OMG 402 - Operations ManagementSpring 1997

CLASS 6:

Process Design and Performance Measurement

Harry Groenevelt

March 1997 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

March 1997 3

Servers (s)

systemqueue

departuresarrivals ( 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!

March 1997 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

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

March 1997 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’?

March 1997 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 =

March 1997 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))

March 1997 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

March 1997 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

March 1997 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 packetEven with seemingly plenty of booths we observe long queues. Why?

Impact of Variability: An ExampleCheck-in for an Operations Management Convention in Morocco

Original Physical Arrangement:

March 1997 12

arrivals step off of tour buses from the hotel

Impact of Variability: An ExampleCheck-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?

March 1997 13What systems can be modeled this way?

Limited Space SystemsM/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

CustomerArrival

Departures

Departures

Leave Without Service

March 1997 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?)

March 1997 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.

March 1997 16(see: Frontiers of Electronic Commerce by Profs. Kalakota and Whinston)

Limited Space SystemsExample: Local Internet Service Provider (ISP)

N trunk lines

(all customers ‘arrive’ by same-number dialup)

national ISP andInternet Backbone

calls may queuefor a modem here

SwitchTerminal

serverRouter

Modem Farm

modem 1

modem 2

modem 3

modem s

March 1997 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

March 1997 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?

March 1997 19(28 modems, 35 trunk lines, average session length: 20 minutes)

Performance measures trade-offNow hold the number of trunk lines constant and

increase arrival rate:

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10 11 12

Arrival Rate (1/min)

Ave

rag

e W

ait

to L

og

On

(m

inu

tes)

0%

20%

40%

60%

80%

100%

Pe

rce

nt

Re

ceiv

ing

Bu

sy S

ign

als

Fraction Busy Signals(see scale on the right)

Avg Wait to Log On(see scale on the left)

March 1997 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!

March 1997 21

Subscriptions to AOL, 1994-1997

012345678

Jun-94 Dec-94 Jun-95 Dec-95 Jun-96 Dec-96

Nu

mb

er o

f S

ub

scri

ber

s (m

illio

ns)

January, 1997

December, 1996(flat-rate accessintroduced)

source: Jupiter Communications and the Los Angeles Times

Capacity Strategy: America Online

March 1997 22

Capacity Strategy: Expansionist

012345678

Jun-94 Dec-94 Jun-95 Dec-95 Jun-96 Dec-96

Nu

mb

er o

f S

ub

scri

ber

s (m

illio

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Modem Capacity

Nr of Subscribers

March 1997 23

012345678

Jun-94 Dec-94 Jun-95 Dec-95 Jun-96 Dec-96

Nu

mb

er o

f S

ub

scri

ber

s (m

illio

ns)

Capacity Strategy: Wait-and-See

Modem Capacity

Nr of Subscribers

March 1997 24

Capacity Strategy

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

March 1997 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

March 1997 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

March 1997 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