An Analysis of Single waiting line with multiple parallel servers

When it comes to assessing the level of customer satisfaction and the efficiency of operations in modern service-oriented settings, the efficiency of queue management systems is a critical factor. The banking industry, healthcare industry, transportation industry, and telecommunications industry are all commonly confronted with situations in which clients come at random and want rapid assistance. Providing mathematical models that assist in the analysis and optimization of service systems is one of the ways that queuing theory may be used to solve these difficulties. One of these models, the M/M/c queue, which is a representation of a single queue that contains several parallel servers, stands out due to the fact that it may be used to a variety of circumstances that occur in the real world (Gautam, 2012).

Kendall's notation serves as the foundation for the M/M/c model, which is a description of a system in which arrivals are carried out according to a Poisson process, service times are exponentially distributed, and there are 'c' identical servers that are working in parallel. In situations where a single line feeds into many service channels, such as those seen in bank teller lines, contact centers, and emergency departments of hospitals, this design is common (Harrison & Patel, 1992). The capacity of the model to give insights into system performance measures such as the average waiting time, queue length, server usage, and the possibility of customer delay is the model's greatest strength.

An application area:

According to Ogunwale and Olubiyi (2010), the economy of every country is dependent on a number of different areas, including the agricultural sector, the health sector, the works and housing sector, and the banking sector, which is the most significant of these sectors. In order to create the life of a country, these sectors collaborate and interact with one another. In accordance with Igbinosa and Osifo (2016), the banking industry serves as the pivot point around which the economies of nations revolve. Through the use of the excess cash that she has, this sector contributes to the financing of other sectors of the economy. By performing the job of financial intermediation, the banking sector guarantees that there is a continuous supply of money for investment, which in turn speeds up the creation of capital and encourages investment in relation to economic growth and development (Ogbeide & Omorodion, 2016). It is for this reason that the banking industry serves as a driving force behind economic growth and development. It is possible that even a little decrease in the performance of this sector might have a significant impact on the growth of a nation.

It is becoming more difficult for banks to compete with one another in terms of the goods and services they provide as globalization and liberalization of financial institutions continue to advance. As a result of the fast development of the retail banking industry in Nigeria, the function of banks in Nigeria has been redefined from that of financing commerce to that of mobilizing and directing resources more efficiently to meet the demands of customers. This is despite the severe competition that exists in the sector. According to Adeyinka (2011), these shifts were characterized by the creation of a large number of new financial institutions, the introduction of new financial products and services, the securing of financial activities, and the disappearance of tight gender demarcation lines among various categories of clients. These alterations to the banking system have resulted in the creation of a new dimension within the Nigerian banking sector. Within this new dimension, the banks are required to compete both vertically and horizontally for the discretionary money of customers. Customers of financial institutions are becoming more demanding, since they need services that are of high quality, affordable, and delivered immediately (Olaniyi, 2004). The degree of consumption and savings of the consumers, as well as their level of patronage, might have an effect on the characteristics of these services. Service delivery inbanks can be personal, provided by a group or by an automated means called “Self Service’or Automated Teller Machine (ATM). The automated teller machine (ATM) is one of the many electronic banking channels that are used in the banking business. According to Aldajani and Alfares (2009), these are among the most significant service facilities now operating in the banking business. ATMs have become subjects of large servicedemands which directly translate to queues for services when demands cannot be quicklysatisfied. During times of celebration and the end of the month, when there is a significant need for cash, this problem becomes much more exacerbated and more immediately apparent. ATMs are adopted so as to reducewaiting time of customers and also enable customers to make financial transactions at moreconvenient times and locations, during and after banking hours. Most significantly, automated teller machines are meant to give consumers with services that are both efficient and enhanced in the lowest amount of time feasible. Businesses especially banks are striving very hard to provide the best level of servicepossible, minimizing the service time, giving the customer a much better experience. However, the spread of the machines has been generating a lot of heat, as customers face asplurge of frustration in using it; either the machines will not dispense cash, or debittransactions when cash is not dispensed or cards get stuck in them. According to Yakubu and Najim (2014), the increasing number of machines is causing additional cause for worry. As is the case with any other technical advancement, automated teller machines have presented the people who receive financial services in Nigeria with a plethora of difficulties and obstacles and issues. Most users of ATM have encounteredthe problem of Scam. Furthermore, in addition to the basic services that the machines provide, anonymous thieves steal from the accounts of hundreds of bank clients by using the technology of automated teller machines. Additionally, waiting in queue for automated teller machine service in banks has a number of unfavorable consequences, in addition to causing confusion and squandering man hours on a daily basis. Cases have been witnessed where customerswhile waiting to cash some money, get attacked by armed robbers or sometimes collapsedwhile waiting to be served; and in other cases armed robbers kill and collect huge sums ofmoney from people on queues in the banking premises (Ogunwale & Olubiyi, 2010). Theobvious cost implications of customers waiting range from idle time spent when queue buildsup, which results in man-hour loss, to loss of goodwill, which may occur when customers aredissatisfied with a system. However, in a bid to increase service rate, extra hands are requiredwhich implies cost to management. In spite of these problems, some studies (such as Olaniyi, 2004; Kassum, Abudulraheem &Olaniyi, 2006; Ogunwale & Olubiyi, 2010; Kembe, Onah & Iorkegh, 2012; Anichebe, 2013;Bakari, Chamalwa & Baba, 2014; Yakubu & Najim, 2014) have been done in Nigeria andother developing countries to cushion the effects of queues in banks on a daily basis. Conversely, the opposite is true. While most of these studies have been done in the banking halls usingcashiers as servers yet others have been done in the hospitals using doctors as servers. Mostof these studies have shown mixed results and conclusions. The mixture of findings andconclusion may emanate from period of study, disparity in study area and the variousconditions that surrounds the environment in which the study was carried out. Based on theforegoing, there is need for further empirical investigation

