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ISE 195 Introduction to Industrial & Systems Engineering

ISE 195 Introduction to Industrial & Systems Engineering. Queuing Theory (Topics in ISE 484). Ref: Applied Management Science, 2 nd Ed., Lawrence & Pasternack Slides based on textbook slides. Queuing is Familiar. Queuing Occurs in Service Systems. Introduction.

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ISE 195 Introduction to Industrial & Systems Engineering

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  1. ISE 195Introduction to Industrial & Systems Engineering

  2. Queuing Theory(Topics in ISE 484) Ref: Applied Management Science, 2nd Ed., Lawrence & Pasternack Slides based on textbook slides

  3. Queuing is Familiar

  4. Queuing Occurs in Service Systems

  5. Introduction • Queuing is the study of waiting lines, or queues. • The objective of queuing analysis is to design systems that enable organizations to perform optimally according to some criterion. • Possible Criteria • Maximum Profits or Minimize Costs • Desired Service Level.

  6. Application Areas for Queuing • Transportation Systems • Cars waiting in traffic • Planes waiting on runways • Factories • Jobs waiting for machines • Machines waiting for repairs • Service Systems • Customers at retails stores or restaurants • Paperwork in a business process • Patients at a hospital emergency room or waiting for an operating room

  7. Essential Tradeoff in Queuing • What is the value of improved service to a customer? • Higher levels of service either require more resources (people, machines, technology) at higher cost • Saving money on resources often leads to longer waiting times, and lower levels of service

  8. Introduction • Analyzing queuing systems requires a clear understanding of the appropriate service measurement. • Possible service measurements • Average time a customer spends in line. • Average length of the waiting line. • The probability that an arriving customer must wait for service.

  9. Elements of the Queuing Process • A queuing system consists of three basic components: • Arrivals: Customers arrive according to some arrival pattern. • Waiting in a queue: Arriving customers may have to wait in one or more queues for service. • Service: Customers receive service and leave the system.

  10. Fundamental Queuing Insights • Over long periods of time, customers cannot arrive at a rate that is higher than the overall rate of service, or queues get large! • Queuing theory helps us estimate the waiting time • How many resources are enough to guarantee reasonable queues? • During short busy periods, rate of arrivals can outpace rate of service, but service will suffer. • How will we manage busy periods?

  11. Ways to Manage Queues • Segment the customers • Split off customers that can be handled quickly and uniformly • Inform customers of what to expect • Divert the customer’s attention while waiting • Encourage customers to come during slow times • Train servers to be friendly

  12. The Arrival Process • There are two possible types of arrival processes • Deterministic arrival process. • Random arrival process. • The random process is more common in businesses.

  13. The Arrival Process • Under three conditions the arrivals can be modeled as a Poisson process • Orderliness : one customer, at most, will arrive during any time interval. • Stationarity: for a given time frame, the probability of arrivals within a certain time interval is the same for all time intervals of equal length. • Independence : the arrival of one customer has no influence on the arrival of another.

  14. The Poisson Arrival Process (lt)ke- lt k! P(X = k) = Where l = mean arrival rate per time unit. t = the length of the interval. e = 2.7182818 (the base of the natural logarithm). k! = k (k -1) (k -2) (k -3) … (3) (2) (1). Note: Arrivals according to Poisson Process have exponential inter-arrivals which turns out to provide quite nice theoretical results.

  15. HANK’s HARDWARE • Arrival Process • Customers arrive at Hank’s Hardware according to a Poisson distribution. • Between 8:00 and 9:00 A.M. an average of 6 customers arrive at the store. • Hank would like to know what is the probability that k customers will arrive between 8:00 and 8:30 in the morning (k = 0, 1, 2,…)?

  16. HANK’s HARDWARE 0 Illustration of the Poisson distribution. • Input to the Poisson distribution l = 6 customers per hour.t = 0.5 hour.lt = (6)(0.5) = 3. X = Number of Arrivals in lt = 3 time units P(X = k) 1 2 3 4 5 6 7 8 k k 1 0 2 3 (lt) e- lt k ! 1 0 = P(X = k ) = 2 0.224042 0.224042 0.149361 0.049787 3 1! 0! 2! 3!

  17. Waiting Line Characteristics • Factors that influence the modeling of queues • Line configuration • Jockeying • Balking • Priority • Tandem Queues • Homogeneity

  18. Line Configuration • A single service queue. • Multiple service queue with single waiting line. • Multiple service queue with multiple waiting lines. • Tandem queue (multistage service system).

  19. Waiting Line Configurations Multiple ServerMultiple Lines Multiple ServerSingle Line Single Server Tandem Queue

  20. Jockeying and Balking • Jockeying occurs when customers switch lines once they perceive that another line is moving faster. • Balking occurs if customers avoid joining the line when they perceive the line to be too long.

  21. Priority Rules • These rules select the next customer for service. • There are several commonly used rules: • First come first served (FCFS). • Last come first served (LCFS). • Estimated service time. • Random selection of customers for service.

  22. Tandem Queues • These are multi-server systems. • A customer needs to visit several service stations (usually in a distinct order) to complete the service process. • Examples • Patients in an emergency room. • Passengers prepare for the next flight.

