1. 1 Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow SMU EMIS 5300/7300 NTU SY-521-N Queuing Theory Basic Concepts and Models updated 11.07.01

2. 2 Basic Concept Service facilities are usually designed so that their capacity is less than the maximum demand Whenever demand exceeds capacity, a waiting line, or a queue is formed; that is, the customers do not get service immediately upon request but must wait On other occasions the service facility will be idle. Management is interested in finding the appropriate level of service

3. 3 x 1 x 2 x 3 … x n Arriving Customers Waiting for Service or Queue System ServicingServiced 1 2 K...... Departing Customers Service Facility

4. 4 Examples of Queuing Theory Application Determining the capacity of an emergency room in a hospital Determining the number of runways at an airport Determining the number of elevators in a building Determining the number of traffic lights and their frequency of operation Determining the number of flights between two cities

5. 5 Examples of Queuing Theory Application Determining the number of first-class seats in an airplane Determining the size of a restaurant Determining the number of employees in a storeroom, in a typing pool, or in a nursing team

6. 6 The Structure of a Queuing System The customers and their source - Customers are defined as those in need of service. Customers can be people, airplanes, machines, or raw materials. The customers are generated from a population or a source. For example, a hospital’s ‘population’ would be the sick requiring hospitalization. The arrival process - The manner in which customers show up at the service facility is called the arrival process.

7. 7 The Structure of a Queuing System The service facility and the service process - The service is provided by a service facility (or facilities). This may be a person (a bank teller, a barber), a machine (elevator, gasoline pump), or a space (airport runway, parking lot, hospital bed), to mention just a few. A service facility may include one person or several people operating as a team. The queue - Whenever an arriving customer finds that the service facility is busy, a queue, or waiting line, is formed.

8. 8 Some Examples of Queuing Systems SystemQueueService BankPeopleTellers TelephoneCallersSwitchboard LibraryBooks to be shelvedLibrarian’s assistant FreewayAutomobilesTollbooth AirplanePeopleSeats, flights AirportCircling planesRunways

9. 9 Some Examples of Queuing Systems 0 100 200 300 400 500 cost ($) 1 2 3 4 5 6 7 8 9 10 min total cost cost of providing service cost of waiting optimal levellevel of service

10. 10 Costs Involved in a Queuing Situation The facility cost - the cost of providing a service includes: 1. Cost of construction (capital investment) as expressed by interest and amortization. 2. Cost of operation: labor, energy, and materials required for operations. 3. Cost of maintenance and repair. 4. Other costs: insurance, taxes, rental of space, and other fixed costs.

11. 11 Costs Involved in a Queuing Situation The cost of waiting customers - the cost of waiting time is more difficult to assess. It involves several components. For example, a waiting customer may get impatient and leave, thus resulting in a loss of revenue and possible loss of repeat business due to his or her dissatisfaction. There may also be ‘ill-will’ cost incurred.

12. 12 Management Objectives Management may hold either or both of the following objectives when making decisions about an appropriate service level. Cost minimization - In cases where it is possible to ascribe a cost to the waiting time (usually when a company is serving its own employees or equipment), management will provide a service level such that the total cost of waiting and service is minimized.

13. 13 Management Objectives Achieving a specified performance level (service goal) - Instead of (sometimes in addition to) minimizing costs, management will strive to achieve a certain level of service. For example: - Telephone companies want to repair 99% of all inoperative telephones within 24 hours. - Fast-food restaurants advertise that you will not have to wait more than three minutes. - Banks try to avoid having more than six cars in any lane of their drive in windows at a time. - Service facilities should be in use at least 60% of the time.

14. 14 The Process The managerial application of waiting line theory involves the use of computed measures of performance for selecting an alternative solution to a queuing problem, usually among small numbers of alternatives. The entire process involves three steps: Establish the measures of performance (or the operating characteristics) of the queuing system. Compute the measures of performance (result variables) Conduct an analysis

15. 15 Establish the Measures of Performance In this step a model of the problem is formulated and the measures of performance are decided upon. Examples of such measures are: The average waiting time per customer The average number of customers in the waiting line The utilization (busy period) of the service facility, or else its idle time.

