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My study uses a multiphase queuing system of three phases with either one or two servers. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. The arrival rate follows a Poisson distribution, and service times follow an exponential distribution.
After the 1940s queueing theory became an area of research interest to mathematicians. In other words, such factors as different sizes of customer purchases, the amount of change the cashier must count out, and different forms of payment alter the number of persons that can be served over time. Given these assumptions, it is further assumed that arrivals at a service facility conform to some probability distribution. General queueing The screen below shows the possible input parameters to the package in the case of a general queueing model i.
Documentation - Finally, the sensitivity analysis of the model is performed.
Sometimes, it is a pleasant experience, but many times it can be extremely frustrating for both the customer and the store manager. Given the intensity of competition today, a customer waiting too long in line is potentially a lost customer. Queues are basic to both external customer-facing and internal business processes, which include staffing, scheduling and inventory levels. For this reason, businesses often utilize queuing theory as a competitive advantage. Fortunately, Six Sigma professionals — through their knowledge of probability distributions, process mapping and basic process improvement techniques — can help organizations design and implement robust queuing models to create this competitive advantage. The Cost of Waiting in Line The problem in virtually every queuing situation is a trade-off decision. The manager must weigh the added cost of providing more rapid service i. For example, if employees are spending their time manually entering data, a business manager or process improvement expert could compare the cost of investing in bar-code scanners against the benefits of increased productivity. Likewise, if customers are walking away disgusted because of insufficient customer support personnel, the business could compare the cost of hiring more staff to the value of increased revenues and maintaining customer loyalty. The relationship between service capacity and queuing cost can be expressed graphically Figure 1. Initially, the cost of waiting in line is at a maximum when the organization is at minimal service capacity. As service capacity increases, there is a reduction in the number of customers in the line and in their wait times, which decreases queuing cost. The optimal total cost is found at the intersection between the service capacity and waiting line curves. Figure 1: Service Capacity vs. Cost Source: Richard B. Chase and Nicholas J. Aquilano, Production and Operations Management, 1973, page 131. Queuing Theory Queuing theory, the mathematical study of waiting in lines, is a branch of operations research because the results often are used when making business decisions about the resources needed to provide service. At its most basic level, queuing theory involves arrivals at a facility i. The number of arrivals generally fluctuates over the course of the hours that the facility is available for business Figure 2. Figure 2: Number of Arrivals at Facility Customers demand varying degrees of service, some of which can exceed normal capacity Figure 3. The store manager or business owner can exercise some control over arrivals. For example, the simplest arrival-control mechanism is the posting of business hours. Other common techniques include lowering prices on typically slow days to balance customer traffic throughout the week and establishing appointments with specific times for customers. The point is that queues are within the control of the system management and design. A good rule of thumb to remember the two distributions is that time between arrivals is exponentially distributed and the numbers of arrivals per unit of time is Poisson distributed. The Servicing or Queuing System: The servicing or queuing system consists of the line s and the available number of servers. Factors to consider include the line length, number of lines and the queue discipline. Queue discipline is the priority rule, or rules, for determining the order of service to customers in a waiting line. Others include a reservations first, treatment via triage i. An important feature of the waiting structure is the time the customer spends with the server once the service has started. This is referred to as the service rate: the capacity of the server in numbers of units per time period i. Another important aspect of the servicing system is the line structure. The simplest type of waiting line structure is the single-channel, single-phase. Here, there is only one channel for arriving customers and one phase of the service system. An example is the drive-through window of a dry-cleaning store or bank. Exit: There are two possible outcomes after a customer is served. The customer is either satisfied or not satisfied and requires re-service. Waiting Line Models and Equations Table 1 shows the four types of commonly used waiting line models, along with key properties and examples. If a practitioner knows the arrival rate λ and the service rate µ of their customers, they can easily calculate the answers to these questions using the formulas in Table 2. Table 2: Equations for Line Calculations Key Questions Equation What is the utilization of the worker s? To understand how Rite Aid could make such a guarantee, I visited a few stores in the Atlanta metro area, where I live. As a good Six Sigma practitioner, I carefully watched the process of how prescriptions were filled, talked with the employees and took notes. Prescription filling is not a serial process. While the technician is handling customers drop off and pick up , the pharmacist fills prescriptions. Table 3 reflects the high-level, three-step process and key activities. At peak times, I observed upward of 20 customers per hour arriving λ at the pharmacy. The guarantee applies to orders of three prescriptions or fewer and prescriptions that do not require prior authorization. Occasionally, the medication was not in inventory. Other times, prescriptions required prior-authorization by the insurance company. Sometimes, customers simply asked the technician for general store information. On one occasion, I observed a customer applying for more than three prescriptions. These conditions combined to bring the number of qualified customers down to 16. Technicians were able to service a customer in about three minutes including drop off and pick up or a service rate µ of 20. The ratio of refills to new prescriptions averaged 70:30. Clearly, new prescriptions took a little longer than the refills. Assuming Poisson arrivals and exponential service for this case, my first question was to understand the utilization rate Ï of the customer-facing technician. This seemed about right considering that the technician must also answer the phone and handle miscellaneous questions regarding insurance and other products. Utilization rates of 100 percent or more should serve as red flags to Six Sigma professionals to re-assess the process and staffing levels. Based on my observations, these figures made sense considering peak times. The next set of questions relate to waiting time. However, this metric did not explain the whole story. I needed to determine the average wait time for customers in the system time spent waiting in line, plus service time. I could have also arrived at this answer by summing up 12 minutes plus 3 minutes service rate. From my cursory analysis using queuing model equations, I was able to see how Rite Aid could make such a bold guarantee. If Rite Aid wanted to reduce the waiting time to, say, 10 minutes, they could either reduce the arrival rate i. Putting Theory to Use Queuing theory is not just some esoteric branch of operations research used by mathematicians. It is a practical operations management technique that is commonly used to determine staffing, scheduling and inventory levels, and to improve customer satisfaction. By understanding queues and learning how to manage them through simple models and equations, Six Sigma practitioners can help improve customer-facing and internal processes to give organizations a competitive advantage. Sure, we get rid of waste Muda but getting rid or managing unevenness and overburden Muri, Mura is essential for improvement of our processes. Where do you find most of these, in the queue. Great explanation and application of the queuing theory technique. In the Rite Aid example, my nagging question is whether 15 minutes is the best guarantee, given the average of 15 for system time. Assuming this is typical across the chain, should they have opted for something on the high side of the average? Sir, I am working on a dissertation study as a requirement of my graduation for BS. My study uses a multiphase queuing system of three phases with either one or two servers. My problem is, I am not sure if I am doing right. I plan to sum up all the average waiting time for each phases to get the total waiting time of the system. Same as the other performance measure. Am I doing right?
Various measures of performance are derived. For example in a limbo of the flow of people through supermarket checkouts input data like the amount of shopping people have collected is represented by a statistical probability distribution and results relating to factors such as customer waiting times, queue lengths, etc are also represented by probability distributions. Complex queuing systems are almost always met using. The guarantee applies to orders of three prescriptions or fewer and prescriptions that do not require prior authorization. Queueing theory is the mathematical study of waiting lines, or. Computer Solution of the Multiple-Server Model with QM for Windows Exhibit 13. No work is lost in either con.