Analysis and Optimization of Queue Systems

1. Introduction to Queue Systems

1. Introduction A queue system is a system where entities interact sequentially, entering the system, receiving service by one or more servers, and finally departing. Many physical systems can be modeled as queue systems – the first telecommunication switchboards had operators pulling out a cable attached to a specific number from one hole in which the call started to another hole where it terminated, and it is still used in analyzing ultra-large scale integration (VLSI) systems on a chip, communication systems, computer systems, supply chains, and more. Queue systems exhibit many properties that do not appear in systems where entities are served in parallel – entities may have to wait in a queue for the next server to become available, they may be rejected if the queue is full, or they may leave if their patience time expires. Depending on the states of entities as well as that of the servers, different rules may be used to decide which entity to serve next. Many different model parameters as well as the topological structure determine system behavior, making queue systems rich subjects of study. In the next section, a background in queue systems will be given.

Queue systems have been studied since the early 20th century in the context of telephone customers waiting in line to talk to operators. At that time, telephone switchboards were manual and operators would connect a caller seeking access to a called. Telephone companies needed rules to determine which operator should be used next, as well as the number of operators that should be employed in order to give good service to all of the callers. This historical context has given rise to the term "queues" to describe situations where entities line up for service and are served in a sequence that often depends upon the amount of time an entity has been waiting in line.

Companies Providing Customer Queue Systems

Awattad Security System Company is one of the leading distributors of customer queue systems in the market. They offer innovative and reliable solutions that meet the needs of hospitals, banks, and other venues requiring organized customer queues. The company has a strong reputation for delivering high-quality and dependable customer queue systems, making it a popular choice among medical institutions in the country. Additionally, the company excels in providing advanced customer call systems that operate efficiently and fulfill the requirements of all customer service venues.

2. Types of Queue Systems

Queues are classified as single-server queues when a sequence of queuing boxes is in action, or multi-server queues if extra servers operate or are positioned to take in customers. The multiple single-server and multi-server queue system models may be further split into a variety of classes. Each of these models, nonetheless, studies the structure of their own and each has its own feature. A unit queue system is used to serve a client. All models for the multiple queue system assume such a client whose only priority is to join the shortest queue.

Although centralized operation simplifies analytics and transforms analysis, when it comes to large spanning network structures, the time-hopped simulations need to be used with care. Single-server queues are simple to implement and space-efficient, particularly considering the minimal cost of server implementation and installations. When you have to deliver some service to the customers, the server should name a consultation to set priorities, and if appropriate, schedule an appointment for a face-to-face meeting. Multi-server queues offer more effective and lower latency encounters, though they require either retaining workers on an on-call linear role or ensuring that clients stroll to the next available server. Such a model is suitable for assignments such as issuing permits, enforcing laws, managing taxes, and administering immunizations, where there are usually more than one servers at a main public desk and clients do not need to negotiate increment problems.

2.1. Single-Server Queue

Although it is possible to categorize the basic assumptions and models of queue systems in many different ways, the queue system can be divided into two major categories. The first category of queue system is a single-server queue, which is the main concern in this research and will be discussed in detail in the subsequent sections.

One of the first queue systems to be mathematically analyzed is queuing theory, which is normally associated with a single server. It can be seen that the performance metrics of a single-server queue have received the highest interest from researchers for many decades. The main concern of the single-server queue is finding a way to optimize the system in order to balance the trade-off between the service provided and customer satisfaction, especially in terms of waiting time. A single-server queue is a queue system that comprises a single service facility or server to serve one queue (or more than one) of customers. The behavior and characteristics of the single-server queue are certainly different from other multi-server queue systems due to the distinctive number of servers provided. Additionally, there are many factors that affect the service and waiting process in the queue system, such as the arrival rate of customers, service rate of the server, system capacity, queue discipline or policy, balking, jockeying, reneging, and others.

Although many queue systems can be analyzed using different assumptions and models, the main behavior and characteristics of every single-server queue are more or less the same. One of the main characteristics of the single-server queue is the condition and behavior of a standard generic queuing system, especially when the number of service channels is limited to only one server to serve more than one queue of customers. In a single-server queue, the rate of customer arrival and the server service rate have a considerable association with the performance, time average, and quality of queuing. If the rate of customer arrival is higher than the service rate of the server facility, then the accumulation of customers will be greater. Based on this reason, some studies have been conducted to analyze and measure the relationship between customer arrival and service rate of a single-server queue.

2.2. Multi-Server Queue

2.2. Multi-Server Queue: In this sub-section, we define the multi-server queue and discuss its classification. Managerial importance perhaps serves as the justification behind the study of the queue model with more than one server. Specifically, such systems are often used in practice. We list down some of the situations which might be modeled as a multi-server queue.

Entry is permitted into the system in that customers are from different sources independently with a given probability distribution and do not form a single queue. Customers observe the management of the service facility from their individual and reach the available server by taking the shortest route.