16. 16 Compute the Measures of Performance Once the problem has been formulated, one of two solution methods is employed to find the measures of performance: For problems in which certain theoretical statistical distributions can describe the data, formulas or equivalent tables can be used. For other problems, Monte Carlo simulation is used. The measures of performance are then computed for every course of action under consideration

17. 17 The Analysis In queuing analysis, there are usually only a small number of alternatives to be evaluated. For example, in a decision about the number of elevators to be constructed in a new building, 10 possibilities would be a realistic consideration, but not 5000. The number of feasible alternatives in a service system is usually small because of human, technical, and legal constraints. Alternatives may differ in the size of the facility, the number of facilities, the speed of service, the priorities given to customers or in the operating procedures. For each alternative, the measures of effectiveness must be computed.

18. 18 The Analysis The alternative solutions are then compared on the basis of their overall effectiveness. One approach here is the use of the total cost curve. The major problem in this step may be cost assessment. A queuing system usually involves several measures of performance, and it is necessary to establish a common denominator (such as a total cost or a total utility) to quantitatively compare the alternatives. In some cases a qualitative comparison of the alternatives is performed and no attempt is made to perform a cost analysis.

19. 19 The Analysis In a limited number of cases the comparative analysis leads to an optimal solution - for example, a decision regarding the choice of the proper number of identical service facilities. In such cases, an explicit dollar value for the cost of waiting must be specified. Most frequently the analysis involves the assessment of performance levels under different system configurations. For example, if waiting time per customer is important it is useful to know, for each alternative configuration, how long customers must wait.

20. 20 Different Arrangement of Service Facilities a. single service facility b. multiple, parallel identical facilities (single queue) waiting line

21. 21 Different Arrangement of Service Facilities c. multiple, parallel non-identical facilities express line regular lines

22. 22 Different Arrangement of Service Facilities d. series of facilities

23. 23 Different Arrangement of Service Facilities e. combination of facilities

24. 24 Queue Discipline A queue is formed whenever customers arrive and the facility is busy. The characteristics of the queue depend on rules and regulations that are termed the queue discipline. The queue discipline describes the policies that determine the manner in which customers are selected for service. Examples of some common disciplines are: A priority system - Priority is given to selected customers. For example, those with five items or less in a supermarket can go to the express lane. The handicapped and passengers with reservations board airplanes first.

25. 25 Queue Discipline Emergency (preemptive priority) systems - This is a system in which an important customer not only has a priority in entrance but can even interrupt a less important customer in the middle of his or her service. For example, in an emergency case in a hospital, the doctor may leave the regular patient in the middle of the treatment. That is, the regular patient is preempted by the emergency one.

26. 26 Queue Discipline Last-in, first-served (LIFS) - Last arrivals are served first. This system is commonly used with parts and materials in a warehouse since it reduces handling and transportation. First-in, first-served (FIFS) - Customers that arrive first are served first.

27. 27 The Behavior in a Queue Some interesting observations of human behavior in queues are: Balking - customers refusing to join a queue, usually because of its length Reneging - Customers tiring and leaving the queue before they are served. Jockeying - Customers switching between waiting lines (a common scene in a supermarket)

28. 28 The Behavior in a Queue Combining, dividing - Combining or dividing queues at certain lengths (e.g., in a supermarket when a counter is closed or opened). Cycling - Returning to the queue immediately after obtaining service. (Children taking turns at a playground or ore cars at a mine). Note: In this text we assume that a customer enters the system, stays in the line (if necessary), receives service, and leaves. If a customer behaves otherwise (according to any of the above observations), the queuing system becomes very complex, requiring simulation for analysis.