1. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Waiting Line Models

2. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Module Objectives Explain the importance of waiting line models to service operations management. Give a real example of a decision modeled with waiting line methods. Demonstrate the measures important in waiting line situations: Understand the basic elements of waiting line systems (and the assumptions implied). Identify system features that can be predicted by waiting line formulas. Calculate expected system performance for given sets of assumptions. Understand the difference in standard sets of queuing formulas. Understand the context of where queuing formulas apply. Show how waiting line models can be solved.

3. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. New York City Justice System Severe backlog Need to reduce average wait to 24 hours Analysis –Identification of system –Data gathering –Modeling – display estimated performance Used Monte Carlo simulation for waiting line problem due to complexity

4. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Example Waiting Line Systems ArrivalsQueueService Barber shopCustomersAssigned numbers Barbers Car washCarsLineWashers Computer center JobsStacked jobsCPU Shipping dock ShipsEmpty or loadCranes Telephone system callslinesSwitchboard

5. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Waiting Line Systems COMPONENTS –Arrivals Calling population  limit K Distribution –Discipline FIFOLIFOpriority –Services Number of servers Distribution –Departures

6. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Waiting Line Decisions Steady State: –performance at equilibrium Transient State: –temporary characteristics on way to equilibrium Total Cost –Balance Cost of providing service Cost of waiting

7. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Source of Waiting Line System Performance Differences Variance –In arrival pattern –In service times If constant system, calculations simple If variance, system will have idle periods balanced with periods where work is waiting Need to overbuild if variance exists

8. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Impact of Busy System If rate of arrival  rate of service capacity –System is degenerate (cannot get work done) If rate of arrival is close to service capacity –System will experience long waiting lines at some times should variance exist

9. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. System Measures P n probability of n in the system W q average time waiting for service L q average number waiting for service Waverage time in system (including service) Laverage number in system (including being served)  system utilization rate

10. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Waiting Line Model Assumptions Arrival distribution Service distribution Number of servers Queue discipline Maximum number of customers in system Number of potential customers

11. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Arrival Distributions MExponential (or Poisson) DDeterministic (constant) E k Erlang (shape parameter k) GIGeneral independent Others = number of arrivals per unit time

12. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Service Distributions MExponential (or Poisson) DDeterministic (constant) E k Erlang (shape parameter k) GSGeneral service Others  = Mean service time

13. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Other Assumptions Snumber of servers (clones) Queue discipline –FIFO (FCFS)first in, first out –LIFOlast in, first out –SIROrandom –GDgeneral distribution (other rules) Max Line Length Calling Population

14. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Standard Models M/M/1:FCFS/  /  M/D/1:FCFS/  /  constant service M/GS/1:FCFS/  /  normal service M/M/s:FCFS/  /  multiple servers M/M/1:FCFS/k/  max k in system M/M/s:FCFS/k/  /multiple servers, max k others

15. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Summary Many service operations involve waiting lines Waiting line models good to understand counterintuitive behavior of such systems For complex systems, use simulation