QMD: Waiting-line QMD: Waiting-line analysisanalysis

OverviewOverview TerminologyTerminology Characteristics of Waiting-linesCharacteristics of Waiting-lines Operating characteristicsOperating characteristics M/M/1 model – single channelM/M/1 model – single channel

• Example: video storeExample: video store M/M/S model – multi-channelM/M/S model – multi-channel

• Example: Multiplex theaterExample: Multiplex theater Psychology of waitingPsychology of waiting

‘‘The other line The other line always moves faster.’always moves faster.’

‘‘If you change lines, If you change lines, the one you left will the one you left will start to move faster start to move faster than the one you’re in.’than the one you’re in.’

© 1995 Corel Corp.

Thank you for holding. Hello...are you there?

You’ve Been There Before!You’ve Been There Before!

Population Service Facility

Waiting Line

Waiting Line SystemWaiting Line System

Situation Arrivals Servers Service Process

Waiting Line ExamplesWaiting Line Examples

QueueQueue: Waiting line: Waiting line ArrivalArrival: 1 person, machine, part, : 1 person, machine, part,

etc. that arrives and demands etc. that arrives and demands

serviceservice Queue disciplineQueue discipline: Rules for : Rules for

determining the order that arrivals determining the order that arrivals

receive servicereceive service ChannelChannel: Number of servers: Number of servers PhasePhase: Number of steps in service: Number of steps in service

Waiting Line TerminologyWaiting Line Terminology

Behavior of the arrivalsBehavior of the arrivals• join the queue, and wait join the queue, and wait

until serveduntil served• No balking; refuse to join No balking; refuse to join

the linethe line• No reneging; leave the No reneging; leave the

lineline

Population Service FacilityWaiting Line

Pattern of arrivalsrandomscheduled

Arrival Characteristics of aArrival Characteristics of a Waiting Line System Waiting Line System

Input source Service

facilityWaiting

line

Service system

© 1995 Corel Corp.

Line was too long!

BalkingBalking

RenegingReneging

Input source Service

facilityWaiting

line

Service system

© 1995 Corel Corp.

I give up!

Line Characteristics of aLine Characteristics of a Waiting Line System - continued Waiting Line System - continued

Length of the Length of the queuequeue limitedlimited unlimitedunlimited

Service priorityService priority FIFOFIFO otherother

Population Service FacilityWaiting Line

Waiting Line ArrangementsWaiting Line ArrangementsService facilities

(a) Single line

Waiting Line ArrangementsWaiting Line ArrangementsService facilities

(a) Single line Service facilities

(b) Multiple lines

Waiting Line ArrangementsWaiting Line ArrangementsService facilities

(a) Single line Service facilities

(b) Multiple lines

Number of channels single multiple

Number of phases in service system single multiple

Service Facility Characteristics of aService Facility Characteristics of a Waiting Line System - continued Waiting Line System - continued

Population Service FacilityWaiting Line

Service Facility ArrangementsService Facility Arrangements

Service Facility ArrangementsService Facility Arrangements

Service facility

(a) Single channel, single phase

ArrivalsServed units

Service facility

Queue

Service system

Dock

Waiting ship lineShips at sea

Ship unloading system Empty ships

Single-Channel, Single-Phase SystemSingle-Channel, Single-Phase System

Service Facility ArrangementsService Facility Arrangements

(b) Single channel, multiple phase

Service facility 1

Service facility 2

Cars& food

Single-Channel, Multi-Phase SystemSingle-Channel, Multi-Phase System

ArrivalsServed units

Service facility

Queue

Service system

Pick-up

Waiting carsCars in area

McDonald’s drive-through

Pay

Service facility

Service Facility ArrangementsService Facility Arrangements

(c) Multiple channel, single phase

Service facility 1

Service facility 2

Arrivals

Served units

Service facilityQueue

Service system

Service facility

Example: Bank customers wait in single line for one of several tellers.

