15.760 Class #8: Basic Concepts in Queueing System Performance = f(System parameters) Arrival rate ( λ )Output/throughput rate ( ( λ � )) Service rate ( µ )Inventory Level/Queue Size/ Service time ( M )Line length Number of servers ( S )Waiting Time/Cycle Time ( W ) Queue/Buffer capacity ( R )Capacity or Server utilization ( ρ ) Capacity or Server utilization ( ρ )Probability that Queue is full ( Pfull ) Number of Service classes ( K )Fish Processing Example Input Rate (Tons per month) 4800 3600 600 0 4 8 12 Time (Months)Fish Processing Example (con’t) λ = 3600 then 4800 then 600 µ=3000, R= unlimited Freezer Inventory (Exhibit 3) Average Inventory = 1/3 x (2400/2) + 1/3 x (9600 + 2400)/2 + 1/3 x (9600/2) = (1200 + 6000 + 4800)/3 = 4000 tonsFish Processing Example (con’t) λ = 3600 then 4800 then 600 µ=3000, R= 2400 Freezer Inventory (Exhibit 4) Average Inventory = 1/3 x (2400/2) + 1/3 x 2400 + 1/12 x (2400/2) + 1/4 x 0 = (1200/3 + 2400/3 + 1200/12) = 1300 tonsFish Processing Example (con’t) λ = 3600 then 4800 then 600 µ=3300, R= 2400 Freezer Inventory (Exhibit 5) Average Inventory = 1/3 x (1200/2) + .8/12 x 1800 + 3.2/12 x (2400) + .89/12 x (2400/2) = (200 + 120 + 640 + 89) = 1049 tonsFish Processing Example (con’t) λ = 3600 then 4800 then 600 Average Thruput Capacity Inventory per month Utilization µ=3000, R=∞ 4000 tons 3000 (.63) 100% µ=3000, R= 2400 1300 tons 2400 (.63) 80% µ=3300, R= 2400 1049 tons 2600 (.63) 79% Tradeoffs: Cost of processing capacity vs. Cost of Storage Capacity vs. Value of output (net of holding costs)Basic Concepts in Queueing: Capacity Utilization in Stochastic Systems Suppose avg arrival rate = 1/minute Avg service rate = 1.33/minute (or avg service time = 45 seconds) 0 1 2 3 4 5 6 7 8 9 10 time 1 1 .5 1 1 .5 .5 .5 1 .5 arrivalsBasic Concepts in Queueing: Capacity Utilization in Stochastic Systems System Performance = f(System parameters) Capacity or Server utilization ρ = λ / µ (arrival rate/service rate) ρ = λ (S x µ) ρ = λ x MBasic Concepts in Queueing: Little’s Law System Performance = f(System parameters) Conservation of Flows in Stochastic Systems L = λ x W Avg Length of the Queue = Arrival rate x Avg Waiting time 600 MBA’s = 300/year x 2 yearsBasic Concepts in Queueing: Nonlinearities in Congestion in Stochastic Systems If service times and interarrival times have exponentialdistributions, then L= ρ2/(1-ρ) W= ρ2/λ(1-ρ) ∆WA Total wait in the queue ∆ ρ ∆ ρA B ∆WB 0 1 (Arrival Rate / Service Rate = ρ) = “congestion”Basic Concepts in Queueing: Nonlinearities in Congestion in Stochastic Systems System Performance = f(System parameters) With exponential (λ) interarrivals, and service times with mean = M and std dev = σ, Then W= λ (M2 +σ2) 2(1-ρ)Volatility Amplification in the Supply Chain: “The Bullwhip Effect” Order Order Order InfoInfo Info Retailer Wholesaler Distributor Factory How does production control work in the Beer Game? Information lags Delivery lags SOLUTIONS: Over- and underordering Countercyclical Markets Countercyclical Technologies Lumpiness in ordering Misperceptions of feedback Collaborative channel mgmt. (Cincinnati Milacron & Boeing)Chain accumulationsManagement of Queues The Physics of Waiting Lines – Number and type of servers – Waiting time, service time, and system time – Queue discipline – Number of people in queue – System utilization 0 Delay Cost Capacity Cost Total Cost Cost ∆ ρ ∆ ρ A B ∆WA ∆WB 0 Total time in the queue 1 1 Congestion (Arrival Rate / Service Rate) Congestion (Arrival Rate / Service Rate)Management of Queues SERVERSThe Psychology of Waiting Lines CUSTOMERS WAITING LINE Propositions 1. Unoccupied time feels longer than occupied time 2. Process waits feel longer than in process waits 3. Anxiety makes waits seem longer 4. Uncertain waits seem longer than known, finite waits 5. Unexplained waits are longer than explained 6. Unfair waits are longer than equitable waits 7. The more valuable the service, the longer the customer will wait 8. Solo waits feel longer than group