Adrian E. Conway

RECAL—a new efficient algorithm for the exact analysis of multiple-chain closed queuing networks

RECAL, a Recursion by Chain Algorithm for computing the mean performance measures of product-form multiple-chain closed queuing networks, is presented. It is based on a new recursive expression that relates the normalization constant of a network with r closed routing chains to those of a set of networks having (r - 1) chains. It relies on the artifice of breaking down each chain into constituent subchains that each have a population of one. The time and space requirements of the algorithm are shown to be polynomial in the number of chains. When the network contains many routing chains, the proposed algorithm is substantially more efficient than the convolution or mean value analysis algorithms. The algorithm, therefore, extends the range of queuing networks that can be analyzed efficiently by exact means.

Efficient decomposition methods for the analysis of multi-facility blocking models

Three new decomposition methods are developed for the exact analysis of stochastic multi-facility blocking models of the product-form type. The first is a basic decomposition algorithm that reduces the analysis of blocking probabilities to that of two separate subsystems. The second is a generalized M-subsystem decomposition method. The third is a more elaborate and efficient incremental decomposition technique. All of the algorithms exploit the sparsity of locality that can be found in the demand matrix of a system. By reducing the analysis to that of a set of subsystems, the overall dimensionality of the problem is diminished and the computational requirements are reduced significantly. This enables the efficient computation of blocking probabilities in large systems. Several numerical examples are provided to illustrate the computational savings that can be realized.

Exact computation of blocking probabilities in state-dependent multi-facility blocking models

In this paper, we develop a new general-purpose recursive algorithm for the exact computation of blocking probabilities in multi-facility blocking models with some forms of state-dependent arrival rates. The recursion is cast in terms of the partition function of a product-form model. A dynamic scaling procedure is also proposed to avoid numerical overflow or underflow. The recursion is sufficiently general to be applied to a variety of different state-dependent arrival processes and service rate functions.

Blocking formulae for the Engset model

In this paper, we present simple recursive algorithms for computing call and time congestion in the classical Engset model with M sources and N servers. The first recursion has the complexity of O(MN) and gives the blocking probabilities for all intermediate values of M and N. The second recursion assumes a particular value of M and has the complexity of O(N). It gives the blocking probabilities for all intermediate values of N. Both recursions are similar to the well-known recurrence for computing the Erlang loss function.

Mean-value analysis of multi-facility blocking models with state-dependent arrivals

Abstract A new mean-value type of algorithm is developed for analyzing multi-facility blocking models with state-dependent arrival rates. It can be applied to a broad class of blocking systems with simultaneous resource possession including, for example, circuit-switched networks. The underlying recursion is cast in terms of blocking probabilities and marginal state probabilities. The developments made here generalize previous results that were restricted to the case of constant arrival rates.

An efficient algorithm for semi-homogeneous queueing network models

The class of product-form semi-homogeneous queueing networks is introduced as a generalization of the class of homogeneous networks, which has been considered by Balbo et al for the performance modeling of local area networks. In semi-homogeneous networks, the relative traffic intensity at the various shared resources may depend on the routing chain to which a customer belongs. We develop an efficient algorithm for the exact analysis of this class of networks. It is based on the equations which form the foundation of RECAL, a general purpose exact algorithm for multiple-chain closed queueing networks. The complexity of the algorithm is shown to be of order less than exponential in (P-1)1/2, where P is the number of processors (workstations) in the network. It is therefore, in general, more efficient than a direct application of either convolution, MVA or RECAL to the class of semi-homogeneous queueing networks. The algorithm presented here may be situated between the algorithms of Balbo et al and the general purpose algorithms, both in terms of its generality and efficiency.