Service Performance Analysis and Improvement for a Ticket Queue with Balking Customers. Long Gao. joint work with Jihong Ou and Susan Xu

Transcription

1 Service Performance Analysis and Improvement for a Ticket Queue with Balking Customers joint work with Jihong Ou and Susan Xu THE PENNSYLVANIA STATE UNIVERSITY MSOM, Atlanta June 20, 2006

2 Outine Introduction 1 Introduction Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions 2 3 Comparison of Ticket and Physical Queues Managerial Insights 4

3 Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions Physical Queue: To Queue or Not to Queue? Physical Queues Physical queues can be really stressful and exhausting.

4 What Is a Ticket Queue? Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions Examples banks, hospitals, restaurants(page or beeper), Disney theme park(pager), Department of Motor Vehicles (DMV), the Immigration Department...

5 Advantages of Ticket Queues Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions For customers, reduce stress level, more comfortable environment eliminate cut-in-front-of-line phenomenon better security and privacy For service provider, improve customer satisfaction, increase market share reduce staff stress level, increase their efficiency combined with CRM to generate more sales (pharmacies) collect business intelligence

6 Problems of Ticket Queues Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions However, balking occurs frequently if the job is not time-sensitive or can be done elsewhere Balking customers create empty ticket positions Because there is no visible waiting lines, neither the system nor other customers know until the numbers are called Customers nature tendency is to assume all positions are real, ignore balking customers, overestimate their waiting time Problems Customers: lower service rate; Management: underutilization of service capacity.

7 Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions Difference between Physical and Ticket Queues Physical queues: complete information of N (actual number of customers) observe N by simple counting joining or balking based on the queueing position N Ticket queues: partial information of N observe the ticket number and panel display number, compute the ticket position D, D = ticket number panel display number joining or balking based on the ticket position D, an upper bound of N D is observable and N is unobservable

8 Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions Industry Practices and Academic Research Several companies develop and sell ticket queue management hardware and software: Q-Matic: Customer Flow Management, sold 30, 000 installations Q-Nomy: Q-Flow Systems Surprisingly, we are not aware of academic literature analyzing ticket queue systems It is perceived that the ticket queue and the physical queue behave the same

9 Motivations & Research Questions Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions Extensive research on queues with balking customers that have no information or complete information of N ; no research on ticket queues that has partial information of N How to model and evaluate ticket queues? distributions of D and N balking rate and system utilization E[N D], conditional expected waiting time (given his ticket position) What is the performance difference between the ticket queues and physical queues? What is the value of providing customers with better information? How to improve the service performance?

10 Outine Introduction 1 Introduction Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions 2 3 Comparison of Ticket and Physical Queues Managerial Insights 4

11 Assumptions Introduction Single server Poison arrivals with rate λ Exponential service time with rate µ for a joining customer Service time of a balking customer is 0 Balk if D K, where K is customer s patience level Challenge We need a model that carries complete information of both D and N.

12 A natural way is to represent system states by binary sequences: e.g., x = (1, 1, 1, 0, 0, 0, 1, 0) Problem: not workable, dimension can be arbitrarily long Our compact way to model states: x = (1, 1, 1, 0, 0, 0, 1, 0 ) } {{ } }{{} 4 2 = (1, 1, 4, 2) = (n 1, n 2, n 3, n 4 )

13 Formal Definition Introduction L = actual number of customers in the system; L is a realization of N n l =number of tickets issued between l th to (l + 1) th actual customers, l = 1, 2,..., L; State of the Markov chain is an L dimensional vector: S = {0} { } L 1 (n 1,..., n L ) N L : n l < K, n l 1, L = 1,..., K l=1

14 Example: Ticket Queue with K = 4 Challenge: infinite number of states, no close form solutions!

15 A Two-Step Solution Procedure Step 1. Aggregate all states with x = L and n L K into a super state S L. MC with super states has a finite state space and can be modeled as a quasi birth-death (QBD) process and admits matrix product form solution. Step 2. Back to the original MC obtain steady state probabilities of states x = L and n L K recursively, using the known probabilities derived in Step 1.

