Volume 32, Issue 4

A group service system for three queues is considered. At each time t = 1, 2, . . ., with some probability, a customer enters the system, selects randomly two queues, and goes to the shorter one. At each moment such that there is at least one customer in each queue, each queue performs instantly the service of one customer. By means of Lyapunov functions, a criterion for the ergodicity of the Markov chain corresponding to this queuing system is established. The limiting joint distribution of queue lengths is found, and the connection with the problem of balanced allocations of particles into cells is described. In the corresponding problem of balanced allocation of particles, the limiting distribution of the range is found, i. e. the difference between the maximal and minimal numbers of particles in cells.

Explicit recurrent formulas for the numbers of sequences containing a given pattern given number of times are constructed. These formulas depend on the length of the sequence, the length of the pattern and its period only. By means of these results one may find the distribution of statistics of the NIST overlapping matching test for binary sequences and arbitrary pattern parameters.

A class of one-dimensional discrete power series distributions containing negative binomial distributions is considered. Properties of distributions of this class are investigated. Limit theorems generalizing similar theorems for the negative binomial distributions are proved. The proofs are based both on elementary asymptotic methods and on a modification of saddle-point method.

We study matrices over quotient rings modulo univariate polynomials over a two-element field. Lower bounds for the fraction of the invertible matrices among all such matrices of a given size are obtained. An efficient algorithm for calculating the determinant of matrices over these quotient rings and an algorithm for generating random invertible matrices (with uniform distribution on the set of all invertible matrices) are proposed and analyzed. An effective version of the latter algorithm for quotient rings modulo polynomials of form x r − 1 is considered and analyzed. These methods may find practical applications for generating keys of cryptographic schemes based on quasi-cyclic codes such as LEDAcrypt.

Reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs are considered. For such a circuit implementing a map f:Z2n→Z2n, $f\colon \mathbb Z_2^n \to \mathbb Z_2^n,$we study the Shannon complexity function L ( n , q ) under the condition that the number of additional inputs is q = O ( n 2 ). For this range of q , it is shown that L(n,q)≍n2n/log2n. $L(n,q) \asymp n2^n \mathop / \log_2 n.$We show that L(n,q)≍n2n/log2(n+q) $L(n,q) \asymp n2^n \mathop / \log_2 (n+q)$for all q≲n2n−⌈n/ϕ(n)⌉, $q \lesssim n2^{n-\lceil n \mathop / \phi(n)\rceil},$where ϕ(n)→∞andn/ϕ(n)−log2n→∞asn→∞. $\phi(n) \to \infty {\text {and}} \,n \mathop / \phi(n) - \log_2 n \to \infty \,{\text {as}}\, n \to \infty.$

An approach to the construction of efficient algorithms for the exact computation of distributions of statistics by means of the Markov chains is described. The Pearson statistic, the number of empty cells for random allocations of particles, and the Kolmogorov – Smirnov statistic are considered as examples. Possibilities of extending the approach are discussed, in particular to the computation of the joint distributions of statistics.