Multi-Channel, Single Phase SystemMulti-Channel, Single Phase System

Decision AreasDecision Areas

Arrival rates Number of service

channels Number of phases Service time Priority rule Line arrangement

Average queue length, Average queue length, LLqq

Average number in system, Average number in system, LLss

Average queue time, Average queue time, WWqq

Average time in system, Average time in system, WWss

System utilization, System utilization,

Performance MeasuresPerformance Measuresoror

Operating CharacteristicsOperating Characteristics

Simple (M/M/1)Simple (M/M/1)•Example: Information booth at mallExample: Information booth at mall

Multi-channel (M/M/S)Multi-channel (M/M/S)•Example: Airline ticket counterExample: Airline ticket counter

Types of Queuing ModelsTypes of Queuing Models

Constant Service (M/D/1)Example: Automated car wash

Limited PopulationExample: Department with only 7 drills

Type:Type: Single-channel, single- Single-channel, single-phase systemphase system

QueueQueue: Unlimited; single line: Unlimited; single line Queue disciplineQueue discipline: FIFO (FCFS): FIFO (FCFS) Service rate > arrival rateService rate > arrival rate

Simple (M/M/1) Model Simple (M/M/1) Model CharacteristicsCharacteristics

Queuing ModelsQueuing Models

- ‘lambda’- ‘lambda’• arrival arrival raterate of the customers of the customers

- ‘mu’- ‘mu’• service service raterate for a single server for a single server

• If the service time is 15 minutes per If the service time is 15 minutes per customer, what is the service rate??customer, what is the service rate??

Simple (M/M/1) Model EquationsSimple (M/M/1) Model Equations

sLAverage number of customers in the system

2

qLAverage number of customers in the queue

Simple (M/M/1) Model EquationsSimple (M/M/1) Model Equations

1sW

qW

Average time customers are in the system

Average time customers are in the queue

Simple (M/M/1) Model EquationsSimple (M/M/1) Model Equations

Utilization factor – percent of time that the single server is busy; also the probability that a customer will wait

Example: Video StoreExample: Video Store

The manager of a video store is The manager of a video store is interested in providing good service. interested in providing good service. On a Friday or Saturday night, on On a Friday or Saturday night, on average 30 customers per hour average 30 customers per hour arrive at the counter to check out a arrive at the counter to check out a video. The customers are served at video. The customers are served at an average rate of 35 customers per an average rate of 35 customers per hour from a single cash register.hour from a single cash register.

Example: Video StoreExample: Video Store

Determine the operating Determine the operating characteristics for the video store.characteristics for the video store.• Average number of customers in lineAverage number of customers in line• Average number of customers in the Average number of customers in the

systemsystem• Average wait time in lineAverage wait time in line• Average time in the systemAverage time in the system• Server utilizationServer utilization

Average number of customers in lineAverage number of customers in line

2

qL

Average number of customers in the systemAverage number of customers in the system

sL

Average wait time in lineAverage wait time in line

qW

Average time in the systemAverage time in the system

1sW

Server utilizationServer utilization

Example: Video StoreExample: Video Store

• What service rate would be What service rate would be required to have customers required to have customers average only 5 minutes in the average only 5 minutes in the system?system?

What service rate would be required to have What service rate would be required to have customers average only 5 minutes in the customers average only 5 minutes in the

system?system?

1sW

GO TO EXCEL OM

Type:Type: Multichannel system Multichannel system Queue disciplineQueue discipline: FIFO : FIFO

(FCFS)(FCFS) Service rates > arrival Service rates > arrival

raterate

Multichannel (M/M/S) Model Multichannel (M/M/S) Model CharacteristicsCharacteristics

Multichannel M/M/S EquationsMultichannel M/M/S Equations

Probability of zero customers in the system:

Average number of customers in the system:

S

S

Sn

PSnS

n !

1

!

1

1

1

0

0

02!1P

SSL

S

s

Average number of customers waiting for service:

sq LL

Multichannel M/M/S EquationsMultichannel M/M/S Equations

Average time a person or unit spends in the queue

Average time a unit spends in the system:

s

S

s

LP

SSW

1

!102

q

sq

LWW

1

Example: Movie TheaterExample: Movie Theater

A multiplex movie theater has 3 A multiplex movie theater has 3 concession clerks serving customers on a concession clerks serving customers on a first come, first served basis. The service first come, first served basis. The service time per customer is exponentially time per customer is exponentially distributed with an average of 2 minutes distributed with an average of 2 minutes per customer. Concession customers per customer. Concession customers arrive at a rate of 81 customers per hour. arrive at a rate of 81 customers per hour. 10 minutes of previews run in the lobby. If 10 minutes of previews run in the lobby. If the average time in the concession area the average time in the concession area exceed 10 minutes, customers become exceed 10 minutes, customers become dissatisfied.dissatisfied.

