On open problems in polling systems

In the present paper we address two open problems concerning polling systems, viz., queueing systems consisting of multiple queues attended by a single server that visits the queues one at a time. The first open problem deals with a system consisting of two queues, one of which has gated service, while the other receives 1-limited service. The second open problem concerns polling systems with general (renewal) arrivals and deterministic switch-over times that become infinitely large. We discuss related, known results for both problems, and the difficulties encountered when trying to solve them.

[1]  R. D. van der Mei,et al.  Delay in polling systems with large switch-over times , 1999 .

[2]  Lajos Takács.  Two Queues Attended by a Single Server , 1968, Oper. Res..

[3]  H. Saunders,et al.  Probability, Random Variables and Stochastic Processes (2nd Edition) , 1989 .

[4]  Hideaki Takagi,et al.  Queuing analysis of polling models , 1988, CSUR.

[5]  Oliver C. Ibe,et al.  Analysis of polling systems with mixed service disciplines , 1990 .

[6]  Geert-Jan van Houtum,et al.  A STATE-DEPENDENT POLLING MODEL WITH k-LIMITED SERVICE , 2009, Probability in the Engineering and Informational Sciences.

[7]  I. N. Sneddon,et al.  Boundary value problems , 2007 .

[8]  Toshihisa Ozawa,et al.  Alternating Service Queues with Mixed Exhaustive and K-Limited Services , 1990, Perform. Evaluation.

[9]  Sem C. Borst,et al.  Polling Models With and Without Switchover Times , 1997, Oper. Res..

[10]  Emm Erik Winands,et al.  Polling, production & priorities , 2007 .

[11]  D. Lee.  A two-queue model with exhaustive and limited service disciplines , 1996 .

[12]  Jacques Resing,et al.  Polling systems and multitype branching processes , 1993, Queueing Syst. Theory Appl..

[13]  W. D. Ray,et al.  Stochastic Models: An Algorithmic Approach , 1995 .

[14]  Chatschik Bisdikian.  A Queueing Model with Applications to Bridges and the DQDB (IEEE 802.6) MAN , 1993, Comput. Networks ISDN Syst..

[15]  Erik M. M. Winands,et al.  Branching-type polling systems with large setups , 2011, OR Spectr..

[16]  Onno J. Boxma,et al.  Workloads and waiting times in single-server systems with multiple customer classes , 1989, Queueing Syst. Theory Appl..

[17]  Duan Shin Lee.  ANALYSIS OF A TWO QUEUE MODEL WITH BERNOULLI SCHEDULES , 2003 .

[18]  Athanasios Papoulis,et al.  Probability, Random Variables and Stochastic Processes , 1965 .

[19]  V. M. Vishnevskii,et al.  Mathematical methods to study the polling systems , 2006 .

[20]  Ivo J. B. F. Adan,et al.  Closed-form waiting time approximations for polling systems , 2011, Perform. Evaluation.

[21]  Erik M. M. Winands,et al.  On polling systems with large setups , 2007, Oper. Res. Lett..

[22]  Toshihisa Ozawa,et al.  Waiting time distributions in a two-queue model with mixed exhaustive and gated-type K-limited services , 1997, PMCCN.

[23]  Robert D. van der Mei,et al.  Towards a unifying theory on branching-type polling systems in heavy traffic , 2007, Queueing Syst. Theory Appl..

[24]  Moshe Sidi,et al.  Polling systems: applications, modeling, and optimization , 1990, IEEE Trans. Commun..

[25]  T. Olsen.  LIMIT THEOREMS FOR POLLING MODELS WITH INCREASING SETUPS , 2001, Probability in the Engineering and Informational Sciences.

[26]  Onno Boxma,et al.  Two queues with alternating service and switching times , 1987 .

[27]  Onno Boxma,et al.  Boundary value problems in queueing system analysis , 1983 .