Queuing Theory
Queuing Theory is generated from the service industries such as shops and retail dealers that need to pay much attention to the feeling of the customers and, at the mean time, to the cost with that the service was offered. As a retailer manager, one of the important things he or she might focus on is that the queue line length which could not be too long or too short. If the queue line is too long, the customer would be impatient and complain about the service quality the shop offers while if the shop gives too many counters to deal with the customers transaction further to reduce the length of the queue, it is definitely to increase the cost of the operation. Queue Theory is a kind of tool that could help the managers who need to analyze the queue line and estimate the cost of controlling it to understand the situation and make a decision on it. The prerequisite of Queue Theory is that the customers, services and other factors in the systems are discrete. In other words they are independent with each other since the rate of the customer coming and the rate of the service provided would not affect each other. Then these factors could meet the demand of Poisson Distribution. There are four models about Queuing Theory according to our textbook: MM1- Single-Server Queuing Model, MMS- Multiple-Server Queuing Model, MD1- Constant-Service-Time Model and Limited-Population Model. They are very useful in the different areas in the business. The first model single-server Queuing Model is the model the paper plan to explain a little detailed than the other three. In the real situation, you could easily imagine that there is a queue line with a large amount of people waiting for checking out in the grocery store or waiting for the service provided by the bank counter. A single channel and waiting line are the basic features for the MM1 Model. And this model has several other factors that need to be paid attention. First, first-in and first-out queue discipline, second average rate of arrivals and the rate of service provided are independent and do not change over times. Third, the population coming is unlimited and very larger, which fits to a Poisson probability distribution. Four, the service rate is faster than the arrival rate.1 These factors are the prerequisite of MM1 Model. A detailed example would be given to demonstrate how the Theory is used in the real problem. This example is from the question of our textbook in page 758 D.9. The main situation of the problem is following: “Neve Commercial Bank the only local bank is in the town of York, Pennsylvania. Normally, an average of 10 customers per hour arrive at the bank to transact business. There is currently one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the exponential distribution. If a single teller is used, find: a) The average time in the line. b) The average number in the line. c) The average time in the system. d) The average number in the system. e) The probability that the bank is empty.” The solution of these problems are these: The third question is inclined to be calculated at the first time for it is helpful to solve the other questions. According to the formulation, the average time in the system means that, in my opinion, how much time the entire system might be idle for the service rate is faster than the arrival rate. It is calculated as 1/(μ - λ) which equals (1604-10)=15 , then Utilization rate is 10/15 = 2/3, so we could get the result of the first question, is 1/5×2/3=2/15 . Well question four, the average number in the system, means that the customers are in the processing during a period of time. It is calculated as λ / (μ - λ) which equals (10604-10) = 2, then the question two, The average number waiting in the line, could be calculated by Utilization rate timing the average number in the system, which is 2/3 × 2 = 4/3, The last question is the easiest one, 1-2/3 = 1/3. Assuming that “CEO Benjamin of Neve is considering adding a second teller (who would work at the same rate as the first) to reduce the waiting time for customers.” Let’s say the result of this condition. The results are listed as the following table: | One teller | Two teller | The average time in the line | 2/15 | 1/60 | The average number in the line | 4/3 | 1/6 | The average time in the system | 1/5 | 1/20 | The average number in the system | 2 | 1/2 | The probability that the bank is empty. | 1/3 | 2/3 |
So we could easily see the situation that each number of Queuing Theory drops so much if additional teller is added. And especially the average waiting time in the line is only one of eighth compared with the initial situation.
Knowing these data could help the managers in the service and industry to estimate the assumption more accurately with the help of queuing theory. As a competent manager, it is useful to master the knowledge on all of these.

Simulation Model
Simulation is another useful and powerful mathematical tool that the managers should master to forecast the future situation based on the data in the past, which they have already collected. And thousands of large companies have been using Simulation Model to help them to determine the operations decisions. In the real world, the situation and the trend might appear again if there are less factors influencing the system so much. What happened in the past have the possibility to occur in future. As the competent manager, it is their ability to reduce the likeliness of risk and to avoid some loss with their foresightedness and knowledge. Based on the past data, the manager should have the ability to see the future trend through analyzing the data collected in the past. Of course, one of the most important things in this process is that the data collected in the past must be true or the real that could truly reflect the past. So the data collection is also the basic foundation for Simulation Model. With the help of the past data and the mathematic Simulation Model, a manager could do a lot such as reducing the turnover of inventory, the likelihood of losing customer, and investment on special process. An excellent example of using of Simulation Model is in our textbook on page 788. Under the help of Monte Carlo Simulation, the inventory managers could know the average rate of loss sales and make a decision to keep the inventory at which level could meet the demand of sale such as maintaining the product supply up to 8 every day. Based on the Model, you could also forecast what might occur in future. It gives you a more detailed vision about the trend.
Monte Carlo Simulation could help you solve the problem with one variables and also the problem with two variables. For example in the textbook, there is an example called “An Inventory Simulation with two varibles”. This example has two variables, one is the number of Ace Drill demanded every day and the other is intervals for reorder lead time. Both of these factors are of the importance for the sale of this shop since the intervals for lead time would cause the shop to loss sales in the condition of lacking of the inventory. So a shopper keeper, of course, need to place much more attention to both of these factors in the following days in order to keep the shop with enough inventory to sell. According to Monte Carlo Simulation, there are stable steps to simulate the problem which listed in the textbook clearly so that it is no requirement to rewrite them here. However, foreseeing a trend in the Monte Carlo Simulation model is one of the significant abilities that a manager should have.