Analysis of Single Server Unreliable and Removable Server Queueing model with  Partial Breakdown and Balking Behavior

One way to improve customer service and resource distribution is to use queuing theory as a scientific basis. This is very important in many fields, like healthcare, industry, telecommunications, and finance, because it’s main goal is to find a balance between operating efficiency and cost. Queuing models can help organizations figure out why there is congestion and how to make systems that provide reliable and quick service while cutting costs and wait times? In order to work around real issues, we have greatly expanded the M/M/1 queueing model, which is a basic tool in operations management. It's a big step forward in this field that working vacation models let servers keep running at a lower capacity during downtime instead of stopping all service. Any business, company, or group in the real world doesn't want to lose customers because it can hurt their brand and revenue. Problems with service often cause customers to leave a business. An important reason for these kinds of problems is that the service system breaks down, making customers wait longer. Problems with hardware, software, technical breakdowns, repair needs, or unplanned downtime are some of the reasons why things can go wrong. ISPs, airlines, public transit, retail checkout systems, hospital emergency rooms, banks, and industrial machinery problems are all real-life examples of situations where services go down because of system failures.

When a server has a partial breakdown, it can still provide some level of service even though it isn't fully working. When a server has a full breakdown, on the other hand, it stops working totally until it is fixed or replaced. The repair time is the amount of time needed to fix a computer that has broken down and can't serve customers for a short time. In accordance with the queueing theory, the amount of time required to get a server ready to serve customers once more is referred to as the setup time. The process may include a number of steps, including the configuration of the server, the addition of any data or software that is required, the modification of the settings, and the verification that the system is prepared to process requests.

According to the queueing theory, an unreliable server is one that has the potential to malfunction, experience hardware difficulties, or experience program issues, and thus stop functioning or being available at any given moment. A server that is unreliable or unstable can have a significant influence on the operation of queues as well as on performance metrics for the system, such as throughput, waiting times, and line length measurement. A removable server is a server that may be temporarily switched off for reasons such as maintenance, user weariness, energy-saving measures, or hardware difficulties. This type of server is included in the concept of queues. The activation or deactivation of these servers is determined by the requirements of the system. When a server is restarted, it requires some time to be set up before it can begin functioning normally again. During this stage, tasks like as initializing the system, adding any data or software that it requires, modifying settings, and ensuring that it is prepared to handle requests are carried out. The conduct of customers who are very impatient is a significant issue for commercial organizations, whether they are public or private. People tend to get restless and impatient when they are required to wait in queue for services. When we are waiting, a lot of us experience feelings of anxiety and irritability. When it comes to the study of queues, making assumptions about duration of wait time makes the model more practical and adaptive.

The purpose of this section is to provide a comprehensive review of major research on the M/M/1 model, which is equipped with an unstable server and a working vacation, as well as with impatient customers and repair services. The concept of working vacation was originally presented by Servi and Finn (2002) within the context of a single-server queueing architecture. In order to ascertain the steady-state distributions of system size and sojourn duration, they approached the problem using the PGF method. It was investigated how these findings may be used to the evaluation of the performance of gateway routers in fiber communication networks. Liu et al. (2007) conducted an investigation on the stochastic decomposition of waiting time and queue length. They did this by utilizing the queueing model developed by Servi and Finn (2). It was the matrix geometric approach that they utilized in order to investigate the proposed model.

Yadin and Naor (1963) improved the cost structure of a single-server Markovian queue with partially shut down and setup states by analyzing the system features. An M/M/1/∞ queue with a single unreliable server that could malfunction while in use was studied by Liou (2013), taking into account consumer impatience during outages. In a single-server queueing system that was difficult to manage and unreliable, Singh and Jain (2013) examined how several performance measures evolved over time using Laplace transforms. The idea of a waiting server, as well as malfunctions and repairs during peak hours, was first presented by Gupta and Kumar (2021). To examine the behavior of an unreliable and removable server, Wang and Hsieh (1995) created an economic cost model. A study by Economou et al. (2011) looked at how customers might balk in a single-server line with widely dispersed service and vacation periods. Table 1 presents an overview of the literature on the evaluation of a model with working vacation, an anxious client, an unreliable server, and repair. Scholars employ many methodologies to identify both transient and steady-state solutions for their models.

Table 1 Literature on unreliable server with working vacation and customer’s impatience

The article is organized as follows: section 2 gives a full description of the model, section 3 discusses mathematical formulation, section 4 discusses steady state probabilty, section 5 provides graphical illustrations, section 6 concludes the chapter.

This article examines the notion of setup with repair, as well as an M/M/1 queueing model with an unreliable and removable server that fails intermittently while on vacation. An exponential distribution with rate  reflects the rate of service during busy periods and a rate , where , during working vacations, whereas a Poisson distribution with rate λ explains the arrival of customers. When a customer comes, they have the option of joining the queue with probability b or leaving the system with probability 1-b. During idle time, the server takes a single working vacation using the option ψ. Customers receive service at a lesser rate than during a typical busy period when the server encounters a partial breakdown, also known as a soft failure, while on vacation because no physical repair is required. During vacation, the server closes down at an exponential rate (α). Because the server is removable, it is shut off at this point. When the server is not operating, customers must wait till it is turned on. Restarting the server requires a specific period of time, known as setup time, with rates m that follows an exponential distribution. Because the server is considered unreliable, it may fail during setup activation with a probability of q=p-1; otherwise, it will return to the busy phase with a probability of p. The failed or broken server is sent to be fixed at a rate δ with exponential distributed

Here are the balance equations for each server state:

1.     For busy state

2.     For working vacation state

3.     For closed down state

4.     For repair state

4.     Steady State Probabilities and some System Performances (Transient Analysis)

Consider probability generating function (PGF)   

When we simplify the equation above and apply the equation (9), we obtain

On simplifying above equation, we get

Using the recurrence relation (1), (2), (3), (4), (5), (6), (7), and (8), we are able to obtain the following:

Table 1: Impact of repair rate r on various system performances

Figure 1:Shows graphical illustration of repair rate r on Expected queue length E(L3) in the when server in repair state

Figure 1 clearly displays the influence of the repair rate r on the expected queue length E (L3) while the server is in repair mode. It has been noticed that when r increases, the value of E (L) decreases steadily.

This article examines an M/M/1 queueing system with a removable and unreliable server, partial breakdowns during work vacations, and the setup, repair, and balking operations. Finally, it has been determined that breakdown, slow setup, balking, and repair time inject uncertainty and unpredictability into the queueing system by directly influencing system performance, customer satisfaction, and operating expenses. Effective manipulation of these factors can result in smoother functioning and greater overall efficiency. The model may be expanded by incorporating the idea of bulk arrival and Threshold Policy for Service.