DECLARATION 2 PUBLICATIONS
3.1 Queuing Mathematical Theory
Queuing theory is one of the most efficient and widely used mathematical techniques to perform research on waiting lines and queues in commercial manufacturing applications [42]. Queuing theory was developed by A. K. Erlang in 1903 in his attempt to solve a congestion problem during a telephone call [44]. His widely used approach towards estimating the queue length and waiting line has always been a success in an advanced manufacturing environment. Queuing theory has been used to estimate the average waiting time of products/services in a queue, the expected time spent within a system, average queue length, the required numbers of customers attended to at a time [47].
Queues and waiting lines are commonly experienced in industries, banking systems, colleges, telephone calls, traffic, hospitals, post offices, and schools [59]. A queue is also experienced among customers waiting to lodge online complaints. Clients would have to wait for so long before they have been attended to by the operator on duty [46] [60] [60]. A queue is a common manufacturing problem that requires adequate control. It delays the manufacturing process and can result in a high cost of production. A queue has significant impact on the throughput rate in an advanced manufacturing environment.
3.1.1 Features of the Queuing Mathematical Theory
The three main components considered when describing the queuing system were the queue, rate of arrival, and the service facility.
a) The queue (waiting line)
b) Rate of arrival (the rate at which customer enter the system) c) Service (service facility given to customers)
The above components of the queuing theory required adequate examination before an effective mathematical model was developed to analyze queuing problems in an advanced manufacturing environment.
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3.1.2 Advantages of Queuing Mathematical Modeling Theory.
The following benefits were derived from applying the queuing mathematical modeling theory in solving problems in an advanced manufacturing environment.
a) Queuing mathematical modeling theory was useful in the situation whereby the manufacturer was required to create a balanced relationship between the optimization of the service costs and the waiting costs.
b) Queuing mathematical modeling theory was used to illustrate, analyze and provide a clear understanding of how a waiting length can be and provide an adequate solution to manage the queue.
c) Queuing mathematical modeling theory was used to develop classical models used to determine the pattern of arriving customers/products in an advanced manufacturing environment.
3.1.3 Limitation of Queuing Theory.
The queuing theory also has its limiting factors which are presented below.
a) Queuing mathematical models are very complex and difficult to analyze due to the presence of uncertainties in the models. The uncertainties are as a result of:
i. The type of probability distribution that was used in the model.
ii. The process parameters are unknown in some cases, although the probability distribution may be known.
iii. If conditions (i) and (ii) are known, the probability distribution can be determined without having an adequate understanding of the real outcome.
iv. There are limitations in some cases when the first in, first-out (FIFO) condition was not applicable. The queuing mathematical model analysis became more complex when FIFO was not applicable.
3.1.4 Applications of Queuing Theory in an Advanced Manufacturing Environment.
Queuing theory was a viable decision-making tool in an advanced manufacturing environment. It can be widely used for a variety of applications. The applications of queuing theory are listed below:
a) Inventory control and analysis.
b) Controlling of a production line.
c) Controlling congestion.
d) Take-off and landing of aircraft at busy airports.
46 e) Assembling lines for parts control.
f) Issuing and returning tools in assembling plants.
3.1.5 Newton Raphsonβs Iteration.
The application of Newton-Raphsonβs iteration was based on finding an approximate value for the root of a valued function of x [57]. The implementation of the Newton-Raphson equation for numerical analysis in the research gave an effective result with minimal error.
Newton-Raphson equation was used an iterative tool that was suitable for simulation purposes when finding zeros of an arbitrary function. Newton-Raphson equation has a distinct advantage over some other mathematical tools that can also be used to simulate results in the research carried out on the performance optimization of waiting time using queuing theory in an advanced manufacturing environment. Newton-Raphsonβs Iteration equation was modeled using the steps described below.
The derivation of the Newton-Raphson iteration equation was expressed in the form below:
Given that the root of the arbitrary equation was ππ, and π₯π₯0 represented the approximate value of ππ, β denoted the approximate value of π₯π₯0 from the initial value. Where
ππ = π₯π₯0 + β,β = ππ β π₯π₯0 (3-1) β is negligible and its linear approximation was illustrated as:
0 = (ππ) = (π₯π₯0 + β) β (π₯π₯0) + βππβ²(π₯π₯0) The analysis was valid if, ππβ²(π₯π₯0) was approximately equals to zero.
β β ππππ(π₯π₯β²(π₯π₯0)
0) (3-2) ππ= π₯π₯0+β β π₯π₯0βππππ(π₯π₯β²(π₯π₯0)
0) (3-3) Therefore, the estimated value π₯π₯1 of r gives:
π₯π₯1=π₯π₯0βππππ(π₯π₯β²(π₯π₯0)
0) (3-4)
π₯π₯2 also follows the same trend as π₯π₯1: π₯π₯2 =π₯π₯1βππππ(π₯π₯β²(π₯π₯1)
1) (3-5)
For a required number of π₯π₯, π₯π₯ππ is the next approximate value.
47 Therefore, π₯π₯ππ+1 are given by:
π₯π₯ππ+1= π₯π₯ππβππππ(ππβ²(ππππ)
ππ) (3-6) 1.6 Renewal Reward Theorem.
The renewal theory is a probability theory that was used to generalize a process that had the characteristics of Poisson such that the model had a random operating time [191]. The renewal reward mathematical process utilized identical time that was evenly distributed with finite mean and was independent of each other. The renewal reward theorem was suitable for the research as it had a function that was effectively related to the expected number of parts arriving from a conveyor with the expected reward outcome which represented the throughput rate. The implementation of the renewal reward theorem along with the queuing mathematical theory was suitable in deriving useful mathematical models that was effective for calculating the throughput rate in an advanced manufacturing environment. The renewal theorem was suitable for analyzing the parts that randomly arrived from a conveyor.
3.2 Mathematical Modeling and Optimization of the Waiting Time using Queuing