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Queueing systems on a circle. (English) Zbl 0772.60076

Summary: Consider a ring on which customers arrive according to a Poisson process. Arriving customers drop somewhere on the circle and wait there for a server who travels on the ring. Whenever this server encounters a customer, he stops and serves the customer according to an arbitrary service time distribution. After the service is completed, the server removes the client from the circle and resumes his journey.
We are interested in the number and the locations of customers that are waiting for service. These locations are modeled as random counting measures on the circle. Two different types of servers are considered: The polling server and the Brownian (or drunken) server. It is shown that under both server motions the system is stable if the traffic intensity is less than 1. Furthermore, several earlier results on the configuration of waiting customers are extended, by combining results from random measure theory, stochastic integration and renewal theory.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G57 Random measures
60H05 Stochastic integrals
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References:

[1] Asmussen S (1987)Applied Probability and Queues. Wiley & Sons, New York · Zbl 0624.60098
[2] Coflman Jr. EG, Gilbert EN (1986) A Continuous Polling System with Constant Service Times.IEEE Trans. Inf. Th. it-32 (4):584-591
[3] Coffman Jr EG, Gilbert EN (1987) Polling and Greedy Servers on a Line.Queueing Systems 2:115-145 · Zbl 0653.90021 · doi:10.1007/BF01158396
[4] Daley DJ, Vere-Jones D (1988)An Introduction to the Theory of Point Processes. Springer-Verlag, New York · Zbl 0657.60069
[5] Eisenberg M (1972) Queues with periodic service and change-over times.Oper. Res. 20:440-451 · Zbl 0245.60073 · doi:10.1287/opre.20.2.440
[6] Franken P, König D, Arndt U, Schmidt V (1981)Queues and Point Processes. Akademie-Verlag, Berlin · Zbl 0564.60089
[7] Fuhrmann S, Cooper RB (1985a) Stochastic Decomposition in theM/G/1 Queue with Generalized Vacations.Oper. Res. 33:1117-1119 · Zbl 0585.90033 · doi:10.1287/opre.33.5.1117
[8] Fuhrmann S, Cooper RB (1985b) Application of Decomposition Principle inM/G/1 Vacation Model to Two Continuum Cyclic Queueing Models?Especially Token-Ring LANs.AT&T Technical Journal 64 (5):1091-1098
[9] Gutjahr R (1977) Eine Bemerkung zur Existenz unendlich vieler ?Leerpunkte? in Bedienungssystemen mit unendlich vielen Bedienungsgeräten.Math. Operationsforsch. Statist. Ser. Optimization 8:245-251 · Zbl 0401.60103
[10] Kroese DP, Schmidt V (1992) A Continuous Polling System with General Service Times.Ann. of Appl. Probab 2(4):906-927 · Zbl 0772.60075 · doi:10.1214/aoap/1177005580
[11] Liptser RS, Shiryayev AN (1978)Statistics of Random Processes. II: Applications. Springer Verlag, New York · Zbl 0369.60001
[12] Mack C, Murphy T, Webb NL (1957) The efficiency ofN machines uni-directionally patrolled by one operative when walking time and repair times are constant.J. Roy. Statist. Soc. (B) 19:166-172 · Zbl 0090.35301
[13] Stolyar A, Coffman Jr. EG (1991) Polling on a Line: General Service Times. Preprint
[14] Takagi H (1986)Analysis of Polling Systems. The MIT Press, Cambridge Massachusetts · Zbl 0647.01001
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