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Some comments on single-server queuing methods and some new results

Published online by Cambridge University Press:  24 October 2008

Julian Keilson
Julian Keilson
Affiliation:
Sylvania Electronic Systems, Waltham, Massachusetts
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Extract

1. Introduction. Over the past few years, the theory of single-server queues has grown substantially. Much of the effort expended has been applied to the study of transient multivariate queuing processes in continuous time, i.e. to the study of joint time-dependent distributions of two or more system variables given some known initial system configuration. A variety of methods have been employed in these studies to accommodate general interarrival and service time distributions, among them Erlang's method of fictitious phases, the method of supplementary variables, the extended imbedded chain method, and the semi-Markov method. In the first part of this paper an informal review of these transient studies and of the methods employed is given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 2 , April 1964 , pp. 237 - 251
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Copson, E. T.An introduction to the theory of functions of a complex variable (Oxford University Press, 1935).Google Scholar
(2)Cox, D. R.The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Proc. Cambridge Philos. Soc. 51 (1955), 313–319.CrossRefGoogle Scholar
(3)Cox, D. R. and Smith, W. L.Queues (Methuen; London, 1961).Google Scholar
(4)Daniels, H. E.Saddlepoint approximation in statistics. Ann. Math. Statist. 25 (1954), 631.CrossRefGoogle Scholar
(5)Fabens, A. J.The solution of queuing and inventory models by semi-Markov processes. J. Roy. Statist. Soc. Ser. B, 23 (1961), 113–127.Google Scholar
(6)Finch, P. D.The effect of the size of the waiting room on a simple queue. J. Roy. Statist. Soc. Ser. B, 1 (1958), 182–186.Google Scholar
(7)Gaver, D. P.The influence of servicing times in queuing processes. Operations Res. 2 (1954), 134–149.Google Scholar
(8)Gaver, D. P.Imbedded Markov chain analysis of a waiting-line process in continuous time. Ann. Math. Statist. 30 (1959), 698–720.CrossRefGoogle Scholar
(9)Gaver, D. P.A waiting line with interrupted service including priorities. J. Roy. Statist. Soc. Ser. B, 24 (1962), 73–90.Google Scholar
(10)Heathcote, C. R.On the queuing process M/G/1. Ann. Math. Statist. 32 (1961), 770–773.CrossRefGoogle Scholar
(11)Jaiswal, N. K.Time dependent solution of the bulk-service queuing problem. Operations Res. 8 (1960), 773.Google Scholar
(12)Jaiswal, N. K.Preemptive resume priority queue. Operations Res. 9 (1961), 732–742.CrossRefGoogle Scholar
(13)Jaiswal, N. K.Time-dependent solution of head-of-the-line priority queue. J. Roy. Statist. Soc. Ser. B, 24 (1962), 91–101.Google Scholar
(14)Keilson, J. The homogeneous random walk on the half-line and the Hilbert problem. Bulletin de L'Institut International de Statistique 113 (Paris, 1962).Google Scholar
(15)Keilson, J.The use of Green's functions in the study of bounded random walks with application to queuing theory. J. Math. Phys. 41 (1962), 42–52.Google Scholar
(16)Keilson, J.Non-stationary Markov walks on the lattice. J. Math. Phys. 41 (1962), 205–211.Google Scholar
(17)Keilson, J.The general bulk queue as a Hilbert problem. J. Roy. Statist. Soc. Ser. B, 24 (1962), no. 2.Google Scholar
(18)Keilson, J.A simple random walk and an associated asymptotic behaviour of the Bessel functions. Proc. Cambridge Philos. Soc. 58 (1962), 708–709.CrossRefGoogle Scholar
(19)Keilson, J.Queues subject to service interruption. Ann. Math. Statist. 33 (1962), 1314–1322.CrossRefGoogle Scholar
(20)Keilson, J. and Kooharian, A.On time dependent queuing processes. Ann. Math. Statist. 31 (1960), 104–112.CrossRefGoogle Scholar
(21)Keilson, J. and Kooharian, A.On the general time dependent queue with a single server. Ann. Math. Statist. 33 (1962), 767–791.CrossRefGoogle Scholar
(22)Kendall, D. G.Some problems in the theory of queues. J. Roy. Statist. Soc. Ser. B, 13 (1951), 151–185.Google Scholar
(23)Kendall, D. G.Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Statist. 24 (1953), 338–354.CrossRefGoogle Scholar
(24)Lindley, D. V.The theory of queues with a single server. Proc. Cambridge Philos. Soc. 48 (1952), 227–289.CrossRefGoogle Scholar
(25)Luchak, G.The solution of the single-channel queuing equations characterized by a time-dependent Poisson-distributed arrival rate and a general class of holding times. Operations Res. 4 (1956), 711–732.Google Scholar
(26)Luchak, G.The distribution of the time required to reduce to some pre-assigned level a single-channel queue characterized by a time-dependent Poisson-distributed arrival rate and a general class of holding times. Operations Res. 5 (1957), 205–209.Google Scholar
(27)Luchak, G.The continuous time solution of the equations of the single channel queue with a general class of service-time distributions by the method of generating function. J. Roy. Statist. Soc. Ser. B, 20 (1958), 176–181.Google Scholar
(28)Muskhelishvili, N. T.Singular integral equations (Noordhoff; Groningen, 1946).Google Scholar
(29)Prabhu, N. U.Some results for the queue with Poisson arrivals. J. Roy. Statist. Soc. Ser. B, 22 (1960), 104–107.Google Scholar
(30)Smith, W. L.Regenerative stochastic processes. Proc. Roy. Soc. London, Ser. A, 232 (1955), 6–31.Google Scholar
(31)Takacs, L.Investigation of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hungary, 6 (1955), 101–129.CrossRefGoogle Scholar
(32)Takacs, L.Transient behavior of single server queuing processes with recurrent input and exponentially distributed service times. Operations Res. 8, (1960), 231–245.Google Scholar
(33)Takacs, L.The probability law of the busy period for two types of queuing processes. Operations Res. 9 (1961), 402–450.Google Scholar
(34)Takacs, L.A single-server queue with Poisson input. Operations Res. 10 (1962), 388–394.CrossRefGoogle Scholar
(35)Takacs, L.A single-server queue with recurrent input and exponentially distributed service times. Operations Res. 10 (1962), 395–399.Google Scholar
(36)Takacs, L.Introduction to the theory of queues (New York: Oxford University Press, 1962).Google Scholar
(37)Wishart, D. M. G.A queuing system with χ2 service-time distribution. Ann Math. Statist. 27 (1956), 768–779.CrossRefGoogle Scholar
(38)Wishart, D. M. G.A queuing system with service time distribution of mixed chi-squared type. Operations Res. 7 (1959), 174–179.Google Scholar
(39)Wishart, D. M. C.Queuing systems in which the discipline is ‘Last-come, first served’. Operations Res. 8 (1960), 591–599.CrossRefGoogle Scholar