# Optimal control of queueing systems with non-collaborating servers

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## Abstract

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.

## Keywords

Tandem queueing networks Flexible servers Non-collaborative systems Finite buffers Throughput optimality Markov decision processes## Mathematics Subject Classification

90B22 90C40## Notes

### Acknowledgments

This work was supported by the National Science Foundation under Grant CMMI-0856600. The research of the third author was also supported by the National Science Foundation under Grant CMMI-0969747. The authors thank the associate editor and two anonymous referees for their helpful comments and suggestions.

## References

- 1.Ahn, H., Duenyas, I., Lewis, M.: Optimal control of a two-stage tandem queuing system with flexible servers. Probab. Eng. Inf. Sci.
**16**(4), 453–469 (2002)CrossRefGoogle Scholar - 2.Ahn, H., Duenyas, I., Zhang, R.Q.: Optimal stochastic scheduling of a two-stage tandem queue with parallel servers. Adv. Appl. Probab.
**31**(4), 1095–1117 (1999)CrossRefGoogle Scholar - 3.Ahn, H., Duenyas, I., Zhang, R.Q.: Optimal control of a flexible server. Adv. Appl. Probab.
**36**(1), 139–170 (2004)CrossRefGoogle Scholar - 4.Ahn, S., Righter, R.: Dynamic load balancing with flexible workers. Adv. Appl. Probab.
**38**(3), 621–642 (2006)CrossRefGoogle Scholar - 5.Andradóttir, S., Ayhan, H.: Throughput maximization for tandem lines with two stations and flexible servers. Oper. Res.
**53**(3), 516–531 (2005)CrossRefGoogle Scholar - 6.Andradóttir, S., Ayhan, H., Down, D.G.: Server assignment policies for maximizing the steady-state throughput of finite queueing systems. Manag. Sci.
**47**(10), 1421–1439 (2001)CrossRefGoogle Scholar - 7.Andradóttir, S., Ayhan, H., Down, D.G.: Dynamic server allocation for queueing networks with flexible servers. Oper. Res.
**51**(6), 952–968 (2003)CrossRefGoogle Scholar - 8.Andradóttir, S., Ayhan, H., Down, D.G.: Dynamic assignment of dedicated and flexible servers in tandem lines. Probab. Eng. Inf. Sci.
**21**(4), 497–538 (2007)CrossRefGoogle Scholar - 9.Andradóttir, S., Ayhan, H., Down, D.G.: Queueing systems with synergistic servers. Oper. Res.
**59**(3), 772–780 (2011)CrossRefGoogle Scholar - 10.Andradóttir, S., Ayhan, H., Down, D.G.: Optimal assignment of servers to tasks when collaboration is inefficient. Queueing Syst.
**75**(1), 79–110 (2013)CrossRefGoogle Scholar - 11.Argon, N.T., Andradóttir, S.: Partial pooling in tandem lines with cooperation and blocking. Queueing Syst.
**52**(1), 5–30 (2006)CrossRefGoogle Scholar - 12.Arumugam, R., Mayorga, M.E., Taaffe, K.M.: Inventory based allocation policies for flexible servers in serial systems. Ann. Oper. Res.
**172**(1), 1–23 (2009)CrossRefGoogle Scholar - 13.Bartholdi, J., Eisenstein, D.: A production line that balances itself. Oper. Res.
**44**(1), 21–34 (1996)CrossRefGoogle Scholar - 14.Brown, G.G., Geoffrion, A.M., Bradley, G.H.: Production and sales planning with limited shared tooling at the key operation. Manag. Sci.
**27**(3), 247–259 (1981)Google Scholar - 15.Gargeya, V.B., Deane, R.H.: Scheduling in the dynamic job shop under auxiliary resource constraints: a simulation study. Int. J. Prod. Res.
**37**(12), 2817–2834 (1999)CrossRefGoogle Scholar - 16.Hasenbein, J.J., Kim, B.: Throughput maximization for two station tandem systems: a proof of the Andradóttir-Ayhan conjecture. Queueing Syst.
**67**(4), 365–386 (2011)CrossRefGoogle Scholar - 17.Hillier, F.S., Boling, R.W.: The effect of some design factors on the efficiency of production lines with variable operation times. J. Ind. Eng.
**17**(1), 651–657 (1966)Google Scholar - 18.Hopp, W.J., Tekin, E., Van Oyen, M.P.: Benefits of skill chaining in serial production lines with cross-trained workers. Manag. Sci.
**50**(1), 83–98 (2004)CrossRefGoogle Scholar - 19.Hopp, W.J., Van Oyen, M.P.: Agile workforce evaluation: a framework for cross-training and coordination. IIE Trans.
**36**(10), 919–940 (2004)CrossRefGoogle Scholar - 20.Kim, H.W., Yu, J.M., Kim, J.S., Doh, H.H., Lee, D.H., Nam, S.H.: Loading algorithms for flexible manufacturing systems with partially grouped unrelated machines and additional tooling constraints. Int. J. Adv. Manuf. Technol.
**58**(5), 683–691 (2012)CrossRefGoogle Scholar - 21.Mandelbaum, A., Stolyar, A.L.: Scheduling flexible servers with convex delay costs: heavy-traffic optimality of the generalized \(c\mu \)-rule. Oper. Res.
**52**(6), 836–855 (2004)CrossRefGoogle Scholar - 22.Mayorga, M.E., Taaffe, K.M., Arumugam, R.: Allocating flexible servers in serial systems with switching costs. Ann. Oper. Res.
**172**(1), 231–242 (2009)CrossRefGoogle Scholar - 23.Puterman, M.L.: Markov Decision Processes. Wiley, New York (1994)CrossRefGoogle Scholar
- 24.Schiefermayr, K., Weichbold, J.: A complete solution for the optimal stochastic scheduling of a two-stage tandem queue with two flexible servers. J. Appl. Probab.
**42**(3), 778–796 (2005)CrossRefGoogle Scholar - 25.Sennott, L.I., Van Oyen, M.P., Iravani, S.: Optimal dynamic assignment of a flexible worker on an open production line with specialists. Eur. J. Oper. Res.
**170**(2), 541–566 (2006)Google Scholar - 26.Tsai, Y.C., Argon, N.T.: Dynamic server assignment policies for assembly-type queues with flexible servers. Nav. Rese. Logist.
**55**(3), 234–251 (2008)CrossRefGoogle Scholar - 27.Van Oyen, M.P., Gel, E., Hopp, W.J.: Performance opportunity for workforce agility in collaborative work systems. IIE Trans.
**33**(9), 761–777 (2001)CrossRefGoogle Scholar