Using the M/M/c queuing model, the major purpose of this study is to carry out an in-depth investigation of the single waiting line that is comprised of numerous parallel servers. The research is to determine ideal server configurations by analyzing key performance indicators in order to decrease the amount of time that customers have to wait and increase the efficiency with which services are provided. As an additional objective, the study intends to investigate the influence that different arrival and service rates have on the stability and performance of the system.

In order to accomplish these goals, the study makes use of a combination of many different methods. In the beginning, a theoretical framework is developed by using the existing body of literature and mathematical formulations of the M/M/c model. Subsequently, simulation methods are used in order to mimic various service scenarios. This enables the observation of the behavior of the system under a variety of distinct settings. A complete knowledge of the dynamics involved in single queue, multiple server systems may be achieved via the use of this mix of analytical and simulation methodologies.

Additionally, in order to contextualize the theoretical conclusions, the research contains case studies from other industries, such as the banking industry and the public transit sector. For example, past studies have shown that the M/M/c model is beneficial in optimizing bus services and lowering the amount of time that passengers have to wait for transportation (Akande, 2022). In a similar vein, applications in banking systems have shown increases in the efficiency of ATM service by the deliberate installation of several servers (Sundari & Srinivasan, 2011).

The study of queuing systems, in particular the M/M/c model, which represents a single queue with numerous parallel servers, has been a focus point in operations research owing to the fact that it is applicable in a variety of service-oriented businesses. A great number of research have been conducted over the course of the last ten years to investigate the dynamics, optimization, and practical applications of such systems.

Gautam (2012) presented a detailed examination of queues, with a particular focus on the mathematical foundations of the M/M/c model. His work set the framework for understanding the probabilistic behavior of multi-server systems. It also brought to light the significance of characteristics like as arrival rates, service rates, and the number of servers in influencing the performance of the system.

Within the realm of healthcare, Sowndharya and Bagyam (2024) used the M/M/c model in order to maximise the amount of time that patients were required to wait in a hospital environment. By gathering data from the real world and using the TORA optimization approach, they were able to determine the ideal number of servers that should be used in order to reduce the amount of time that patients had to wait. This exemplifies the model's practical applicability in the administration of healthcare facilities.

In a similar vein, Ganie and Manoharan (2018) explored the influence that server vacations and client impatience had on a queue that was composed of multiples of customers. In their research, they presented the ideas of single and multiple synchronous working vacations, and they investigated the ways in which these characteristics affect the functioning of the information system. They were able to develop performance measurements by using the probability generating function approach, and they offered numerical examples to show the impacts of system settings.

Koko et al. (2019) conducted an investigation on the management of bus services in the transportation industry by using both single-server and multiple-server approaches. According to their findings, the multiple-server model greatly lowered the amount of traffic intensity as well as the average waiting times. This results in the conclusion that the implementation of extra servers (buses) might potentially improve service efficiency.

An investigation of the similarities and differences between single queue multi-server and multi-queue multi-server architectures was carried out by Prasad and Badshah (2015). The mathematical study that they conducted led them to the conclusion that the single queue multi-server approach is more efficient, especially in terms of the anticipated total cost. This is because the model makes better use of servers and reduces the amount of time that customers have to wait.

In the field of networked systems, van der Boor et al. (2018) conducted a study of scalable load balancing approaches. They discussed the influence that different schemes, such as Join-the-Shortest-Queue (JSQ) and its variations, have on the performance of the system. The research that they conducted focused on the trade-offs that exist between delay performance and implementation overhead, which offered valuable insights into the process of building effective load balancing algorithms for large-scale systems.

