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

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.

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