Minimizing Queuing Delays and Number of Messages in Mobile Phone Location

In this paper, Goodman et. al made two contributations.

First, they established a model of paging a single mobile user in $N$ cells within $D$ rounds. They assume $p_1=\ldots=p_N=\frac{1}{N}$ and paging strategy is to arbitrarily partition the cells into equal-sized subsets. The also assumes the calls arrives at Poisson distribution and the bottle neck of the paging process is at the radio channels. They construct the $M/M/1$ model, that is, multiple calls arrives at multiple users, while only a single user sent by a single call can be paged at a time. Through analysis, they established the distribution of the queue length in while calls are waiting to page.

Second, they designed an algorithm, of complexity $\Theta(N^2 D)$, using dynamical program, such that a paging strategy can be generated through the $p_i$s and the expected number of paged cells is minimized.

Hilbert’s Tenth Problem Is Unsolvable

This is a comprehensible proof to the intractability to the Hilbert’s 10th Problem

Hilbert 10th problem – decide if there is an integer solution to a Diophantine equation.

Unsolvable.

Proof of Recursive Unsolvability of Hilbert’s Tenth Problem

The USSR version of Hilbert Tenth Problem.

Hilbert 10th problem – decide whether there is a integer solution to an Diophantine equation.

This is a short proof showing the problem is unsolvable.