of the
QBD process.
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• In the context of the first passage time analysis, G = Gvvi = G(0,1) captures the behavior
of the process in the absence of boundary states, providing insights into its performance
without considering boundary effects.
• G^V0) (K, x) is the conditional probability that enters the level 0 from 1 at time t = 0.
• G(0V0j (K, x) is the first conditional probability returning to level 0.
• KiV denotes the anticipated duration for the first passage between levels и and и — 1 when
the process is in state (и, v) at time t = 0.
• ^ is a column vector composed of the entries K1v, representing the expected first passage
times for different states.
• ^2V stands for the average number of customers who receive service during the initial
passage between levels и and и — 1 when the process begins in state (и, v) at time t = 0.
• Я2 is a column vector composed of the entries ^2V, signifying the average number of service
completions during the first passage time for different states.
• K(1,0) represents the average duration for the first passage from level 1 to 0 within the QBD
process.
• й21,0) signifies the average number of completed services during the initial passage from
level 1 to 0.
• Я(0,0) denotes the average time taken for the first return to level 0 within the QBD process.
• к2°,0) represents the average number of completed services during the initial return to level
0.
The G matrix can be computed using the following expression, utilizing the previously determined
rate matrix R obtained through the Logarithmic Reduction Algorithmic technique:
G = —[ A\ + RA2] 1А2
For the boundary states, specifically 1 and 0, we can establish equations satisfied by G(1,0)(z, s)
and G(0,0)(z, s), respectively:
G(1,0)(z,s) = z [sI — Ai]—1B10 + [si — A1]—1A0G(z,s)G(1,0)(z,s).
G(0,0)(z,s) = z [sI — B00]—1 B01G(1,0)(z,s).
Since G, G(1,0)(z, s), and G(0,0)(z, s) are stochastic in nature, we can readily compute moments as
follows.
Я1
Я2
я"’
<'0)
»f0)
ds G(z, s)|s=0,z=1 =
dZG (z, s)|s=0,z=1 = —
— dS(G(1,0) (z, s) |s=0,z=1
dZ(G(1,0) (z, s) |s=0,z=1 =
— JS(G(0,0) (z, s) |s=0,z=1
dZG (0,0)(z, s)|s=0,z=1 =
— [ A0 (G + 1) + A1] 1 e
[ A0 (G + 1)+ A1]—1 A2e
= — [ A1 + A0G] 1 [ А0я1 +
= — [ А1 + A0G] 1 [B10 e + A0
B
00
e + В01Я1
(1,0)
—B—01 B01 я21,0).
1
e]
Я2 ]
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7. Performance Measures
When a system reaches a steady-state, it signifies that the system has achieved stability and
performance measures can be derived and examined. These performance measures play a vital
role in evaluating the various aspects of system performance and determining its efficiency and
effectiveness. By analyzing these measures, we can gain valuable insights into the system's behav-
ior and identify areas that require improvement to enhance overall performance. Performance
measures serve as quantitative indicators that shed light on important system characteristics such
as throughput, response time, resource utilization, and reliability. They provide a comprehensive
view of how well the system is functioning and can help in assessing its overall effectiveness
in meeting desired objectives. By closely monitoring and analyzing performance measures,
decision-makers can identify potential bottlenecks, inefficiencies, or areas of improvement within
the system. This enables them to make informed decisions and take appropriate actions to
optimize system performance, increase productivity, and enhance customer satisfaction.
• Probability the server is idle .
Pi = Ey=i x0j0 + Ei=i x100.
• Probability the server is busy with packing.
Pbp = S=0 Ef=0 Xj1.
• Probability the server is busy with rework.
PBR = П=0 Ej=0 XiJ2.
• Probability the server is in breakdown while busy with packing.
PBDP = G=0 Ej=0 xij3.
• Probability the server is in breakdown while busy with rework.
PBDP = G=0 Ej=0 xJ4.
• Probability the server is on vacation.
Pv = LT=0 Ef=0 XJ5.
• Expected system size
pSystem xl[(1 R) ]e1.
8. Cost Analysis
Let us introduce a cost associated with different system management metrics for our model of
interest. We can then formulate a cost function, TC, which takes these metrics into account.
ТС = CH * Esystem + Pv * CV + Pi * Ci + Pbp * CBP + Pbr * CBR + Pbdp * CBDP + Pbdr *
CBDR + ц1 * Cl + ц2 * C2 + y * C3
where
• TC-Total cost of the system per unit time.
