Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

Some Problems in the Theory of Queues

David G. Kendall
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 13, No. 2 (1951), pp. 151-185
Published by: Wiley for the Royal Statistical Society
Stable URL: http://www.jstor.org/stable/2984059
Page Count: 35
  • Read Online (Free)
  • Download ($29.00)
  • Subscribe ($19.50)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Some Problems in the Theory of Queues
Preview not available

Abstract

The paper opens with a general review of some points in congestion theory, and continues with a simplified account of the Pollaczek-Khintchine "equilibrium" theory for the single-counter queue fed by an input of the Poisson type and associated with a general service-time distribution. It is pointed out that although the stochastic process describing the fluctuations in queue-size is not (in general) Markovian, it is possible to work instead with an enumerable Markov chain if attention is directed to the epochs at which individual customers depart (these epochs forming a sequence of regeneration points). The ergodic properties of the chain are investigated with the aid of Feller's theory of recurrent events; it is found to be irreducible and aperiodic, and the further classification of the states depends on the value of the traffic intensity, ρ (measured in erlangs). When $\rho < 1$ the states are all ergodic and the system ultimately settles down to a regime of statistical equilibrium, the transition-probabilities converging to the values given by the equilibrium solution. When $\rho > 1$ the states are all transient, and when ρ has the critical value of unity the states are all recurrent-and-null. The paper closes with some further general comments, with a re-interpretation of Borel's theory of "busy periods" in terms of the Galton-Watson "branching process" of stochastic population theory and with a sketch of an argument leading to the distribution of the length in time of a busy period.

Page Thumbnails

  • Thumbnail: Page 
[151]
    [151]
  • Thumbnail: Page 
152
    152
  • Thumbnail: Page 
153
    153
  • Thumbnail: Page 
154
    154
  • Thumbnail: Page 
155
    155
  • Thumbnail: Page 
156
    156
  • Thumbnail: Page 
157
    157
  • Thumbnail: Page 
158
    158
  • Thumbnail: Page 
159
    159
  • Thumbnail: Page 
160
    160
  • Thumbnail: Page 
161
    161
  • Thumbnail: Page 
162
    162
  • Thumbnail: Page 
163
    163
  • Thumbnail: Page 
164
    164
  • Thumbnail: Page 
165
    165
  • Thumbnail: Page 
166
    166
  • Thumbnail: Page 
167
    167
  • Thumbnail: Page 
168
    168
  • Thumbnail: Page 
169
    169
  • Thumbnail: Page 
170
    170
  • Thumbnail: Page 
171
    171
  • Thumbnail: Page 
172
    172
  • Thumbnail: Page 
173
    173
  • Thumbnail: Page 
174
    174
  • Thumbnail: Page 
175
    175
  • Thumbnail: Page 
176
    176
  • Thumbnail: Page 
177
    177
  • Thumbnail: Page 
178
    178
  • Thumbnail: Page 
179
    179
  • Thumbnail: Page 
180
    180
  • Thumbnail: Page 
181
    181
  • Thumbnail: Page 
182
    182
  • Thumbnail: Page 
183
    183
  • Thumbnail: Page 
184
    184
  • Thumbnail: Page 
185
    185