  23. Homogeneity • A homogeneous customer population is one in which customers require essentially the same type of service. • A non-homogeneous customer population is one in which customers can be categorized according to: • Different arrival patterns • Different service treatments (such as hospital)

  24. The Service Process • In most business situations, service time varies widely among customers. • When service time varies, it is treated as a random variable. • The exponential probability distribution is used sometimes to model customer service time. • Simple model; useful for analysis purposes.

  25. HANK’s HARDWARE • Service time • Hank’s estimates the average service time to be 1/m = 4 minutes per customer. • Service time follows an exponential distribution. • Define: • X = Amount of time it takes to serve the next customer • What is the probability that it will take less than 3 minutes to serve the next customer? • What is Pr( X < 3 minutes )?

  26. HANK’s HARDWARE 3 minutes = .05 hours • Using Excel for the Exponential Probabilities • The mean number of customers served per minute is ¼ = ¼(60) = 15 customers per hour. • P(X < .05 hours) = 1 – e-(15)(.05) = ? • From Excel we have: • EXPONDIST(.05,15,TRUE) = .5276 • In ISE 484 you will learn the details of these calculations; for now we are simply illustrating how the results would be used

  27. HANK’s HARDWARE • What is your recommendation to HANK’s management based on this? • Management wants to have 95% of customers served in less than 3 minutes • You recommend a detailed study of the service process to implement some lean techniques to improve the level of customer service • Notice, this is just the service time, and doesn’t include any of the queuing time

  28. The Exponential Distribution • Characteristics • The “Lack of Memory” property. • No additional information about the time left for the completion of a service, is gained by recording the time elapsed since the service started. • For Hank’s, the probability of completing a service within the next 3 minutes is (0.52763) independent of how long the customer has been served already. • The Exponential and the Poisson distributions are related to one another. • If customer arrivals follow a Poisson distribution with mean rate l, their interarrival times are exponentially distributed with mean time 1/l.

  29. Performance Measures in Queuing Systems • Performance can be measured by focusing on: • Customers in queue. • Customers in the system. • Performance is measured for a system in steady state.

  30. Steady State Performance Measures • P0 = Probability that there are no customers in the system. • Pn = Probability that there are “n” customers in the system. • L = Average number of customers in the system. Lq = Average number of customers in the queue. W = Average time a customer spends in the system. Wq = Average time a customer spends in the queue. • Pw = Probability that an arriving customer must wait • for service. r = Utilization rate for each server (the percentage of time that each server is busy).

  31. MARY’s SHOES • Customers arrive at Mary’s Shoes every 12 minutes on the average, according to a Poisson process. • Service time is exponentially distributed with an average of 8 minutes per customer. • Management is interested in determining the performance measures for this service system.

  32. MARY’s SHOES - m –l = 7.5 – 5 = 2.5 per hr. P(X<10min) = 1 – e-2.5(10/60) = .565 Pw = l/m =0.6667 r = l/m =0.6667 • Solution • Input l = 1/12 customers per minute = 60/12 = 5 per hour. m = 1/ 8 customers per minute = 60/ 8 = 7.5 per hour. • Performance Calculations • P0 = 1 - (l/m) = 1 - (5/7.5) = 0.3333 • Pn = [1 - (l/m)](l/m)n = (0.3333)(0.6667)n • L = l/(m - l) = 2 • Lq = l2/[m(m - l)] = 1.3333 • W = 1/(m - l) = 0.4 hours = 24 minutes • Wq = l/[m(m - l)] = 0.26667 hours = 16 minutes

  33. Excel Queueing Spreadsheet

  34. MARY’s SHOES • Recommendations to management • 16 minutes of time in queue seems excessive • Question 1: Is the model correct? • Compare model predictions to reality • How does the 16 minute waiting time compare to the goals for customer service for Mary’s shoes? • Make recommendations to improve waiting time • More resources • Better processes • Better information systems

  35. Tandem Queuing Systems Meats Beverage • In a Tandem Queuing System a customer must visit several different servers before service is completed. • Examples • All-You-Can-Eat restaurant

  36. Tandem Queuing Systems • In a Tandem Queuing System a customer must visit several different servers before service is completed. Meats Beverage • Examples • All-You-Can-Eat restaurant

  37. Tandem Queuing Systems • In a Tandem Queuing System a customer must visit several different servers before service is completed. Meats Beverage • Examples • All-You-Can-Eat restaurant • A drive-in restaurant, where first you place your order, then pay and receive it in the next window. • A multiple stage assembly line.

  38. Applications of Queuing • Determine number of servers • Examine line configurations • Evaluate efficiency of process • Determine storage requirements • Various cost benefit analyses • Bound performance of complex systems

  39. Other Applications • In ISE 483 (Integrated Systems for Manufacturing) • Use Queuing models to decide number of machines, workers in a system • Extend queuing models to include impact of downtime (breakdowns, maintenance) • In ISE 471 (Simulation) • Use queuing models to get a quick “guess” at what the result of a simulation should be close to • Helps in debugging and validating models

  40. Questions? Ref: Applied Management Science, 2nd Ed., Lawrence & Pasternack Slides based on textbook slides

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