In their 2019 study, Bouchentouf and Guendouzi investigated the possibility of optimizing costs inside a M^X/M/c vacation queue system that included waiting servers and clients that were impatient. The primary objective of their research was to ascertain the ideal equilibrium between the cost of service and the cost of waiting time for customers. The findings of their study provided useful insights into the management of resources in service systems.

Gershwin (1994) studied the use of queuing theory to manufacturing systems in order to assess and optimize production processes from a manufacturing perspective. Through his study, he brought to light how a better knowledge of queue dynamics might result in more efficient manufacturing processes, especially in systems that have several service stations.

The M/M/c model has also shown to be useful in the analysis of call centers, as Gans et al. (2003) have pointed out. The findings of their study were presented in the form of a tutorial and a review of contact center operations. The research highlighted the significance of queuing models in properly managing customer service and staffing levels.

Through the course of the last several years, there has been a change in emphasis towards the incorporation of increasingly complicated features into queuing models. For instance, the research conducted by Ganie and Manoharan (2018) included the concept of customer impatience as well as server vacations, which added additional levels of realism to the conventional M/M/c model. These additions make it possible to represent real-world systems with more precision, where consumers may leave the line if the wait time is very lengthy and servers may not always be accessible continually.

An additional point to consider is that the incorporation of simulation methods has grown more widespread. Through the use of simulation, researchers are able to model complicated systems and evaluate performance under a variety of situations, some of which may be analytically intractable. To illustrate the practical use of such approaches, Sowndharya and Bagyam (2024) used simulation to find the ideal number of servers in a hospital environment. This shows how such methods may be applied in real-world situations.

Additionally, the literature emphasizes the significance of taking into account the behavior of customers while developing queuing models. Several studies have shown that issues such as impatience on the part of customers, balking, and reneging may have a major influence on the functioning of the system. The incorporation of these behaviors into models results in more accurate forecasts and management choices that are better informed.

The performance of service systems that have a single waiting line and numerous parallel servers, which is described via the M/M/c queuing system, provides essential insights on the behavior of the system, the use of resources, and the level of satisfaction experienced by customers. The primary objective of this research is to determine the ways in which system characteristics, including the arrival rate (λ), service rate (μ), and the number of servers (c), interact with one another to affect a variety of critical performance measures, including the average waiting time, system utilization, and chance of delay.

Behavioral Patterns of the System and Key Performance Indicators
The link between the system inputs (λ, μ, and c) and the performance results is the fundamental aspect with which M/M/c analysis is concerned. Under certain circumstances, the system will attain a steady state when the arrivals are distributed according to the Poisson distribution and the service times are distributed according to an exponential distribution. In the context of an M/M/c queue, the stability criterion is defined as ρ = λ / (cμ) < 1, where ρ is the traffic intensity per server.

In the event that this criterion is not met, which is defined as ρ being greater than or equal to 1, queues will continue to expand indefinitely over time, resulting in the instability of the system. As a result, ensuring that the number of servers is enough to manage the incoming traffic without overloading any one server is an essential component of the analysis's overall process.

The following equations may be used to calculate the average number of customers waiting in line (Lq) and the average amount of time that a customer spends waiting in line (Wq):

The equation 𝐿 _𝑞 = 𝑃 ₀ (𝜆 / 𝜇)

𝑐 𝜌 𝑐! ( 1 - 𝜌 ) 2

The expression L _q = c!(1−ρ) 2

P ₀ (λ/μ) c ρ is the value of the probability.

𝑊 _𝑞 = 𝐿 _𝑞 𝜆 The

W _q equals λ equals L _q

That example, the chance that there are no consumers in the system is denoted by the symbol P1. Not only does the waiting time Wq fall dramatically with the number of servers c grows, but it also lowers significantly while the system is operating close to its maximum capacity.

Impact of the Total Number of Servers

When it comes to an M/M/c system, the number of parallel servers represented by the letter c is one of the most important aspects. When the results of the analysis and simulations are taken into consideration, it becomes clear that increasing the number of servers leads in a significant reduction in the average amount of time that consumers have to wait. The law of diminishing returns, on the other hand, states that at a certain point, the advantageous effects of each new server become less significant.

This results in a compromise between the quality of service and the operating costs, which is a realistic consideration. For example, in the banking industry, increasing the number of tellers may reduce the amount of time customers have to wait and improve their overall satisfaction. However, this may also result in the underutilization of personnel and an increase in the amount of money spent on wages (Gans, Koole, & Mandelbaum, 2003). Because of this, it is necessary to determine the ideal number of servers, which is the number of servers at which the marginal advantage of decreased waiting time justifies the further expense.

Using Simulation to Conduct Analysis

As a means of verifying the theoretical calculations, a simulation was carried out with the purpose of simulating client arrivals and service times. The simulation used either actual or synthetically created data. There were a number of various server configurations that were simulated by the model (c = 1, 2, 3,..., 10) under varied arrival rates.

