Kelly and Jackson networks with interchangeable, cooperative servers

In open Kelly and Jackson networks, servers are assigned to individual stations, serving customers only where they are assigned. We investigate the performance of modified networks where servers cooperate. A server who would be idle at the assigned station will serve customers at another station, speeding up service there. We assume interchangeable servers: the service rate of a server at a station depends only on the station, not the server. This gives work conservation, which is used in various ways. We investigate three levels of server cooperation, from full cooperation, where all servers are busy when there is work to do anywhere in the network, to one-way cooperation, where a server assigned to one station may assist a server at another, but not the converse. We obtain the same stability conditions for each level and, in a series of examples, obtain substantial performance improvement with server cooperation, even when stations before modification are moderately loaded.

APPROXIMATING AND STABILIZING DYNAMIC RATE JACKSON NETWORKS WITH ABANDONMENT

In this paper, we generalize the Gaussian Variance Approximation (GVA), developed by Massey and Pender [16], to Jackson networks with abandonment. We approximate the queue length process with a multivariate Gaussian distribution and thus, we are able to estimate the mean and covariance matrix of the entire network with more accuracy than the associated fluid and diffusion limits of Mandelbaum, Massey, and Reiman [14]. We also show how the GVA method can be used to construct staffing schedules that approximately stabilize salient performance measures such as the probability of delay and the abandonment probabilities for the entire network. Unlike the work of Feldman et al. [5] which uses Monte Carlo simulation to stabilize the delay probabilities, our method does not require simulation and only requires the numerical integration of ${1 \over 2}(N^2 + 3N)$ differential equations for an N-dimensional network, which is more computationally efficient. Lastly, to confirm our approximations are accurate, we perform several numerical experiments for a wide range of parameter settings.

Nonergodic Jackson networks with infinite supply–local stabilization and local equilibrium analysis

Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes. Our results complement the results known in the literature for the subcases of Jackson networks with either infinite supply nodes or unstable nodes by providing an analysis of the significantly more challenging case of networks with both types of nonstandard node present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.

Jackson networks in nonautonomous random environments

We investigate queueing networks in a random environment. The impact of the evolving environment on the network is by changing service capacities (upgrading and/or degrading, breakdown, repair) when the environment changes its state. On the other side, customers departing from the network may enforce the environment to jump immediately. This means that the environment is nonautonomous and therefore results in a rather complex two-way interaction, especially if the environment is not itself Markov. To react to the changes of the capacities we implement randomised versions of the well-known deterministic rerouteing schemes 'skipping' (jump-over protocol) and `reflection' (repeated service, random direction). Our main result is an explicit expression for the joint stationary distribution of the queue-lengths vector and the environment which is of product form.