We consider a tandem queue with coupled processors, which is subject to global breakdowns. When the network is in the operating mode and both queues are non empty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the non-empty station. Moreover, arrival rates depend on the state of the network. The system is described by a Markov modulated random walk in the quarter plane representing the number of jobs in the two stations and the state of the network. By applying the generating function approach, we first apply the power series approximation method to obtain power series expansions of the generating function of the stationary queue lengths for both network states. Then, we also provide a way to derive the generating function of the stationary queue lengths for both network states in terms of the solution of a Riemann-Hilbert boundary value problem. Numerical results are obtained to show insights in the system performance.
A tandem queue with two coupled processors was analyzed in @cite_27 , while later computational issues, as well as asymptotic results were discussed and presented in @cite_15 @cite_25 , respectively. There, it was shown that the problem of finding the bivariate generating function of the stationary joint queue length distribution is reduced to a Riemann-Hilbert boundary value problem.
@cite_27 @cite_25 @cite_15
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We consider a tandem queue with coupled processors and analyze the two-dimensional Markov process representing the numbers of jobs in the two stations. A functional equation for the generating function of the stationary distribution of this two-dimensional process is derived and solved through the theory of Riemann-Hilbert boundary value problems. In Resing and Ormeci [16] it is shown that the two-stage tandem queue with coupled processors can be solved using the theory of boundary value problems. In this paper we consider the issues that arise when calculating performance measures like the mean queue length and the fraction of time a station is empty. It is assumed that jobs arrive at the first station according to a Poisson process and require service at both stations before leaving the system. The amount of work that a job requires at each of the stations is an independent, exponentially distributed random variable. When both stations are nonempty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the nonempty station. We study the two-dimensional Markov process representing the numbers of jobs at the two stations. The problem of finding the generating function of the stationary distribution can be reduced to two different Riemann-Hilbert boundary value problems, where both problems yield a complete analytical solution. We discuss the similarities and differences between the two problems, and relate them to the computational aspects of obtaining performance measures. This paper presents a novel technique for deriving asymptotic expressions for the occurrence of rare events for a random walk in the quarter plane. In particular, we study a tandem queue with Poisson arrivals, exponential service times and coupled processors. The service rate for one queue is only a fraction of the global service rate when the other queue is non-empty; when one queue is empty, the other queue has full service rate. The bivariate generating function of the queue lengths gives rise to a functional equation. In order to derive asymptotic expressions for large queue lengths, we combine the kernel method for functional equations with boundary value problems and singularity analysis. We derive in this paper closed formulae for the joint probability generating function of the number of customers in the two FIFO queues of a generalized processor-sharing (GPS) system with two classes of customers arriving according to Poisson processes and requiring exponential service times. In contrast to previous studies published on the GPS system, we show that it is possible to establish explicit expressions for the generating functions of the number of customers in each queue without calling for the formulation of a Riemann-Hilbert problem. We specifically prove that the problem of determining the unknown functions due to the reflecting conditions on the boundaries of the positive quarter plane can be reduced to a Poisson equation. The explicit formulae are then used to derive some characteristics of the GPS system (in particular the tails of the probability distributions of the numbers of customers in each queue). We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability β, and a customer of queue 2 is served with probability 1?β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z 1,z 2) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z 1,z 2) in β. The first coefficient of this power series corresponds to the priority case β=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Pade approximation. We consider the double queue that arises when arriving customers simultaneously place two demands handled independently by two servers. It is assumed that the customer arrivals form a Poisson process with mean 1, the servers have exponential service times with rate @math , and @math , which insures stability of the queue. Let @math be the respective lengths of the @math - and @math -queues, and @math at equilibrium. In a previous paper with the same title we obtained a formula for the generating function @math . We use this to derive the asymptotic behavior of @math as @math . The asymptotic results are employed to study the interdependence of @math . We derive limit laws for the expectation and distribution for either of these random variables conditioned on the other. It is well known that head-of-line blocking limits the throughput of an input-queued switch with first-in-first-out (FIFO) queues. Under certain conditions, the throughput can be shown to be limited to approximately 58.6 . It is also known that if non-FIFO queueing policies are used, the throughput can be increased. However, it has not been previously shown that if a suitable queueing policy and scheduling algorithm are used, then it is possible to achieve 100 throughput for all independent arrival processes. In this paper we prove this to be the case using a simple linear programming argument and quadratic Lyapunov function. In particular, we assume that each input maintains a separate FIFO queue for each output and that the switch is scheduled using a maximum weight bipartite matching algorithm. We introduce two maximum weight matching algorithms: longest queue first (LQF) and oldest cell first (OCF). Both algorithms achieve 100 throughput for all independent arrival processes. LQF favors queues with larger occupancy, ensuring that larger queues will eventually be served. However, we find that LQF can lead to the permanent starvation of short queues. OCF overcomes this limitation by favoring cells with large waiting times. This paper deals with two parallel processors interacting only during breakdown. The system is analyzed in the case where the load is not lost, and in steady state. Generating functions are obtained by solving a functional equation in two complex variables with the aid of the theory of boundary value problems for regular functions. We provide power series approximations for a structured batch arrival single server retrial system with two infinite capacity weighted fair orbit queues, i.e., the re-transmission rate of an orbit depends on the state of the other orbit queue. We consider both exponential and arbitrary distributed service times. In both cases we obtain power series expansions of the generating functions of the stationary joint orbit queue-length distributions, and provide a recursive approach to calculate their coefficients. We also show how to obtain the generating function of the stationary joint orbit queue-length distribution with the aid of a Riemann boundary value problem. Power series approximations are also provided for the model with two independent Poisson streams of jobs with single arrivals. Numerical illustrations are performed and show the accuracy of our approach. This paper considers a special case of the coupled-tasks scheduling problem on one processor. The general problems were analyzed in depth by Orman ad Potts [1]. In this paper, we cosider that all processing times are equal to 1, the gap has exact length L, we have precedence constraits, compatibility constraits are introduced and the criterion is to minimize the scheduling length. We use this problem to study the problem of data acquisittion and data treatment of a torpedo under the water. We show that this problem is NP-complete and we propose an ρ-approximation algorithm where ρ ≤ (L+6) 6. