The optimal allocation of servers on an interval. (English. Ukrainian original) Zbl 0923.60094

Zakusilo, O. K.; Dotsenko, S. I.

Theory Probab. Math. Stat. 55, 75-78 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 73-76 (1996).

Let \(0\leq x_1 <x_2 <\ldots <x_{n-1} <x_n \leq 1\) be coordinates of some apparatuses. Let \(\xi( x_1,x_2,\ldots,x_n)\) be the distance of the random variable \(\xi\) uniformly distributed on \([0,1]\) to the proximate apparatus: \(\xi( x_1,x_2,\ldots,x_n) = \min_i | \xi -x_i |.\) The authors prove that \(E \xi( x_1,x_2,\ldots,x_n)\) attains its minimum if \(x_i =(i-{1\over 2})/n\), \(i=1,2,\ldots,n.\) Moreover, in this case \(\xi( x_1,x_2,\ldots,x_n)\) is the stochastic minimum, that is \(F_{\xi^\ast}(x) \geq F_\xi(x)\), where \(F_{\xi^\ast}(x)\) is the distribution function of \(\xi( x_1,x_2,\ldots,x_n)\) if \(x_i =(i-{1\over 2})/n\), \(i=1,2,\ldots,n\) and \(F_\xi(x)\) is the distribution function of \(\xi( x_1,x_2,\ldots,x_n)\) if \(x_i\) differ from \((i-{1\over 2})/n\) at least for one \(i.\) The analogous problem for a ring is considered, too.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60K25	Queueing theory (aspects of probability theory)
68M20	Performance evaluation, queueing, and scheduling in the context of computer systems
90B22	Queues and service in operations research

Keywords:

distance; random variable; uniform distribution; stochastic minimum; distribution function