...PHILIPPINE WOMEN’S UNIVERSITY Waiting Line Theory A Narrative Report 2014 MBA PROGRAM Introduction Each organization has different illustrations of waiting lines or commonly called “queue”. No wonder why we also need to study waiting line because it is part of our everyday routine as well. It merely affects the performance and profit of the company. Waiting line or queue refers to a busy service facility or server so service is momentarily occupied or being used. One great example is during enrollment. At our very first step at the school gate, you will fall in line to have your bag checked by the security personnel. You will also need to fall in line at your college department to seek advise of what subjects needed to take. You might wait for your adviser attending to other students. At the time you have your subjects needed to enrol, you will need to fall in line at Registrar’s office for subjects encoding. Once you get your assessment form, you will be redirected to Accounting Office for payment and expect a new line again. Then, you will need to update your school ID card after payment, it means another line. Waiting line or queue is a repetitive scenario in our everyday lives. We can’t deny but we are also used in waiting. The organization itself also encounter waiting line or queue. One example is when an airplane has to wait in line for fueling, inspection, a particular gate, a specific flight route, an assigned crew, food loading, verified passenger count...
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...Queuing Theory Queuing Theory • Queuing theory is the mathematics of waiting lines. • It is extremely useful in predicting and evaluating system performance. • Queuing theory has been used for operations research. Traditional queuing theory problems refer to customers visiting a store, analogous to requests arriving at a device. Long Term Averages • Queuing theory provides long term average values. • It does not predict when the next event will occur. • Input data should be measured over an extended period of time. • We assume arrival times and service times are random. • • • • Assumptions Independent arrivals Exponential distributions Customers do not leave or change queues. Large queues do not discourage customers. Many assumptions are not always true, but queuing theory gives good results anyway Queuing Model Q W λ Tw Tq S Interesting Values • Arrival rate (λ) — the average rate at which customers arrive. • Service time (s) — the average time required to service one customer. • Number waiting (W) — the average number of customers waiting. • Number in the system (Q) — the average total number of customers in the system. More Interesting Values • Time in the system (Tq) the average time each customer is in the system, both waiting and being serviced. Time waiting (Tw) the average time each customer waits in the queue. Tq = Tw + s Arrival Rate • The arrival rate, λ, is the average rate new customers arrive measured in arrivals per time period....
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...Table of contents 1. Introduction 2. Arrival Pattern of customers 3. Service Patterns 4. System Capacity 5. Number of Service Channels 6. Queue Discipline 7. Queuing Cost 8. The Four Models 9. Model-1(Single Channel Queuing Model) 10. Model-2 (Multiple-Channel Queuing) 11. Model-3 (Constant-Service-Time) 12. Model-3 (Constant-Service-Time) 13. Simulation 14. Conclusion Abstract This report is about queuing theory, it’s application and analysis. Queuing theory has a vast number of applications starting from the simplest day to day life examples to complicated computer algorithms. To further explain the queuing theory analysis we have used simulation of an example from our case study. We have done an in depth analysis of the four queuing theory models and chosen one of them for the simulation. The results can be helpful in improving the overall performance of the manufacturing facility. Introduction According to U. Narayan Bhat waiting line are a phenomena through which businesses and facilities can be helped in an orderly manner. There are several ways to forma queue (waiting line), for instance when people wait to get a boarding pass from an airline counter, there can be 3 service stations (airline counters) and hence 3 waiting lines, or there can be one service station and hence one queue. These days we mostly see one counter for airline services as this benefit the passengers and airline best. This conclusion...
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...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...
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...ramen regardless of the number of customers. In short, they are more on the quality than the quantity; not on the profit side but rather on the quality side. But because they really want to serve more customers especially those ramen lovers who came from far places, they want to solve these queuing problems. Service time distribution Arrivals Customer 3 Customer 2 Customer 1 Service Facility Queue Fig. 1 Queuing System Configuration Assumptions of the model: Since Tamagoya Noodle House uses a Single-Channel, Single-Phase model in order to avoid confusion of customer’s order. The model we used assumes that seven conditions exist: 1. Arrivals are served on a First-in, First-out basis. Though some of customers who ordered less and or senior citizens were prioritized to be served first. 2. Every customer waits to be served regardless of the length of time and so there is no balking and reneging. 3. Arrival of customers is independent of the preceding arrivals, but the average no. of arrivals does not change over time (arrival rate). 4. Arrival of customers is described by Poisson probability distribution and come from infinite or very large population. 5. Service times...
