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 Exit Servicing Systems Customer Arrival and Distribution

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

Time between arrivals f(t) = λe - λt f(t) = The probability that the next arrival will come in (t) minutes or more

Minutes (t)Probability that the next arrival will come in t minutes or more Probability that the next arrival will come in t minutes or less Time between arrivals

Number of arrivals per unit of time (T) = 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

Infinite Potential –Length is not limited by any restrictions Limited Capacity –Length is limited by space or legal restriction Length

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 µ = Service Rate 1/µ = Average Service Time 1/λ = Average time between arrivals р = Utilization rate: ratio of arrival rate to service rate ( ) L q = Average number waiting in line L s = Average number in system W q = Average time waiting in line W s = Average total time in system n = number of units in system S = number of identical service channels P n = Probability of exactly n units in system P w = 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

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

Current Copy Machine: = 4 people in the system hours waiting in the system Exercise

Upgraded Copy Machine: people in system hours Exercise

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. 2 nd Edition. New York: McGraw-Hill/Irwin, Print. Newell, Gordon Frank. Applications of Queueuing Theory. 2 nd Edition. London: Chapman and Hall, 1982.