Feedback invariant discipline and insensitivity in closed queueing networks

Motivated by control theory concepts, we investigate a closed queueing network by adding a feedback path to each node. A feedback invariant service discipline is introduced. This approach differs from the conventional queueing theory approach and provides a new criteria and an intuitive explanation for the insensitivity property in a closed queueing network.     

A unified view of product-form approximation techniques for general closed queueing networks

Two major approximate techniques have been proposed for the analysis of general closed queueing networks, namely the aggregation method and Marie's method. The idea of the aggregation technique is to replace a subsystem (a subnetwork) by a flow equivalent single-server with load-dependent service rates. The parameters of the equivalent server are obtained by analyzing the subsystem in isolation as a closed system with different populations. The idea of Marie's method is also to replace a subsystem by an equivalent exponential service station with load-dependent service rates. However, in this case, the parameters of the equivalent server are obtained by analyzing the subsystem in isolation under a load-dependent Poisson arrival process. Moreover, in Marie's case, the procedure is iterative.

In this paper we provide a general and unified view of these two methods. The contributions of this paper are the following. We first show that their common principle is to partition the network into a set of subsystems and then to define an equivalent product-form network. To each subsystem is associated a load-dependent exponential station in the equivalent network. We define a set of rules in order to partition any general closed network with various features such as general service time distributions, pupulation constraints, finite buffers, state-dependent routing. We then show that the aggregation method and Marie's method are two ways of obtaining the parameters of the equivalent network associated with a given partition. Finally, we provide a discussion pertaining to the comparison of the two methods with respect to their accuracy and computational complexity.     

Sensitivity results in open, closed and mixed product form queueing networks

General formulas are proposed to quantify the effects of changing the model parameters in the so-called BCMP network [F. Baskett et al., J. ACM 22 (2) (April 1975) 248–260]. These formulas relate the derivative of the expectation of any function of both the state and the paramaters of the network with respect to any model parameter (i.e., arrival rate, mean service demand, service rate, visit ratio, traffic intensity) to known functions of the state variables. Applications of our results to sensitivity analysis and optimization problems are given.     

Characteristics of optimal workload allocation for closed queueing networks

We consider the problem of allocating a given workload among the stations in a multi-server product-form closed queueing network to maximize the throughput. We first investigate properties of the throughput function and prove that it is pseudoconcave for some special cases. Some other characteristics of the optimal workload and its physical interpretation are also provided. We then develop two computational procedures to find the optimum workload allocation under the assumption that the throughput function is pseudoconcave in general. The primary advantage of assuming pseudoconcavity is that, under this assumption, satisfaction of first order necessary conditions is sufficient for optimality. Computational experience with these algorithms provides additional support for the validity of this assumption. Finally, we generalize the solution procedure to accommodate bounds on the workloads at each station.     

ACAL—An efficient adaptive chain oriented algorithm for the analysis of multiple chain closed queueing networks

This paper presents an Adaptive Chain Oriented Algorithm (ACAL) for the analysis of closed product form queueing networks with multiple chains. The algorithm calculates the joint queue length distributions as well as the mean performance values. It is shown to be more efficient than existing ones, e.g. MVAC, RECAL or DAC, in dealing with networks with a large number of chains and a small number of nodes. In addition, it has an adaptive nature, which further improves the efficiency of ACAL. The adaptive nature also distinguishes ACAL from existing algorithms.     

Arrival first queueing networks with applications in kanban production systems

In this paper, we introduce a new class of queueing networks called arrival first networks. We characterise its transition rates and derive the relationship between arrival rules, linear partial balance equations, and product form stationary distributions. This model is motivated by production systems operating under a kanban protocol. In contrast with the conventional departure first networks, where a transition is initiated by service completion of items at the originating nodes that are subsequently routed to the destination nodes (push system), in an arrival first network a transition is initiated by the destination nodes of the items and subsequently those items are processed at and removed from the originating nodes (pull system). These are similar to the push and pull systems in manufacturing systems.

Our characterisation provides necessary and sufficient conditions for the network to possess linear traffic equations, and sufficient conditions for the network to have a product form stationary distribution. We apply our results to networks operating under a kanban mechanism and characterise the rate at which items are pulled as well as the routing and blocking protocols that give rise to a product form stationary distribution.     

On the nonconcavity of throughput in certain closed queueing networks

Analytic queueing network models are being used to analyze various optimization problems such as server allocation, design and capacity issues, optimal routing, and workload allocation. The mathematical properties of the relevant performance measures, such as throughput, are important for optimization purposes and for insight into system performance.We show that for closed queueing networks of m arbitrarily connected single server queues with n customers, throughput, as a function of a scaled, constrained workload, is not concave. In fact, the function appears to be strictly quasiconcave. There is a constraint on the total workload that must be allocated among the servers in the network. However, for closed networks of two single server queues, we prove that our scaled throughput is concave when there are two customers in the network and strictly quasi-concave when there are more than two customers. The mathematical properties of both the scaled throughput and reciprocal throughput are demonstrated graphically for closed networks of two and three single server queues.     

Optimal buffer allocation in finite closed networks with multiple servers

In this study, we present an optimal buffer allocation procedure for closed queueing networks with finite buffers. The performance measures are evaluated using the expanded mean value analysis, and the solution procedure is incorporated into a nonlinear optimization scheme to arrive at the sub-optimal buffer space vector. The effectiveness of the method is demonstrated through several numerical experiments. Discussions on convergence and computational complexity are also included.     

