Abstract
A threshold start-up policy is appealing for manufacturing (service) facilities that incur a cost for keeping the machine (server) on, as well as for each restart of the server from its dormant state. Analysis of single product (customer) systems operating under such a policy, also known as the N-policy, has been available for some time. This article develops mathematical analysis for multiproduct systems operating under a cyclic exhaustive or globally gated service regime and a threshold start-up rule. It pays particular attention to modeling switchover (setup) times. The analysis extends/unifies existing literature on polling models by obtaining as special cases, the continuously roving server and patient server polling models on the one hand, and the standard M/G/1 queue with N-policy, on the other hand. We provide a computationally efficient algorithm for finding aggregate performance measures, such as the mean waiting time for each customer type and the mean unfinished work in system. We show that the search for the optimal threshold level can be restricted to a finite set of possibilities.
Similar content being viewed by others
References
E. Altman, P. Konstantopoulos and Z. Liu, Stability, monotonicity and invariant quantities in general polling systems, Queueing Systems 11 (1992) 35–57.
S.C. Borst, A globally gated polling system with a dormant server, Probab. Engrg. Inform. Sci. 9 (1995) 239–254.
O.J. Boxma and W.P. Groenendijk, Pseudo-conservation laws in cyclic-service systems, J. Appl. Probab. 24 (1987) 949–964.
O.J. Boxma, J.A. Weststrate and U. Yechiali, A globally gated polling system with server interruptions, and applications to the repairman problem, Probab. Engrg. Inform. Sci. 7 (1993) 187–208.
P.J. Burke, Delays in single-server queues with batch input, Oper. Res. 23 (1975) 830–833.
R.B. Cooper, Introduction to Queueing Theory, 3rd ed. (CEE Press, 1990).
M. Eisenberg, The polling system with a stopping server, Queueing Systems 18 (1994) 387–431.
S.W. Fuhrmann and R.B. Cooper, Stochastic decompositions in the M/G/1 queue with generalized vacations, Oper. Res. 33 (1985) 1117–1129.
Y. Günalay and D. Gupta, Threshold start-up control policy for polling systems, Working paper, McMaster University, Hamilton, Ontario, Canada (1996).
D.P. Heyman and M.J. Sobel, Stochastic Models in Operations Research, Vol. I: Stochastic Processes and Operating Characteristics (McGraw-Hill, New York, 1982).
D.P. Heyman and M.J. Sobel, Stochastic Models in Operations Research, Vol. II: Stochastic Optimization (McGraw-Hill, New York, 1984).
A.G. Konheim, H. Levy and M.M. Srinivasan, Descendant set: An efficient approach for the analysis of polling systems, Trans. Commun. 42 (1993) 1245–1253.
Z. Liu, P. Nain and D. Towsley, On optimal polling policies, Queueing Systems 11 (1992) 59–83.
J.A.C. Resing, Polling systems and multitype branching processes, Queueing Systems 13 (1993) 409–426.
W.E. Smith, Various optimizers for single-stage production, Naval Res. Logist. Quart. 3 (1956) 59–66.
M.M. Srinivasan and D. Gupta, When should a server be patient?, Mangm. Sci. 42 (1996) 437–451.
M.M. Srinivasan, S.C. Niu and R.B. Cooper, Relating polling models with nonzero and zero switchover times, Queueing Systems 19 (1995) 149–168.
H. Takagi, Analysis of Polling Systems (MIT Press, Cambridge, MA, 1986).
H. Takagi, Queuing analysis of polling models: An update, in: Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (Elsevier/North-Holland, 1990) pp. 267–318.
H. Takagi, Queueing Analysis of Polling Models: Progress in 1990–1993 (Institute of Socio-Economic Planning, University of Tsukuba, Japan, 1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Günalay, Y., Gupta, D. Threshold start-up control policy for polling systems. Queueing Systems 29, 399–421 (1998). https://doi.org/10.1023/A:1019152601966
Issue Date:
DOI: https://doi.org/10.1023/A:1019152601966