A simple algebraic approximation to the Erlang loss system

We use Palm calculus to derive a simple, intuitive system of two linear-quadratic equations and two unknowns, whose algebraic solution yields Harel’s (1988) upper bound to the Erlang loss probability. We then derive a sequence of progressively stronger systems of equations, which eventually become exact. We provide two example applications.     

A critically loaded multiclass Erlang loss system

We consider a generalization of the classical Erlang loss model to multiple classes of customers. Each of the J customer classes has an associated Poisson arrival process and an arbitrary holding time distribution. Classj customers requireM _j servers simultaneously. We determine the asymptotic form of the blocking probabilities for all customer classes in the regime known as critical loading, where both the number of servers and offered load are large and almost equal. Asymptotically, the blocking probability of classj customers is proportional toM _j . This asymptotic result provides an approximation for the blocking probabilities which is reasonably accurate. We also consider the behavior of the Erlang fixed point (reduced load approximation) for this model under critical loading. Although the blocking probability of classj customers given by the Erlang fixed point is again asymptotically proportional toM _j , the Erlang fixed point typically gives the wrong limit. Next we show that under critical loading the throughputs have a pleasingly simple form of monotonicity with respect to arrival rates: the throughput of classi is increasing in the arrival rate of classi and decreasing in the arrival rate of classj forji. Finally, we compare two simple control policies for this system under critical loading.     

Monotonicity and error bounds for networks of Erlang loss queues

Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are

1. classical circuit switch telephone networks (loss networks) and
2. present-day wireless mobile networks.

Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to

• upper bounds for loss probabilities and
• analytic error bounds for the accuracy of the approximation for various performance measures.

The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:

• pure loss networks as under (i)
• GSM networks with fixed channel allocation as under (ii).

The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning.     

On the Erlang Loss Model with Time Dependent Input

We consider the M(t)/M(t)/m/m queue, where the arrival rate λ(t) and service rate μ(t) are arbitrary (smooth) functions of time. Letting p_n(t) be the probability that n servers are occupied at time t (0≤ n≤ m, t > 0), we study this distribution asymptotically, for m→∞ with a comparably large arrival rate λ(t) = O(m) (with μ(t) = O(1)). We use singular perturbation techniques to solve the forward equation for p_n(t) asymptotically. Particular attention is paid to computing the mean number of occupied servers and the blocking probability p_m(t). The analysis involves several different space-time ranges, as well as different initial conditions (we assume that at t = 0 exactly n₀ servers are occupied, 0≤ n₀≤ m). Numerical studies back up the asymptotic analysis. AMS subject classification: 60K25,34E10 Supported in part by NSF grants DMS-99-71656 and DMS-02-02815     

Estimating Effective Capacity in Erlang Loss Systems under Competition

We consider an Erlang loss system (modem bank) with two streams of arriving customers, where arrival rates vary by time-of-day. We can observe one of the traffic streams (our customers), but we do not know how many servers the system has, or the characteristics of the other stream. Using detailed sample-path data, we construct a maximum likelihood estimator that makes good use of the data, but is slow to evaluate. As an alternative, we present an estimation system based on traffic data summarized by hour. This estimation system is much faster, and tends to produce good lower bounds on the size of the system and competing traffic.     

Multiphase free discontinuity problems: Monotonicity formula and regularity results

The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding state functions. Our main results state that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. The proof relies on a multiphase monotonicity formula and on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.     

New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems

The authors establish a general monotonicity formula for the following elliptic system
△ui＋fi（x,ui,…,um）=0 in Ω,
where Ω belong to belong to R^n is a regular domain, （fi（x, u1,... ,um）） = △↓F（x,→↑u）, F（x,→↑u ） is a given smooth function of x ∈ R^n and →↑u = （u1,…,um） ∈ R^m. The system comes from understanding the stationary case of Ginzburg-Landau model. A new monotonicity formula is also set up for the following parabolic system
δtui-△ui-fi（x,ui,…,um）=0 in（ti,t2）×R^n,
where t1 〈 t2 are two constants, （fi（x,→↑u ） is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonieity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied.     

On Monotonicity and Order-Preservation for Multidimensional G-Diffusion Processes

In this article, we prove a comparison theorem for multidimensional stochastic differential equations driven by G-Brownian motion (G-SDEs, for short). Moreover, we obtain respectively the sufficient conditions and necessary conditions of the monotonicity and order-preservation for two multidimensional G-diffusion processes. Finally, we give some applications.     

