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On the input and output processes for a general birth-and-death queueing model

Published online by Cambridge University Press:  01 July 2016

Bent Natvig
Bent Natvig*
Affiliation:
University of Sheffield
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Abstract

The steady-state input and output processes are considered for a birth-and-death queueing model with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and an arbitrary queueing discipline. Let an index n indicate that the quantity in question depends on the system state but not on time t. The instantaneous arrival rate is λ, the probability of balking (i.e., not trying to obtain service) being ξn. The instantaneous departure rate, μn, of customers having joined the system is the sum of the rate of service completions and the rate of defections before service completion. Three cases are considered. We start by ignoring balking customers; in the first case treating a lost customer neither as an input nor as an output, then secondly as both. Finally, balking and lost customers are considered both as inputs and outputs.

Keywords

GENERAL BIRTH-AND-DEATH QUEUEING MODELSTATE-DEPENDENTBALKINGDEFECTIVE CUSTOMERSLOSSSTEADY STATEINPUT PROCESSOUTPUT PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 3 , September 1975 , pp. 576 - 592
Copyright
Copyright © Applied Probability Trust 1975 

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References

Ancker, C. J. and Gafarian, A. V. (1963a) Some queueing problems with balking and reneging. I. Operat. Res. 11, 88–100.Google Scholar
Ancker, C. J. and Gafarian, A. V. (1963b) Some queuing problems with balking and reneging II. Operat. Res. 11, 928–937.CrossRefGoogle Scholar
Boes, D. C. (1969) Note on the output of a queuing system. J. Appl. Prob. 6, 459–461.Google Scholar
Burke, P. J. (1956) The output of a queuing system. Operat. Res. 4, 699–704.Google Scholar
Cox, D. R. and Miller, H. D. (1970) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Daley, D. J. (1972) A bivariate Poisson queueing process that is not infinitely divisible. Proc. Camb. Phil. Soc. 72, 449–450.CrossRefGoogle Scholar
Daley, D. J. (1975) Queueing output processes. Private communication.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Dwass, M. and Teicher, H. (1957) On infinitely divisible random vectors. Ann. Math. Statist. 28, 461–470.Google Scholar
Erdélyi, A. et al. (1954) Tables of Integral Transforms 1. McGraw-Hill, New York.Google Scholar
Finch, P. D. (1959) The output process of the queueing system M/G/1. J. R. Statist. Soc. B 21, 375–380.Google Scholar
Hadidi, N. (1969) On the service time distribution and the waiting time process of a potentially infinite capacity queueing system. J. Appl. Prob. 6, 594–603.Google Scholar
Haight, F. A. (1957) Queueing with balking. Biometrika 44, 360–369.Google Scholar
Homma, T. (1955) On a certain queuing process. Rep. Statist. Appl. Res. Un. Japan Sci. Engrs. 4, 14–37.Google Scholar
Karlin, S. and McGregor, J. (1957) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
Kendall, D. G. and Reuter, G. E. H. (1957) The calculation of the ergodic projection for Markov chains and processes with a countable infinity of states. Acta Math. 97, 103–144.Google Scholar
Milne, R. K. (1971) Stochastic analysis of multivariate point processes. Ph. D. Thesis, Australian National University.Google Scholar
Mirasol, N. M. (1963) The ouput of an M/G/∞ queuing system is Poisson. Operat. Res. 11, 282–284.Google Scholar
Natvig, B. (1975) On a queuing model where potential customers are discouraged by queue length. Scand. J. Statist. 2, 34–42.Google Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.Google Scholar
Reynolds, J. F. (1968) The stationary solution of a multiserver queueing model with discouragement. Operat. Res. 16, 64–71.Google Scholar
Shanbhag, D. N. (1972) Letter to the Editor. J. Appl. Prob. 9, 470.Google Scholar
Shanbhag, D. N. (1973) Characterization for the queueing system M/G/∞ . Proc. Camb. Phil. Soc. 74, 141–143.Google Scholar
Shanbhag, D. N. and Tambouratzis, D. G. (1973) Erlang's formula and some results on the departure process for a loss system. J. Appl. Prob. 10, 233–240.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Takács, L. (1969) On Erlang's formula. Ann. Math. Statist. 40, 71–78.Google Scholar