The document discusses operations management concepts including inventory build-up and Little's Law. It provides examples of inventory build-up diagrams showing the relationship between input and capacity rates over time and how excess demand leads to increasing inventory levels. It introduces Little's Law which establishes a relationship between average inventory, average throughput rate, and average flow time in a queueing system. Specifically, it states that average inventory equals average throughput rate multiplied by average waiting time.

1. Operations Management
Lecture 3
Inventory build-up and little’s law
1

2. Reminders
 Office hours: by appointment: dror.hermel@Rotman.Utoronto.Ca
 Midterm Exam: Feb. 11th, 7PM-9 PM (2 hrs)
 If you have conflicts please you already let me know!
 Tutorials: We will have optional attendance review session (tutorial):
– TBA
– Location: TBA.
 Assignment # 1 is due Sunday January 26 at 11:59pm.
 Any question related to assignments and utormat issues must be
consulted with Tas (Tianshu Lu and others).
 Case study groups sign up
2

3. 3
Today’s Lecture:
Little’s Law, Inventory Build-up &
Capacity Analysis
 We introduce the effect of variability
 Inventory Build-up diagram
 Little’s Law

4. 4
What is Variability?
 Definition:
– The randomness associated with a process.
– The extent at which measurements of the
performance of a process differ from each other.
 In reality, all processes exhibit variability as
nothing is completely predictable!

5. 5
Types of Variability:
Thanksgiving Example
 Predictable variability refers to “knowable”
changes in input and/or capacity rates
– Mean demand for turkey will go up close to
Thanksgiving
 Unpredictable variability refers to “unknowable”
changes in input and/or capacity rates
– Supply of turkey changes each year
– Exact demand for turkeys each day
 Both types of variability exist simultaneously
– Turkey sales will go up during Thanksgiving, but we
we do not know the exact demand for turkey

6. 6
Where does it come from?
Security Checking
Variability comes from
– Variable input rate
– Variable capacity rate

7. 7
Predictable Variability
 Predictable variability can be controlled by
making changes to the system
– We could increase or decrease the demand for
turkeys by increasing or decreasing the price
 Other examples of predictable variability
– Medical procedures take longer to perform in July
– Students in office hours right before the midterm
– Demand for uber during rush hours

8. Short Run Analysis: Funnel Analogy
8
Buffer
• In the short run, the input
rate can be larger than the
capacity rate for a period of
time
 A properly sized buffer is
needed to store units waiting
to be processed (build-up
inventory)

9. Operational Challenge
Mismatch between demand and supply
• In any process, the input and output rates will vary
over time
• A key operational challenge is matching supply and
demand
– i.e., matching the input and output rates
• For a variety of reasons, a perfect match is not
possible
– What are some of these reasons?
9

10. Short-run vs. Long-run Averages
• Since the input and output rates may vary over
time, both the short-run average and the long-
run average rates provide useful information.
10
• Long-run average input rate must be less
than the long-run average capacity rate
• Long-run average throughput rate
= Long-run average input rate
• Short-run average input rate can be
greater than the short-run average
capacity rate
But what would
this lead to?
Why?
Why?

11. Security Screening Example Revisited
• What is the capacity rate?
Note: In this example, the capacity rate is given. In practice, it
may not be obvious. Finding the capacity rate will involve
drawing a process flow map, identifying activities, times,
resources, etc, and finding the bottleneck
• What is the (average) size of the line?
• How long do passengers wait (flow time)?
11

12. Inventory Build-Up Diagram
12
Time
Input rate
(passengers/15 min slot)
Capacity rate
(passengers/15 min slot)
Excess
Demand
Excess
Capacity
INVENTORY
BUILD-UP
6:15 7 15 0 8 0
6:30 10 15 0 5 0
6:45 8 15 0 7 0
7:00 12 15 0 3 0
7:15 9 15 0 6 0
7:30 16 15 1 0 1
7:45 14 15 0 1 0
8:00 19 15 4 0 4
8:15 22 15 7 0 11
8:30 17 15 2 0 13
8:45 13 15 0 2 11
9:00 11 15 0 4 7
9:15 12 15 0 3 4
9:30 8 15 0 7 0
9:45 10 15 0 5 0
10:00 7 15 0 8 0
TOTAL 195 240

13. Inventory Build-Up Diagram
13
0
2
4
6
8
10
12
14
6:15 6:30 6:45 7:00 7:15 7:30 7:45 8:00 8:15 8:30 8:45 9:00 9:15 9:30 9:45 10:00
Inventory Build-Up
• What is the “average inventory” in the buffer?

