ISM 270

ISM 270

Service Engineering and Management

Lecture 7: Forecasting and Managing Service Capacity

Announcements Project Proposal Due today Homework 4 due next week $15 check for ‘Responsive Learning Technologies’ Final four weeks:

Capacity Planning Outsourcing Capacity Management Game Project Presentations

Today

Capacity Management Queueing Models Introduction to R

Managing Waiting Lines – Queueing Models

Essential Features of Queuing Systems

DepartureQueue

discipline

Arrival process

Queueconfiguration

Serviceprocess

Renege

Balk

Callingpopulation

No futureneed for service

Arrival Process

Static Dynamic

AppointmentsPriceAccept/Reject BalkingReneging

Randomarrivals withconstant rate

Random arrivalrate varying

with time

Facility-controlled

Customer-exercised

control

Arrival process

Distribution of Patient Interarrival Times

1 2 3 4 5 6 7 8 9 10

11

12

13

0

10

20

30

40

Patient interarrival time, minutes

Rel

ativ

e fr

equ

ency

, %

Temporal Variation in Arrival Rates

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

0

0.5

1

1.5

2

2.5

3

3.5

Hour of day

Ave

rag

e ca

lls

per

ho

ur

1 2 3 4 560

70

80

90

100

110

120

130

140

Day of week

Per

cen

tag

e o

f av

erag

e d

aily

ph

ysic

ian

vis

its

Poisson and Exponential Equivalence

Poisson distribution for number of arrivals per hour (top view)

One-hour

1 2 0 1 interval

Arrival Arrivals Arrivals Arrival

62 min.40 min.

123 min.

Exponential distribution of time between arrivals in minutes (bottom view)

Queue Configurations

Multiple Queue Single queue

Take a Number Enter

3 4

8

2

6 10

1211

5

79

Queue Discipline

Queuediscipline

Static(FCFS rule)

Dynamic

selectionbased on status

of queue

Selection basedon individual

customerattributes

Number of customers

waitingRound robin Priority Preemptive

Processing timeof customers

(SPT rule)

Queuing Formulas

Single Server Model with Poisson Arrival and Service Rates: M/M/1

1. Mean arrival rate:2. Mean service rate:3. Mean number in service:4. Probability of exactly “n” customers in the system:5. Probability of “k” or more customers in the system:6. Mean number of customers in the system:

7. Mean number of customers in queue:

8. Mean time in system:

9. Mean time in queue:

Pn

n ( )1

P n k k( )

sL

qL

1sW

qW

Queuing Formulas (cont.)

Single Server General Service Distribution Model: M/G/1

Mean number of customers in queue for two servers: M/M/2

Relationships among system characteristics (Little’s Law for ALL queues):

)1(2

222

qL

2

3

4

qL

qq

ss

qs

qs

LW

LW

WW

LL

1

1

1

Congestion as 10.

0 1.0

100

10

8

6

4

2 0

With:

Ls 1Then:

Ls

0 00.2 0.250.5 10.8 40.9 90.99 99

Single Server General Service Distribution Model : M/G/1

)1(2

222

qL

1. For Exponential Distribution:

22

1

)1()1(2

2

)1(2

/ 22222

qL

2. For Constant Service Time: 2 0

)1(2

2

qL

3. Conclusion:

Congestion measured by Lq is accounted for equally by variability in arrivals and service times.

