### Abstract

We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

Original language | English |
---|---|

Pages (from-to) | 944-967 |

Number of pages | 24 |

Journal | Journal of Applied Probability |

Volume | 55 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sep 2018 |

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### Keywords

- Customer routeing
- dynamic programming
- fluid queue
- individual optimum, social optimum
- linear quadratic regulator
- Nash equilibrium
- polling system
- Ricatti equation

### Cite this

*Journal of Applied Probability*,

*55*(3), 944-967. https://doi.org/10.1017/jpr.2018.59

}

*Journal of Applied Probability*, vol. 55, no. 3, pp. 944-967. https://doi.org/10.1017/jpr.2018.59

**Optimal routeing in two-queue polling systems.** / Adan, I. J.B.F.; Kulkarni, V. G.; Lee, N.; Lefeber, E.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Optimal routeing in two-queue polling systems

AU - Adan, I. J.B.F.

AU - Kulkarni, V. G.

AU - Lee, N.

AU - Lefeber, E.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

AB - We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

KW - Customer routeing

KW - dynamic programming

KW - fluid queue

KW - individual optimum, social optimum

KW - linear quadratic regulator

KW - Nash equilibrium

KW - polling system

KW - Ricatti equation

UR - http://www.scopus.com/inward/record.url?scp=85056763501&partnerID=8YFLogxK

U2 - 10.1017/jpr.2018.59

DO - 10.1017/jpr.2018.59

M3 - Article

AN - SCOPUS:85056763501

VL - 55

SP - 944

EP - 967

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 3

ER -