14. Calculating “Average Inventory”
14
Time
Input rate
(passengers/15
min slot)
Capacity rate
(passengers
/ 15 min slot)
Excess
Demand
Excess
Capacity
INVENTORY
BUILD-UP
6:15 7 15 0 8 0
6:30 10 15 0 5 0
6:45 8 15 0 7 0
7:00 12 15 0 3 0
7:15 9 15 0 6 0
7:30 16 15 1 0 1
7:45 14 15 0 1 0
8:00 19 15 4 0 4
8:15 22 15 7 0 11
8:30 17 15 2 0 13
8:45 13 15 0 2 11
9:00 11 15 0 4 7
9:15 12 15 0 3 4
9:30 8 15 0 7 0
9:45 10 15 0 5 0
10:00 7 15 0 8 0
195 240 3.1875
Empty Buffer
(No Queue)
Buffer NOT
empty
Average
Inventory
= 3.1875

15. Consider another example …
15
Time
Input rate
(passengers/15
min slot)
Capacity rate
(passengers
/ 15 min slot)
Excess
Demand
Excess
Capacity
INVENTORY
BUILD-UP
6:30 17 30 0 13 0
7:00 20 30 0 10 0
7:30 25 30 0 5 0
8:00 33 30 3 0 3
8:30 39 30 9 0 12
9:00 24 30 0 6 6
9:30 20 30 0 10 0
10:00 17 30 0 13 0
195 240 2.625
0
2
4
6
8
10
12
14
7:00 8:00 9:00 10:00 8:30 9:00 9:30 10:00
Inventory Build-Up
Average
Inventory
We now aggregate input (arrival) and output (max
capacities) of two consecutive 15-minute time slots.

16. … and another …
16
Time
Input rate
(passengers/15
min slot)
Capacity rate
(passengers
/ 15 min slot)
Excess
Demand
Excess
Capacity
INVENTORY
BUILD-UP
7:00 37 60 0 23 0
8:00 58 60 0 2 0
9:00 63 60 3 0 3
10:00 37 60 0 23 0
195 240 0.75
Average
Inventory
0
0.5
1
1.5
2
2.5
3
3.5
7:00 8:00 9:00 10:00
Inventory Build-Up

17. … and another
17
Time
Input rate
(passengers/15
min slot)
Capacity rate
(passengers
/ 15 min slot)
Excess
Demand
Excess
Capacity
INVENTORY
BUILD-UP
8:00 95 120 0 25 0
10:00 100 120 0 0 0
195 240 0
Average
Inventory
0
0.2
0.4
0.6
0.8
1
7:00 8:00
Inventory Build-Up

18. Estimating Process Measures
• Process measures change over time
– Depending on the mismatch between input rate and
the capacity rate that inevitably occurs over time
• We are interested in averages of these quantities
• “Average” values of process measures can be
misleading
• It is often convenient to assume continuous input
and output processes
18

19. Inventory In Process Analysis
 Flow units (eg. customers, raw materials,
vehicles) awaiting their turn to enter the process
as well as units that are currently being served.
 They are waiting in a queue or a buffer.
 Flow units can be extracted from the buffer or
serviced in a variety of ways.
– The way in which flow units are served is called the
service policy (FCFS, LCFS, Priority Service).
 How do we track the queued inventory?
19

20. Definitions
• Instantaneous Flow Rates
20
Ri(t) The input rate to the process at time t
Ro(t) The output rate of the process at time t
∆R(t) = Ri(t) – Ro(t) Instantaneous inventory accumulation at time t
• Inventory Level
• Flow Time
I(t) The number of units within the process
boundaries at time t
T(t) The time that a unit which enters (leaves) the
process at time t spends (has spent) within the
process
This can be defined in many ways

21. Inventory and Flow Dynamics
• Let (t1,t2) denote an interval
of time starting at t1 and
ending at t2
• Suppose ∆R(t) is constant
over (t1,t2) and equals ∆R.
Then,
21
t1 t2
I(t1)
I(t2)
I(t)
t
∆R *(t2-t1)
)
(
)
(
)
( 1
2
1
2 t
t
R
t
I
t
I 




2
Inventory
Ending
Inventory
Starting
Inventory
Average


Ending
Inventory
Starting
Inventory
Change in
Inventory

22. Inventory Build-Up Diagram
Capacity rate = 100/hr
22
10AM
50
200
Input Rate
12PM 2PM 6PM
10AM
100
200
Inventory
(or Backlog)
12PM 2PM 6PM
Assumes inventory
level changes in
“discrete amounts”
Assumes inventory
level changes in
“continuous amounts”

