A feedback fluid queue with two congestion control thresholds

Feedback fluid queues play an important role in modeling congestion control mechanisms for packet networks. In this paper we present and analyze a fluid queue with a feedback-based traffic rate adaptation scheme which uses two thresholds. The higher threshold B ₁ is used to signal the beginning of congestion while the lower threshold B ₂ signals the end of congestion. These two parameters together allow to make the trade-off between maximizing throughput performance and minimizing delay. The difference between the two thresholds helps to control the amount of feedback signals sent to the traffic source. In our model the input source can behave like either of two Markov fluid processes. The first applies as long as the upper threshold B ₁ has not been hit from below. As soon as that happens, the traffic source adapts and switches to the second process, until B ₂ (smaller than B ₁) is hit from above. We analyze the model by setting up the Kolmogorov forward equations, then solving the corresponding balance equations using a spectral expansion, and finally identifying sufficient constraints to solve for the unknowns in the solution. In particular, our analysis yields expressions for the stationary distribution of the buffer occupancy, the buffer delay distribution, and the throughput.     

The N-policy of a discrete time Geo/G/1 queue with disasters and its application to wireless sensor networks

We analyze an N-policy of a discrete time Geo/G/1 queue with disasters. We obtain the probability generating functions of the queue length, the sojourn time, and regeneration cycles such as the idle period and the busy period. We apply the queue to a power saving scheme in wireless sensor networks under unreliable network connections where data packets are lost by external attacks or shocks. We present various numerical experiments for application to power consumption control in wireless sensor networks. We investigate the characteristics of the optimal N-policy that minimizes power consumption and derive practical insights on the operation of the N-policy in wireless sensor networks.     

A discrete weighted Helmholtz decomposition and its application

We propose a discrete weighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a weighted $L^2$ inner product and stable uniformly with respect to the jumps in the discontinuous weight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Maxwell system in non-homogeneous media by a non-overlapping domain decomposition preconditioner, i.e., the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain, and more importantly, it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media. Numerical experiments are presented to validate the effectiveness of the non-overlapping domain decomposition preconditioner.     

On an application of convexity to discrete systems

We prove the following result: Let A be a symmetric matrix, f be gradient (or certain subgradient) of a convex function, and {y_i} be a sequence defined by y_{i + 1} = f(Ay_i), y₀ arbitrary. Then the only possible periods of {y_i} are 1 or 2.     

A kind of nonnegative matrices and its application on the stability of discrete dynamical systems

In this paper we introduce a new kind of nonnegative matrices which is called (sp) matrices. We show that the zero solutions of a class of linear discrete dynamical systems are asymptotically stable if and only if the coefficient matrices are (sp) matrices. To determine that a matrix is (sp) matrix or not is very simple, we need only to verify that some elements of the coefficient matrices are zero or not. According to the result above, we obtain the conditions for the stability of several classes of discrete dynamical systems.     

The finite capacity multi-server queue with inhomogeneous arrival rate and discrete service time distribution — and its application to continuous service time problems

The finite capacity multi-server queueing model with inhomogeneous arrival rate and discrete service time distribution is developed. The system is formulated as an inhomogeneous Markov chain in discrete time. An algorithm is described to solve the model numerically. A method is then proposed for using this model to approximate the time dependent behaviour of multi-server queues with inhomogeneous arrival rate and continuous service time distribution. Empirical results are presented to show that this approximation will produce results that are accurate enough for most practical purposes.     

An intermittently served discrete time queue with applications to meteor scatter communications

A discrete time single server queue with service interruptions is analyzed in the steady-state under general assumptions. The main motivation for the study is the performance evaluation of a communication protocol using ionized layers created by meteors. The analysis yields the joint distribution of the queue size and the remaining duration of the current operative or inoperative period. The solution takes a particularly simple form in the case where the operative periods have a rational generating function.On short-term visits to the University of Newcastle upon Tyne and AT&T Bell Laboratories.Work done while the author was visiting the AT&T Bell Laboratories.     

