General Queueing Networks Research Articles

Global Optimization of Relay Placement for Seafloor Optical Wireless Networks

Optical wireless communication is a promising technology for underwater broadband access networks, which are particularly important for high-resolution environmental monitoring applications. This paper focuses on a deep-sea monitoring system, where an underwater optical wireless network is deployed on the seafloor. We model such an optical wireless network as a general queueing network and formulate an optimal relay placement problem, whose objective is to maximize the stability region of the whole system, i.e., the supremum of the traffic volume that the network is capable of accommodating. The formulated optimization problem is further shown to be non-convex, so that its global optimization is non-trivial. In this paper, we develop a global optimization method for this problem and we provide an efficient algorithm to compute an optimal solution. Through numerical evaluations, we show that a significant performance gain can be obtained by using the derived optimal solution.

Reward maximization in general dynamic matching systems

We consider a matching system with random arrivals of items of different types. The items wait in queues—one per item type—until they are “matched.” Each matching requires certain quantities of items of different types; after a matching is activated, the associated items leave the system. There exists a finite set of possible matchings, each producing a certain amount of “reward.” This model has a broad range of important applications, including assemble-to-order systems, Internet advertising, and matching web portals. We propose an optimal matching scheme in the sense that it asymptotically maximizes the long-term average matching reward, while keeping the queues stable. The scheme makes matching decisions in a specially constructed virtual system, which in turn controls decisions in the physical system. The key feature of the virtual system is that, unlike the physical one, it allows the queues to become negative. The matchings in the virtual system are controlled by an extended version of the greedy primal–dual (GPD) algorithm, which we prove to be asymptotically optimal—this in turn implies the asymptotic optimality of the entire scheme. The scheme is real time; at any time, it uses simple rules based on the current state of the virtual and physical queues. It is very robust in that it does not require any knowledge of the item arrival rates and automatically adapts to changing rates. The extended GPD algorithm and its asymptotic optimality apply to a quite general queueing network framework, not limited to matching problems, and therefore are of independent interest.

Optimal routing in general finite multi-server queueing networks.

The design of general finite multi-server queueing networks is a challenging problem that arises in many real-life situations, including computer networks, manufacturing systems, and telecommunication networks. In this paper, we examine the optimal routing problem in arbitrary configured acyclic queueing networks. The performance of the finite queueing network is evaluated with a known approximate performance evaluation method and the optimization is done by means of a heuristics based on the Powell algorithm. The proposed methodology is then applied to determine the optimal routing probability vector that maximizes the throughput of the queueing network. We show numerical results for some networks to quantify the quality of the routing vector approximations obtained.

Strong and weak stochastic bounds for multidimensional Markov chains

We study queueing networks similar to Jackson networks, modelled by a multidimensional Markov chain. The performance analysis may be very difficult or intractable, if there is no specific solution form. We explain how stochastic comparisons of Markov chains can be used to overcome this problem. We build new queueing which are easier to analyse and providing stochastic bounds upper or lower for the original model. In this paper, we propose different queueing systems in the sense of the strong and weak stochastic ordering for a general queueing network model in order to compute performance measure bounds as blocking probabilities. We discuss the accuracy of the bounds under different input parameter values.

Performance evaluation for general queueing networks in manufacturing systems: Characterizing the trade-off between queue time and utilization

Performance evaluation plays a key role in manufacturing system design and productivity improvement. Characterizing performance objectively is the first step. Inspired by the underlying structure of tandem queues, we have derived an approximate model to characterize the system performance. The model decomposes system queue time and variability into bottleneck and non-bottleneck parts while capturing the dependence among workstations. Compared the new model with prior approaches, the new model not only is more accurate but also requires less information. The property of manufacturing system performance is given based on the insight from the model.

