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OMG 402 - Operations Management Spring 1997

Understand queuing relationships, modeling distributed queues, impact of variability, and performance measurement trade-offs. Learn about capacity strategy, queuing management, and Limited Space systems. Explore examples and calculations using QMacros.

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OMG 402 - Operations Management Spring 1997

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

  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. Recap: Basic Queuing Relationships m customers/hr/server Servers (s) 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! departures arrivals (l customers/hr) queue system

  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/(m–l) • Average number in system = l /(m-l) = r/(1–r)

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

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

  11. Impact of Variability: An Example Check-in for an Operations Management Convention in Morocco Original Physical Arrangement: check-in booths arrivals step off of tour buses from the hotel queue in lobby of convention hall • 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? departures

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

  13. 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 Leave Without Service Server 1 Departures N in System? Queue CustomerArrival Server s Departures What systems can be modeled this way?

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

  17. N trunk lines N–s lines lines logged onto s modems (virtual queue) 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

  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. Performance measures trade-off Now hold the number of trunk lines constant and increase arrival rate: 5 100% 4 80% Avg Wait to Log On 3 60% (see scale on the left) Percent Receiving Busy Signals Average Wait to Log On (minutes) 2 40% Fraction Busy Signals (see scale on the right) 1 20% 0 0% 0 1 2 3 4 5 6 7 8 9 10 11 12 Arrival Rate (1/min) (28 modems, 35 trunk lines, average session length: 20 minutes)

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

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

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

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

  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. 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. 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

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