Queuing System Cost Tradeoff

Let: Cw = Cost of one customer waiting in queue for an hour

Cs = Hourly cost per serverC = Number of servers

Total Cost/hour = Hourly Service Cost + Hourly Customer Waiting Cost

Total Cost/hour = Cs C + Cw Lq

Note: Only consider systems where

C

General Queuing Observations

1. Variability in arrivals and service times contribute equally to congestion as measured by Lq.

2. Service capacity must exceed demand.

3. Servers must be idle some of the time.

4. Single queue preferred to multiple queue unless jockeying is permitted.

5. Large single server (team) preferred to multiple-servers if minimizing mean time in system, WS.

6. Multiple-servers preferred to single large server (team) if minimizing mean time in queue, WQ.

Managing Capacity and Demand

Segmenting Demand at a Health Clinic

60

70

80

90

100

110

120

130

140

1 2 3 4 5

Day of week

Perc

enta

ge o

f ave

rage

dai

ly

phys

icia

n vi

sits

Smoothing Demand by AppointmentScheduling

Day Appointments

Monday 84Tuesday 89Wednesday 124Thursday 129Friday 114

Hotel Overbooking Loss Table

Number of Reservations Overbooked

No- Prob-

shows ability 0 1 2 3 4 5 6 7 8 9

0 .07 0 100 200 300 400 500 600 700 800 900

1 .19 40 0 100 200 300 400 500 600 700 800

2 .22 80 40 0 100 200 300 400 500 600 700

3 .16 120 80 40 0 100 200 300 400 500 600

4 .12 160 120 80 40 0 100 200 300 400 500

5 .10 200 160 120 80 40 0 100 200 300 400

6 .07 240 200 160 120 80 40 0 100 200 300

7 .04 280 240 200 160 120 80 40 0 100 200

8 .02 320 280 240 200 160 120 80 40 0 100

9 .01 360 320 280 240 200 160 120 80 40 0

Expected loss, $ 121.60 91.40 87.80 115.00 164.60 231.00 311.40 401.60 497.40 560.00

Daily Scheduling of Telephone Operator Workshifts

0

5

10

15

20

25

30

Time

Nu

mb

er o

f o

per

ato

rs

Scheduler program assigns tours so that the number of operators present each half hour adds up to the number required

Topline profile

12 2 4 6 8 10 12 2 4 6 8 10 120

500

1000

1500

2000

2500

Time

Cal

ls

12 2 4 6 8 10 12 2 4 6 8 10 12

LP Model for Weekly Workshift Schedule with Two Days-off Constraint

Objective function: Minimize x1 + x2 + x3 + x4 + x5 + x6 + x7

Constraints: Sunday x2 + x3 + x4 + x5 + x6

3 Monday x3 + x4 + x5 + x6 + x7 6

Tuesday x1 + x4 + x5 + x6 + x7 5

Wednesday x1 + x2 + x5 + x6 + x7 6 Thursday x1 + x2 + x3 + x6 + x7 5 Friday x1 + x2 + x3 + x4 + x7

5 Saturday x1 + x2 + x3 + x4 + x5 5

xi 0 and integer

Schedule matrix, x = day offOperator Su M Tu W Th F Sa 1 x x … … … … ... 2 … x x … … … … 3 … ... x x … … … 4 … ... x x … … … 5 … … … … x x … 6 … … … … x x … 7 … … … … x x … 8 x … … … … … xTotal 6 6 5 6 5 5 7Required 3 6 5 6 5 5 5Excess 3 0 0 0 0 0 2

Seasonal Allocation of Rooms by Service Class for Resort Hotel

First class

Standard

Budget

Per

cent

age

of c

apac

ity a

lloca

ted

to d

iffer

ent s

ervi

ce c

lass

es

60%

50%30%

20%

50%

Peak Shoulder Off-peak Shoulder (30%) (20%) (40%) (10%)Summer Fall Winter Spring

Percentage of capacity allocated to different seasons

30%20% 20%

10% 30%

50% 30%

Demand Control Chart for a Hotel

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0

50

100

150

200

250

300

350

Days before arrival

Res

erva

tio

ns

Expected Reservation Accumulation

2 standard deviation control limits

Yield Management Using the Critical Fractile Model

P d x

C

C C

F D

p Fu

u o

( )( )

Where x = seats reserved for full-fare passengers d = demand for full-fare tickets p = proportion of economizing (discount) passengers Cu = lost revenue associated with reserving one too few seatsat full fare (underestimating demand). The lost opportunity is the difference between the fares (F-D) assuming a passenger, willingto pay full-fare (F), purchased a seat at the discount (D) price. Co = cost of reserving one to many seats for sale at full-fare(overestimating demand). Assume the empty full-fare seat wouldhave been sold at the discount price. However, Co takes on twovalues, depending on the buying behavior of the passenger whowould have purchased the seat if not reserved for full-fare. if an economizing passenger if a full fare passenger (marginal gain)Expected value of Co = pD-(1-p)(F-D) = pF - (F-D)

CD

F Do

( )

Statistical Analysis in R

Homework 4 is designed to introduce you to analysis using R