1. OMG 402 - Operations Management Spring 1997 CLASS 6: Process Design and Performance Measurement Harry Groenevelt

2. 2 Agenda Recap –Basic Queuing Relationships –Modeling a Distributed Queue The Impact of Variability ‘Limited Space’ Systems and Performance Measure Trade-offs Capacity Strategy and Queuing Management Summary of Insights

3. 3 Servers (s) system queue departures arrivals ( customers/hr)  customers/hr/server Recap: Basic Queuing Relationships avg. # in system = (avg. # in queue) + (avg. # in service) avg. # in service = (# of servers) * (utilization) avg. time in system = (avg. time in queue) + (avg. service time) … and remember Little’s Law!

4. 4 Recap: Basic Queueing Relationships M/M/1 Queue: –Special service time and inter-arrival time distributions (a ‘memoryless’ process) –Single Server –Average time in system = 1/(  – ) –Average number in system = /(  - ) =  /(1–  )

5. 5 Recap: Basic Queuing Relationships M/M/s Queue: –Again, memoryless arrival and service times –‘s’ servers –Results via QMacros G/G/s Queue: –When arrivals or service times are not of this ‘special’ type –Results via QMacros

6. 6 Recap: Modeling a Distributed Queue The On-Call Computer Consultant Customer arrives by telephone. Information for Queue: appointment book of the consultant Physical Queue: Customers’ Offices What is the ‘server’?

7. 7 Recap: Modeling a Distributed Queue The On-Call Computer Consultant When does service begin and end? Customer point of view: Consultant (server) point of view: For QMacros, service time =

8. 8 Recap: Impact of variability (G/G/s) Using QMACROS –For arrival process specify: Arrival Rate Coefficient of Variation of inter-arrival time distribution (cv(A)) –For service time distribution specify: Service Rate Coefficient of Variation of service time distribution (cv(S))

9. 9 Recap: Impact of variability Reminder: if X is a random variable with mean  and std. dev , then its Coefficient of Variation = cv(X) =  /  For exponential random variables: –Coefficient of Variation = 1 For deterministic random variables: –Coefficient of Variation = 0

10. 10 Recap: approximate G/G/1 formula An approximation for average wait in queue that works well for ‘congested’ systems: W q (G/G/1) = 0.5 * (cv(A) 2 +cv(S) 2 ) * W q (M/M/1) Use QMacros to analyze G/G/s

11. 11 check-in booths queue in lobby of convention hall departures arrivals step off of tour buses from the hotel Typical tasks at check-in: ask for name and check for registration look up registration number check off list hand over packet Even with seemingly plenty of booths we observe long queues. Why? Impact of Variability: An Example Check-in for an Operations Management Convention in Morocco Original Physical Arrangement:

12. 12 arrivals step off of tour buses from the hotel Impact of Variability: An Example Check-in for an Operations Management Convention in Morocco Revised Arrangement: A-G H-P Q-Z check-in booths departures arrivals check pre- registration information on posted computer printouts What are the advantages of this system? What are the disadvantages?

13. 13 What systems can be modeled this way? Limited Space Systems M/M/s/N (‘limited space’) system –Same as M/M/s system, except: –Assumes only N positions available –An arriving customer who finds all N positions occupied leaves without waiting and without receiving service N in System? Queue Server 1 Server s Customer Arrival Departures Leave Without Service

14. 14 Limited Space Systems: Performance Measures Fraction Not Served: fraction of arrivals not served because they found all N positions in the system occupied Throughput: the rate at which customers are served by the system Load Factor: arrival rate/total capacity (how is this different from utilization?)

15. 15 Limited Space Systems: Performance Measures Throughput = Arrival Rate * (1– Fraction Not Served) All other performance measures (time in system, etc.) are for served customers only, and satisfy all the relationships we’ve seen. Similar measures for M/M/s/I system with impatient customers.

16. 16 (see: Frontiers of Electronic Commerce by Profs. Kalakota and Whinston) Limited Space Systems Example: Local Internet Service Provider (ISP) N trunk lines (all customers ‘arrive’ by same-number dialup) national ISP and Internet Backbone calls may queue for a modem here Switch Terminal server Router Modem Farm modem 1 modem 2 modem 3 modem s

17. 17 Limited Space System: Local ISP Each caller uses one trunk line and one modem Arriving caller waits on a trunk line if all S modems are used Arriving caller busied out if N trunk lines used lines logged onto s modemsN–s lines (virtual queue) N trunk lines

18. 18 Performance measure trade-offs Consider the system with high utilization (i.e., AOL at peak hours!) As we decrease the number of trunk lines: What happens to fraction not served (busied out)? What happens to average wait in queue?

19. 19 (28 modems, 35 trunk lines, average session length: 20 minutes) Performance measures trade-off Now hold the number of trunk lines constant and increase arrival rate: 0 1 2 3 4 5 0123456789101112 Arrival Rate (1/min) Average Wait to Log On (minutes) 0% 20% 40% 60% 80% 100% Percent Receiving Busy Signals Fraction Busy Signals (see scale on the right) Avg Wait to Log On (see scale on the left)

20. 20 Performance measures trade-off As demand increases but capacity does not keep pace: –wait in queue increases but is limited by available space; –percentage busied-out (not served) increases up to 100%. –When load factors are high, customers must go somewhere!

21. 21 Subscriptions to AOL, 1994-1997 0 1 2 3 4 5 6 7 8 Jun-94Dec-94Jun-95Dec-95Jun-96Dec-96 Number of Subscribers (millions) January, 1997 December, 1996 (flat-rate access introduced) source: Jupiter Communications and the Los Angeles Times Capacity Strategy: America Online

22. 22 Capacity Strategy: Expansionist 0 1 2 3 4 5 6 7 8 Jun-94Dec-94Jun-95Dec-95Jun-96Dec-96 Number of Subscribers (millions) Modem Capacity Nr of Subscribers

23. 23 0 1 2 3 4 5 6 7 8 Jun-94Dec-94Jun-95Dec-95Jun-96Dec-96 Number of Subscribers (millions) Capacity Strategy: Wait-and-See Modem Capacity Nr of Subscribers

24. 24 Capacity Strategy What drives ‘expansionist’ vs. ‘wait-and- see’ strategies?

25. 25 Queuing Management The firm’s view: Manage demand as well as capacity Balance cost of service with cost of waiting (“economic optimization” at LL Bean) Use customer waiting time –co-production –sales

26. 26 Queuing Management The psychology of queuing: There’s more to a line than its wait (Larson) –perceived waiting time & the environment –justice –information and expectations

27. 27 Management of Queues: Summary of Insights High utilization causes congestion, high WIP and long lead times Variability causes congestion, high WIP and long lead times Multiple performance measures are often necessary to gauge true performance. Cost must be balanced with service, and the entire customer experience must be managed