Applied Queueing Theory pp 22-45 | Cite as

# Models of Single-channel Processes

## Abstract

It is customary to begin expositions of queueing theory with a presentation of various single-channel queueing models, and in particular to set out in considerable detail the mathematics of the most straightforward of all: that is the model *M/M/I*: (∞/FIFO). Now it should be clearly understood that the reasons for this are, above all, pedagogical. The practical significance of single-channel queueing models of any sort is relatively slight. When a single-channel queueing system is found to be in trouble, the problem is usually to evaluate the merits of making it a two-channel system. The pedagogical merits of single-channel models are however not to be ignored. In spite of their limited direct practical applicability we shall hold to tradition and consider them first.

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### Notes on Sources and References

- 1.It almost seems invidious to quote examples of work of this type: it would be wise therefore to quote examples which are, from the theoretical point of view, of unquestioned excellence, such as: Smith, W. L.
*On the Distribution of Queueing Times*(1953).*Proc. Cambridge Phil. Soc*.,**49**, pp. 449–461. Takacs, L.*Investigation of Waiting-time Problems by Reduction to Markov Processes*(1955),*Acta Math. Acad. Sci. Hung*.,**6**, pp. 101–129.Google Scholar - 2.See Feller (c), p. 305.Google Scholar
- 3.Cox and Smith (g), pp. 61–64: unfortunately the result given in this reference is wrong due to an early error in their derivation. It has been corrected in the present version.Google Scholar
- 4.See Feller (c), Chapter 11, for a discussion of the properties of probability generating functions.Google Scholar
- 5.Statistical equilibrium, or steady-state, or stationary conditions, are discussed most rigorously in Feller (c), pp. 356 ff. It is questionable whether, in most real world processes, such a thing as a steady-state exists. However it is possible for an engineer to make a great deal of practical progress by using formulas for the properties of gases derived from the models of statistical mechanics (which also assume steady-state conditions). Similarly the operational research practitioner (who is more likely in his daily activities to behave, and be called upon to behave, like an engineer than any sort of scientist) can make much headway by using steady-state formulas. The reader should remember that it would take a long time indeed for a real, live queueing-process to reach a steady-state even when the input and servicing parameters were constant, except at low traffic intensities. The fact that so many queueing-processes involving humans do, quite obviously, appear to be in statistical equilibrium when one knows that the mean input rate is not at all constant, and the service-time distribution is always changing shape like an amoeba, is due to the presence of ‘
*beneficient ghosts*’ which, being effects not allowed for in most mathematical models, yet tend to preserve stability. An example of such a*ghost*would be an inverse relationship between arrival rate and service-time. In practice, when designing operating systems of the queueing type, it is usually necessary to invent such ghosts if they cannot be relied upon to materialize. For example, one can observe at many London railway terminals (and it was also possible to do so at the old BEA Air Terminal in Kensington) an old man who flags passing taxis into the feeder taxi-rank when there are vacancies. By this means the ranks are kept full. I believe that the taxi-drivers reward the old men in some small way for this admirable piece of private enterprise. If they did not exist, it would be necessary for someone to invent them (the railway company, at twice the cost?). These are the ghosts, not to be found in any of the more familiar and beloved queueing models, which alone ensure that reality conforms, albeit approximately, to theory. By such means is the theoretical convenience which is statistical equilibrium realized in the operations of the world around us.Google Scholar - 6.See Syski (Chapter 2, note 4), pp. 13–14 for a discussion of measures of traffic.Google Scholar
- 7.Not always true. Automatic recording equipment often can most conveniently record busy-periods, and it is necessary to work backwards from that distribution to the desired ones such as queueing-time.Google Scholar
- 8.See: Palm, C.
*Research on Telephone Traffic Carried by full-availability Groups*(1957),*Tele*(English Edn.).Google Scholar - 9.Riordan, J.
*Delay Curves for Calls Served at Random*(1953).*Bell System Tech. J*., 32, pp. 100–119. See Appendix 3. The chart A3-7, although reproduced from another source, is Riordan’s master-chart which incorporates the results of his researches.Google Scholar - 10.Cox and Smith (g), p. 58.Google Scholar
- 11.This is the derivation given by Kendall in his paper (Ch. 2, note 5).Google Scholar
- 12.See Saaty (a), pp. 164–166 for a fuller exposition.Google Scholar
- 13.See Morse (b), pp. 72–82. Morse’s mathematical methods are different, but the difficulties are clear enough. The calculations are obviously so involved that the only reasonable way of going forward is by numerical approximation and a good digital computer.Google Scholar
- 14.The original paper is in this case the best account of all. See: Crommelin, C. D.
*Delay Probability Formulae When the Holding Times are Constant*(1932),*P.O. Elec. Engrs. J*.,**25**, pp. 41–50.Google Scholar - 15.Burke, P. J.
*Equilibrium Delay Distribution for One Channel with Constant Holding Time, Poisson Input and Random Service*(1959),*Bell System Tech. J*., July. The charts from Burke’s paper are reproduced in this book as A3-1 to A3-4 in Appendix 3. The paper itself is worth reading in full.Google Scholar - 16.See note 1 to this chapter.Google Scholar