OPSM 301: Operations Management

Session 19:

Flow variability

Koç University

Zeynep Aksinzaksin@ku.edu.tr

Announcements

Midterm 2-December 14 at 18:30 CAS Z48, CAS Z08– Does not include Midterm 1 topics– LP, Inventory, Variability (Congestion+Quality)– LP: from course pack– Inventory Ch6 excluding 6.7, Ch 7.1, 7.2, 7.3– Chapter 8 excluding 8.6 and 8.8 (this week)– Chapter 9 (next week)

Components of the Queuing System Visually

Customers Customers come income in

Customers are Customers are servedserved

Customers Customers leaveleave

Flow Times with Arrival Every 4 Secs(Service time=5 seconds)

Customer Number

Arrival Time

Departure Time

Time in Process

1 0 5 5

2 4 10 6

3 8 15 7

4 12 20 8

5 16 25 9

6 20 30 10

7 24 35 11

8 28 40 12

9 32 45 13

10 36 50 14

0 10 20 30 40 50

Time

1

2

3

4

5

6

7

8

9

10

Cust

omer

Num

ber

What is the queue size? Can we apply Little’s Law?What is the capacity utilization?

Customer Number

Arrival Time

Departure Time

Time in Process

1 0 5 5

2 6 11 5

3 12 17 5

4 18 23 5

5 24 29 5

6 30 35 5

7 36 41 5

8 42 47 5

9 48 53 5

10 54 59 5

0 10 20 30 40 50 60

Time

1

2

3

4

5

6

7

8

9

10

Cust

omer

Num

ber

Flow Times with Arrival Every 6 Secs (Service time=5 seconds)

What is the queue size?What is the capacity utilization?

Customer Number

Arrival Time

Processing Time

Time in Process

1 0 7 7

2 10 1 1

3 20 7 7

4 22 2 7

5 32 8 8

6 33 7 14

7 36 4 15

8 43 8 16

9 52 5 12

10 54 1 11

0 10 20 30 40 50 60 70

Time

1

2

3

4

5

6

7

8

9

10

Cu

sto

mer

Queue Fluctuation

0

1

2

3

4

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

Time

Nu

mb

er

Effect of Variability

What is the queue size?What is the capacity utilization?

Customer Number

Arrival Time

Processing Time

Time in Process

1 0 8 8

2 10 8 8

3 20 2 2

4 22 7 7

5 32 1 1

6 33 1 1

7 36 7 7

8 43 7 7

9 52 4 4

10 54 5 7 0 10 20 30 40 50 60 70

1

2

3

4

5

6

7

8

9

10

Effect of Synchronization

What is the queue size?What is the capacity utilization?

Conclusion

If inter-arrival and processing times are constant, queues will build up if and only if the arrival rate is greater than the processing rate

If there is (unsynchronized) variability in inter-arrival and/or processing times, queues will build up even if the average arrival rate is less than the average processing rate

If variability in interarrival and processing times can be synchronized (correlated), queues and waiting times will be reduced

To address the “how much does variability hurt” question: Consider service processes

This could be a call center or a restaurant or a ticket counter

Customers or customer jobs arrive to the process; their arrival times are not known in advance

Customers are processed. Processing rates have some variability.

The combined variability results in queues and waiting. We need to build some safety capacity in order to reduce

waiting due to variability

Why is there waiting?

the perpetual queue: insufficient capacity-add capacity

the predictable queue: peaks and rush-hours-synchronize/schedule if possible

the stochastic queue: whenever customers come faster than they are served-reduce variability

A measure of variability

Needs to be unitless Only variance is not enough Use the coefficient of variation C or CV= /

Interpreting the variability measures

Ci = coefficient of variation of interarrival times

i) constant or deterministic arrivals Ci = 0

ii) completely random or independent arrivals Ci =1

iii) scheduled or negatively correlated arrivals Ci < 1

iv) bursty or positively correlated arrivals Ci > 1

Specifications of a Service Provider

ServiceProvider

Leaving Customers

Waiting Customers

Demand Pattern

Resources

• Human resources

• Information system

• other...

Arriving Customers

Satisfaction Measures

Reneges or abandonments

Waiting Pattern

Served Customers

Service Time

Distribution of Arrivals

Arrival rate: the number of units arriving per period– Constant arrival distribution: periodic, with exactly

the same time between successive arrivals– Variable (random) arrival distributions: arrival

probabilities described statistically• Exponential distribution for interarrivals

• Poisson distribution for number arriving

• CV=1

Service Time Distribution

Constant– Service is provided by automation

Variable– Service provided by humans– Can be described using exponential distribution CV=1

or other statistical distributions

The Service Process

Customer Inflow (Arrival) Rate (Ri) ()– Inter-arrival Time = 1 / Ri

Processing Time Tp (unit load)– Processing Rate per Server = 1/ Tp (µ)

Number of Servers (c)– Number of customers that can be processed simultaneously

Total Processing Rate (Capacity) = Rp= c / Tp (cµ)

Operational Performance Measures

Flow time T= Tw + Tp (waiting+process)

Inventory I = Iw + Ip

Flow Rate R = Min (Ri, RpStable Process = Ri < Rp,, so that R = Ri

Little’s Law: I = R T, Iw = R Tw, Ip = R Tp

Capacity Utilization = Ri / Rp < 1

Safety Capacity = Rp – Ri

Number of Busy Servers = Ip= c = Ri Tp

waiting processing() Ri

e.g10 /hr

R ()

10 /hr

10 min, Rp=12/hrTw?

Summary: Causes of Delays and Queues

High Unsynchronized Variability in– Interarrival Times– Processing Times

High Capacity Utilization = Ri / Rp, or Low Safety Capacity Rs = Rp – Ri, due to

– High Inflow Rate Ri

– Low Processing Rate Rp = c/ Tp (i.e. long service time, or few servers)

The psychology of waiting

waiting as psychological punishment keep the customer busy keep them entertained keep them informed break the wait up into stages in multi-stages, its the end that matters

The psychology of waiting

waiting as a ritual insult sensitivity training make initial contact

waiting as a social interaction prevent injustice improve surroundings design to minimize crowding get rid of the line whenever possible

Reducing perceived wait

Understand psychological thresholds Distract customers (mirrors, music, information) Get customers out of line (numbers, call-back) Inform customers of wait (over-estimate) Keep idle servers out of sight Maintain fairness (FCFS) Keep customers comfortable

Is a queue always bad?

queues as a signal for quality doctors business schools restaurants

other people demand similar things the advantage of being in

A solution: Add capacity to remove a persistent line?

You want others to be there to signal quality

Risks of being in versus out: its an unstable proposition!

Don’t want to relate everything to price

The challenge: matching demand and supply

changing number of servers changing queue configuration changing demand managing perceptions