Applied Queueing Theory pp 6-21 | Cite as

# Models of Queueing Processes

## Abstract

There is no science of queueing-processes. A *Theory* of *Queues* exists which is concerned with the study of symbolic, mathematical models of queueing-processes, but it appears to be founded on a surprisingly slight body of empirical knowledge. Not many experimental results concerning queueing-processes have so far appeared. This odd-from the scientific viewpoint-state of affairs has consequences for the applicability and usefulness of the results of the theory, as we shall see. But before looking into the reasons for this emphasis, and its consequences, we should consider two preliminary questions: what is a symbolic-or mathematical-model?1† And what is its purpose? The following factors might be considered in a model of the queueing-process at Paltries railway station

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## Notes on Sources and References

- 1.I See Chapter 3 on models in: Churchman, C. W., Ackoff, R. and Arnoff, L.
*Introduction to Operations Research*(1957), John Wiley and Sons, New York.Google Scholar - 2.There is an excellent discussion of measures of effectiveness in: Morse, P. M. and Kimball, G.
*Methods of Operations Research*(1955).Google Scholar - 3.See also: Tocher, K. D.
*The Role of Models in Operational Research*(1961), J. Roy. Statist. Soc., Pt. 2.Google Scholar - 4.For a brief historical account which indicates without attempting to do so the genesis of this situation, read: Saaty (a), Chapter 1, section 1-ai]6_The reader should also try to see: Syski, R.
*Introduction to Congestion Theory in Telephone Systems*(1960), Oliver and Boyd, Edinburgh. The first chapter in particular.Google Scholar - 5.Kendall, D. G.
*Some Problems in the Theory of Queues*(1951), J. Roy. Statist. Soc., Ser. B,**13**, No. 2. This is a brilliant expository paper, in which for the first time Kendall hinted at his concept of the*imbedded Markov chain*, subsequently developed by him and other writers.Google Scholar - 6.A clear and simple statement of assumptions is to be found in: Fry, T. C.
*Probability and Its Engineering Uses*, Second Edition 1965), D. Van Nostrand Co. Inc., New York.Google Scholar - 7.There are many papers which discuss problems of this type. Four examples are: Bowen, E. G. and Pearcey, T.
*Delays in the Flow of Air Traffic*(1948), J. Roy. Aero. Soc.,**52**, P. 447.Google Scholar - Pearcey, T.
*Delays in the Landing of Air Traffic*(1948), J. Roy. Aero. Soc.,**52**, p. 450.Google Scholar - Bell, E. G.
*Operational Research into Air Traffic Control*(1949), J. Roy. Aero. Soc.,**53**, p. 331.Google Scholar - Galliher, H. P. and Wheeler, R. C.
*Non-stationary Queueing Probabilities for Landing Congestion of Aircraft*(1958), Operat. Res.,**6**, No. 2.Google Scholar - 8.Cox and Smith (g), p. 14.Google Scholar
- 9.Read the comments in Cox and Smith (g), pp. 106–107 and p. 141, last paragraph.Google Scholar
- 10.Such processes have been modelled and studied, but are not discussed further in this book. A good paper is: Gumbel, H.
*Waiting Lines with Heterogeneous Servers*(1960), Operat. Res.,**8**, No. 4.Google Scholar - 11.The FIFO rule is sometimes referred to as ‘first come, first served’ and this would seem, in fact, a much more precise description than ‘first in, first out’ if one were mistaken in thinking of customers being first in and first out of the whole
*queueing system*, that is including the servers. FIFO, however, means first into the queue—or*waiting-line*—and first out of it*into*a service channel. In a multi-channel process, the first customer into a channel does not necessarily leave it sooner than another customer, who has entered a different channel at a later time, leaves*his*channel. Consequently queue-discipline rules like FIFO or SIRO refer to the way customers are selected from the queue to obtain service, and have nothing at all to do with the order in which they depart from the system. In the case of single-channel queueing systems there is, of course, no difference unless pre-emptive priority rules are in operation.Google Scholar