University:
Massachusetts Institute of Technology
Course:
15.992 | S-Lab: Laboratory for Sustainable Business
Academic year:

2023
Views:

258

Pages:

13
Author:

Kaitlynn Hoover

"Improving Service by Informing Customers About Anticipated Delays." Motivation • Investigate alternative ways to manage a service system, eg. Call Centers. • Use Birth-and-Death (BD) stochastic process models to model 2 types of service systems – Conventional queues allowed with no info – Queues with delay or state information • Of value to both customers and service providers Two alternative queuing systems • First: provide waiting room but no info on state or queuing time – No balking but customer may renege • Second: provide waiting room but info on either state or queuing time – Higher balking rate relative to renege – Information about anticipated delays increases customer satisfaction, resulting in more repeat business – Increasing capability for service providers to provide delay info (Rappaport 1996) M/M/s/r Model 1 Poisson Arrival, λ r waiting space FCFS s servers exponential svc time with mean μ^-1 System state not known by customers Independent α and β If a server is not immediately available customer balk with probability β Then, customer waits till T is reached before reneging Æ Model with time dependent reneging Model 1 – (See explanation and variable definitions in section 2, page 194 of the Whitt paper.) – Characterize by 0 ≤ k ≤ s −1 ⎧λ λk = ⎨ ⎩λ (1 − β ) s ≤ k ≤ s + r − 1 1 ≤ k ≤ s −1 ⎧k µ µk = ⎨ ⎩ s µ + (k − s )α s ≤ k ≤ s + r – Pk state probabilities are easy to calculate! M/M/s/r Model 2 r waiting space Poisson Arrival, λ FCFS s servers exponential svc time with mean μ^-1 System state now communicated to customers upon arrival Dependent α and β Balking now depends on state of system State dependent balking replaces reneging after waiting Æ Model with mainly state dependent balking plus some reneging Model 2 • Case 1: Required waiting time is given as state information • If waiting time > T, customer balks • If not all servers are occupied, customer is served immediately • If all servers are occupied, customer either balks or stays with probability qk ≡ P(T > S k ) 0 ≤ k ≤ r −1 Where Sk : time from arrival until first served where state at time is k. Model 2 • To find the state dependent probability of joining in an exact manner, ∞ qk = ∫ e 0 −α t g k (t )dt = Ee −α S k ⎛ sµ ⎞ =⎜ ⎟ + s µ α ⎝ ⎠ k +1 • To find a reasonable approximation of the state dependent probability of joining, q k ≡ P(T > ES k ) = e −α ( k +1) / sµ , k ≥ 0 Model 2 • To add state dependent reneging to generalize model 2, define • δ'j : renege rate of customer with j-1 customers ahead in queue • Total renege rate, δ = ∑ δ ′ • BD process can be characterized by k k ⎧λ λk = ⎨ ⎩λ (1 − β )qk − s ⎧k µ µk = ⎨ ⎩ sµ + δ k − s j =1 j 0 ≤ k ≤ s −1 s ≤ k ≤ s + r −1 1≤ k ≤ s s +1 ≤ k ≤ s + r Finding Performance Measures • Step 1: Find the steady state distribution • Step 2: Calculate probability of completing service and the mean, variance and full distribution of the conditional response time given that service is completed. • Step 3: Calculate probability of customer reneging and the mean, variance and full distribution of the conditional time to renege given that customer reneges. Stochastic Comparisons • Consider Models 1 and 2 with all basic parameters fixed • In reality parameters will change, as information increases customer satisfaction, arrival rates will increase, leading to increase in the number of servers, leading to higher service satisfaction • Use existing tools for comparison – Likelihood ratio ordering Numerical Example • • Economies of scale: All performance measures improve as s increases Two systems do not differ much, differences reduce as s gets larger Critique • No clear literature reviews and contributions • Assume first paper? • Use of k as system state and others..confusing