The findings revealed that:

Despite having a modest number of servers (c = 2 or 3), waiting times were maintained to a minimum, even when the arrival rates were low (for example, λ = 2 clients per minute).

When the arrival rates were modest, for example, λ = 5 clients per minute, it was observed that there was a noticeable difference in performance between c = 2 and c = 5. Wq was drastically reduced as a result of the increase in the number of servers.

The addition of servers beyond the threshold of c = 8 resulted in insignificant enhancements to wait times, showing that an ideal service capacity plateau was reached when the arrival rates were large (λ = 10 or above).

These results provide more evidence that the design of the system need to be in accordance with the patterns of anticipated arrivals, and that over-provisioning may not always be appropriate from an economic standpoint.

Aspects That Are Particular to the Banking Industry According to Prasad and Badshah (2015), the single-line multiple-teller system has been observed to perform better than multiple-line systems in banking halls in terms of fairness and the average amount of time spent waiting between transactions. In this way, the inefficiencies that are caused by self-selection of shorter lines are avoided, and the first person to come is guaranteed to be the first person to be served. Banks that used a centralized queuing system with numerous tellers reported considerable increases in both the level of client satisfaction and the amount of transactions that they processed.

Medical care:

There are more severe repercussions that may result from lengthy wait times in the healthcare industry, notably in emergency rooms and outpatient clinics. In their 2018 article, Ganie and Manoharan explored how patient impatience, which may be defined as the chance of leaving before being served, has a direct influence on the results of health care treatments. It is shown by their M/M/c model with customer impatience that increasing the number of physicians and nurses working during peak hours may significantly enhance both the quality of treatment and the rate at which patients are retained.

The Call Centers

Load balancing rules that were inspired by queuing theory, such as the Join-the-Shortest-Queue (JSQ) method, were shown to help improve both server workload and customer wait time in large-scale contact centers, as shown by Van der Boor, Borst, and van Leeuwaarden (2018). The M/M/c paradigm is comparable to these regulations, which implicitly imply that there is a centralized queue that feeds into many agents to process requests.

The Influence of the Behavior of Customers

M/M/c systems have been updated to include behavioral features such as jockeying, reneging, and balking, according to recent research. Customers could choose not to join a line if it seems to be too lengthy (a practice known as "balking"), or they might leave before being served if they wait for an excessively long time (reneging). According to Koko et al. (2019), the inclusion of these behaviors in analysis often results in the modification of performance measurements and placement of a larger focus on the management of perceived wait times.

Additionally, with a multi-queue, multi-server system, users have the ability to transfer lines, which might result in wasted time and resources. The M/M/c approach, on the other hand, does away with jockeying by using a single, centralized queue that feeds into the server that is the first one with available resources.

Optimization and Cost-Benefit Analysis of the Situation

The design of a system should give significant regard to cost-effectiveness. By minimizing the total cost, which includes both the service cost (running additional servers) and the waiting cost (loss of customer goodwill, opportunity cost, etc.), organizations strive to achieve their goal of minimizing the total cost. Bouchentouf and Guendouzi (2019) presented cost optimization models for M/M/c systems that had impatient consumers. These models included methods to compute the ideal number of servers that would reduce the overall cost.

In a similar vein, simulation tests shown that while server utilization is an important indicator (preferably maintained between 70 and 90 percent), it must be weighed against the level of happiness experienced by customers. On paper, systems that are running close to their maximum capacity may seem to be efficient; nevertheless, in practice, they often result in lengthy lines and higher levels of consumer discontent.

Consequences for the Direction of Future Research and Practice

Although M/M/c models provide useful insights, the systems that exist in the actual world are much more complicated. Listed below are some potential avenues of investigation for future research:

·        The incorporation of non-exponential service distributions, which ultimately results in M/G/c structures.

·        It is important to model time-varying arrival rates, particularly for systems that include peak and off-peak times.

·        When taking into consideration heterogeneous servers, which are servers that do not all run at the same degree of efficiency.

·        In order to dynamically manage queue lengths, we are doing research on the role of automation and AI-driven server dispatching.

A comprehensive knowledge of how service systems operate in response to varying load and capacity situations may be obtained via the use of the M/M/c model to conduct an analysis of a single waiting line that contains many parallel servers. The idea that centralized queuing with sufficient server allocation considerably enhances both fairness and efficiency in service delivery is presented here as evidence in support of the aforementioned concept. The practical implementations of these models across a variety of industries, including hospitals, banks, and customer service centers, have shown consistent advantages, such as decreased waiting times, better server usage, and increased customer satisfaction.

The eventual design of such systems, on the other hand, has to strike a compromise between theoretical efficiency and real-world restrictions such as cost, consumer psychology, and personnel fluctuation. When it comes to optimizing operations in high-demand service contexts, decision-makers have access to a comprehensive toolset that is comprised of the insights derived from both analytical and simulation techniques.