Finite-dimensional discrete linear stochastic accelerated-time systems and their application to quadratic stochastic dynamical systems

The limiting behavior of the trajectories {x ⁽ⁿ⁾ } of linear discrete stochastic systems of the form (K, P ^an+b ) _nN , whereK is the standard simplex in ^N ,P: ^{N N} is a linear operator,PK K,a ft,b ,a+b>0, is described. An application to a class of quadratic stochastic dynamical systems is considered.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 709–718, May, 1996.     

A series of a discrete group and its application to the Dirichlet series associated with automorphic forms

Starting from the Poincaré-Selberg series, another series is defined for a discrete group; the relationship is established between this series and the Dirichlet series for which the coefficients are products of the Fourier coefficients of the automorphic forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 3–7, 1976.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

Analysis of a system with minimal repair and its application to replacement policy

The periodic replacement with minimal repair at failures is studied by many authors, however, there is not a clear definition for minimal repair. This paper defines a minimal repair in the term of the failure rate and devices some probability quantities and reliability properties. As an application of these results, the replacement model where a system is replaced at time T or at nth failure are considered and the optimum policies are discussed.     

A fuzzy based threshold policy for a single server retrial queue with vacations

In this paper we deal with a single server retrial queue with vacations. The server serves the customers until the system becomes empty, then it takes a vacation. The system consists of two types of costs. The blocking cost is considered whenever a customer is blocked either because of the server is busy or off. There is also a cost each time the server is turned on. The problem is to find an effective policy for turning on the dormant server. We propose a Fuzzy Based Threshold Policy (FBTP) to control the server, substitute for conventional threshold policies. The FBTP is based on four input parameters, an inference stage and it is tuned up using a stochastic List Based Threshold Accepting (LBTA) algorithm. Simulation models are developed to validate the fuzzy controller. Numerical experiments are provided to show that the proposed method is superior to crisp threshold policies.     

A primogenitary linked quad tree data structure and its application to discrete multiple criteria optimization

A data structure called the primogenitary linked quad tree is developed. Each node in the data structure has a pointer to its parent, a pointer to its immediate existing younger sibling, a pointer to its eldest existing son, and an integer as its successorship to its parent. To access any other son of a node, the first-born existing son must be accessed first. The siblings of the same parent are managed as a linked list. This data structure is an extension or enhancement of the traditional quad tree data structure. The primogenitary linked quad tree is applied to discrete multiple criteria optimization for the identification, storage, and retrieval of nondominated criterion vectors. Algorithms managing this data structure are developed and implemented. Major advantages of using the primogenitary linked quad tree instead of the traditional quad tree are savings in memory or storage space and savings in execution time. Examples are provided to demonstrate the application. A computational experiment is conducted to test the performances of the data structure and the algorithms. Computational results show that this data structure uses only a small fraction of the CPU time used by the traditional quad tree to perform the same task. Using this data structure, the identification, storage and retrieval of nondominated criterion vectors become an easy task for discrete multiple criteria optimization problems with many criteria and hundreds of thousands criterion vectors. This data structure can also be used for storage and retrieval of data with composite keys in other applications.     

Nonlinear discrete inequality in two variables with delay and its application

Delay discrete integral inequalities with n nonlinear terms in two variables are discussed, which generalize some existing results and can be used as powerful tools in the analysis of certain partial difference equations. An application example is also given to show boundedness of solutions of a difference equation.     

State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic

Recently Gamarnik and Zeevi (Ann. Appl. Probab. 16:56–90, 2006) and Budhiraja and Lee (Math. Oper. Res. 34:45–56, 2009) established that, under suitable conditions, a sequence of the stationary scaled queue lengths in a generalized Jackson queueing network converges to the stationary distribution of multidimensional reflected Brownian motion in the heavy-traffic regime. In this work we study the corresponding problem in multiclass queueing networks (MQNs).     

Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.     

The existence of HGDDs with block size four and its application to double frames

In this note, we complete the existence spectra of HGDDs with block size four as well as double frames with block size three.