Some results for the queues Mx/M/c and GIx/ G/c

We develop for the queue M^x/M/c an upper bound for the mean queue length and lower bounds for the delay probabilities (that of an arrival group and that of an arbitrary customer in the arrival group). An approximate formula is also developed for the general bulk-arrival queue GI^x/G/c. Preliminary numerical studies have indicated excellent performance of the results.     

关于环Fpm+uFpm+u2Fpm上循环码的注记

本文研究了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码分类.通过建立环F_p^m+uF_p^m+u²F_p^m到环F_p^m+uF_p^m的同态,给出了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码的新分类方法.应用这种方法,得到了环F_p^m+uF_p^m+u²F_p^m长度为p^s的循环码的码词数.     

Large-time asymptotics for the Gt/Mt/st+GIt many-server fluid queue with abandonment

We previously introduced and analyzed the G _t /M _t /s _t +GI _t many-server fluid queue with time-varying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper, we establish an asymptotic loss of memory (ALOM) property for that fluid model, i.e., we show that there is asymptotic independence from the initial conditions as time t evolves, under regularity conditions. We show that the difference in the performance functions dissipates over time exponentially fast, again under the regularity conditions. We apply ALOM to show that the stationary G/M/s+GI fluid queue converges to steady state and the periodic G _t /M _t /s _t +GI _t fluid queue converges to a periodic steady state as time evolves, for all finite initial conditions.     

Une question sur les équations xm ? ym = Rzn (A Question on the Equations xm ? ym = Rzn)

Let m and n be integers at least two and R be a nonzero natural number. In this paper, we study the problem of the determination of the proper solutions of the Diophantine equation x ^m – y ^m equals; Rz ⁿ . We raise a question concerning the existence of any proper nontrivial solution of this equation, in case some precise conditions are satisfied by the triple (m, n, R). We prove some results about it.     

M/G/1/MLPS compared to M/G/1/PS

Multilevel processor sharing scheduling disciplines have recently been resurrected in papers that focus on the differentiation between short and long TCP flows in the Internet. We prove that, for M/G/1 queues, such disciplines are better than the processor sharing discipline with respect to the mean delay whenever the hazard rate of the service time distribution is decreasing.     

Ueber das volle Formensystem der tern?ren biquadratischen Formf=x13x2+x23x3+x33x1

Ohne ZusammenfassungHieizu cme Tale     

On the asymptotic behavior of xn+1=anxn/xn−1

We investigate the asymptotic behavior of solutions of a separable difference equation of the form

     

Expansion in SLd(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL _d (Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of π _q (G) with respect to the generating set π _q (S) form a family of expanders, where π _q is the projection map Z→Z/q Z.     

Solving Ck/Cm/1/N queues by using characteristic roots in matrix analytic methods

In this paper, we study a C_k/C_m/1/N open queueing system with finite capacity. We investigate the property which shows that a product of the Laplace Stieltjes Transforms of interarrival and service times distributions satisfies an equation of a simple form. According to this equation, we present that the stationary probabilities on the unboundary states can be written as a linear combination of vector product-forms. Each component of these products is expressed in terms of roots of an associated characteristic polynomial. As a result, we carry out an algorithm for solving stationary probabilities in C_k/C_m/1/N systems, which is independent of N, hence greatly reducing the computational complexity.     

Proof of the impossibility of the fermat equation Xp + Yp = Zp for special values of p and of the more general equation bXn + cYn = dZn

This paper continues the search to determine for what exponents n Fermat's Last Theorem is true. The main theorem and Corollary 1 consider the set of prime exponents p for which mp + 1 is prime for certain even integers m and prove the truth of FLT in Case 1 for such primes p. The remaining theorems prove the inequality of the more general Fermat equation bXⁿ + cYⁿ = dZⁿ.     

The Lyapunov matrix equation SA+A1S=S1B1BS

The matrix equation $\text{SA+A}{}^1\text{S=S}{}^1\text{B}{}^1\text{BS}$ is studied, under the assumption that (A, B¹) is controllable, but allowing nonhermitian S. An inequality is given relating the dimensions of the eigenspaces of A and of the null space of S. In particular, if B has rank 1 and S is nonsingular, then S is hermitian, and the inertias of A and S are equal. Other inertial results are obtained, the role of the controllability of (A¹, B¹S¹) is studied, and a class of D-stable matrices is determined.     

The least squares problem of the matrix equation A1X1B1^T ＋ A2X2B2^T = T

The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.     

Resolvable decomposition of Kn1

In this paper, we prove that $\text{K}{}_{\mathrm{n}}{}^1$ admits a resolvable decomposition into TT₃ or C₃ if and only if n ≡ 0 (mod. 3), n ≠ 6.     

Some remarks on the Schr?dinger equation with a potential in LrtLsx

We study the dispersive properties of the linear Schr?dinger equation with a time-dependent potential V(t,x). We show that an appropriate integrability condition in space and time on V, i.e. the boundedness of a suitable L^r_tL^s_x norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials V ∈L^r_tL^s_x. Support. The authors are partially supported by the Research Training Network (RTN) HYKE and by grant HPRN-CT-2002-00282 from the European Union. The third author is supported also by INDAM     

Approximation of M/M/s/K retrial queue with nonpersistent customers

We consider the M/M/s/K retrial queues in which a customer who is blocked to enter the service facility may leave the system with a probability that depends on the number of attempts of the customer to enter the service facility. Approximation formulae for the distributions of the number of customers in service facility, waiting time in the system and the number of retrials made by a customer during its waiting time are derived. Approximation results are compared with the simulation.