OMG 402 - Operations ManagementSpring 1997

CLASS 4:

THE IMPACT OF VARIABILITY

Harry Groenevelt

March 1997 2

Agenda

• The Puff Line• Sources of Variability• Introduction to Queues• The Physics of Queues

– Impact of high utilization– Economies of scale– Quantifying the impact of variability

• Summary of Insights

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Sources of Variability

In any process there is variability in: demand from external or internal customers

processing time within the system

The impact of variability can be especially severe in service systems, which cannot build inventory to prepare for the ‘peak’

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Sources of Demand Variability: Examples

• IRS service center: seasonal variability• Mutual fund service center: early evening peak• Variability created by other processes within the

firm:– production batches (Shot-Peening, ...)

– transfer batches (e.g., filling a cart, truck or boat)

– surges at airline ‘hubs’

– others….

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Servers (s)

systemqueue

departuresarrivals ( customers/hour)

customers/hr./server

Introduction to Queues: Notation

• arrival rate () in customers/hour

• service rate () in customers/hour(avg. time for one customer = ____)

• # servers = s;

• Therefore, capacity = ____

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Introduction to Queues: Examples

service customer queue ‘service’facility arrives.. location process

health clinic front waiting treatmentdesk room

on-call computerconsultant

AOL‘modem farm’

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average number in queueaverage number in system

wait in queue

systemqueue

wait in system

prob(waiting time > t)

Introduction to Queues: Performance Measures

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Performance Measures for QueuesThroughput = rate of customers served

Utilization () = throughput / capacity = proportion of all server time spent working = avg. number being served / number of servers

Load Factor = arrival rate / capacity

For the system on page 5,Throughput = ______ utilization = load factor = _____

But … when is throughput arrival rate, utilization load factor?

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‘memoryless’ arrivals ‘memoryless’ servers one server

Introduction to Queues:Specifying Variability• Must specify variability of arrivals and service times.• One example: the M/M/1 queue

– Time between arrivals exponentially distributed (‘Poisson’ arrival process)

– Service times exponentially distributed– All customers served in order of arrival– Arrival and service rates constant

(stationary system)

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Avg. time between arrivals = 1/ = 0.2 hours = 12 minutes

Avg. service time = 1/ = 10 minutes

Physics of Queues: Example‘Rapid Oil Change’:

– one service bay– Poisson arrivals with rate 5 cars/hour ()– Exponential service times, mean = 10 min. (1/ )

0

0.02

0.04

0.06

0.08

0.1

0 10 20 30 40 50 60

time between arrivals (minutes)

Pro

babi

lity

Den

sity

0

0.02

0.04

0.06

0.08

0.1

0 5 10 15 20 25 30

time for one oil change (minutes)

Pro

babi

lity

Den

sity

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utilization 0 1

avg.numberin system

Physics of Queues: Impact of High Utilization For an M/M/1 system

average time in system = 1/(–) = 1/(6 cars/hour – 5 cars/hour) = 1 hour

load factor = / = = 5/6

average number in system = /(–) = /(1–)= (5/6) / (1–5/6) = 5 cars.

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Physics of Queues: Utilization

As utilization approaches 1, average time in system, wait in queue, number in system and number in queue all rise dramatically.

This effect seen in any system, including:– M/M/s (multiple servers with ‘snake’ line)– G/G/s (multiple servers with ‘snake’ line,

General arrival or service distribution)

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Physics of Queues: Economies of Scale• M/M/s Example: Rapid Oil Change

Same as in first example except:– 2 service bays (twice as many)– Poisson arrivals with rate 10 per hour (twice as high)

What is this system’s utilization?

• Equations for M/M/s are not as simple, so we implement them in Excel...

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M/M/s

Parameters

Arrival Rate (1/hr) 5 10Service Rate (1/hr) 6 6

Nr of Servers 1 2

Results

Load Factor 0.833333333 0.833333333Fraction Not Served 0 0

Thruput (1/hr) 5 10Utilization 0.833333333 0.833333333

Remaining Results for All Customers All CustomersAvg Nr in System 5 5.454545455Avg Nr in Queue 4.166666667 3.787878788

Time in System (hr) 1 0.545454545Wait in Queue (hr) 0.833333333 0.378787879

Pr{Wait=0} 0.166666667 0.242424242t (hr) Pr{Wait<=t} Pr{Wait<=t}

0.083333333 0.233296321 0.358725966

M/M/s example results (from QMACROS):single bay

two bays

Physics of Queues

Note the difference between 1 and 2

bay systems!

March 1997 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10 11 12 13Number of Servers

Pr{

Wai

t >

5 m

inut

es}

Pr{Wait > 5 minutes}(see left scale)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(Avg

) W

ait

in Q

ueue

(hr

s)

(Avg) Wait in Queue (hrs)

(see right scale)

Physics of Queues• M/M/s example: Economies of Scale

– Vary number of servers and raise arrival rate proportionally (load factor always 5/6 = 0.8333)

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The Physics of Queues: Impact of Variability

G/G/s model (same as M/M/s model, except):– General service time distribution

– General inter-arrival time distribution

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Impact of variability

• G/G/s model– 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))

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Impact of variability

• Reminder: if X is a random variable with mean and standard deviation , then its Coefficient of Variation

= cv(X) = / • For exponential random variables:

– Coefficient of Variation = 1

• For deterministic random variables:– Coefficient of Variation = ________

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• An approximation good for ‘congested’ systems:

• What happens as arrivals become more ‘lumpy’?

• As service times become more variable?

• Similar effect for G/G/s (try in QMACROS)

Impact of variability: G/G/1

1

12)()( 22

queuein wait average ScvAcv

March 1997 20

Summary of Insights

• High utilization causes congestion, high WIP and long lead times

• Variability causes congestion, high WIP and long lead times

• At the same utilization, a larger system will perform better than a smaller system

- or -smaller systems must have lower utilization to perform as well as larger systems