Bounding Aggregations for Transient and Stationary Performance Analysis of Subnetworks

We consider large queueing networks for which transient and stationary probability distributions are very difficult or impossible to obtain due to the state space explosion problem. In performance analysis, we need in general to study only a part of the network (a node or a path). Thus, we propose to define bounding systems that lead to compute bounds on performance measures of the considered subsystem. The original large state space is mapped into a smaller space to overcome the state space explosion problem, and bounds both on stationary and on transient performance measures are computed from these reduced-size models. This approach provides an interesting solution for complex networks since we have a trade-off between the quality of the bounds and the state space size, thus the computational complexity. As an application, we study a general multi-server queueing network, with finite capacity queues. We define bounding systems to compute blocking probabilities. The influence of parameters on the precision of the computed bounds are studied through some numerical examples in order to give more insights into the proposed approach.

Buffer and throughput trade-offs in M/ G/1/ K queueing networks: A bi-criteria approach

The optimal design of real-life systems modeled as finite queueing networks is a difficult stochastic optimization problem. Besides being non-linear and including integer variables, different objectives often conflict with each other. For example, one conflicting pair of objectives includes first minimizing the overall number of buffers and then maximizing throughput. In this paper, we present an original methodology to solve a buffer allocation and throughput trade-off problem in single server general queueing networks. An approximation of the complete set of all best solutions, known as the Pareto optimal or non-inferior set, is derived by a special version of a multi-objective genetic algorithm (MOGA). The applied MOGA proves to be suitable for the stochastic trade-off problem. A comprehensive set of computational results attest to the efficiency and efficacy of the proposed methodology. We were able to show from a medium-sized mixed network that the squared coefficient of service time variation plays an important role in buffer allocation, which indicates the importance of using a general service model. Moreover, after the analysis of the solutions in the decision variable space, we confirm that the buffer allocation is highly dependent on the target throughput, which sometimes can be sacrificed in favor of using fewer of the usually expensive buffer spaces.

Performance optimization of open zero-buffer multi-server queueing networks

Open zero-buffer multi-server general queueing networks occur throughout a number of physical systems in the semi-process and process industries. In this paper, we evaluate the performance of these systems in terms of throughput using the generalized expansion method (GEM) and compare our results with simulation. Secondly, we embed the performance evaluation in a multi-objective optimization setting. This multi-objective optimization approach results in the Pareto efficient curves showing the trade-off between the total number of servers used and the throughput. Experiments for a large number of settings and different network topologies are presented in detail.

Departure process of a single server queueing system with Markov renewal input and general service time distribution

The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to take these into consideration can lead to serious underestimation of the performance measures. We consider herein a single server queueing system with a Markov renewal process (MRP) for its arrival process and a general service time distribution, and derive the distribution function and correlation coefficient of the departure process. Since the departure process also often corresponds to an arrival process in downstream queues, the results obtained here can be used to derive a better approximation of the performance measures of a non-product form general queueing network.

Performance analysis of an ingress switch in a JumpStart optical burst switching network

We consider an ingress optical burst switching (OBS) node employing the JumpStart signaling protocol. The switch serves a number of users, each connected to the switch with a fiber link that supports multiple wavelengths. Each wavelength is associated with a 3-state Markovian burst arrival process which permits short and long bursts to be modeled. We model the ingress switch as a closed multi-class non-product-form queueing network, which we analyze approximately by decomposition. Specifically, we develop new techniques to analyze the queueing network, first assuming a single class of customers, and subsequently multiple classes of customers. These analytical techniques have applications to general queueing networks beyond the one studied in this paper. We also develop computationally efficient approximate algorithms to analyze an ingress switch in the limiting case where the number of wavelengths is large. The algorithms have a good accuracy, and they provide insight into the effect of various system parameters on the performance of an ingress OBS switch.

The [formula omitted] tandem queue with losses

Tandem queues of the B M A P / G / 1 / N ¯ → ⋅ / P H / 1 / M type are good models for different fragments of communication systems and networks, so their investigation is interesting for theory and applications. These queues may play an important role for the validation of different decomposition algorithms designed for investigating more general queueing networks. Exact analytic analysis of this kind of queues for the cases of infinite and finite input buffers is implemented. Possible correlation and group arrivals are taken into account by means of considering the Batch Markovian Arrival Process ( BMAP) as input stream to the system. The Markov chain embedded at service completion epochs at the first service stage and the process of system states at arbitrary time are investigated. Loss probabilities at the first and second stages are calculated. Numerical results are presented to demonstrate the feasibility of the presented algorithms and describe the performance of the queueing model under study. The necessity of taking the input correlation into account is illustrated.

