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.

On the Relative Waiting Times in the GI/M/s and the GI/M/1 Queueing Systems

Uri Yechiali
Operational Research Quarterly (1970-1977)
Vol. 28, No. 2, Part 1 (1977), pp. 325-337
DOI: 10.2307/3009188
Stable URL: http://www.jstor.org/stable/3009188
Page Count: 13
  • Subscribe ($19.50)
  • Cite this Item
On the Relative Waiting Times in the GI/M/s and the GI/M/1 Queueing Systems
Preview not available

Abstract

The expected steady-state waiting time, Wq(s), in a GI/M/s system with interarrival-time distribution H(·) is compared with the mean waiting time, Wq, in an "equivalent" system comprised of s separate GI/M/1 queues each fed by an interarrival-time distribution G(·) with mean arrival rate equal to 1/s times that of H(·). For H(·) assumed to be Exponential, Gamma or Deterministic three possible relationships between H(·) and G(·) are considered: G(·) can be of the "same type" as H(·); G(·) can be derived from H(·) by assigning new arrivals to the individual channels in a cyclic order; and G(·) may be obtained from H(·) by assigning customers probabilistically to the different queues. The limiting behaviour of the ratio R = Wq/Wq(s) is studied for the extreme values (1 and 0) of the common traffic intensity, ρ. Closed form results, which depend on the forms of H(·) and G(·) and on the relationships between them, are derived. It is shown that Wq is greater than Wq(s) by a factor of at least (s + 1)/2 when ρ approaches one, and that R is at least s(s!) when ρ tends to zero. In the latter case, however, R goes to infinity (!) in most cases treated. The results may be used to evaluate the effect on the waiting times when, for certain (non-queueing) reasons, it is needed to partition a group of s servers into several small groups.

Page Thumbnails

  • Thumbnail: Page 
325
    325
  • Thumbnail: Page 
326
    326
  • Thumbnail: Page 
327
    327
  • Thumbnail: Page 
328
    328
  • Thumbnail: Page 
329
    329
  • Thumbnail: Page 
330
    330
  • Thumbnail: Page 
331
    331
  • Thumbnail: Page 
332
    332
  • Thumbnail: Page 
333
    333
  • Thumbnail: Page 
334
    334
  • Thumbnail: Page 
335
    335
  • Thumbnail: Page 
336
    336
  • Thumbnail: Page 
337
    337