69.196.182.208: /* Real number representations */ Removed extraneous comment & replaced N with x on the left of the equality.

2013-10-21T20:22:14Z

Real number representations: Removed extraneous comment & replaced N with x on the left of the equality.

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In [[queueing theory]], a discipline within the mathematical theory of probability, '''Ross's conjecture''' gives a lower bound for the average waiting-time experienced by a customer when arrivals to the queue do not follow the simplest model for random arrivals. It was proposed by Sheldon M. Ross in 1978 and proved in 1981 by Tomasz Rolski.<ref name="rolski" /> Equality can be obtained in the bound; and the bound does not hold for finite buffer queues.<ref>{{citation
| last = Heyman | first = D. P.
| issue = 1
| journal = Journal of Applied Probability
| jstor = 3213936
| mr = 644439
| pages = 245–249
| title = On Ross's conjectures about queues with non-stationary Poisson arrivals
| volume = 19
| year = 1982
| doi = 10.2307/3213936}}.</ref>

==Bound==

Ross's conjecture is a bound for the mean delay in a queue where arrivals are governed by a [[doubly stochastic Poisson process]] or by a non-stationary [[Poisson process]].<ref name="rolski">{{citation
| last = Rolski | first = Tomasz
| doi = 10.2307/1426787
| issue = 3
| journal = Advances in Applied Probability
| jstor = 1426787
| mr = 615953
| pages = 603–618
| title = Queues with non-stationary input stream: Ross's conjecture
| volume = 13
| year = 1981}}.</ref><ref name="ross">{{citation
| last = Ross | first = Sheldon M.
| issue = 3
| journal = Journal of Applied Probability
| jstor = 3213122
| mr = 0483101
| pages = 602–609
| title = Average delay in queues with non-stationary Poisson arrivals
| volume = 15
| year = 1978
| doi = 10.2307/3213122}}.</ref> The conjecture states that the average amount of time that a customer spends waiting in a queue is greater than or equal to
::<math>\frac{\lambda \operatorname E (S^2)}{2 \{1-\lambda \operatorname E (S) \}}</math>
where ''S'' is the service time and λ is the average arrival rate (in the limit as the length of the time period increases).<ref name="rolski" />

==References==

{{Reflist}}

[[Category:Probabilistic inequalities]]
[[Category:Queueing theory]]
{{probability-stub}}

Ostrowski numeration - Revision history

69.196.182.208: /* Real number representations */ Removed extraneous comment & replaced N with x on the left of the equality.