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...Great Zimbabwe University Faculty of Agriculture and Natural Sciences Department of Mathematics and Computer Science Student: Sigwadhi Teddy M149125 Research Project (HSOR 460) Proposal Presentation in partial fulfilment of BSc. 4th Year Special Honours Degree in Operations Research and Statistics Supervisor: Mr. R. Mawonike Research Topic Queuing theory based approach to the analysis of sales checkout at Montagu Spar supermarket Location: Avenues Area, Harare, Zimbabwe Background of the study • Zimbabwe is an important emerging country among the developing countries. • The Spar Montagu has been chosen to be the research object primarily because of its clientele which have different buying behaviors. There are a mix of customers, low to high class customers and it has been seen to provide interesting results on the busy and non busy periods. • The main purpose of this project is to study the application of queuing theory and to evaluate the parameters involved in the service unit for the sales checkout operation in Spar Montagu supermarket Background of the study continued… • Queuing theory is the theory of waiting lines and service provision • A mathematical model is to be developed to analyse the performance of the checking out service unit • Two parameters need to be determined from the data collected in the supermarket through the mathematical model to the service point. • One parameter is the customer arrival rate to the service point per hour • The other is the...
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...Introduction Being in a queue (waiting line) is an inevitable fact of our daily life, such as waiting for checkout at a supermarket, or waiting to make a bank deposit. Queuing theory, started with research by Agner Krarup Erlang, is used to examine the impact of management decisions on these waiting lines (Anderson et.al, 2009). A basic Queuing Model structure consists of three main characteristics, namely behaviour of arrivals, queue discipline, and service mechanism (Hillier and Lieberman, 2001). In this assignment, New England Foundry’s queuing problem will be solved in Excel, and then, time and cost savings will be identified. First of all, current and new situation will be analysed in order to demonstrate the queuing model by using Kendall’s Notation (for the current queuing problem, queuing model is M/M/s). After that, arrival rate, queue size, and service rate will be defined, and added-in Excel file (Queuing models.xlsx). The results will be discussed at the end. Description New England Foundry (NEF) produces four different types of woodstoves for home use and additional products that are used with these four stoves. Due to the increase in energy prices, George Mathison president of the company wants to change the layout to increase the production of their bestselling type of Warmglo III. NEF has several operations in order to produce woodenstoves which are illustrated as a flow diagram in Figure 1. Current State Analysis Current layout offers one counter...
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...Assignment 5 1. A LAN has a data rate of 4 Mbps and a propagation delay between two stations at opposite ends of 20 μs. For what range of PDU sizes does stop-and-wait give an efficiency of at least 50%? r = 4 d = 20 E > 0.5 The equation should look like: Range = 2 * 20 * 10^-6 * 4 * 10^-6 which is = 0.00000000016 or 160 bits 2. A disadvantage of the contention approach for LANs is the capacity wasted due to multiple stations attempting to access the channel at the same time. Suppose that time is divided into discrete slots, with each stations attempting to transmit with probability p during each slot. What fraction of slots are wasted due to multiple simultaneous transmission attempts? The fraction of slots wasted due to multiple transmission attempts is equal to the probability that there will be 2 or more transmission attempts in a slot. So: Probability[2 or more attempts] = 1 – Probability[no attempts] – Probablity[exactly 1 attempt] = 1 - (1-P)^N - N × (P × (1-P)^N-1) 3. A simple medium access control protocol would be to use a fixed assignment time division multiplexing (TDM) scheme. Each station is assigned one time slot per cycle for transmission. For the bus, the length of each slot is the time to transmit 100 bits plus the ene-to-end propagation delay. For the ring, assume a delay of 1 bit time per station, and assume that a round-robin assignment is used. Stations monitor all time slots for reception. Assume a propagation...