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. Most existing discussions regarding the time-dependent distribution of queue length was undertaken in the context of isolated intersections. However, computing queue length distributions for a signalized network with generic topology is very challenging because such process involves convolution and nonlinear transformation of random variables, which is analytically intractable. To address such issue, this study proposes a stochastic queue model considering the strong interdependence relations between adjacent intersections using the probability generating function as a mathematical tool. Various traffic flow phenomena, including queue formation and dissipation, platoon dispersion, flow merging and diverging, queue spillover, and downstream blockage, are formulated as stochastic events, and their distributions are iteratively computed through a stochastic network loading procedure. Both theoretical derivation and numerical investigations are presented to demonstrate the effectiveness of the proposed approach in analyzing the delay and queues of signalized networks under different congestion levels. This paper introduces a scheduling problem with coupled-tasks in presence of a compatibility graph on a single processor. We investigate a specific configuration, in which the coupled-tasks possess an idle time equal to 2. The complexity of these problems will be studied according to the presence or absence of triangles in the compatibility graph. As an extended matching, we propose a polynomial-time algorithm which consists in minimizing the number of non-covered vertices, by covering vertices with edges or paths of length two in the compatibility graph. This type of covering will be denoted by 2-cover technique. According on the compatibility graph type, the 2-cover technique provides a polynomial-time rho-approximation algorithm with rho=13 12 (resp. rho=10 9) in absence (resp. presence) of triangles. An important class of queueing networks is characterized by the following feature: in contrast with ordinary units, a disaster may remove all work from the network. Applications of such networks include computer networks with virus infection, migration processes with mass exodus and serial production lines with catastrophes. In this paper, we deal with a two-stage tandem queue with blocking operating under die presence of a secondary flow of disasters. The arrival flows of units and disasters are general Markovian arrival processes. Using spectral analysis, we determine the stationary distribution at departure epochs. That distribution enables us to derive the distribution of the number of units which leave the network at a disaster epoch, We calculate the stationary distribution at an arbitrary time and, finally, we give numerical results and graphs for certain probabilistic descriptors of the network. In this paper, we describe and analyze a joint scheduling, routing and congestion control mechanism for wireless networks, that asymptotically guarantees stability of the buffers and fair allocation of the network resources. The queue-lengths serve as common information to different layers of the network protocol stack. Our main contribution is to prove the asymptotic optimality of a primal-dual congestion controller, which is known to model different versions of transmission control protocol well The Laplace transform of the probability distribution of the end-to-end delay in tandem networks is obtained where the first and or second queue are G-queues, i.e. they have negative arrivals. For the most general case the method is based on the solution of a boundary value problem on a closed contour in the complex plane, which itself reduces to the solution of a Fredholm integral equation of the second kind. We also consider the dependence or independence of the sojourn times at each queue in the two special cases where only one of the queues is a G-queue, the other having no negative arrivals. We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x-1 log P(W > x) -+ -0* as x - o0 for 0* > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Girtner-Ellis condition for the cumulant generating function of the associated partial sums, i.e. n-1 log Eexp (OSn) --+ (0) as n - oo, plus regularity conditions on the decay rate function 0. The asymptotic decay rate 0* is the root of the equation 0(0) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates. We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the set-valued operator. In addition, relaxations and not necessarily vanishing proximal parameters are allowed. Weak and strong almost sure convergence properties of the iterates is established under mild conditions on the underlying stochastic processes. Leveraging these results, we also establish the almost sure convergence of the iterates of a stochastic variant of a primal-dual proximal splitting method for composite minimization problems. This paper describes a novel approach of packing sparse convolutional neural networks for their efficient systolic array implementations. By combining subsets of columns in the original filter matrix associated with a convolutional layer, we increase the utilization efficiency of the systolic array substantially (e.g., 4x) due to the increased density of nonzeros in the resulting packed filter matrix. In combining columns, for each row, all filter weights but one with the largest magnitude are pruned. We retrain the remaining weights to preserve high accuracy. We demonstrate that in mitigating data privacy concerns the retraining can be accomplished with only fractions of the original dataset (e.g., 10 for CIFAR-10). We study the effectiveness of this joint optimization for both high utilization and classification accuracy with ASIC and FPGA designs based on efficient bit-serial implementations of multiplier-accumulators. We present analysis and empirical evidence on the superior performance of our column combining approach against prior arts under metrics such as energy efficiency (3x) and inference latency (12x). We apply a new method of analysis to the asymmetric rendezvous search problem on the line (ARSPL). This problem, previously studied in a paper of Alpern and Gal (1995), asks how two blind, speed one players placed a distance d apart on the line, can find each other in minimum expected time. The distance d is drawn from a known cumulative probability distribution G, and the players are faced in random directions. We show that the ARSPL is strategically equivalent to a new problem we call the double linear search problem (DLSP), where an object is placed equiprobably on one of two lines, and equiprobably at positions ±d. A searcher is placed at the origin of each of these lines. The two searchers move with a combined speed of one, to minimize the expected time before one of them finds the object. Using results from a concurrent paper of the first author and J. V. Howard (1998), we solve the DLSP (and hence the ARSPL) for the case where G is convex on its support, and show that the solution is that conjectured in a paper of Baston and Gal (1998).