1. The Impact of Variability on Process Performance: Throughput Losses Chapter 8

2. Buffer or Suffer Principle
Table 8.1 Sandwich Vendor
Demand and capacity take on values of 0, 1, or 2 within a 5 minute time window
Scenario
Demand
Capacity
Flow Rate A B 1 C 2 D E F G H I

3. Emergency Room Crowding and Ambulance Diversion
Pictures from various Newspapers; put together by L. Green

4. Analyzing Loss Systems
Resources
3 trauma bays (m=3)
Demand Process
One trauma case comes in every 3 hours
(a=3 hours)
a is the interarrival time
Exponential interarrival times
Service Process
Patient stays in trauma bay for an average of 2 hours
(p=2 hours)
p is the service time
Can have any distribution
Trauma center moves to diversion status once all servers are busy
incoming patients are directed to other locations
What is Pm, the probability that all m resources are utilized?

5. Throughput Loss for a Queue with One Single Resource
Probability of m units in a system Pm depends on two parameters, u and r.
Implied utilization u – it is possible to have u>100%; some flow units don’t enter the system and don’t contribute to throughput.
r stands for the ratio of the process time to the interarrival time

6. Analyzing Loss Systems: Finding Pm(r)
Define
Example: r= 2 hours/ 3 hours r=0.67
Recall m=3
Use Erlang Loss Table
Find that P3 (0.67)=0.0255
Given Pm(r) we can compute: m
r = p / a

7. Erlang Loss Table Appendix B (1st section)
Probability{all m servers busy}=

8. Implied Utilization Vs Probability of Having all Servers Utilized: Pooling Revisited
Probability that all servers are utilized
0.1
0.2
0.3
0.4
0.5
0.6
m=1
m=2
m=3
m=5
m=10
m=20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
1.1
Implied utilization

9. Customer Impatience & Throughput Loss
Customers wait forever Vs Customers balk at waiting (or are blocked)
Intermediate cases
Customers join a buffer
Customers abandon the queue (renege)
Three improvements for the intermediate cases
Reduce wait times
Increase the maximum number of flow units that can be in the buffer
Avoid customers leaving that have already waited.

10. 8. 1 Flow units arrive at a demand rate of 55 units per hour
8.1 Flow units arrive at a demand rate of 55 units per hour. It takes, on average, six minutes to serve a flow unit. Service is provided by seven servers.
What is the probability that all seven servers are utilized?
How many units are served every hour?
How many units are lost every hour?

11. 8.3 A small video store has nine copies of the DVD The Saw, in its store. There are 15 customers every day who request this movie for their children. If the movie is not on the shelf, they leave and go to a competing store. Customers arrive evenly distributed over 24 hours; the average rental duration is 36 hours.
What is the likelihood that a customer going to the video store will find the movie available?
Assume each rental is $5. How much revenue does the store make per day from the movie?
Assume each child that is not able to obtain the movie will receive a $1 bill. How much money would the store have to give out to children every day?
Assume demand for the movie will stay the same for another six months. What would be the payback time (not considering interest rates) for purchasing an additional copy of the movie at $50?