Buffer or Suffer Principle

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Presentation transcript:

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

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

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

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?

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

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

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

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

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.

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?

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?