23. Another Inventory Build-Up Example
23
0
200
400
I(t)
Inventory in week t
1 2 3
Week Input Rate Throughput Rate Inventory
0 400
1 900 800 500
2 900 1200 200
3 900 1000 100
Week

24. Under the continuous
assumption:
The average inventory?
“Area under the curve”
Average Inventory
Average inventory depends on
whether inventory is assumed
to change in discrete steps, or
continuously
24
0
200
400
I(t)
1 2 3 Week
Under the discrete
assumption:
The average inventory over
weeks 0 to 3 is 300
Under the continuous
assumption:
The average inventory?
??????

25. 25
Inventory Buildup: Cranberry Example
During harvesting season, a processing factory works around the clock.
 Farmers deliver their loads of cranberries from 12am to 12pm (last
truck arrives at 11 am) at a constant rate of 2 tons/hour.
 The fruits are dumped into a big storage bin and processed at a rate
of 1 ton/hour.
• Draw an Inventory buildup diagram.
(Assume the flow unit is cranberries and that the they arrive
at the station at a constant pace all day).
• What is the average inventory of cranberries in the
factory?

26. 26
Inventory Buildup: Cranberry Example
12:00am
Time
12:00pm 12:00am
12
Average
6
Tons Average Inventory in the System
(Area Under the Curve) / (Time)

27. 27
Inventory Buildup: Cranberry Example Continued
Suppose the storage bin has room to hold only 6 tons of cranberries.
Once this space is filled, the farmers’ trucks must wait to dump their
contents
 Notice there are now two buffers:
» Cranberry buffer in the storage bin
» Truck buffer
Questions:
• What would happen if there was no truck buffer?
• What is the average inventory of cranberries
• In the bin?
• In the trucks?
• In both?
• Do we lose any cranberries?

28. 28
Inventory Buildup: Cranberry Example
12:00am
Time
12:00pm 12:00am
12
Capacity
6
Tons Average Inventory in the System
(Area Under the Curve) / (Time)
Trucks
Storage bin

29. 29
Inventory Buildup: Cranberry Example Continued
Now let us change the flow unit to a “truck”
 Assumptions:
– Each truck carries 1 ton of cranberries, i.e., two trucks arrive
every hour between 12:00am to 12:00pm.
– The storage bin has a capacity of 6 tons.
– At the start of every hour, the processor takes 1 ton of
cranberries from the storage bin.
• Draw an Inventory buildup diagram of trucks.
• At what time will the trucks likely start to wait to unload?
• What is the “average” inventory of trucks waiting?

30. 30
Inventory Buildup: Cranberry Example Continued
0
1
2
3
4
5
6
7
0:00
1:00
2:00
3:00
4:00
5:00
6:00
7:00
8:00
9:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
Number
of
trucks
waiting
Time
Inventory of trucks

31. 31
Little’s Law: Introduction & Capacity Analysis
 Establishes a relationship between:
 Average Inventory
 Average Throughput Rate
 Average Flow Time
Average Flow Time (T)
[hr]
Average Inventory (I)
[units]
Average Throughput
Rate
(R) [units/hr]
... ... ...
... ... ...

32. 32
Little’s Law:
Coffee Example
 Time to make one coffee: 30 seconds
What is the throughput rate in minutes?
 60 customers in the system (Inventory)
Waiting time:
Waiting Time = 60 customers * 0.5 (min/customer)
Waiting Time = Inventory / Throughput Rate
Inventory = Throughput Rate x Waiting Time
Waiting Time = 60 customers / 2 (customer/min)
Coffee
Shop
... ...
Average Waiting Time (T) [hr]
... ...
...

33. 33
Little’s Law:
Big Reveal & Key Equation
Inventory = Throughput Rate x Waiting Time
I = R x T
 We are talking about an inventory of flow units:
– Could be customers in a restaurant
– Claims in an insurance company
– Materials in a manufacturing industry
 This equation can be re-arranged to solve for
other quantities depending on the question!

