Improving Service by Informing Customers About Anticipated Delays

This paper investigates the effect upon performance in a service system, such as a telephone call center, of giving waiting customers state information. In particular, the paper studies two M/M/s/r queueing models with balking and reneging. For simplicity, it is assumed that each customer is willing to wait a fixed time before beginning service. However, customers differ, so the delay tolerances for successive customers are random. In particular, it is assumed that the delay tolerance of each customer is zero with probability β, and is exponentially distributed with mean α-1 conditional on the delay tolerance being positive. Let N be the number of customers found by an arrival. In Model 1, no state information is provided, so that if N ≥ s, the customer balks with probability β; if the customer enters the system, he reneges after an exponentially distributed time with mean α-1 if he has not begun service by that time. In Model 2, if N - s + k μ s, then the customer is told the system state k and the remaining service times of all customers in the system, so that he balks with probability β + (1 - β)(1 - qk), where qk = P(T > Sk), T is exponentially distributed with mean α-1, Sk is the sum of k + 1 independent exponential random variables each with mean (sμ)-1, and μ-1 is the mean service time. In Model 2, all reneging is replaced by balking. The number of customers in the system for Model 1 is shown to be larger than that for Model 2 in the likelihood-ratio stochastic ordering. Thus, customers are more likely to be blocked in Model 1 and are more likely to be served without waiting in Model 2. Algorithms are also developed for computing important performance measures in these, and more general, birth-and-death models.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[2]  Carl M. Harris,et al.  Fundamentals of queueing theory , 1975 .

[3]  J. Kalbfleisch Statistical Inference Under Order Restrictions , 1975 .

[4]  L. Delbrouck A Unified Approximate Evaluation of Congestion Functions for Smooth and Peaky Traffics , 1981, IEEE Trans. Commun..

[5]  W. Whitt Comparing counting processes and queues , 1981, Advances in Applied Probability.

[6]  W. Whitt,et al.  Resource sharing for efficiency in traffic systems , 1981, The Bell System Technical Journal.

[7]  Daniel P. Heyman,et al.  Stochastic models in operations research , 1982 .

[8]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[9]  Carl M. Harris,et al.  Fundamentals of queueing theory (2nd ed.). , 1985 .

[10]  W. Whitt,et al.  Blocking when service is required from several facilities simultaneously , 1985, AT&T Technical Journal.

[11]  S. Resnick Extreme Values, Regular Variation, and Point Processes , 1987 .

[12]  Ward Whitt,et al.  Ordinary CLT and WLLN Versions of L = λW , 1988, Math. Oper. Res..

[13]  Ronald W. Wolff,et al.  Stochastic Modeling and the Theory of Queues , 1989 .

[14]  Gennadi Falin,et al.  A survey of retrial queues , 1990, Queueing Syst. Theory Appl..

[15]  P. Kolesar,et al.  The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals , 1991 .

[16]  S. Zachary,et al.  Loss networks , 2009, 0903.0640.

[17]  Randolph W. Hall,et al.  Queueing Methods: For Services and Manufacturing , 1991 .

[18]  W. Whitt The pointwise stationary approximation for M 1 / M 1 / s , 1991 .

[19]  WhittWard The Pointwise Stationary Approximation for Mt/Mt/s Queues Is Asymptotically Correct As the Rates Increase , 1991 .

[20]  Onno Boxma,et al.  Multiserver queues with impatient customers , 1993 .

[21]  Shirley Taylor Waiting for Service: The Relationship between Delays and Evaluations of Service , 1994 .

[22]  Moshe Shaked,et al.  Stochastic orders and their applications , 1994 .

[23]  Kin K. Leung,et al.  An inversion algorithm to compute blocking probabilities in loss networks with state-dependent rates , 1995, TNET.

[24]  Ward Whitt,et al.  Sensitivity to the Service-Time Distribution in the Nonstationary Erlang Loss Model , 1995 .

[25]  Ward Whitt,et al.  Numerical Inversion of Laplace Transforms of Probability Distributions , 1995, INFORMS J. Comput..

[26]  Ward Whitt,et al.  Stationary-Process Approximations for the Nonstationary Erlang Loss Model , 1996, Oper. Res..

[27]  Michael K. Hui,et al.  What to Tell Consumers in Waits of Different Lengths: An Integrative Model of Service Evaluation , 1996 .

[28]  Keith W. Ross,et al.  Multiservice Loss Models for Broadband Telecommunication Networks , 1997 .

[29]  Ward Whitt,et al.  Control and recovery from rare congestion events in a large multi-server system , 1997, Queueing Syst. Theory Appl..

[30]  Ward Whitt,et al.  Predicting Queueing Delays , 1999 .