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Thruput Loss

Thruput Loss. Chapter 7. Learning Objectives. Thruput loss due to limited buffer Thruput loss due to customer impatience Blocking and Starving of resources. Emergency Room Crowding and Ambulance Diversion. December 19, 1987 Ambulances Being Diverted In New York By RONALD SULLIVAN.

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Thruput Loss

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  1. Thruput Loss Chapter 7

  2. Learning Objectives • Thruput loss due to limited buffer • Thruput loss due to customer impatience • Blocking and Starving of resources

  3. Emergency Room Crowding and Ambulance Diversion December 19, 1987 Ambulances Being Diverted In New YorkBy RONALD SULLIVAN

  4. Macro Economic Trends Driving Emergency Room Crowding and Ambulance Diversion • Increase in ER visits (14% from 1997 to 2000) • 40% of patients admitted through the ER • Uninsured are 25% of Texas population • Personal insurance costs about $8000 per person • A major portion of health care costs happen in the last 6 months of life • U.S. health care expenses so high with respect to the rest of the industrialized world yet the U.S, life expectancy is so low. • Decrease in number of emergency departments (8.1% decline since 1994) • Consequences: • Long wait times (see waiting time analysis) • Loss of throughput (requires new analysis) • Data from L. Green; general accounting office 20% of US hospitals are on diversion status for more than 2.4 hours per day

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

  6. Given Pm(r) we can compute: • Time per day that system has to deny access • Flow units lost = 1/a * Pm (r) Analyzing Loss Systems: Finding Pm(r) m • Define r = p / aThis is the utilization if m=1. • Ex: r= 2 hours/ 3 hours r=0.67 • Recall m=3 • Use Erlang Loss Table • Find that P3 (0.67)=0.0255 r = p / a

  7. Erlang Loss Table Probability{all m servers busy}= What a strange formula!

  8. 1/a 1/a 1/a 0 1 2 3 Aside:Towards Agner Erlang’s Formula for m=3 • How does the number of people in the system change? 3/p 1/p 2/p • What happens to the arrival when there already are 3 in the system? • π0, π1, π2, π3 are the probabilities respectively that there are 0,1,2,3 in the system. • If there is nobody in the system (w.p. π0), there will be 1 in the system with rate 1/a. • The rate of switching from 0 in the system to 1 in the system = π0(1/a) • Similarly, • The rate of switching from 1 in the system to 2 in the system = π1(1/a) • The rate of switching from 2 in the system to 3 in the system = π2(1/a)

  9. Aside:Agner Erlang’s Formula for m > 3

  10. Implied utilization vs probability of having all servers utilized: Pooling Revisited 0.6 Probabilitythat all serversare utilized 0.5 0.4 m=1 m=2 0.3 0.2 m=3 m=5 m=10 0.1 m=20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Implied utilization Recall Implied utilization = (Capacity requested by demand) / (Available capacity)

  11. What happens with the buffer size? • In the hospital’s trauma center, there is no buffer space; a patient must be given a bed in the intensive care unit. • In U.K., ambulance is used as a buffer. • In manufacturing or other non-critical service operations, there are buffers to hold queues before servers. • In these cases, the diversion happens when the buffer is full. • Since extra units can be accommodated in the buffer, diversion probability drops sharply with the buffer size. • Hence, the rate of customers served=(Demand rate)x(1-Prob that buffer is full) rises with the buffer size. Probability that system is full Rate of customers served 100 0.5 80 0.4 Increasing levels of utilization 60 0.3 Increasing levels of utilization 40 0.2 20 0.1 0 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 Buffer size Buffer size

  12. Fraction of customer’s lost Average wait time [seconds] at a call center Impatient customers balk from the queue • Impatient customers leave the queue even when there is enough buffer to accommodate them while waiting. • What information do customers have to estimate waiting time • Physical queue; count the number of people in front of you, see if the people in the cashier’s queue have bought a lot in the supermarkets • Virtual queue (i.e., call center); believe what you are told • Buy info when queue is not visible: Disney parks may sell queue info to visitors. • Leave as soon as arrive or after some wait • Also, Abandon rate is decreasing convex in the Service (% of calls answered within 1 min) See McKinsey Quarterly No. 4, 2006 Article “Finding the right level of call center staffing” by T. Oguz and H. Sakhnini

  13. Lack of buffer space causes Blocking and Starvation Empty space for a flow unit Space for a flow unit with a flow unit in the space Resource is blocked Activity completed Inflow Outflow Outflow Inflow Activity not yet completed Resource is starved Starving or Blocking bottleneck resources is a major problem.

  14. 6.5 min/unit 6.5 min/unit 6.5 min/unit 7 min/unit 7 min/unit 7 min/unit 6 min/unit 6 min/unit 6 min/unit Pick your buffer size and configuration Sequential system, unlimited buffers Cycle time=7 minutes; inventory “explodes” Sequential system, no buffers Cycle time=11.5 minutes 3 resources, 19.5 min/ unit each (1) (1) Horizontally pooled system Cycle time=19.5/3 minutes=6.5 minutes Sequential system, one buffer space each Cycle time=10 minutes

  15. Appointment Systems to control Arrivals • Appointment systems attempt to match supply with demand • But: this creates two types of waiting lines (one of them hidden) - currently: 30% of US population cannot get an appointment with MD • What to do with emergency cases? - add cases / cancel appointments - reserve capacity for emergencies • Advanced Access Model: “Do today’s work today” - based on JIT idea (zero inventory) • - need to forecast demand better (flu, back-to-school)

  16. Summary • Thruput loss due to limited buffer • Thruput loss due to customer impatience • Blocking and Starving of resources

  17. 1/a 1/a 1/a 0 1 2 3 Aside:Towards Agner Erlang’s Formula for m=3 • How does the number of people in the system change? 3/p 1/p 2/p • What happens to the arrival when there already are 3 in the system? • π0, π1, π2, π3 are the probabilities respectively that there are 0,1,2,3 in the system. • If there is nobody in the system (w.p. π0), there will be 1 in the system with rate 1/a. • The rate of switching from 0 in the system to 1 in the system = π0(1/a) • Similarly, • The rate of switching from 1 in the system to 2 in the system = π1(1/a) • The rate of switching from 2 in the system to 3 in the system = π2(1/a)

  18. 1/a 1/a 1/a 0 1 2 3 Aside:Towards Agner Erlang’s Formula for m=3 3/p 1/p 2/p • If there is 1 in the system (w.p. π1), there will be 0 in the system with rate 1/p. • The rate of switching from 1 in the system to 0 in the system = π1(1/p) • Somewhat similarly, • The rate of switching from 2 in the system to 1 in the system = π2(2/p) • The rate of switching from 3 in the system to 2 in the system = π3(3/a)

  19. 1/a 1/a 1/a 0 1 2 3 Aside: Equilibrium probabilitiesThe rate of leaving and entering is the same 3/p 1/p 2/p

  20. Aside:Agner Erlang’s Formula for any m

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