Monotonicity and bounds for convex stochastic control models

We consider a general convex stochastic control model. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the transition kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on convex stochastic orderings of probability measures. We derive several interesting sufficient conditions of these ordering concepts, where we make also use of the Blackwell ordering. The structural results are illustrated by partially observed control models and Bayesian information models.     

Approximate analysis of non-stationary loss queues and networks of loss queues with general service time distributions

A Fixed Point Approximation (FPA) method has recently been suggested for non-stationary analysis of loss queues and networks of loss queues with Exponential service times. Deriving exact equations relating time-dependent mean numbers of busy servers to blocking probabilities, we generalize the FPA method to loss systems with general service time distributions. These equations are combined with associated formulae for stationary analysis of loss systems in steady state through a carried load to offered load transformation. The accuracy and speed of the generalized methods are illustrated through a wide set of examples.     

一类积分方程组正解的对称性和单调性

本文讨论积分方程组（？）解的性质,其中G_α是α阶贝塞尔位势核,0≤β〈α（n-α＋β）/n,1/（q＋1）＋1/（r＋1）〉（n-α＋β）/n,1/（r＋1）＋1/（p＋1）〉（n-α＋β）/n.我们用积分形式的移动平面法证明上述积分方程组的正解是径向对称且单调的.     

凸对策核仁的非空性及合成凸对策核仁的单调性

本文给出了核仁与核及最小核心之间的关系 ,且证明了凸对策核仁的存在性和唯一性 ,证明了凸对策的合成对策仍是凸对策 .最后 ,我们讨论了合成凸对策的核仁不满足单调性 .     

Monotonicity of the power functions of modified likelihood ratio criterion for the homogeneity of variances and of the sphericity test

The modified likelihood ratio criterion for testing the homogeneity of variances of p univariate normal populations, and the sphericity test, are both shown in this paper to have a monotone nondecreasing power function.     

带干扰的Erlang(2)风险模型的不破产概率

本文讨论了带干扰的Erlang(2)风险模型,通过构造一个延迟更新过程,我们得到了不破产概率满足的积分-微分方程,进而得到了不破产概率的明确表达式．     

Sharp and simple bounds for the Erlang delay and loss formulae

We prove some simple and sharp lower and upper bounds for the Erlang delay and loss formulae and for the number of servers that invert the Erlang delay and loss formulae. We also suggest simple and sharp approximations for the number of servers that invert the Erlang delay and loss formulae. We illustrate the importance of these bounds by using them to establish convexity proofs. We show that the probability that the M/M/s queue is empty is a decreasing and convex function of the traffic intensity. We also give a very short proof to show that the Erlang delay formula is convex in the traffic intensity when the number of servers is held constant. The complete proof of this classical result has never been published. We also give a very short proof to show that the Erlang delay formula is a convex function of the (positive integer) number of servers. One of our results is then used to get a sharp bound to the Flow Assignment Problem.     

Monotonicity of input-output mappings in inverse coefficient and source problems for parabolic equations

This article presents a mathematical analysis of input-output mappings in inverse coefficient and source problems for the linear parabolic equation ut=(kx(x)ux)+F(x,t), (x,t)∈ΩT:=(0,1)×(0,T]. The most experimentally feasible boundary measured data, the Neumann output (flux) data f(t):=−k(0)ux(0,t), is used at the boundary x=0. For each inverse problems structure of the input-output mappings is analyzed based on maximum principle and corresponding adjoint problems. Derived integral identities between the solutions of forward problems and corresponding adjoint problems, permit one to prove the monotonicity and invertibility of the input-output mappings. Some numerical applications are presented.     

Explicit Expressions for the Ruin Probabilities of Erlang Risk Processes with Pareto Individual Claim Distributions

In this paper we first consider a risk process in which claim inter-arrival times and the time untilthe first claim have an Erlang (2) distribution.An explicit solution is derived for the probability of ultimateruin,given an initial reserve of u when the claim size follows a Pareto distribution.Follow Ramsay,Laplacetransforms and exponential integrals are used to derive the solution,which involves a single integral of realvalued functions along the positive real line,and the integrand is not of an oscillating kind.Then we showthat the ultimate ruin probability can be expressed as the sum of expected values of functions of two differentGamma random variables.Finally,the results are extended to the Erlang(n) case.Numerical examples aregiven to illustrate the main results.     

Optimal prices for finite capacity queueing systems

We prove a lower bound on the optimal price for a fairly large class of blocking systems with general arrival and service processes, determine optimal price expressions for M/M/1/m and M/GI/s/s systems, and investigate how optimal prices change with changes in the size of the waiting room and service capacity.