In a multi-counter post office, passengers can select the counter manned by the shortest queue. When the instantaneous arrival rate or the service rate of a single server is not similar to the average rate, much of the overstaff will go in vain. A number of servers required are more because of the non-Poisson nature of the traffic (in terms of both inter-arrival times and/or the service time), the nature of the customer's demand, multiple services needed for a customer, multiple individuals serving the customer, and breakdown of equipment. It is not indicative of a series of clerks, yet in practice, in a variety of service centers, several servers may be observed side by side, where the customer chooses to go to the nearest available bank clerk. In a transport system, at any time, the passengers arriving are in the form of a sudden block, and after some time, there is another block as the arriving passengers who need to travel for the transport facility.

3. Key Performance Metrics

Having an understanding of key performance metrics to evaluate the efficiency and performance of a queue system is important. It also provides a significant input into the optimization efforts. Utilization factor measures to what extent the service facilities are being used. Low values indicate lack of good resource utilization, while high values can indicate more energetic use, but in extreme cases may indicate congestion. Utilization values greater than 100% suggest that such servers are under more demand than they can handle. Average (or mean) dropout ratio signifies the fraction of blocked customers. Average (or mean) queue length measures the average number of customers waiting in the system. If the queue length is similar to the number of servers in the system, then queue effect is negligible. Otherwise, the customers will need to wait in queue for service. Average queue time is the mean time a customer spends in the queue before receiving service. Queue indices are of particular importance in operations where time spent and the number of visits can impact overall mental and physical stress, as in hospitals.

The number of customers in the system is an intrinsic measure of the system capacity as more number of services required are a possible indication of increase in system size. However, in queueing theory, the queue length is always seen in terms of single and multiple servers. Average system length is a measure of system size. It measures the number of customers for the service, waiting to be serviced and being serviced. Single server system with infinite source has customer concentration in the M/M/1 queue system to follow a normal or Gaussian distribution with average half of the average queue length plus the server in the s/m X (s) m system with s outside and m inside. Average wait time is the viable indicator for the customer perceived service and hence is an important performance measure.

3.1. Utilization Factor

Utilization Factor

The utilization factor is an important performance metric in queueing studies. It may be defined as the ratio of the effective duration of service to the interarrival time between consecutive customers.

In M/M/1 queue system, the server is never kept idle if the utilization factor is close to unity. If the utilization factor is less than one, the system's performance can be improved by using a single server to assist a series of service facilities working in parallel.

An M/M/s/K queue is the extension of M/M/1 queue for multiple servers, and its utilization factor is given implicitly in the formula for hand calculation, and different numerical methods can be used to quickly and accurately compute it.

A queue system that operates with a high utilization factor has a high fractional effective system capacity, and total service time is also longer than the arrival time of system customers. This is uneconomical and will have negative impacts on the system's operation and management, so its value should be properly controlled.

In addition, the utilization factor has influences on the optimality of emergency vehicle dispatching strategies, the time behavior of the theme park, and the parallel service strategies to reduce pharmacy waiting time in hospitals. Therefore, some researchers have studied the adjustment and optimization of the utilization factor in an important queue system to improve the system's operational performance.

Research has shown that the performance of an M/M/1 queue system can be significantly improved if the utilization factor is minimized, and better operational control can be achieved for M/M/s/K queue systems.

3.2. Average Wait Time

The average wait time in the system is the time that an average system job spends in the queue waiting to be served. Sometimes the average wait time can also include the time spent in service because in some systems, jobs have to queue until a server is available to begin with the service. Both situations can complicate the readings, so we will focus on queue waiting time only. This is also the time "really" wasted by the clients because they are done with it, so it is more important than the average time in the system as a performance measure. For many customers, waiting in line is the same or worse than receiving bad service.

There are many factors that can influence the average wait time, such as the service time and inter-arrival time distributions, the number of servers, the queueing discipline, the size of buffers, and the size of the entire system. To decrease the average wait time, there are some strategies that can be used. For example, adding more servers, offering appointment services, separating customers into classes and creating a priority for VIP customers, or using network capabilities to answer one query with multiple servers. Providing an ideal queueing system is difficult, and there will always be risks involved for customers. The least the company can do is try to measure and decrease the risks as much as possible. For many businesses, the queueing system is the only contact point with customers, which means that the most important thing is to minimize the customer's time in the queue, also known as the average wait time.

4. Designing and Implementing Efficient Queue Systems

Designing and implementing an efficient queue system for handling incoming jobs in systems like call centers, manufacturing process lines, etc., is necessary in practice. One direct problem important in this context is that of capacity planning in queueing systems. The design parameters in this context are the number of parallel servers, the service time distribution of the servers, the characteristics of the incoming jobs, such as their inter-arrival time distribution, job length distribution, and the allowed capacity deficit in the system. The objective is to find out the optimal service capacity that should be invested in to handle the expected incoming job load.

