On multi-class multi-server queueing and spare parts management

Multi-class multi-server queuing problems are a generalization of the wellknown M/M/k situation to arrival processes with clients of N types that require exponentially distributed service with different averaged service time. Problems of this sort arise naturally in various applications, such as spare parts management, for example. In this paper we give a procedure to construct exact solutions of the stationary state equations. Essential in this procedure is the reduction of the problem for n = the number of clients in the system > k to a backwards second order difference equation with constant coefficients for a vector in a linear space with dimension depending on Nand k, denoted by d(N,k). Precisely d(N,k) of its solutions have exponential decay for n 00. Next, using this as input, the equations for n ::; k can be solved by backwards recursion. It follows that the exact solution does not have a simple product structure as one might expect intuitively. Further, using the exact solution several interesting performance measures related to spare parts management can be computed and compared with heuristic approximations. This is illustrated with numerical results

Heavy-tailed Distributions In Stochastic Dynamical Models

Heavy-tailed distributions are found throughout many naturally occurring phenomena. We have reviewed the models of stochastic dynamics that lead to heavy-tailed distributions (and power law distributions, in particular) including the multiplicative noise models, the models subjected to the Degree-Mass-Action principle (the generalized preferential attachment principle), the intermittent behavior occurring in complex physical systems near a bifurcation point, queuing systems, and the models of Self-organized criticality. Heavy-tailed distributions appear in them as the emergent phenomena sensitive for coupling rules essential for the entire dynamics

Many-Sources Large Deviations for Max-Weight Scheduling

In this paper, a many-sources large deviations principle (LDP) for the transient workload of a multi-queue single-server system is established where the service rates are chosen from a compact, convex and coordinate-convex rate region and where the service discipline is the max-weight policy. Under the assumption that the arrival processes satisfy a many-sources LDP, this is accomplished by employing Garcia's extended contraction principle that is applicable to quasi-continuous mappings. For the simplex rate-region, an LDP for the stationary workload is also established under the additional requirements that the scheduling policy be work-conserving and that the arrival processes satisfy certain mixing conditions. The LDP results can be used to calculate asymptotic buffer overflow probabilities accounting for the multiplexing gain, when the arrival process is an average of \emph{i.i.d.} processes. The rate function for the stationary workload is expressed in term of the rate functions of the finite-horizon workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page