Optimal control of queueing systems with non-collaborating servers

We study the dynamic server allocation problem for tandem queueing systems with an equal number of stations and servers. The servers are flexible, yet non-collaborative, so that at most one server can work at a station at any time. The objective is to maximize the long-run average throughput. We show that if each server is the fastest at one station, then a dedicated server assignment policy is optimal for systems of arbitrary size and with general service requirement distributions. Otherwise, the optimal policy is more complex as servers must divide their time between stations. For Markovian systems with two stations and two servers, we characterize the optimal policy completely. For larger Markovian systems, we use our results for two-station systems to propose four heuristic server assignment policies and provide computational results that show that our heuristics are near-optimal. We also compare collaborative and non-collaborative settings to evaluate the benefits of dynamic server allocation, as opposed to collaboration, in systems with flexible servers. We conclude that the loss in the long-run average throughput due to lack of collaboration is mitigated by the similarity of the tasks in the system, and cross-training can still be beneficial in non-collaborative systems.

[1]  Li Zhuang,et al.  Approximate decomposition for closed queueing network models of FMSs with a block-and-wait and state-dependent routing mechanism , 1993 .

[2]  Sigrún Andradóttir,et al.  Dynamic Server Allocation for Queueing Networks with Flexible Servers , 2003, Oper. Res..

[3]  Saif Benjaafar,et al.  Modeling and analysis of flexible queueing systems , 2004 .

[4]  J. George Shanthikumar,et al.  Optimal server allocation in a system of multi-server stations , 1987 .

[5]  Zvi Rosberg,et al.  Optimal control of service in tandem queues , 1982 .

[6]  Sigrún Andradóttir,et al.  Dynamic server allocation for unstable queueing networks with flexible servers , 2012, Queueing Syst. Theory Appl..

[7]  Sigrún Andradóttir,et al.  Optimal assignment of servers to tasks when collaboration is inefficient , 2013, Queueing Syst. Theory Appl..

[8]  Edward G. Coffman,et al.  Feedback Queueing Models for Time-Shared Systems , 1968, J. ACM.

[9]  Sigrún Andradóttir,et al.  Profit maximization in flexible serial queueing networks , 2014, Queueing Syst. Theory Appl..

[10]  Leonard Kleinrock,et al.  Time-shared Systems: a theoretical treatment , 1967, JACM.

[11]  T. M. Farrar.  Optimal use of an extra server in a two station tandem queueing network , 1993 .

[12]  Wallace J. Hopp,et al.  Performance Opportunity for Workforce Agility in Collaborative and Noncollaborative Work Systems , 2001 .

[13]  Andreas Brandt,et al.  Waiting times for M/M systems under state-dependent processor sharing , 2008, Queueing Syst. Theory Appl..

[14]  Bara Kim,et al.  Throughput maximization for two station tandem systems: a proof of the Andradóttir–Ayhan conjecture , 2011, Queueing Syst. Theory Appl..

[15]  Donald D. Eisenstein,et al.  A Production Line that Balances Itself , 1996, Oper. Res..

[16]  Diane P. Bischak.  Performance of a manufacturing module with moving workers , 1996 .

[17]  Urtzi Ayesta,et al.  Sojourn times in a processor sharing queue with multiple vacations , 2012, Queueing Systems.

[18]  Scott Jordan,et al.  Throughput in Processor-Sharing Queues , 2007, IEEE Transactions on Automatic Control.

[19]  Sigrún Andradóttir,et al.  Dynamic assignment of dedicated and flexible servers in tandem lines , 2007 .

[20]  Walter W Garvin,et al.  Introduction to Linear Programming , 2018, Linear Programming and Resource Allocation Modeling.

[21]  Vidyaranya B. Gargeya,et al.  Scheduling in the dynamic job shop under auxiliary resource constraints: A simulation study , 1999 .

[22]  B. Hajek.  Optimal control of two interacting service stations , 1982, 1982 21st IEEE Conference on Decision and Control.

[23]  Izak Duenyas,et al.  Optimal stochastic scheduling of a two-stage tandem queue with parallel servers , 1999, Advances in Applied Probability.

[24]  Wallace J. Hopp,et al.  Agile workforce evaluation: a framework for cross-training and coordination , 2004 .

[25]  U. Rieder,et al.  Markov Decision Processes , 2010 .

[26]  Wayne J. Davis,et al.  Scheduling a Flexible Manufacturing System with Tooling Constraints: An Actual Case Study , 1995 .

[27]  Izak Duenyas,et al.  Optimal control of a flexible server , 2004, Advances in Applied Probability.

[28]  K. Glazebrook.  On stochastic scheduling with precedence relations and switching costs , 1980, Journal of Applied Probability.

[29]  Ger Koole,et al.  Asigning a Single Server to Inhomogeneous Queues with Switching Costs , 1997, Theor. Comput. Sci..

[30]  Qian Wang,et al.  Balancing staffing and switching costs in a service center with flexible servers , 2007, Eur. J. Oper. Res..

