Presentation based on:Whitt, W. "Improving Service by Informing Customers About Anticipated Delays." Management Science 45 (2MotivationTwo alternative queuing systemsM/M/s/r Model 1Model 1M/M/s/r Model 2Model 2Model 2Model 2Finding Performance MeasuresStochastic ComparisonsLikelihood Ratio OrderingStochastic ComparisonNumerical ExampleCritiquePresentation based on:Whitt, W. "Improving Service by Informing Customers About Anticipated Delays." Management Science 45 (2), 1999.Presentation by Huanran WangThis summary presentation is based on: Whitt, Ward. "Improving Service by Informing Customers About Anticipated Delays." Management Science 45, no. 2 (1999): 192-207.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 providersTwo 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 1s serversexponential svc time with mean μ^-1 Poisson Arrival, λr waiting spaceFCFSSystem state not known by customersIndependent α and βIf a server is not immediately available customer balk with probability βThen, customer waits till T is reached before renegingÆ Model with time dependent renegingModel 1– (See explanation and variable definitions in section 2, page 194 of the Whitt paper.)– Characterize by– Pk state probabilities are easy to calculate!01(1 ) 111()kkkssksrkkssks sksrλλλβµµµα≤≤−⎧=⎨−≤≤+−⎩≤≤−⎧=⎨+−≤≤+⎩M/M/s/r Model 2s serversexponential svc time with mean μ^-1 Poisson Arrival, λr waiting spaceFCFSSystem state now communicated to customers upon arrivalDependent α and βBalking now depends on state of systemState dependent balking replaces reneging after waitingÆ Model with mainly state dependent balking plus some renegingModel 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 probabilityWhere Sk : time from arrival until first served where state at time is k. ()0 1kkqPTS kr≡> ≤≤−Model 2• To find the state dependent probability of joining in an exact manner,• To find a reasonable approximation of the state dependent probability of joining,10()kkStkksqegtdtEesααµµα+∞−−⎛⎞===⎜⎟+⎝⎠∫(1)/() ,0kskkqPTES e kαµ−+≡> = ≥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,• BD process can be characterized by01(1 ) 111kkskksksqsksrkksssksrλλλβµµµδ−−≤≤−⎧=⎨−≤≤+−⎩≤≤⎧=⎨++≤ ≤+⎩1kkjjδδ=′=∑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 (see Shakedand Shantikumar 1994)– Likelihood ratio orderingLikelihood Ratio Ordering(See section 4, pages 199-200 of the Whitt paper, particularly the explanation surrounding equations 4.1 and 4.2)Stochastic Comparison(See Theories 4.1, 4.2, 4.3, and 4.4 on pages 200-1 of the Whitt paper)Numerical Example• Economies of scale: All performance measures improve as s increases• Two systems do not differ much, differences reduce as s gets larger(See Table 1 on page 202 of the Whitt paper)Critique• No clear literature reviews and contributions• Assume first paper?• Use of k as system state and