• CH-Customer holding cost in the system per unit time.
• CV - Cost when the server is on vacation per unit time.
• CI - Cost when the server is idle per unit time.
• CBP - Cost when the server is busy with packing per unit time.
• CBR - Cost when the server is busy with rework per unit time.
• CBDP- Cost when the server faces breakdown while packing per unit time.
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• CBDR- Cost when the server faces breakdown while rework per unit time.
• C1 -Cost afforded for packing service by the server per unit time per unit time.
• C2 - Cost afforded for rework service by the server per unit time per unit time.
• C3 - Cost afforded for carrying out the repair process per unit time.
9. Numerical Analysis
In this section, we will delve into the qualitative behavior of the model through a series of
illustrations that include both numerical and graphical representations. By manipulating various
model parameters, such as the arrival process and service time distribution, we aim to gain
a deeper understanding of how these parameters affect the model's behavior. Input data for
these parameters will be drawn from three sets of values available in the literature, allowing us
to examine a wide range of scenarios and explore the model's response to different parameter
settings. Through these illustrations, we will shed light on the dynamics and trends exhibited by
the model as we vary the model parameters, helping us gain insights into its behavior in different
scenarios.
Erlang of order 2 (ERL-A)
-5 5 0 0 0 0 0 0 0 0 0 0 0 0 0
0 -5 5 0 0 0 0 0 0 0 0 0 0 0 0
0 0 -5 5 0 ; D1 — 0 0 0 0 0 ; D2 — 0 0 0 0 0
0 0 0 -5 5 0 0 0 0 0 0 0 0 0 0
0 0 0 0 -5 3 0 0 0 0 2 0 0 0 0
Exponential (Exp-A)
D0 = [-1]; D1 = [0.6] D2 = [0.4]
Hyperexponential (HYP-EXP-A)
D0
- 1.90
0
0
-0.19
; D1
1.026 0.114 D2 — ' 0.684 0.076
0.1026 0.0114 0.0684 0.0076
Given that Varghese et al. [25] has suggested three phase type distributions for the service
process, we will consider these distributions in our analysis. These phase type distributions,
which have been proposed by Chakravarthy [7] and documented in the literature, will serve
as the basis for our examination of the model's behavior. By incorporating these distributions
into our analysis, we aim to gain a deeper understanding of how the model performs under
different service time distribution settings and how it responds to varying parameters associ-
ated with these distributions. This will enable us to assess the qualitative behavior of the model
and uncover any patterns or trends that emerge as we explore these three phase type distributions.
Erlang of order 2 (ERL-S)
ДТ — «2 — (1,0); T) — T2
-2 2
02
Exponential (Exp-A)
«1 — (1); T1 — [-1]
&2 — (1); T2 — [-1]
Hyperexponential (HYP-EXP-A)
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a1 — (0.3,0.7); T1
a2 — (0.4,0.6); T2
-9 3
2 00 ' 1
-12 6
5 10
Illustration 1:
In this analysis, we examine the implications of the reneging rate (Si ) of the customers on the
expected system size (Esystem) for various combinations of service and arrival times. We consider
specific parameter values, including A — 2, pL — 5, p2 — 6, £ — 1, у — 3, S2 — 1, q — 4, p — 0.3,
and q — 0.7. The observations derived from Table 1 to 3 are outlined below.
• As the reneging rate increases, more customers choose to leave the system without complet-
ing their service requests. This results in a lower number of customers in the system at any
given time, leading to an decrease in the expected system size.
• When customers renege at a higher rate, the system experiences a shorter average waiting
time and lower congestion due to customers leaving before being served. This decrease
congestion leads to only few customers remaining in the system, resulting in a lower
expected system size.
Illustration 2:
In this analysis, we examine the effects of the vacation rate (£) and service rate (pi) of the server
on the expected system size (Esystem). We consider various combinations of service and arrival
times and use specific parameter values, including A — 2, pL — 6, у — 3 Sl — 1, S2 — 1 q — 4,
p — 0.3, and q — 0.7. The observations derived from Figure 29-37 are outlined below.
• When both the vacation rate (£) and service rate (pi) increase, it generally leads to a
decrease in the expected system size. This means that, on average, there will be fewer
customers present in the system at any given time.
• An increase in the vacation rate (£) implies that the availability of the server increases.