[31]  Alexander L. Stolyar,et al.  Scheduling Flexible Servers with Convex Delay Costs: Heavy-Traffic Optimality of the Generalized cµ-Rule , 2004, Oper. Res..

[32]  Gerald G. Brown,et al.  Production and Sales Planning with Limited Shared Tooling at the Key Operation. Revision. , 1980 .

[33]  Isi Mitrani,et al.  Sharing a Processor Among Many Job Classes , 1980, JACM.

[34]  Victor Pestien,et al.  Asymptotic Behavior of Large Discrete-Time Cyclic Queueing Networks , 1994 .

[35]  Hyoung-Ho Doh,et al.  Loading algorithms for flexible manufacturing systems with partially grouped unrelated machines and additional tooling constraints , 2012 .

[36]  Sigrún Andradóttir,et al.  Technical Note - Queueing Systems with Synergistic Servers , 2011, Oper. Res..

[37]  Saligrama R. Agnihothri,et al.  Workforce cross-training decisions in field service systems with two job types , 2003, J. Oper. Res. Soc..

[38]  Josef Weichbold,et al.  A complete solution for the optimal stochastic scheduling of a two-stage tandem queue with two flexible servers , 2005 .

[39]  Maria E. Mayorga,et al.  Allocating flexible servers in serial systems with switching costs , 2009, Ann. Oper. Res..

[40]  Dimitrios G. Pandelis.  OPTIMAL CONTROL OF FLEXIBLE SERVERS IN TWO TANDEM QUEUES WITH OPERATING COSTS , 2007, Probability in the Engineering and Informational Sciences.

[41]  Saifallah Benjaafar,et al.  Machine Sharing in Manufacturing Systems: Total Flexibility versus Chaining , 1998 .

[42]  Sigrún Andradóttir,et al.  Flexible servers in tandem lines with setup costs , 2012, Queueing Syst. Theory Appl..

[43]  Arne A. Nilsson,et al.  On Optimal Scheduling Algorithms for Time-Shared Systems , 1981, JACM.

[44]  Diwakar Gupta,et al.  Control of a Single-Server Tandem Queueing System with Setups , 1998, Oper. Res..

[45]  Izak Duenyas,et al.  Stochastic scheduling of parallel queues with set-up costs , 1995, Queueing Syst. Theory Appl..

[46]  James MacGregor Smith,et al.  Queue decomposition & finite closed queueing network models , 2015, Comput. Oper. Res..

[47]  Seyed M. R. Iravani,et al.  Optimal dynamic assignment of a flexible worker on an open production line with specialists , 2006, Eur. J. Oper. Res..

[48]  Sigrún Andradóttir,et al.  Server Assignment Policies for Maximizing the Steady-State Throughput of Finite Queueing Systems , 2001, Manag. Sci..

[49]  Mark E. Lewis,et al.  OPTIMAL CONTROL OF A TWO-STAGE TANDEM QUEUING SYSTEM WITH FLEXIBLE SERVERS , 2002 .

[50]  Sigrún Andradóttir,et al.  Partial Pooling in Tandem Lines with Cooperation and Blocking , 2006, Queueing Syst. Theory Appl..

[51]  Maria E. Mayorga,et al.  Inventory based allocation policies for flexible servers in serial systems , 2009, Ann. Oper. Res..

[52]  Yi-Chun Tsai,et al.  Dynamic server assignment policies for assembly‐type queues with flexible servers , 2008 .

[53]  Sem C. Borst,et al.  The equivalence between processor sharing and service in random order , 2003, Oper. Res. Lett..

[54]  Sigrún Andradóttir,et al.  Throughput Maximization for Tandem Lines with Two Stations and Flexible Servers , 2005, Oper. Res..

[55]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[56]  Sigrún Andradóttir,et al.  Queueing Systems with Synergistic Servers , 2011 .

[57]  Yi-Chun Tsai,et al.  Dynamic control of a flexible server in an assembly-type queue with setup costs , 2012, Queueing Syst. Theory Appl..

[58]  Sigrún Andradóttir,et al.  Design Principles for Flexible Systems , 2013 .

[59]  Sigrún Andradóttir,et al.  Compensating for Failures with Flexible Servers , 2007, Oper. Res..

[60]  Gerard Hooghiemstra,et al.  The M/G/1 processor sharing queue as the almost sure limit of feedback queues , 1990 .

[61]  John A. Buzacott,et al.  A two-stage tandem queue attended by a moving server with holding and switching costs , 1997, Queueing Syst. Theory Appl..

[62]  Wallace J. Hopp,et al.  Benefits of Skill Chaining in Serial Production Lines with Cross-Trained Workers , 2004, Manag. Sci..

[63]  Rhonda Righter,et al.  Dynamic load balancing with flexible workers , 2006, Advances in Applied Probability.

[64]  J. MacGregor Smith,et al.  System capacity and performance modelling of finite buffer queueing networks , 2014 .

[65]  Lawrence M. Wein,et al.  Dynamic Scheduling of a Two-Class Queue with Setups , 2011, Oper. Res..