16 Curse of Dimensionality Unfortunately, the cardinality of state space grows exponentially! For example, for K = 10, the cardinality of state space is 4619; for K = 20, it is nearly 10 million! We need to develop a heuristic method that is computationally efficient while giving a good approximation of the ticket queue

17 Reason: intermixing of joining/balking customers State reduction idea: Separate joining and balking customers into two queues Only keep track of total number of customers in each queue Give joining customers higher priority Example: for D = 5 and N = 3, {113, 122, 131, 212, 221, 311} = {32} Condition for similar stochastic behavior: intermixing of joining and balking customers occurs rarely in the ticket queue Such phenomena were indeed observed in our simulations

18 Approximation: K = 4 = Complexity The approximation system is polynomial O(K 2 ) E.g., for K = 20, reduce by a factor of 29, 000.

19 Accuracy of Approximation P(D): Exact P(D a ): Approximation P b : Exact P b a : Approximation D Dist. of D and D a, K = 9, ρ = 0.1, , K Balking Prob. P b and P a b, ρ = 0.1, 0.3,..., 0.9 Conclusion: Approximation is of high quality in terms of computational efficiency, solution accuracy, and robustness.

20 Outine Introduction Comparison of Ticket and Physical Queues Managerial Insights 1 Introduction Physical Queues and Ticket Queues Industry Practices and Academic Research Motivations & Research Questions 2 3 Comparison of Ticket and Physical Queues Managerial Insights 4

21 Comparison of Ticket and Physical Queues Managerial Insights Comparison of Ticket and Physical Queues We compare balking prob. of physical queue: a new customer balks if N K ticket queue: a new customer balks if D K The physical queue with the same K can be modeled as M/M/1/K queue (complete information) Since D N, ticket queue has a higher balking rate (incomplete information) The performance gap between physical and ticket queues quantifies the impact of information loss in ticket queue

22 Comparison of Ticket and Physical Queues Managerial Insights balking Prob. Difference of Ticket and Physical Queue Balking Rate Diff. (TAD) of Ticket Queue and M/M/1/K Balking rate difference decreases in K for fixed ρ and increases in ρ for fixed K The two systems show most significant differences in balking rate when customers are impatient and traffic is heavy. It can as high as 6%!

23 ρ Introduction Comparison of Ticket and Physical Queues Managerial Insights Partition of the Parameter Space (K, ρ) Significant 0.03 Moderate 0.01 Insignificant Significant :TAD > 3% Moderate :TAD 1~3% Insignificant :TAD <1% K Balking Rate Diff. (TAD) of Ticket Queue and M/M/1/K Significant region: heavy traffic and impatient customers; impact of info loss is most severe Moderate region: moderate traffic and relatively patient customers; impact of info loss is moderate Insignificant region: light traffic and patient customers, impact of info loss is minimum

24 Managerial Insights Comparison of Ticket and Physical Queues Managerial Insights While the Immigration Department is a case in insignificant region, banks and DMVs are more likely to be in significant or moderate region, and therefore require effective management of the ticket queues In the stochastic sense, the ticket queue is actually less crowed than the physical queue by the head count, but appears busier by the ticket count The ticket queue tends to exacerbate the already poor service of the physical queue To influence customer behavior and improve service, the hidden information must be communicated to customers in a clear and quantifiable way

25 Improvement to Ticket Queue Balking Rate: Before and After Improvement P a b vs. P b Ticket Plus Queue M/M/1/K K, ρ={0.1,0.3,0.5,0.7,0.9,1} Comparison of Ticket and Physical Queues Managerial Insights Improvement: provide E[W D], the conditional expected waiting time based on D, his ticket position New information corrects over estimation E[N D = d]/µ d µ The new patience level K satisfies E[N D = K ] K: 0.3 Ticket Plus Queue: Approximation M/M/1/K K K K Conclusion: improved system and physical queue perform virtually identically in several first moment measures (such as expected waiting time)

26 Contributions We introduce the first analytical model of ticket queue We develop efficient and effective evaluation tools that can help management to quantify service performance, benchmark performance gap with physical queue, and implement improvement when it is called for We obtain insights on the impact of information loss in the ticket queue on key service performance measures and propose a remedy to correct it

27 Future Research Modeling: parallel agents, both balking and reneging Improving customer service: nonlinearity disutility (Quality of Service guarantee), reneging (dynamic information) Understanding customer psychology: empirical studies