1. Waiting Line and Queuing Theory Kusdhianto Setiawan Gadjah Mada University

2. Waiting Line Costs All costs associated with and consequences that come from idleness or the unwillingness to wait from customer can be categorized as waiting cost, inlcuding customer dissatisfaction and lost goodwill

3. Objectives Finding the best level of service for an organization that minimizes total expected cost Total expected cost: the sum of service plus waiting time Trade-off: the cost of providing good service and the cost of customer waiting time (more difficult to predict)

4. Example of Waiting Cost Number of Team Working 1234 AAverage number of ships arriving per shift 5555 BAverage time each ship wait to be unloaded (hours) 7432 CTotal ship hours lost per shift (AxB) 35201510 DEstimated cost per hour of idle ship time 1000 EValue of ship’s lost time or waiting cost (CxD) 35,00020,00015,00010,000 FTeam Salary or Service Cost 6,00012,00018,00024,000 GTotal Expected Cost41,00032,00033,00034,000

5. Characteristic of A Queuing System Arrival Characteristics (calling population) The Queue or waiting line The Service Facility All of those must be determined before we develop a model

6. Arrival Characteristics Size of the Calling Population Unlimited (infinite) Limited (finite) Pattern of Arrivals at the system Scheduled Random; when they are independent of one another and cannot be predicted exactly The number of arrivals per unit of time can be estimated by a probability distribution known as the POISSON DISTRIBUTION

7. Poisson Distribution (Discrete Arrival) P(X)= Probability of X Arrival X= number of arrivals per unit of time λ= average arrival rate (2 customer/hour, four trucks/hour, etc.) e= 2,7183

8. Poisson Probability X P

9. Waiting Line Characteristics The length of line can be either Limited (restaurant) Unlimited (tollbooth) Queue Dicipline (the rule by which customers in the line are to receive service FIFO Preempt FIFO (Emergency Unit in a hospotal) Highest Priority

10. Service Facility Characteristics Basic Queuing System Configuration Number of channels/servers Single-Channel System Multi-Channel System Number of phases/service stops Single-Phase System Multiphase System Service Time Distribution Constant (it takes the same amount of time to take care of each customer) Random (following negative exponential probability distribution)

11. Service Facility Arrivals Departure After Service Facility Arrivals Departure After Service Facility Single-Channel, Single-Phase System Single-Channel, Multiphase System

12. Service Facility Arrivals Departure After Service Multichannel, Single-Phase System Service Facility Service Facility Departure After Service Departure After Service

13. Facility Arrivals Departure After Service Multichannel, Multiphase System Service Facility Departure After Service Facility Service Facility

14. Exponential Distribution of Service Time f(x) Service Time (Minutes) Average Service Time of 20 Minutes Average Service Time of 1 hour

15. Identifying Model Using Kendall Notation KENDALL NOTATION Arrival distribution/Service Time/Number of Service Channel open M= Poisson distribution for number of occurences (or exponential times) D= Constant (deterministic) rate G= General distribution with mean and variance known Example: Single-Channel Model= M/M/1 Two-Channel Model= M/M/2 Three-Channel, Poisson Arrival, Constant service time = M/D/3

16. M/M/1 : Single-Channel, Poisson Arrivals, Exponential Service Time Asssumptions of the Model 1.Arrivals are served as FIFO basis 2.Every arrival waits to be served regardless of the length of the line (no balking or reneging) 3.Arrivals are independent of preceding arrivals, but the average number of arrivals (the arrival rate) doesn’t change over time 4.Arrivals are describe by a poisson probability distribution and come from an infinite or very large population 5.Service time also vary from one customer to the next and are independent of one another, but their average rate is known 6.Service time occur according to the negative exponential probability distribution 7.The average service rate is greater than the average arrival rate

17. M/M/1 Continued λ= mean number of arrivals per time period (ex: per hour) µ = mean number of people or items served per time period Click here to view the modelshere