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...Boise Cascade Corporation, Case 5.3 The mill manager of Boise Cascade Corporation seeks to understand how many scaling stations should be open to minimize the wait times for arriving trucks and the ineffectual time of the scalers. The number of trucks arriving during any given hour needs to be determined and the Poisson Probability Distribution has been applied. Having two open scale stations that cover 57.60% of the data would be the best option. Still, having four open scale stations there is a 99.93% chance that zero trucks would have to wait, but there would be a 3.74% chance that a truck will even need the extra station and you would likely be paying scalers for complete downtime. Boise Cascade Corporation | | | | | | | | | | | | | | | | Mean | 12 | | | | | | | | # of Trucks | | | | | | | | | 0 | 0.000006 | 0.000006 | | | | | | | 1 | 0.000074 | 0.000080 | | | | | | | 2 | 0.000442 | 0.000522 | | Time Window | Minutes | | | | 3 | 0.001770 | 0.002292 | | 7am-8am | 6 | 0.04582 | 4.58% | | 4 | 0.005309 | 0.007600 | | 8am-9am | 12 | 0.57597 | 57.60% | | 5 | 0.012741 | 0.020341 | | 9am-10am | 18 | 0.96258 | 96.26% | 3.74% | 6 | 0.025481 | 0.045822 | | 10am-11am | 24 | 0.99931 | 99.93% | | 7 | 0.043682 | 0.089504 | | 11am-12pm | 30 | 1.00000 | | | 8 | 0.065523 | 0.155028 | | 12pm-1pm | 36 | 1.00000 | | | 9 | 0.087364 | 0.242392 | | 1pm-2pm | 42 | 1.00000 | | | 10 | 0.104837 | 0.347229...
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...(a) First, determine the Queuing Models 1. Queuing Models Poisson Arrivals Standard (Infinite Queue) Exponential Service Times Single Server (M/M/1) Standard M/M/1 Model P0 = 1 – P (Probability of 0 customer in the system) P (n ≧k ) = ρk (Probability of Pn = P0ρn (Probability of exactly n customers in the system) Ls = λ / ( μ – λ ) (Mean no. of customers in the system) Lq = ρλ / ( μ – λ ) (Mean no. of customers in queue) Lb = λ / ( μ – λ ) (Mean no. of customers in queue for a busy system) Ws = 1 / ( μ – λ ) (Mean time customer spends in the system) Wq = ρ / ( μ – λ ) (Mean time customers spends in the queue) Wb = 1 / ( μ – λ ) (Mean time customers spends in queue for a busy system) Now, in our case, Λ (mean arrival rate) = 15/hr μ (mean service rate per busy server) Old machine’s mean service time= 3mins, 60/3 =20/hr New machine’s mean service time= 2mins, 60/2=30/hr Wq =ρ/ ( μ – λ ) (Mean time customers spends in the queue), Where ρ = λ / μ Old machine: (15/20)/ 20-15 = 0.1 hr = 9mins New machine: (15/30)/ 30-15= 0.03333 hr = 2mins According to the case, the average wage of the people who bring the documents to be copied is $8/hr, If the company rent the old machine and there are 15 arrivals/hr, There total mean time spend in the queue is ( 15 x 9 )mins = 135mins = 2.25hr, The total wage = $8 x 2.25 = $18 If the company rent the new machine and there are 15 arrivals/hr, The total mean time...
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...Operational Laws and Mean Value Analysis Nirdosh Bhatnagar 1. Introduction Operational technique can be used to analyse both computer and networking systems. We …rst introduce basics of operational analysis. This technique is then extended to open and closed queueing networks. The closed queueing network considered has single servers at its queueing centers. 2. Operational Analysis Basics The word operational means measurable. The analysis presented in this chapter assumes that the performance metrics of a computer can be directly inferred by measurable quantities. The measurable quantities are called operational quantities. The laws which relate operational quantities are called operational laws. Operational laws for computer systems will be studied in this note. We …rst introduce some notation. Notation: M =Number of queueing centers in the network. T = Finite observation period. Ai = Number of arrivals at queue i (device i) during observation period. Bi = Total busy period of queue i during observation period. Ci = Number of service completions at queue i during observation period. Di = Total service demand by a customer at device i. Qi = Queue length at device i (including the job in service) Ri = Response time per visit to the ith device. Si = Average service time per customer visit to queue i during observation period. Ui = Utilization of queue i during observation period. Vi = Average number of visits to queue i by a customer before it leaves the system during observation...
