Queueing Theory Models Training Presentation By: Seth Randall Topics • • • • • What is Queueing Theory? How can your company benefit from it? How to use Queueing Systems and Models? Examples & Exercises How can I learn more? What is Queueing Theory? • The study of waiting in lines (Queues) • Uses mathematical models to describe the flow of objects through systems Can queuing models help my firm? • Increase customer satisfaction • Optimal service capacity and utilization levels • Greater Productivity • Cost effective decisions Examples • • • • How many workers should I employ? Which equipment should we purchase? How efficient do my workers need to be? What is the probability of exceeding capacity during peak times? Brainstorm • Can you identify areas in your firm where queues exist? • What are the major problems and costs associated with these queues? Queueing Systems and Models Customer Arrival and Distribution Servicing Systems Customer Exit Customer Arrivals • Finite Population : Limited Size Customer Pool • Infinite Population: Additions and Subtractions do not affect system probabilities. Customer Arrivals • Arrival Rate λ = mean arrivals per time period • Constant: e.g. 1 per minute • Variable: random arrival 2 ways to understand arrivals • Time between arrivals – Exponential Distribution f(t) = λe- λt • Number of arrivals per unit of time (T) – Poisson Distribution ( T ) n e T PT (n) n! Time between arrivals Exponential Distribution F(t) 1.20 1.00 f(t) = λe- λt 0.80 0.60 0.40 0.20 0.00 0 1 2 3 4 5 6 Time Before Next Arrival f(t) = The probability that the next arrival will come in (t) minutes or more Time between arrivals Minutes (t) Probability that Probability that the next the next arrival will arrival will come in t come in t minutes minutes or less or more 0 1 2 3 4 5 1.00 0.37 0.14 0.05 0.02 0.01 0.00 0.63 0.86 0.95 0.98 0.99 Number of arrivals per unit of time (T) Poisson Distribution 0.25 0.2 Probability of n arrivals in time (T) 0.15 ( T ) n e T PT (n) n! 0.1 0.05 0 0 -0.05 PT (n) 1 2 3 4 5 6 7 8 9 10 Number of arrivals (n) = The probability of exactly (n) arrivals during a time period (T) Can arrival rates be controlled? • • • • Price adjustments Sales Posting business hours Other? Other Elements of Arrivals • Size of Arrivals – Single Vs. Batch • Degree of patience – Patient: Customers will stay in line – Impatient: Customers will leave • Balking – arrive, view line, leave • Reneging – Arrive, join queue, then leave Suggestions to Encourage Patience • • • • • Segment customers Train servers to be friendly Inform customers of what to expect Try to divert customer’s attention Encourage customers to come during slack periods Types of Queues • 3 Factors – Length – Number of lines • Single Vs. Multiple – Queue Discipline Length • Infinite Potential – Length is not limited by any restrictions • Limited Capacity – Length is limited by space or legal restriction Line Structures • • • • • Single Channel, Single Phase Single Channel, Multiphase Multichannel, single phase Multichannel, multiphase Mixed Queue Discipline • How to determine the order of service – – – – – – First Come First Serve (FCFS) Reservations Emergencies Priority Customers Processing Time Other? Two Types of Customer Exit • Customer does not likely return • Customer returns to the source population Notations for Queueing Concepts λ = Arrival Rate Wq = Average time waiting in line µ = Service Rate Ws = Average total time in system 1/µ = Average Service Time n = number of units in system 1/λ = Average time between arrivals S = number of identical service р = Utilization rate: ratio of arrival ) Lq = Average number waiting in line rate to service rate ( Ls = Average number in system channels Pn = Probability of exactly n units in system Pw = Probability of waiting in line Service Time Distribution • Service Rate – Capacity of the server – Measured in units served per time period (µ) Examples of Queueing Functions 2 Lq ( ) Ls Wq Lq Ws Ls Exercise • Should we upgrade the copy machine? – Our current copy machine can serve 25 employees per hour (µ) – The new copy machine would be able to serve 30 employees per hour (µ) – On average, 20 employees try to use the copy machine each hour (λ ) – Labor is valued at $8.00 per hour per worker Exercise Current Copy Machine: 20 Ls 25 20 Ls = 4 people in the system 4 Ws 0.2 hours waiting in the system 20 Exercise Upgraded Copy Machine: 20 Ls 2 people in system 30 20 Ls 2 Ws 0.1 hours 20 Current Machine: – Average number of workers in system = 4 – Average time spent in system = 0.2 hours per worker – Cost of waiting = 4 * 0.2 * $8.00 = $6.40 per hour New Machine: – Average number of workers in system = 2 – Average time spent in system = 0.1 hours per worker – Cost of waiting = 2 * 0.1 * $8.00 = $1.60 per hour Savings from upgrade = $4.80 per hour Conclusion and Takeaways • Queueing Theory uses mathematical models to observe the flow of objects through systems • Each model depends on the characteristics of the queue • Using these models can help managers make better decisions for their firm. How Can I Learn More? • Fundamentals of Queueing Theory – Donald Gross, John F. Shortle, James M. Thompson, and Carl M. Harris • Applications of Queueing Theory – G. F. Newell • Stochastic Models in Queueing Theory – Jyotiprasad Medhi • Operations and Supply Management: The Core – F. Robert Jacobs and Richard B. Chase References • Jacobs, F. Robert, and Richard B. Chase. “Chapter 5." Operations and Supply Management The Core. 2nd Edition. New York: McGraw-Hill/Irwin, 2010. 100-131. Print. • Newell, Gordon Frank. Applications of Queueuing Theory. 2nd Edition. London: Chapman and Hall, 1982.