34. 34
Little’s Law:
Remember the Units
Inventory = Throughput Rate x Waiting Time
I = R x T
All units should match:
– Inventory (flow units)
– Throughput rate (flow units / unit time)
– Waiting time (time)

35. Little’s Law
• Throughput rate: 1 car/min
• 900 cars in the system
• Flow time?
35
car/min
1
cars
900
)
min/car
1
)(
cars
900
(
Time
Flow 

inventory
throughput rate
I = R * T

36. Little’s Law: Example 1
• Patients waiting for an organ transplant are
placed on a list until a suitable organ is available.
We can think of this as a process. Why?
36
Patients matched to
donated organs
INPUT
Patients in need
of a transplant
OUTPUTS
Patients leaving the
list hopefully with a
successful transplant

37. Little’s Law: Example 1
Question (a)
• On average, there are
300 people waiting for
an organ transplant
• On average, patients
wait on the list for 3
years
• Assume that no patients
die during the wait
• How many transplants
are performed per year?
37
300 patients ?? / year
3 years in system
I = R * T
Inventory I = 300 patients
Flow Time T = 3 years
Throughput Rate
R = I/T = 100 patients / year

38. Little’s Law: Example 1
38
Question (b)
• On average, there are
300 people waiting for
an organ transplant
• On average, 100
transplants are
performed per year
• Assume that no patients
die during the wait
• How long do patients
stay on the list?
300 patients 100/year
??? years in system
I = R * T
Inventory I = 300 patients
Throughput R= 100 patients/year
Flow Time
T = I/R = 3 years

39. Insights from Little’s Law
• Throughput rate, flow time, and inventory are
related
• Depending on the situation, a manager can
influence any one of these measures by
controlling the other two
– You cannot independently choose flow time,
throughput and inventory levels!
– Once two are chosen, the third is determined
– For example, if the flow time is fixed, the only way to
reduce inventory is to increase throughput
39

40. Insights from Little’s Law
• How would you reduce wait time for patients
on the transplant waiting list?
– Increase throughput rate
– Decrease number of people on the list (inventory)
• How would you increase throughput rate of
containers at the port
– Decrease flow time
– Increase “inventory”
40

41. 41
Average Process Measures
 When process measures vary over time, average
process measures are useful process characteristics.
 Sometimes, we omit “average” when no confusion will arise.
Coffee
Shop
Average Throughput rate
(R) [units/hr]
... ...
Average Waiting Time (T) [hr]
... ...
...
Average Number of
Waiting Customers (I)
[units]

42. 42
Little’s Law: When the Input Rate Fluctuates
 Recall the cranberries and trucks example
 How long does a truck wait on average?
 We can apply Little’s Law:
– Average inventory = 1.5 trucks
– (Average) throughput rate = 24 trucks/day = 1 truck/hr
– Therefore, by Little’s Law:
Inventory = Throughput Rate x Waiting Time
What is the waiting time for each truck?
T = 1.5 hours

43. 43
Recall Our Focal Questions
 What are the effects of variability on processes?
– In particular, how does variability affect
» Average Throughput rate (R)
» Average Inventory (I)
» Average Flow time (T)
 If the effects are not a good thing, how can we
deal with it? What interventions should we take
to better the performance of the system?

44. Basic Questions
• What are the effects of variability on processes
– In particular, how does variability affect
44
• If the effects are negative, how can we deal with it?
Average Inventory Average Flow Time
Average
Throughput Rate

45. Consider a process with no variability
• Assume that all customers are identical
• Customers arrive exactly 1 minute apart
• The service time is exactly 1 minute for all the customers
45
ATM
Service time
(exactly 1 min)
Input
(1 person/min)
Throughput
Rate?

46. 46
No Variability Process
ATM
Input rate:
1 person/min
Throughput
rate?
Service Time (1 min)
0
Time
(min)
Queue
Length ?
1
2
3
4
5
1 2 3 4 5 6 7 8 9 10 11 12
NO QUEUE

47. Effect of Input Variability (no buffer)
• Assume that customers who find the ATM busy do not wait
47
ATM
Service time
(exactly 1 min)
Random Input
0, 1, 2 customers/min
(with equal probability)
Throughput
Rate?
1 2 3 4 5 6 7
time

48. Effect of Input Variability (no buffer)
• When a process faces input variability, and a buffer
cannot be built, some input may get lost
• Input variability can reduce the throughput
• Lower throughput means
– Lost customers; lost revenue
– Customer dissatisfaction
– Less utilization of resources
• Little’s Law holds
48

49. Dealing with Variability
• When the arrival rate of customers is
unpredictable, what could you do to increase
throughput?
49
Add Buffer
Increase Capacity
(e.g., Add another ATM;
Decrease the time it takes the ATM to serve a customer)

