D-1 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Operations Management Waiting-Line Models Module D

D-1 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Operations Operations ManagementManagement

Waiting-Line ModelsWaiting-Line ModelsModule DModule D


You manage a call center which can answer an average of 20 calls per hour. Your call center gets 17.5 calls in an average hour. On average, what is the time a customer spends on hold waiting for service?

a) on average, customers should not have to wait on hold since capacity is greater than demand.

b) average customer wait will be less than 10 minutes. c) average customer wait will be between 10 and 20 minutes. d) average customer wait will be greater than 20 minutes. e) who knows? There’s no way to tell.

Queues / LinesQueues / Lines

© 1995 Corel Corp.


Two ways to address waiting linesTwo ways to address waiting lines

Queuing theory Certain types of lines can be described mathematically Requires that assumptions are valid for your situation Systems with multiple lines that feed each other are

too complex for queuing theory Simulation

Building mathematical models that attempt to act like real operating systems

Real-world situations can be studied without imposing on the actual system


Why do we have to wait?Why do we have to wait?

© 1995 Corel Corp.

Why do services (and most non-MTS manufacturers) have queues? Processing time and/or arrival time variance Costs of capacity – can we afford to always have more

people/servers than customers? Efficiency – e.g. scheduling at the Doctor’s office


Waiting Costs and Service CostsWaiting Costs and Service Costs

Total expected cost

Cost of waiting time

Cost

Low level of service

Optimal service level

High levelof service

Minimum total cost

Cost of providing service


Costs of QueuesCosts of Queues

Too Little Queue Too Much Queue

Cost of capacity

Wasted capacity

Annoyed customers

Lost customers

Space

Possible opportunity:e.g. wait in the bar for a restaurant table


Bank Customers Teller Deposit, etc.

Doctor’s Patient Doctor Treatmentoffice

Traffic Cars Light Controlledintersection passage

Assembly line Parts Workers Assembly

1–800 software User call-ins Tech support Technical supportsupport personnel

Situation Arrivals Servers Service Process

Waiting Line ExamplesWaiting Line Examples


Service FacilityPopulation

Pattern of arrivals Scheduled Random – estimated by Poisson distribution

Arrival Characteristics

Characteristics of a Waiting Line SystemCharacteristics of a Waiting Line System

Size of the source population Limited Unlimited

Behavior of arrivals Join the queue, wait until served Balk – refuse to join the line Renege – leave the line

Waiting Line

= average arrival rate


Poisson Distributions for Arrival RatesPoisson Distributions for Arrival Rates

=2 =4

0.00

0.05

0.10

0.15

0.20

0.25

0.30

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

Prob

abilit

y

Prob

abilit

y

0.00

0.05

0.10

0.15

0.20

0.25

0.30

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

Prob

abilit

y

= average arrival rate


Service Facility

Waiting LinePopulation

Waiting Line CharacteristicsLength of the queue

Limited Unlimited

Queue discipline FIFO Other



Service FacilityWaiting LinePopulation

Service Characteristics Number of channels

Single Multiple

Number of phases in service system

Single Multiple

Service time distribution Constant Random – estimated by negative exponential distribution


= average service rate


Negative Exponential DistributionNegative Exponential Distribution

Average Service Rate () = 3 customers per hourAverage Service Time = 20 minutes per customer

Average Service Rate () = 1 customer per hour

Probability that Service Time is greater than t=e-t, for t > 0

Time t in Hours

Prob

abili

ty th

at S

ervi

ce T

ime

t

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

Average Service Rate () = 3 customers per hourAverage Service Time = 20 minutes per customer

