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.
Other approaches to analyze two-dimensional queueing models have been developed in @cite_29 (compensation method), and @cite_0 @cite_13 (power series algorithm). In the latter one, power series expansions of steady-state probabilities as functions of a certain parameter of the system (usually the load) were derived. Recently, the authors in @cite_8 studied generalized processor sharing queues and introduced an alternative method, by providing power series expansions of the generating function; see also @cite_30 @cite_31 @cite_1 .
@cite_30 @cite_8 @cite_29 @cite_1 @cite_0 @cite_31 @cite_13
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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 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. Several queueing processes may be modeled as random walks on a multidimensional grid. In this paper the equilibrium distribution for the case of a two-dimensional grid is considered. In previous research it has been shown that for some two-dimensional random walks the equilibrium distribution has the form of an infinite series of products of powers which can be constructed with a compensation procedure. The object of the present paper is to investigate under which conditions such an elegant solution exists and may be found with a compensation approach. The conditions can be easily formulated in terms of the random behaviour in the inner area and the drift on the boundaries. Recently complex function techniques have been developed for the analysis of queueing systems which need for their modelling a two dimensional state space. A variety of computer- and communication networks gives rise to such two-dimensional queueing systems and their analysis is needed for the performance evaluation of these aggregates. The present study reviews these developments 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. We study a processing system comprised of parallel queues, whose individual service rates are specified by a global service mode (configuration). The issue is how to switch the system between various possible service modes, so as to maximize its throughput and maintain stability under the most workload-intensive input traffic traces (arrival processes). Stability preserves the job inflow–outflow balance at each queue on the traffic traces. Two key families of service policies are shown to maximize throughput, under the mild condition that traffic traces have long-term average workload rates. In the first family of cone policies, the service mode is chosen based on the system backlog state belonging to a corresponding cone. Two distinct policy classes of that nature are investigated, MaxProduct and FastEmpty. In the second family of batch policies (BatchAdapt), jobs are collectively scheduled over adaptively chosen horizons, according to an asymptotically optimal, robust schedule. The issues of nonpreemptive job processing and non-negligible switching times between service modes are addressed. The analysis is extended to cover feed-forward networks of such processing systems nodes. The approach taken unifies and generalizes prior studies, by developing a general trace-based modeling framework (sample-path approach) for addressing the queueing stability problem. It treats the queueing structure as a deterministic dynamical system and analyzes directly its evolution trajectories. It does not require any probabilistic superstructure, which is typically used in previous approaches. Probability can be superposed later to address finer performance questions (e.g., delay). The throughput maximization problem is seen to be primarily of structural nature. The developed methodology appears to have broader applicability to other queueing systems. 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 this paper we propose the following approach to the dimensioning of the radio part of the downlink in OFDMA networks. First, we use information theory to characterize the bit-rate in the channel from a base station to its mobile. It depends on the power and bandwidth allocated to this mobile. Then, we describe the resource (power and bandwidth) allocation problem and characterise feasible configurations of bit-rates of all users. As the key element, we propose some particular sufficient condition (in a multi-Erlang form) for a given configuration of bit-rates to be feasible. Finally, we consider an Erlang's loss model, in which streaming arrivals whose admission would lead to the violation of this sufficient condition are blocked and lost. In this model, the blocking probabilities can be calculated using Kaufman-Roberts algorithm. We propose it to evaluate the minimal density of base stations assuring acceptable blocking probabilities for a streaming traffic of a given load per surface unit. We validate this approach by comparison of the blocking probabilities to these simulated in the similar model in which the admission control is based on the original feasibility property (instead of its sufficient condition). Our sufficient bit-rate feasibility condition can also be used to dimension the network with respect to the elastic traffic. This paper is a survey of certain known results concerning networks of queues. The choice of the topics presented has been made with special emphasis on mathematical results which can be applied to the analysis and synthesis of mathematical or simulation models of complex computer systems in which an ensemble of resources is shared among a set of user programs. The subjects covered include the Jackson, and Gordon and Newell theorems; the work-rate theorems of Chang, Lavenberg and Traiger; the Baskett, Chandy, Muntz, Palacios results. We also discuss in a unified manner problems related to Poisson processes in queueing networks. Companion papers (Parts II, III) will present results concerning approximations to queueing networks and some of their applications to computer system performance evaluation. Microsimulation models are normally used to evaluate traffic-adaptive signal control systems. This paper develops an analytical approach for this evaluation based on queuing models. In particular, a queuing model is developed for a simplified adaptive control strategy that is based on rolling horizon scheme, in which a signal serves two movements alternatively. In this strategy, the first movement is served until the queue dissipates, then the second movement is served until the queue dissipates, then the signal goes back to serving the first movement, and this cyclic process repeats. A numerical algorithm is developed in a stochastic context to compute steady-state performance measures such as average delays and expected queue lengths. These results are compared with simulation-based results, and indeed, the analytically derived numerical method predicts well the simulation results In this paper, we study a hybrid scheduling mechanism in discrete-time. This mechanism combines the well-known Generalized Processor Sharing (GPS) scheduling with strict priority. We assume three customer classes with one class having strict priority over the other classes, whereby each customer requires a single slot of service. The latter share the remaining bandwith according to GPS. This kind of scheduling is used in practice for the scheduling of jobs on a processor and in Quality of Service modules of telecommunication network devices. First, we derive a functional equation of the joint probability generating function of the queue contents. To explicitly solve the functional equation, we introduce a power series in the weight parameter of GPS. Subsequently, an iterative procedure is presented to calculate consecutive coefficients of the power series. Lastly, the approximation resulting from a truncation of the power series is verified with simulation results. We also propose rational approximations. We argue that the approximation performs well and is extremely suited to study these systems and their sensitivity in their parameters (scheduling weights, arrival rates, loads ...). This method provides a fast way to observe the behaviour of such type of systems avoiding time-consuming simulations. 