Abstract
We study a queueing system withm exponential servers with distinct service rates. Jobs arrive at the system following an arbitrary point process. Arrived jobs receive service at the first unoccupied server (if any) according to an entry order π, which is a permutation of the integers 1, 2,...,m. The system has a finite buffer capacity. When the buffer limit is reached, arrivals will be blocked. Blocked jobs will either be lost or come back as New arrivals after a random travel time. We are concerned with the dynamic stochastic behavior of the system under different entry orders. A partial ordering is established among entry orders, and is shown to result in some quite strong orderings among the associated stochastic processes that reflect the congestion and the service characteristics of the system. The results developed here complement existing comparison results for queues with homogeneous servers, and can be applied to aid the design of conveyor and communication systems.
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References
R.B. Cooper, Queues with ordered servers that work at different rates, Opsearch 13(1976)69.
Y.L. Deng, On the comparison of point processes, J. Appl. Prob. 22(1985)300.
R.L. Disney, Some multichannel queueing problems with ordered entry — An application to conveyor theory, J. Industrial Eng. 14(1963)105.
T. Kamae, U. Krengel and G. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Prob. 5(1977)899.
A.W. Marshall and I. Olkin,Inequalities: Theory of Majorization and Its Applications (Academic Press, New York, 1979).
W.M. Nawijn, A note on many-server queueing systems with ordered entry, with an application to conveyor theory, J. Appl. Prob. 20(1983)144.
B. Pourbabai and D. Sonderman, Server utilization factors in queueing loss systems with ordered entry and heterogeneous servers, J. Appl. Prob. 23(1986)236.
D. Sonderman, Comparing multi-server queues with finite waiting rooms, I: Same number of servers, II: Different number of servers, Adv. Appl. Prob. 11(1979)439; 448.
D. Stoyan,Comparison Methods for Queues and Other Stochastic Models (Wiley, New York, 1983).
W. Whitt, Comparing counting processes and queues, Adv. Appl. Prob. 13(1981)207.
T. Yanagimoto and M. Okamoto, Partial orderings of permutations and monotonicity of a rank correlation statistic, Ann. Inst. Statist. Math. 21(1969)489.
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Shanthikumar, J.G., Yao, D.D. Comparing ordered-entry queues with heterogeneous servers. Queueing Syst 2, 235–244 (1987). https://doi.org/10.1007/BF01158900
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DOI: https://doi.org/10.1007/BF01158900