Example: Movie theaterExample: Movie theater

What is the average number of customers in the concession area?

What is the average number of customers in the What is the average number of customers in the concession area?concession area?

GO TO EXCEL OM

Multiplex Movie TheaterMultiplex Movie Theater

The distribution is reducing the The distribution is reducing the length of the previews to 8 minutes. length of the previews to 8 minutes. How many servers will be required to How many servers will be required to so that the average wait is no longer so that the average wait is no longer than 8 minutes?than 8 minutes?

Total Cost per HoursTotal Cost per Hours Total cost per Hour= waiting cost + Total cost per Hour= waiting cost +

service costservice cost

TC = CTC = CwwLLss + KC + KCss

• CCww – cost of waiting per hour per customer – cost of waiting per hour per customer

• LLss – number of customers in the system – number of customers in the system

• K – number of serversK – number of servers

• CCss- service cost per hour per server- service cost per hour per server

Total Cost: Video StoreTotal Cost: Video Store Considering adding another cashier for Considering adding another cashier for

Friday and Saturday nights.Friday and Saturday nights.

Part-time cashier will cost $6.00 per Part-time cashier will cost $6.00 per

hourhour

Customers waiting time (cost of Customers waiting time (cost of

goodwill, etc) is $10.00 per hour.goodwill, etc) is $10.00 per hour.

Is it cost effective to hire the cashier?Is it cost effective to hire the cashier?

Total Cost: Video Store M/M/1Total Cost: Video Store M/M/1TC = CTC = CwwLLss + KC + KC

CCw w = $10/hour= $10/hour

LLss = 6 = 6

K = 1K = 1

CCS S = $6.00/hr = $6.00/hr

TC TC = 10(6) + = 10(6) + 1(6) 1(6)

= = $66/hour$66/hour

Total Cost: Video Store M/M/2Total Cost: Video Store M/M/2TC = CTC = CwwL + KCL + KC

CCw w = $10/hour= $10/hour

LLss = 1.05 = 1.05

K = 2K = 2

CCs s = $6.00/hr = $6.00/hr

TC TC = 10(1.05) = 10(1.05) + 2(6)+ 2(6)

= = $22.50/hr$22.50/hr

= Mean number of = Mean number of arrivals per time arrivals per time periodperiod• e.g., 3 units/houre.g., 3 units/hour

= Mean number of = Mean number of people or items people or items served per time periodserved per time period• e.g., 4 units/houre.g., 4 units/hour

1/1/ = 15 = 15 minutes/unitminutes/unit

Remember: Remember: & & Are Rates Are Rates

© 1984-1994 T/Maker Co.

If average service time is 15 minutes, then μ is 4 customers/hour

The Psychology of WaitingThe Psychology of Waiting

Maister’s (1985) Propositions of Maister’s (1985) Propositions of customer satisfaction in queuescustomer satisfaction in queues• Unoccupied time feels longer than Unoccupied time feels longer than

occupied timeoccupied time

• Pre-process wait feels longer than Pre-process wait feels longer than in-process waits.in-process waits.

• Anxiety makes waits feel longerAnxiety makes waits feel longer

The Psychology of WaitingThe Psychology of Waiting

• Uncertain waits seem longer than Uncertain waits seem longer than

certain waitscertain waits

• Unexplained waits seem longer Unexplained waits seem longer

than explained waitsthan explained waits

• Unfair waits seem longer than Unfair waits seem longer than

equitable waitsequitable waits

Psychology of WaitingPsychology of Waiting

•More valuable the service, the More valuable the service, the

longer people will wait.longer people will wait.

•Solo waiting feels longer than Solo waiting feels longer than

group waitinggroup waiting..