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...€250000 for hiring a engineering team, so If we hire three additional engineers it costs around €150000, For example. For 16 days to reduce 14 day company can hire one staff it cost €50000 so it means one staff can reduce 2 days , and 16 days reduce to 10 days company need only 3 staff , it means company can finish the work within 10 days and total cost 3x €50000. So its better to hire 3 engineers and we can reduce the number of working days with less cost. We consider a Level-2 IT service desk with two staff members. Each staff member can handle one service request in 4 working hours on average. Service times are exponentially distributed. Requests arrive at a mean rate of one request every 3 hours according to a Poisson process. What is the average time between the moment a service request arrives at this desk and the moment it is fulfilled? Queueing theory gives us the following formulas for calculating the above parameters for M/M/1 models: Lq—The average number of jobs (e.g. customers) in the queue. Wq—The average time one job spends in the queue. W—The average time one job...
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...waiting times for her guests during peak hours. The data collected during this case study will be presented along with a recommendation to Ms. Shader on which option has the shortest wait time for guests. Introduction First, the researchers determine the average amount of time a guest spends checking in and how this wait time would this change under each of Ms. Shrader’s three options. We will provide her with a recommendation as to which option would best serve her guests. Baseline Data The current system has five clerks each with their own waiting line or five independent queues, each with an arrival time of = 90/5 = 18 per hour. The service rate is one every three minutes or m = 20 per hour. Assuming Poisson arrivals and exponential service times, the...
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...Name: AMS 007 Homework Quiz 2 Thursday, January 30, 2014 1. The following table summarizes results from the sinking of Titanic which had 2,223 passengers: Men Women Boys Girls Survived 332 318 29 27 Died 1360 104 35 18 (a) (2.5 points) If one of the Titanic passengers is randomly selected, find the probability of getting someone who is a girl or boy. (b) (2.5 points) Given that we select a random person who survived, what is the probability of getting boy? (c) (2 points) Suppose that boys who survived, went for cruise at the rate of six per year. Using the Poisson distribution, what would be the mean and standard deviation of the number of boys who survived and went for a cruise in six years? 2. It is found that Viagra users experience headaches with probability 0.15. Suppose we have a group of 105 Viagra users and each of them is independent to experience headaches. (a) (2 points) What is the mean and standard deviation of the number of users that experience headaches? (b) (2 points) Would it be unusual for 19 users to experience headaches? Why or why not? 3. Assume that men’s weight are normally distributed with a mean of 172 and a standard deviation of 29. (a) (3 points) Find the probability that a randomly selected man weights less than 157. (b) (3 points) Find the probability that a randomly selected man weights between 172 and 187. (c) (3 points) Find P20 , which is the weight separating the bottom 20% from the top 80%....
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...By the way, parametric distribution means normal, binomial distribution etc.. Probability Distribution A listing of all the outcomes of an experiment and the probability associated with each outcome. Mean of probability Distribution μ= Σ[xP(x) Variance of probability Distribution σ2= Σ[x-μ2Px] Number of Cars (x) | Probability (x) | (x-μ) | (x-μ)2 | x-μ2P(x) | 0 | 0.1 | 0 – 2.1 | 4.41 | 0.441 (4.41*0.1) | 1 | 0.2 | 1 – 2.1 | 1.21 | 0.242 | 2 | 0.3 | 2 – 2.1 | 0.01 | 0.003 | 3 | 0.3 | 3 – 2.1 | 0.81 | 0.243 | 4 | 0.3 | 4 – 2.1 | 3.61 | 0.361 | | 1.0 (total) | | | σ2=1.290 | To derive μ=00.10+10.20+30.30+40.10 = 2.1 Discrete Random Variable A random variable that can assume only certain clearly separated values. Usually the result of counting something. Clearly separated means e.g. judge scoring points can be 8.8 and 8.9. they are clearly separated. It is NOT clearly separated if the score is 8.34144134134134134 or 8.351512312312. This will be infinite!! Continuous Random Variable A random variable that assume one of an infinitely large number of values, within certain limitations. E.g. The tiems of commercial flights between Singapore and Msia are 4.67 hours, 5.13 hours….so on. The random variable is the number of hours. Limitation rage is between Singapore and Msia. Binomial Probability Distribution * There are only two possible outcome on a trial of an experiment. – Success or failure * The random variable counts the...
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