Modeling Load and Overwork Effects in Queueing Systems with Adaptive Service Rates

Servers in many real queueing systems do not work at a constant speed. They adapt to the system state by speeding up when the system is highly loaded or slowing down when load has been high for an extended time period. Their speed can also be constrained by other factors, such as geography or a downstream blockage. We develop a state-dependent queueing model in which the service rate depends on the system “load” and “overwork.” Overwork refers to a situation where the system has been under a heavy load for an extended time period. We quantify load as the number of users in the system, and we operationalize overwork with a state variable that is incremented with each service completion in a high-load period and decremented at a rate that is proportional to the number of idle servers during low-load periods. Our model is a quasi-birth-and-death process with a special structure that we exploit to develop efficient and easy-to-implement algorithms to compute system performance measures. We use the analytical ...

[1]  Arie Harel,et al.  Sharp and simple bounds for the Erlang delay and loss formulae , 2010, Queueing Syst. Theory Appl..

[2]  Jillian A. Berry Jaeker,et al.  Hurry Up and Wait: Differential Impacts of Congestion, Bottleneck Pressure, and Predictability on Patient Length of Stay , 2012 .

[3]  Avishai Mandelbaum,et al.  ON PATIENT FLOW IN HOSPITALS: A DATA-BASED QUEUEING-SCIENCE PERSPECTIVE , 2015 .

[4]  Christian Terwiesch,et al.  An Econometric Analysis of Patient Flows in the Cardiac Intensive Care Unit , 2012, Manuf. Serv. Oper. Manag..

[5]  Carri W. Chan,et al.  When to use Speedup : An Examination of Intensive Care Units with Readmissions , 2011 .

[6]  Robert J. Batt,et al.  Doctors Under Load : An Empirical Study of State-Dependent Service Times in Emergency Care , 2012 .

[7]  David Y. Sze,et al.  OR Practice - A Queueing Model for Telephone Operator Staffing , 1984, Oper. Res..

[8]  Wallace J. Hopp,et al.  Operations Systems with Discretionary Task Completion , 2006, Manag. Sci..

[9]  Christian Terwiesch,et al.  The Impact of Work Load on Service Time and Patient Safety: An Econometric Analysis of Hospital Operations , 2009, Manag. Sci..

[10]  Stephen G. Powell,et al.  Throughput in Serial Lines with State-Dependent Behavior , 2004, Manag. Sci..

[11]  Carl M. Harris,et al.  Queues with State-Dependent Stochastic Service Rates , 1967, Oper. Res..

[12]  Adam Wierman,et al.  Routing and staffing when servers are strategic , 2014, Oper. Res..

[13]  Serguei Netessine,et al.  When Does the Devil Make Work? An Empirical Study of the Impact of Workload on Worker Productivity , 2014, Manag. Sci..

[14]  J. Michael Harrison,et al.  Dynamic Control of a Queue with Adjustable Service Rate , 2001, Oper. Res..

[15]  Jing Dong,et al.  Service Systems with Slowdowns: Potential Failures and Proposed Solutions , 2015, Oper. Res..

[16]  Peter D. Welch,et al.  On a Generalized M/G/1 Queuing Process in Which the First Customer of Each Busy Period Receives Exceptional Service , 1964 .

[17]  K. Moinzadeh,et al.  The Impact of Discharge Decisions on Health Care Quality , 1998 .

[18]  Johan van Leeuwaarden,et al.  Triangular M/G/1-Type and Tree-Like Quasi-Birth-Death Markov Chains , 2011, INFORMS J. Comput..

[19]  Patrick T. Harker,et al.  Modeling a Phone Center: Analysis of a Multichannel, Multiresource Processor Shared Loss System , 2001, Manag. Sci..

[20]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[21]  Peter L. Rogers,et al.  Measurement and Control , 1993 .

[22]  W. Grassmann.  The convexity of the mean queue size of the M/M/c queue with respect to the traffic intensity , 1983, Journal of Applied Probability.

[23]  Carri W. Chan,et al.  When to Use Speedup: An Examination of Service Systems with Returns , 2014, Oper. Res..

[24]  Dennis C. Dietz.  Practical scheduling for call center operations , 2011 .

[25]  David Y. Sze,et al.  A Queueing Model for Telephone Operator Staffing , 2016 .

[26]  Charles M. Grinstead,et al.  Introduction to probability , 1999, Statistics for the Behavioural Sciences.

[27]  Carri W. Chan,et al.  The Impact of Delays on Service Times in the Intensive Care Unit , 2017, Manag. Sci..

[28]  Tom Burr,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 2001, Technometrics.

[29]  Galit B. Yom-Tov,et al.  Slowdown Services: Staffing Service Systems with Load-Dependent Service Rate , 2013 .

[30]  Ivo J. B. F. Adan,et al.  The snowball effect of customer slowdown in critical many-server systems , 2015, 1502.02856.

[31]  S. L. Hung,et al.  Development of a workforce management system for a customer hotline service , 2000, Comput. Oper. Res..

[32]  James R. Jackson,et al.  Jobshop-Like Queueing Systems , 2004, Manag. Sci..

[33]  T. B. Crabill.  Optimal Control of a Service Facility with Variable Exponential Service Times and Constant Arrival Rate , 1972 .

[34]  Richard Bellman,et al.  Adaptive Control Processes: A Guided Tour , 1961, The Mathematical Gazette.

[35]  Elwood S. Buffa,et al.  AN INTEGRATED WORK SHIFT SCHEDULING SYSTEM , 1976 .

[36]  V. Ramaswami,et al.  Stationary waiting time distribution in queues with phase type service and in quasi-birth-and-death process , 1985, STOC 1985.

[37]  Winfried K. Grassmann.  Transient solutions in markovian queueing systems , 1977, Comput. Oper. Res..

[38]  M. J. Lopez-Herrero,et al.  Analysis of the busy period for the M/M/c queue: an algorithmic approach , 2001, Journal of Applied Probability.

[39]  R StaatsBradley,et al.  Specialization and Variety in Repetitive Tasks , 2012 .

[40]  Ling Tang,et al.  Efficient and Reliable Computation of Birth-Death Process Performance Measures , 2012, INFORMS J. Comput..

[41]  Leslie C. Edie,et al.  Traffic Delays at Toll Booths , 1954, Oper. Res..

[42]  Jorge Nocedal,et al.  Knitro: An Integrated Package for Nonlinear Optimization , 2006 .

[43]  Robert J. Batt,et al.  Early Task Initiation and Other Load-Adaptive Mechanisms in the Emergency Department , 2017, Manag. Sci..

[44]  Stefan Scholtes,et al.  Stress on the Ward: Evidence of Safety Tipping Points in Hospitals , 2015, Manag. Sci..

[45]  Avishai Mandelbaum,et al.  Service times in call centers: Agent heterogeneity and learning with some operational consequences , 2010 .

[46]  J.S.H. van Leeuwaarden,et al.  Quasi-Birth-and-Death Processes with an Explicit Rate Matrix , 2006 .

[47]  Peng Sun,et al.  Diagnostic Accuracy Under Congestion , 2013, Manag. Sci..

[48]  Bradley R. Staats,et al.  Specialization and Variety in Repetitive Tasks: Evidence from a Japanese Bank , 2011, Manag. Sci..

[49]  Markus Schwaninger,et al.  Measurement and control of business processes , 2001 .