50. Effect of Input Variability (with buffer)
• Now assume that customers wait
We can build up an inventory buffer
ATM
Service time
(exactly 1 min)
Random Input
0, 1, 2 customers/min
(with equal probability)
Throughput
Rate?
Buffer
Waiting time

51. 51
Effect of Variability (with buffer)
ATM
Buffer
Waiting time Service time (1 min)
Input Rate:
Random 0,1,2 persons/min
Throughput
rate?
0
Time
(min)
1
2
3
4
5
1 2 3 4 5 6 7 8 9 10 11 12

52. Effect of Input Variability (with buffer)
• If we can build up an inventory buffer,
variability leads to
– An increase in the average inventory in the process
– An increase in the average flow time
– We are not immediately losing customers due to
abandonment (although they may still by unhappy)
– Fewer customers may be unhappy
– More utilization of resources
• Little’s Law holds
52

53. 53
• Definitions:
Balking: When customers arrive and after inspecting
the queue (or process) decide to leave immediately.
The assumption is that the queue is observable.
Customers who balk never enter the buffer!
Reneging: When customers arrive, join the queue
and decide to leave after waiting for some time.
They do not necessarily need to see the queue.
Customers who renege enter the buffer (i.e., the queue)
but do not wait long enough to get service.
Effect of Variability (with buffer)

54. 54
Dealing with Variability
• Input variability can reduce the throughput.
• Lower throughput means
– Lost customers (who are dissatisfied), lost revenue
– Less utilization of resources, we are not as efficient
• When the arrival rate of customers is
unpredictable and contains variability, what could
you do to increase throughput?
– Inventory: Add a buffer before the process.
– Capacity: Staffing, equipment, physical space, etc.
– Information: Decrease the variability via information.

55. The OM Triangle
If a firm is striving to meet the
random demand, then it
can use capacity, inventory,
and information (variability
reduction) as substitutes
You cannot have low
inventory, low capacity, low
information acquisition
effort at the same time.
This is a trade-off.
55
CAPACITY
INVENTORY INFORMATION
(Variability
Reduction)

56. Operations at Dell
• Inventory as “the physical embodiment of bad
information” (a senior exec at Dell)
• Substitute information for inventory
• Less inventory =>higher inventory turns
56
Fast Tech

57. A Single Server Process
A queue forms in a buffer
57
Server
Buffer
Process Boundary
 Long-run average input rate
1/
(Average) Customer inter-arrival
time

Long-run average processing rate of
a single server
1/
Average processing time by one
server
A single phase service system is stable
whenever  < 
K Buffer capacity (for now, let K = )
c
Number of servers in the resource
pool (for now, let c=1)
Note: We are focusing on long-run averages,
ignoring the predictable variability that may
be occurring in the short run. In reality, we
should be concerned with both types of
variability

58. T
Service time Ts
Waiting time Tq
What are we trying to quantify?
Little’s Law holds
Iq = Tq
Is = Ts
I = T
58
Server
Buffer
Throughput
rate = 
I
Is
Iq
Performance Measures
System Characteristics
Tq Average waiting time (in queue)
Iq Average queue length
Ts Average time spent at the server
Is Average number of customers being served
T=Tq+Ts Average flow time (in process)
I=Iq+Is Average number of customers in the process
Utilization

(In a stable system,
 = / < 100%)
Safety
Capacity
 - 

59. Quick “Quiz”
• Average number of persons in the system:
I = Iq + Is
• Question: Is=??? (Express Is in terms of  and )
59
Server
Service rate:  persons/min
(average capacity rate)
Buffer
Arrival rate:
 persons/min
(average input
rate)
Average
throughput rate
 persons/min
Assumption:  < 
• Answer: Is=/

60. Single-Server Queuing Model
60
Server
Service rate:  persons/min
(average capacity rate)
Buffer
Arrival rate:
 persons/min
(average input
rate)
Average
throughput rate
 persons/min
Assumption:  < 
On average, 1 person
arrives every E{a} min.
Thus,  = 1 / E{a}
Time
…
a1 a2 a3 a4 a5 a6 a7 …
Inter-arrival
times:
On average, 1 person can
be served every E{s} min.
Thus,  = 1 / E{s}
Service
times: s1 s2 s3 s4 s5 s6 s7

61. House keeping
61
Assignment # 1 is due this Sunday, January 26,
11:59pm.
Keep checking Quercus announcements.

62. Next lecture
P-K formula: A fairly technical lecture
A tool for measuring operating characteristics of a
system in the presence of both inter-arrival and
service times variability.
62