A Fast Cross-Entropy Method for Estimating Buffer Overflows in Queueing Networks

In this paper, we propose a fast adaptive importance sampling method for the efficient simulation of buffer overflow probabilities in queueing networks. The method comprises three stages. First, we estimate the minimum cross-entropy tilting parameter for a small buffer level; next, we use this as a starting value for the estimation of the optimal tilting parameter for the actual (large) buffer level. Finally, the tilting parameter just found is used to estimate the overflow probability of interest. We study various properties of the method in more detail for the M/M/1 queue and conjecture that similar properties also hold for quite general queueing networks. Numerical results support this conjecture and demonstrate the high efficiency of the proposed algorithm.

Decomposition of general queueing networks with MMPP inputs and customer losses

For nontrivial general (open) queueing networks, decomposition often represents the only feasible solution method besides simulation. The network is partitioned into individual nodes which are analyzed in isolation with respect to approximate internal traffic representations. The quality of the quickly obtained results very much depends on the descriptors for the traffic processes within the network, which may be split and merged before traversing the next queue. Recently, one of the existing decomposition formalisms based on renewal processes as traffic descriptors has been extended in order to include semi-Markov processes (SMPs) and Markov-modulated Poisson processes (MMPPs) with two states. However, due to the restriction to tandem networks, no operations for the splitting and and merging were provided. The numerical procedures for the splitting of SMPs and the superposition of MMPPs proposed in this paper render the extended decomposition framework available for general queueing networks. The correlations in the traffic processes, which are known to have a considerable impact on performance measures, are taken into account to some extent. Moreover, in addition to renewal processes, MMPP inputs—as they frequently arise in computer communication modeling—increase the range of applications of general queueing networks. Numerical experiments on different network configurations illustrate the capabilities and accuracy of the extended decomposition framework.

Approximations for the departure process of the G/G/1 queue with Markov-modulated arrivals

The interarrival time random variables at the nodes of queueing networks are typically correlated. The analysis, by ignoring the correlations, may not be very accurate in many instances, especially when the correlations are high. The G/G/1 queue with Markov-modulated arrivals is one of the few models that can be used to study queues with non-renewal arrival processes. The waiting time of the Markov-modulated G/G/1 queue has been studied widely in the literature. In this paper, we study its departure process. We derive the MacLaurin series for the moments and the lag-1 autocorrelation of the departure process. The coefficients of the MacLaurin series are calculated through simple recursive equations, which are then used to approximate the moments and the lag-1 autocorrelation based on Padé approximation. Several numerical examples are provided which show that our method gives very good estimates. Our results on the departure process can be potentially used to develop better approximation methods for general queueing networks.

Generalized queueing network analysis of integrated supply chains

Supply chain networks are formed from complex interactions between several companies whose aim is to produce and deliver goods to the customers at specified times and places. Computing the total lead time for customer orders entering such a complex network of companies is an important exercise. In this paper we present analytical models for evaluating the average lead times of make-to-order supply chains. In particular, we illustrate the use of generalized queueing networks to compute the mean and variance of the lead time. We present four interesting examples and develop queueing network models for them. The first two examples consider pipeline supply chains and compute the variance of lead time using queueing network approximations available in the literature. This analysis indicates that for the same percentage increase in variance, an increase at the downstream facility has a far more disastrous effect than the same increase at an upstream facility. Through another example, we illustrate the point that coordinated improvements at all the facilities is important and improvements at individual facilities may not always lead to improvements in the supply chain performance. The existing literature on approximate methods of analysis of forkjoin queueing systems assumes heavy traffic and requires tedious computations. We present here two tractable approximate analytical methods for lead time computation in a class of fork-join queueing systems. Our method is based on the results presented by Clarke in 1961. For the case where the 'joining' servers of the queueing system are of the type D/N/1, we present an easy to use approximate method and illustrate its use in evaluating decisions regarding logistics (for instance, who should own the logistics fleet-the manufacturer or the vendor?) and computing simple upper bounds for delivery reliability, that is the probability that customer desired due dates are met.