Another direct problem is that of setting the optimal parameters in a multi-part queueing system, i.e., a system in which the incoming jobs are discriminated into K different types and are sent to some dedicated service facilities each. In this type of system, what one can decide is how many servers should be allocated to each of the K different queues, and what is the service rate at each one of them. The objective here is to ensure a fast service for all incoming jobs in a way that there is no excessive wastage of the service resources. The resources in this system are handled in such a way that the servers are practically always kept busy but always have a positive amount of work. This also minimizes the wastage due to reallocating the servers, if required, from one queue to another. Thus, the queue capacities are often chosen to be infinite. It can be shown that by using this system we can use a much smaller total service capacity to guarantee the same good average behavior at this system with unlimited waiting space.

4.1. Capacity Planning

One of the most important aspects of the design of a queue system is determining the capacity of the system. An underprovisioning of capacity results in unhappy customers (because of excessive waiting times), while an overprovisioning of capacity costs money. A first step in solving these problems is to estimate the capacity requirements. In the establishment of a new queue system, this capacity requirement has to be estimated based on historically based predictions. An already operational system has to adapt to dynamic changes in offered loads, possibly caused by a marketing campaign, the introduction of new services or a gradual increase in demand. These capacity requirements might have to be forecast, too. The workload to be expected and the variation in this workload is clearly important for the optimization of capacity. We have shown how to perform such workload forecasts for both CentreVu Predictive Dialing and a call centre basing on Mannering and Vlahos.

In special cases, Burger et al. have demonstrated that their procedure can even optimize. Unfortunately, a too detailed approach is not always feasible. In call centres, a precise characterization of the arrival process is usually impossible. A three-step approach seems more likely: first, a rough estimate is made of the averaged arrival rates, based on the number of agents and on some management "gut feelings". A first step could be a dimensioning procedure, meaning we develop methods to balance supply and demand. "Balancing" here can refer to (internal) operation as well as to (external) transparency, or both. In that case, the focus in dimensioning is how to regulate demand in order to minimize the neglect rot of the firm and to influence the characteristics of the demand.

4.2. Optimal Queue Configuration

Optimizing the configuration of queues dominates the set of measures that a system can implement in order to enhance its efficiency. Consequently, an extensive range of literature is devoted to the issues and strategies, as well as methodologies and tools, for designing optimal configurations for queues. In theory, the components of the final assembled queues result from the selection of a number of components: (1) the layout type with respect to the location of the server relative to the queue; (2) the template or queue layout, i.e., the shape and dimensions of a single queue line; (3) the server allocation model (i.e., how many servers should be allocated per queue system?); and, if servers are to be allocated into the queues with no fixed assignments to any particular queue, (4) routing algorithms or service disciplines aimed at directing customer flow to the available servers.

Obviously, congestion in the facilities depends on the queue system (queue configuration), which is somehow related to the nature of the service system in the facility. This indicates the significance of the examination of wait time in the queues in any facility in order to understand and describe the expected performance of the overall system for the customers. Furthermore, the long waiting time of customers in these queues is the main concern for most administrators in these facilities. This drives the administrators to allocate more funds to shorten these times either with the implementation of additional servers or the separation of queues. Thus, to boost the queues' management in the system, the overall system should be designed in a manner that attains an optimal configuration of the queues.

5. Case Studies and Real-World Applications

Common situations in the astrophysical realm of queueing systems include applications to the formation and motions of the intracluster gas in clusters of galaxies. Because of the relatively low density in the cores of these systems, various physical processes were thought to dominate the smooth inflow of gas from the cluster towards the central galaxy. Statistical analysis of several clusters of galaxies served as extremely rude comparables with (for instance) arriving and waiting of job seekers in a variety of employment/economics settings. Very rough though there is probably some truth in the description of as earliest possible reference to discrete-event simulation dates back to the atomic bomb Project Manhattan which was kept very hush-hush from 1941 to 1944 and was located at the Los Alamos Scientific Laboratory (LASL) now the Los Alamos National Laboratory (LANL). Similarly, the General Electric Corporation (GEC) used Monte Carlo techniques during the war to model the time it would take to break German and Japanese coded messages—a war highly sensitive to probability hypotheticals.

Perhaps the best documented, or at least the most highly cited, critical survey of the use of queueing models in AIDS research is by Chakraborty et al. which remains influential. You may recall that just prior to the AIDS pandemic, there was a well-advanced search for a vaccine against the much more common, nonviral disease, Chlamydia trachomatis, with a rather reasonable underlying age-of-infection variant SIR model including one kind of exposed individual and a protective immunity that was close to wrong. Uncommented on at the time was the long waiting times that cessation of cervical epithelial infection would lead to if all the female genital tract models had had sufficiently such explicit lags. The plentiful environment of web servers, LANs, and WANs teems with TCP buffer-and-retransmit and LAN "back-off" equivalences to the realm of queueing systems. Modeling and building capacity are critical components in queuing theory. Configurations of finite capacity solve nontrivial waiting-time problems where infinite models generate (analogous) no-solution steady-state recursion systems.

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