Response times in a two-node queueing network with feedback

The study presented in this paper is motivated by the performance analysis of response times in distributed information systems, where transactions are handled by iterative server and database actions. We model system response times as sojourn times in a two-node open queueing network with a processor sharing (PS) node and a first-come-first-served (FCFS) node. External customers arrive at the PS node according to a Poisson process. After departing from the PS node a customer proceeds to the FCFS node with probability p, and with probability 1−p the customer departs from the system. After a visit to the FCFS node, customers are fed back to the PS node. The service requirements at both nodes are exponentially distributed. The model is a Jackson network, admitting a product-from solution for the joint number of customers at the nodes, immediately leading to a closed-form expression for the mean sojourn times in steady-state. The variance of the sojourn times, however, does not admit an exact expression—the complexity is caused by the possibility of overtaking. In this paper we propose a methodology for deriving simple, explicit and fast-to-evaluate approximations for the variance of the sojourn times. Numerical results demonstrate that the approximations are very accurate in most model instances.     

Entropy maximised queueing networks with blocking and multiple job classes

The method of entropy maximisation (MEM) is applied in a state space partitioning mode for the approximation of the joint stationary queue length distribution of an M/M/1/N queue with finite capacity, N( > 1), multiple and distinct classes of jobs, R( > 1), under a complete buffer sharing scheme and mixed service disciplines drawn from the first-come-first-served (FCFS), last-come-first-served with (LCFS-PR) or without (LCFS-NPR) preemption and processor sharing (PS) rules. The marginal and aggregate maximum entropy (ME) queue length distributions and the associated blocking probabilities per class are also determined. These ME results in conjunction with the first moments of the effective flows are used, as building blocks, in order to establish a new product-form approximation for arbitrary exponential open queueing networks with multiple classes of jobs under repetitive-service (RS) blocking with random destination (RD). It is verified that the ME approximation reduces to the exact truncated solution of open multi-class reversible queueing networks. Numerical experiments demonstrate a good accuracy level of ME statistics in relation to simulation. Moreover, recent extentions of MEM for arbitrary GE-type queueing networks with RS-RD blocking and multiple classes of jobs are presented.     

Asymptotic approximations and bottleneck analysis in product form queueing networks with large populations

Asymptotic approximations are constructed for the performance measures of product form queueing networks consisting of single server, fixed rate nodes with large populations. The approximations are constructed by applying singular perturbation methods to the recursion equations of Mean Value Analysis. Networks with a single job class are studied first to illustrate the use of perturbation techniques. The leading term in the approximation is related to bottleneck analysis, but fails to be accurate if there is more than one bottleneck node. A uniform approximation is constructed which is valid for networks with many bottleneck nodes. The accuracy of the uniform approximation is demonstrated for both small and large population sizes. Next, multiclass networks are considered. The leading term in the asymptotic approximation is again related to bottleneck analysis but fails to be valid across “switching surfaces”. Across these the bottleneck nodes of the network change as a function of the fraction of jobs in the different job classes. A boundary layer correction is constructed near the switching surfaces which provides an asymptotic connection across the switching surfaces. Numerical examples are presented to demonstrate the accuracy of the results. We illustrate the asymptotic approach on some simple networks and indicate how to treat more complicated problems.     

Approximation models of feed-forward G/G/1/N queueing networks with correlated arrivals

In this paper we develop approximation models for feed-forward networks of G/G/1/N queues. We use Linear Algebra Queueing Theory (LAQT) techniques to create reduced state space representations for the queue departure processes. Reduced state space departure processes are presented where the first three moments and the correlation decay are mapped to a two state process. A three state process is also presented matching the first five moments and the first three lag autocorrelations. Numerical examples of end-to-end performance for high-speed communications networks with correlated arrival traffic are presented. The results are compared with simulation models and other approximation methods.     

Mean-value analysis and convolution method for queue-dependent servers in closed queueing networks

In this paper, we formulate a recursive relation for the marginal probabilities of a closed network with K customers in terms of the same network with K ? 1 customers. Mean-value analysis (MVA) is an application of this relation, together with Little's formula. It is shown that the convolution method, too, can be based on the same recursive result. This leads to a new convolution algorithm called normalized convolution algorithm (NCA), which like MVA works entirely with probabilities and throughputs rather than with quantities such as normalization contacts. NCA avoids a difficult problem, the occurrence of floating-point over-flows in the original convolution algorithm.We shall also solve a numerical stability problem found in MVA. Finally, we show how MVA and the convolution algorithm can be combined in the same problem to yield a hybrid method retaining the best properties of both methods.     

An approximate MVA algorithm for exponential, class-dependent multiple servers

Mean value analysis (MVA) is an efficient algorithm for determining the mean sojourn time, the mean queue length, and the throughput in a closed multiclass queueing network. It provides exact results for the class of product-form networks. Often different classes have different service requirements in FCFS queues, but such networks are not of product form. There are several possibilities to compute performance measure for such nodes and networks. In this paper we present an approximation formula for multiple-server FCFS queues with class-dependent service times as a Norton flow equivalent product node, where the departure rate of any class depends on the number of customers of all classes in the queue. We will use this approximation in the sojourn time formula of some exact and approximate MVA algorithms.