Average Service Rate () = 1 customer per hour

Probability that Service Time is greater than t = e for t > 0–t


Basic Queuing System DesignsBasic Queuing System Designs

ArrivalsServed unitsService

facility

Queue

Single-Channel, Single-PhaseSingle-Channel, Single-Phase

Service facility

Arrivals

Served units

Service facilityQueue

Service facility

Service facility

Multi-Channel, Multi-PhaseMulti-Channel, Multi-Phase

Arrivals

Served unitsService

facilityQueue

Service facility

Multi-Channel, Single-PhaseMulti-Channel, Single-Phase

e.g. U.S. Post Office

e.g. drive-through bank

e.g. Suds & Suds Laundromat

ArrivalsServed units

Service facility

QueueService facility

Single-Channel, Multi-PhaseSingle-Channel, Multi-Phase

e.g. McDonald’s drive-through



Call Center – Solution (1)Call Center – Solution (1)

= average arrival rate = 17.5 calls/hr

= average service rate = 20 calls/hr

ρ = average utilization of system = / = 87.5%




= average arrival rate = 17.5 calls/hr = average service rate = 20 calls/hrρ = average utilization of system = / = 87.5%

L = average number of customers in service

system (line and being served) = / ( - ) = 7 calls Lq = average number waiting in line = ρL = 6.125 calls


You manage a call center which can answer an average of 20 calls per hour. Your call center gets 17.5 calls in an average hour. On average,what is the time a customer spends on hold waiting for service?


= average arrival rate = 17.5 calls/hr

= average service rate = 20 calls/hr

ρ = average utilization of system = / = 87.5%

W = average time in system

(wait and service) = 1 / ( - ) = .4 hr = 24 min

Wq = average time waiting = ρW = 21 min


Single server equationsSingle server equations = average arrival rate

= average service rate

ρ = average utilization of system = / Pn = probability that n customers are in the system = (1- ρ) ρn

L = average number of customers in service system (line and being served) = / ( - )

Lq = average number waiting in line = ρL

W = average time in system (wait and service) = 1/ ( - )

Wq = average time waiting = ρW


You are opening an ice cream stand that has a single employee (you). You expect to see about 25 customers an hour. It takes you an average of 2 minutes to serve each customer. Customers are served in a FCFS manner.

Your research suggests that if there is a line of more than 4 people that some customers will leave without buying anything. In addition, if customers have to wait more than 6 minutes to get their order filled they are not likely to come back.

How well will this system do at satisfying customers? What assumptions are you making to answer this question?

Izzy’s Ice Cream StandIzzy’s Ice Cream Stand


Izzy’s Ice Cream Stand (2)Izzy’s Ice Cream Stand (2) = 25 customers/hr = 30 customers/hr ρ = / = .833

Pn = probability that n customers are in the system = (1- ρ) ρn

P0 = (1 – .833) x .833 = .167 .167 P1 = (1 – .833) x .833 = .139 .306 P2 = (1 – .833) x .833 = .116 .422 P3 = (1 – .833) x .833 = .097 .519 P4 = (1 – .833) x .833 = .080 .599

Pmore than 4 = 1 – .599 = .401

1

2

3

4

0cumulative

L = / ( - ) = 5 customers

Customers in the system


Izzy’s Ice Cream Stand (3)Izzy’s Ice Cream Stand (3) = 25 customers/hr = 30 customers/hr ρ = / = .833

L = / ( - ) = 25 / (30 – 25) = 5 customersLq = ρL = .833 x 5 = 4.17 customers

W = 1 / ( - ) = 1 / (30 – 25) = .2 hr = 12 minWq = ρW = .833 x 12 min = 10 min

Why not L – 1 ?


Why do we have to wait?Why do we have to wait?

© 1995 Corel Corp.

Why do services (and most non-MTS manufacturers) have queues? Processing time and/or arrival time variance Costs of capacity – can we afford to always have more

people/servers than customers? Efficiency – e.g. scheduling at the Doctor’s office


x

Time t

Prob

abili

ty th

at

Serv

ice

Tim

e

t

Probability

ARRIVALS(Poisson)

SERVICE(Exponential)

CUSTOMERS IN THE

SYSTEM

Prob

abili

ty

n

Documents

D-1 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Operations Management Waiting-Line Models Module D