University:
Massachusetts Institute of Technology
Course:
15.501/516 | Accounting
Academic year:

2023
Views:

292

Pages:

15
Author:

Deborah Gentry

A Simple Process Flow Customers arrive Queue Queue Server Server Patients Doctors Moreno’s Output Clark Customers ATM Phone Calls Operator Web Content Servers Fish Cannery Customers exit Principal Driver: Variability Predictable Seasonal Arrivals Unpredictable Ski Wear Lunch & Dinner @ BK Arrivals Seasonal Servers Workers/Shift Winter Traffic ATM, ER, JobShop WIP Services Machine Breakdowns Car Repairs ER Most Systems have both variabilities, but one usually dominant. Inputs & Outputs INPUTS Arrival rate Service Rate # of Servers Size of Waiting Room Variability OUTPUTS Output Rate ⇑ Inventory Level ⇓ (Queue Size) Waiting Time ⇓ Server Utilization Inventory Build-Up Diagrams Avg. Arrival Rate: 3000/month 3000/month Service Rate: 7200 Inventory Monthly Supply 9600 4800 4800 3600 2400 600 0 0 4 Output Rate: 8 3000/month 12 Average Inventory 9600 Inventory 7200 4800 14400 9600 0 2400*4 2 4 2400 19200 8 12 Total Inventory: 4800 + 9600 + 14400 + 19200 = 48000 0 Average Inventory: 48000/12 = = 4000 Different Supply Pattern… Avg. Arrival Rate: 3000/month Service Rate: 3000/month 9600 Monthly Supply Inventory 7200 4800 4800 3600 2400 600 600 0 0 4 Output Rate: 8 2200/3000/month 12 16 …Different Supply Pattern 9600 Inventory 4800 7200 4800 28800 19200 14400 0 4 2400 8 12 Total Steady State Inventory: Average Steady State Inventory: 16 0 67200 5600 vs. 4000 previously System Configuration Increase in… Output Inventory Service Rate Up down +++ Arrivals Rate UP UP Trade-Off Waiting Room size UP UP Trade-Off Variability Down UP --- # Servers UP Down +++ Unpredictable Variability Queueing Theory Queueing Theory Poisson Arrivals Poisson Service Rates Random Time until next Arrival not function of last arrival λ = Arrival Rate (customers/min) Memoryless µ = Service Rate (= 1/mean service time) M = Number of Servers FCFS (Bank, not Supermarkt) Infinite Waiting Room Long Run Behavior 3 Easy Steps Capacity Utilization: ρ = λ/(Mµ) Average time in Queue: Lq = f(λ/µ,M) (as in Exhibit 3 of Reader) Average time in queue Wq = Lq /λ Lq =λ∗Wq Little’s Law Average number in system L = Lq + λ/ µ Average time in system W = Wq + 1/µ Multi-Stage Process λ1 Mµ Mµ λ2 λ2 = min[λ1, Μµ] Server Server22 Wrap Up Inventory Build-Up Diagrams Predictable Variability Utilization > 1 o.k. Short Run o.k. Varibale Time/Service rates o.k. Queueing Formulas Unpredictable Variability Utilization < 1 only Long Run only Fixed Time/Service rates only Steps in Process Flow Analysis 1. 2. 3. Process Flow Diagram Find Utilization Levels (Bottlenecks) Use Buildup Diagrams or Queueing Formulas FCFS Queuing Discipline SERVERS SERVERS CUSTOMERS CUSTOMERS Frequency Memoryless Property 0 T Time between arrivals (service times)