Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-24T12:25:27.307Z Has data issue: false hasContentIssue false

Regenerative processes in the theory of queues, with applications to the alternating-priority queue

Published online by Cambridge University Press:  01 July 2016

Shaler Stidham Jr.*
Affiliation:
Cornell University

Abstract

Using some well-known and some recently proved asymptotic properties of regenerative processes, we present a new proof in a general regenerative setting of the equivalence of the limiting distributions of a stochastic process at an arbitrary point in time and at the time of an event from an associated Poisson process. From the same asymptotic properties, several conservation equations are derived that hold for a wide class of GI/G/1 priority queues. Finally, focussing our attention on the alternating-priority queue with Poisson arrivals, we use both types of result to give a new, simple derivation of the expected steady-state delay in the queue in each class.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1972 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avi-Itzhak, B., Maxwell, W. L. and Miller, L. W. (1965) Queuing with alternating priorities. Operat. Res. 13, 306318.CrossRefGoogle Scholar
Brown, M. and Ross, S. M. (1971) Asymptotic properties of cumulative processes. Tech. Report No. 120, Department of Operations Research, Cornell University.Google Scholar
Brumelle, S. L. (1971) On the relation between customer and time averages in queues. J. Appl. Prob. 8, 508520.CrossRefGoogle Scholar
Conway, R. W., Maxwell, W. L. and Miller, L. W. (1967) Theory of Scheduling. Addison-Wesley, Reading, Massachusetts.Google Scholar
Cooper, R. B. (1970) Queues served in cyclic order: waiting times. Bell. Syst. Tech. J. 49. 399413.CrossRefGoogle Scholar
Cooper, R. B. and Murray, G. (1969) Queues served in cyclic order. Bell. Syst. Tech., J. 48, 675689.CrossRefGoogle Scholar
Darroch, J. N., Newell, G. F. and Morris, R. W. J. (1964) Queues for a vehicle-actuated traffic light. Operat. Res. 12, 882895.CrossRefGoogle Scholar
Eilon, S. (1969) A simpler proof of L = λW . Operat. Res. 17, 915916.CrossRefGoogle Scholar
Eisenberg, M. (1971) Two queues with changeover times. Operat. Res. 19, 386401.CrossRefGoogle Scholar
Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. 1. John Wiley and Sons, Inc., New York.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. John Wiley and Sons, Inc., New York.Google Scholar
Hawkes, A. G. (1963) Queueing at traffic intersections. Proc. Second Int. Symp. on Theory of Traffic Flow. London.Google Scholar
Jewell, W. S. (1967) A simple proof of L λ W. Operat. Res. 15, 11091116.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. (1956) On the characteristics of the general queuing process with applications to random walk. Ann. Math. Statist. 27, 147161.CrossRefGoogle Scholar
Lighthill, M. J. and Whitham, G. B. (1955) On kinematic waves. Proc. Roy. Soc. A 229, 281345.Google Scholar
Little, J. D. C. (1961) A proof of the queuing formula, L = λ W. Operat. Res. 9, 383387.CrossRefGoogle Scholar
Loève, M. (1963) Probability Theory. D. Van Nostrand Company, Inc., Princeton, N.J.Google Scholar
Maxwell, W. L. (1970) On the generality of the equation L = λ W. Operat. Res. 18, 172173.CrossRefGoogle Scholar
Miller, L. W. (1964) Alternating Priorities in Multiclass Queues. Ph. D. Dissertation, Cornell University.Google Scholar
Nair, S. S. (1969) On certain priority queues. Mim. Series No. 214, Department of Statistics, Purdue University.CrossRefGoogle Scholar
Neuts, M. F. and Yadin, M. (1968) The transient behavior of the queue with alternating priorities, with special reference to the waiting times. Bull. Soc. Math. Belg. 20, 343376.Google Scholar
ROss, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
Smith, W. L. (1955) Regenerative stochastic processes. Proc. Roy. Soc. A 232, 631.Google Scholar
Smith, W. L. (1958) Renewal theory and its ramifications. J. Roy. Statist. Soc. B 20, 43302.Google Scholar
Stidham, S. (1970) On the optimality of single-server queuing systems. Operat. Res. 18, 708732.CrossRefGoogle Scholar
Strauch, R. E. (1970) When a queue looks the same to an arriving customer as to an observer. Management Sci. {Theory) 17, 140141.CrossRefGoogle Scholar
Sykes, J. S. (1971) Simplified analysis of an alternating-priority queue with setup times. Operat. Res. 18, 11821192.CrossRefGoogle Scholar
Takács, L. (1955) Investigation of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hung. 6, 101129.CrossRefGoogle Scholar
Takacs, L. (1968) Two queues attended by a single server. Operat. Res. 16, 639650.CrossRefGoogle Scholar
Wolff, R. W. (1970) Work-conserving priorities. J. Appl. Prob. 7, 327337.CrossRefGoogle Scholar