Download "Lecture 6: Offered Load Analysis. IEOR 4615: Service Engineering Professor Whitt February 10, 2015"
2 What is the Problem? What capacity is needed in a service system? In order to meet uncertain exogenous demand Main Example: Staffing, i.e., how many serivice representatives?
3 Offered Load Analysis Estimate the capacity needed to meet uncertain exogenous demand by determining the capacity that would be used if there were no limit on its availability. NOTE: The demand could be defined directly as the offered load.
4 Definitions The Stochastic Offered Load (SOL): the random amount of capacity needed to meet uncertain exogenous demand if there were no limit on its availability. Offered Load (OL): the expected value of the SOL.
5 When is offered load analysis interesting? 1. When it is not so obvious what the offered load should be. 2. When you can afterwards apply the offered load in a non-obvious way to do a more refined analysis.
10 Real system: Erlang-A (M/M/s + M) model expected patience time = 2 Performance: P(Wait > 0) =.19, P(Wait >.05) =.09, P(Ab) =.006, The offered load analysis works, but we can easily do a more precise analysis with an Erlang-A model solver.
13 Three Time-Varying Concepts Coincide Time-Varying Offered Load (TVOL) Expected Stochastic Offered Load Time-varying mean in M t /GI/ Expected Number of Busy Servers Infinite-Server model with time-varying arrival rate L(t) in Time-Varying Little s Law See second paper by Kim and Whitt (2013).
15 Time-Varying Arrival Rate Long-run average arrival rate = 100 per hour Now λ(t) = sin (t) (new!!) expected service time = 1 hour Want to stabilize performance at similar level P(Wait > 0) 0.20 How many service representatives are needed at each time now?
26 More Accurate Staffing and Performance Prediction: The Modified Offered Load (MOL) Approximation Use steady-state performance of corresponding stationary model with capacity constraints and other details, e.g., customer abandonment, but in a nonstationary way. At time t, make the stationary offered load agree with m(t) by letting λ MOL (t) = m (t)/e[s] where m (t) = E[L(t)] = E[λ(t S e )]E[S] is TVOL, as before -based on M t /GI/ model -having time-varying arrival rate λ(t) Staffing: Let s(t) = maximum s such that P(Wait(t) > 0) 0.2, where Wait(t) is steady-state wait for the stationary model at time t.