Similarly, an increase in the service rate (pi) means that the server can process customer
requests at a faster pace. When both the vacation rate and service rate increase, the server
has a reduced overall availability for serving customers due to more frequent breaks.
• These observations highlight the varying impacts of vacation rate and service rate on the
projected system size across different arrival and service times. Erlang arrivals show the
most significant reduction in system size, followed by exponential arrivals, while hyper
exponential arrivals display a slower rate of decrease.
Table 1: Renege rate (Al) vs Expected System Size - ERL-A
service
Si Erlang Exponential Hyperexponential
1.0 2.689867713 2.764059228 2.776027059
1.1 2.680737416 2.74942792 2.765549157
1.2 2.672576038 2.734797612 2.755075026
1.3 2.665225627 2.720166304 2.744593354
1.4 2.658647922 2.705534995 2.724115453
1.5 2.647557358 2.690903687 2.703637552
1.6 2.640837634 2.676272379 2.683159655
1.7 2.625331925 2.661641071 2.672681749
1.8 2.611886634 2.647009763 2.662203847
1.9 2.591670138 2.632378454 2.651725946
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Table 2: Renege rate *) vs Expected System Size - EXP-A
service
*1 Erlang Exponential Hyperexponential
1.0 2.732538773 2.776538896 2.815784456
1.1 2.721217834 2.766061989 2.794520162
1.2 2.710069838 2.755584087 2.777635494
1.3 2.709039454 2.745106186 2.763847123
1.4 2.698116376 2.734628284 2.752344555
1.5 2.689888555 2.724150383 2.742586259
1.6 2.681785353 2.713672482 2.734194049
1.7 2.673681963 2.703194585 2.726894449
1.8 2.665578647 2.699716679 2.720483688
1.9 2.657475344 2.682238777 2.714806803
Table 3: Renege rate *) vs Expected System Size - HYP-EXP-A
service
*1 Erlang Exponential Hyperexponential
1.0 2.857181106 2.902141938 3.011784445
1.1 2.832316046 2.882431771 2.969464924
1.2 2.813250935 2.866914417 2.945390736
1.3 2.798124579 2.854332628 2.926737224
1.4 2.785797988 2.843905244 2.911866499
1.5 2.775536658 2.835114586 2.899730666
1.6 2.766845678 2.827598651 2.889632399
1.7 2.759379139 2.821103707 2.881091624
1.8 2.752887573 2.815428859 2.873767852
1.9 2.747186259 2.797494693 2.867413281
Figure 2: Vacation rate (n), Service rate (p1)
vs Expected system size - Ek/Ek/1
Figure 3: Vacation rate (n), Service rate (p1)
vs Expected system size - Ek/M/1
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Figure 4: Vacation rate (n), Service rate (ц\)
vs Expected system size - Ek/Hk/1
Figure 5: Vacation rate (n), Service rate (p\)
vs Expected system size - M/Ek/1
Figure 6: Vacation rate (n), Service rate (p\)
vs Expected system size - M/M/1
Figure 7: Vacation rate (n), Service rate (p\)
vs Expected system size - M/Hk/1
Figure 8: Vacation rate (n), Service rate (p\)
vs Expected system size - Hk/Ek/1
Figure 9: Vacation rate (n), Service rate (p\)
vs Expected system size - Hk/M/1
Figure 10: Vacation rate (n), Service rate (p\)
vs Expected system size - Hk/Hk/1
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10. Conclusion
In conclusion, our model encompasses a complex system involving a single server managing
two queues, each susceptible to various uncertainties. We have meticulously detailed the arrival
and service processes, highlighting the critical phases and distribution patterns that govern
them. The model accounts for the inherent unpredictabilities, such as server breakdowns, repairs,
customer reneging, and vacation periods. By examining both infinite and finite capacity arrivals,
we have provided a comprehensive framework for analyzing the performance and reliability of
this intricate system. This model can serve as a valuable tool for optimizing operations, enhancing
service quality, and minimizing disruptions in scenarios where such intricate dynamics are at play.
Broadening the system's scope to accommodate intricate service time patterns mirroring real-
world complexities holds the potential for a more profound comprehension of service dynamics.
Upcoming research endeavors will center on refining scheduling strategies and computational
methods for handling batch arrivals, server disruptions, repair processes, bulk services, and
the involvement of multiple service providers. These initiatives seek to minimize customer
waiting intervals, optimize resource distribution, and elevate overall system effectiveness, with
the ultimate goal of enhancing the applicability of such systems across diverse domains.
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