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 study discrete-time priority queueing systems fed by a large number of arrival streams. We first provide bounds on the actual delay asymptote in terms of the virtual delay asymptote. Then, under suitable assumptions on the arrival process to the queue, we show that these asymptotes are the same. As an application of this result, we then consider a priority queueing system with two queues. Using the earlier result, we derive an upper bound on the tail probability of the delay. Under certain assumptions on the rate function of the arrival process, we show that the upper bound is tight. We then consider a system with Markovian arrivals and numerically evaluate the delay tail probability and validate these results with simulations. We consider the problem of resource allocation in downlink OFDMA systems for multi service and unknown environment. Due to users' mobility and intercell interference, the base station cannot predict neither the Signal to Noise Ratio (SNR) of each user in future time slots nor their probability distribution functions. In addition, the traffic is bursty in general with unknown arrival. The probability distribution functions of the SNR, channel state and traffic arrival density are then unknown. Achieving a multi service Quality of Service (QoS) while optimizing the performance of the system (e.g. total throughput) is a hard and interesting task since it depends on the unknown future traffic and SNR values. In this paper we solve this problem by modeling the multiuser queuing system as a discrete time linear dynamic system. We develop a robust H∞ controller to regulate the queues of different users. The queues and Packet Drop Rates (PDR) are controlled by proposing a minimum data rate according to the demanded service type of each user. The data rate vector proposed by the controller is then fed as a constraint to an instantaneous resource allocation framework. This instantaneous problem is formulated as a convex optimization problem for instantaneous subcarrier and power allocation decisions. Simulation results show small delays and better fairness among users. Generalized Processor Sharing (GPS) is a simple, flexible and fair scheduling mechanism to achieve delay differentiation between several customer classes. The amount of delay differentiation is regulated by the weights given to the classes. In this paper we assume a discrete-time, two-class GPS queueing system. Our goal is to derive the optimal weights in order to minimize a weighted sum of functions of the mean delays of both classes. As analytical results are scarce we use an approximation method. The approximation is based on power series expansions of the mean queue length of each of the queues for certain weights. Pade approximants are used to extrapolate the approximation to the whole domain of possible weights, resulting in a set of approximations. An algorithm is proposed to filter out the infeasible solutions (with regard to monotonicity and other characteristics of the system) and aggregate the others, resulting in a single approximation. The result proves to be an accurate approximation of the optimal weights w.r.t. the cost function. For a load of 90 we have a maximum misprediction of 1 of the cost, in the case of a weighted sum of squares of the mean delays. The main contribution of this article is that power series approximations can be used effectively for optimization purposes. .We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89–148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5–25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks. 1. Introduction. This paper is a continuation of Dai and Lin (2005), in which maximum pressure policies are shown to be throughput optimal for a class of stochastic processing networks. Throughput optimality is an important, first-order objective for many networks, but it ignores some key secondary performance measures like queueing delays experienced by jobs in these networks. In this paper we show that maximum pressure policies enjoy additional optimality properties; they are asymptotically optimal in minimizing a certain workload or holding cost of a stochastic processing network. Stochastic processing networks have been introduced in a series of three papers by Harrison (2000, 2002, 2003). In Dai and Lin (2005) and this paper we consider a special class of Harrison’s model. This class of stochastic processing networks is much more general than multiclass queueing networks that have been a subject of intensive study in the last 20 years; see, for example, Harrison (1988), Williams 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). Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i = 1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a Riemann- Hilbert boundary value problem. Various performance measures are calculated and numerical results are presented. In particular, numerical results demonstrate that the proposed multiple access system with two types of jobs and constant retrial rates provides incentives for the users to respect their contracts. We consider a multiclass multiplexer with support for multiple service classes and dedicated buffers for each service class. Under specific scheduling policies for sharing bandwidth among these classes, we seek the asymptotic (as the buffer size goes to infinity) tail of the buffer overflow probability for each dedicated buffer. We assume dependent arrival and service processes as is usually the case in models of bursty traffic. In the standard large deviations methodology, we provide a lower and a matching (up to first degree in the exponent) upper bound on the buffer overflow probabilities. We introduce a novel optimal control approach to address these problems. In particular, we relate the lower bound derivation to a deterministic optimal control problem, which we explicitly solve. Optimal state trajectories of the control problem correspond to typical congestion scenarios. We explicitly and in detail characterize the most likely modes of overflow. We specialize our results to the generalized processor sharing policy (GPS) and the generalized longest queue first policy (GLQF). The performance of strict priority policies is obtained as a corollary. We compare the GPS and GLQF policies and conclude that GLQF achieves smaller overflow probabilities than GPS for all arrival and service processes for which our analysis holds. Our results have important implications for traffic management of high-speed networks and can be used as a basis for an admission control mechanism which guarantees a different loss probability for each class. A newversion of the Power-Series Algorithm is developed to compute the steady-state distribution of a rich class of Markovian queueing networks. The arrival process is a Multi-queue Markovian Arrival Process, which is a multi-queue generalization of the BMAP. It includes Poisson, fork and round-robin arrivals. At each queue the service process is a Markovian Service Process, which includes sequences of phase-type distributions, setup times and multi-server queues. The routing is Markovian. The resulting queueing network model is extremely general, which makes the Power-Series Algorithm a useful tool to study load-balancing, capacity-assignment and sequencing problems.