Optimal Configurations of General Job Shops

In this paper we study the problem of minimizing the expected number of jobs in a single class general open queueing network model of a job shop. This problem was originally posed by Buzacott and Shanthikumar [2] and solved by them for a special case. We extend their work in this paper. We derive feasibility conditions that simplify the analysis of the problem. We show that the optimal configuration can be completely characterized when both the utilizations of the machine centers are high and there are a large number of servers at each machine center. We also derive conditions under which the optimization problem reduces to solving a concave or a convex program and provide conditions under which the uniform flow line and the symmetric job shop (or variants of these) are optimal configurations for the job shop.

Higher order approximations for the single server queue with splitting, merging and feedback

The performance evaluation of many modern communication and manufacturing systems has been made possible by modeling them as queueing networks. Due to the limitations of exact algorithms to analyze general queueing networks, researchers have been focusing on developing approximations. Most of the approximations are lower order approximations in the sense that only a few parameters of the interarrival and service time distributions are used. This could lead to large errors in many conditions. In this paper, higher order approximations are developed for the single server queue with splitting, merging and feedback phenomena, which can then be integrated along with parametric decomposition to develop higher order approximations for queueing networks. Not only are the higher order approximations more accurate, but they can also be used to estimate the higher moments of the performance measures.

Allocation of interdependent resources for maximal throughput

A general queueing network model of computer and communication systems with interdependent resources is considered. The resources are modeled by a collection of servers that have to be allocated subject to certain constraints. Arrivals are Poisson, services are independent and identically distributed and service completions of different customers cannot be synchronized. A set of necessary conditions for the finiteness of long run average delays is obtained. An adaptive non-preemptive scheduling policy is presented and a set of sufficient conditions under which the policy achieves finite delays, is identified. The necessary and sufficient conditions differ only on the boundary region. The policy does not depend on the knowledge of the arrival rates and has polynomial time complexity for some applications

State space collapse with application to heavy traffic limits for multiclass queueing networks

Heavy traffic limits for multiclass queueing networks are a topic of continuing interest. Presently, the class of networks for which these limits have been rigorously derived is restricted. An important ingredient in such work is the demonstration of state space collapse. Here, we demonstrate state space collapse for two families of networks, first-in first-out (FIFO) queueing networks of Kelly type and head-of-the-line proportional processor sharing (HLPPS) queueing networks. We then apply our techniques to more general networks. To demonstrate state space collapse for FIFO networks of Kelly type and HLPPS networks, we employ law of large number estimates to show a form of compactness for appropriately scaled solutions. The limits of these solutions are next shown to satisfy fluid model equations corresponding to the above queueing networks. Results from Bramson l4,5r on the asymptotic behavior of these limits then imply state space collapse. The desired heavy traffic limits for FIFO networks of Kelly type and HLPPS networks follow from this and the general criteria set forth in the companion paper Williams l41r. State space collapse and the ensuing heavy traffic limits also hold for more general queueing networks, provided the solutions of their fluid model equations converge. Partial results are given for such networks, which include the static priority disciplines.

G-networks with multiple classes of negative and positive customers

In recent years a new class of queueing networks with “negative and positive” customers was introduced by one of the authors [5], and shown to have a nonstandard product form. This model has undergone several generalizations to include triggers or signals which are special forms of customers whose role is to move other customers from some queue to another queue [9, 10, 6, 7]. Positive customers are identical to the usual customers of a queueing network, while a negative customer which arrives to a queue simply destroys a positive customer. We call these generalized queueing networks G-networks. In this paper we extend the basic model of [5] to the case of multiple classes of positive customers, and multiple classes of negative customers. As in other multiple class queueing networks, a positive customer class is characterized by the routing probabilities and the service rate parameter at each service center while negative customers of different classes may have different “customer destruction” capabilities. In the present paper all service time distributions are exponential and the service centers can be of the following types: FIFO (first-in-first-out), LIFO/PR (last-in-first-out with preemption), PS (processor sharing), with class-dependent service rates.