Two-dimensional Markov chains with geometric jumps

Several queueing problems lead to Markov chains with jumps of unbounded length, particularly with geometric behaviour in one or more directions. In the present paper the equilibrium behaviour is analysed for two-dimensional nearest neighbour random walks, which may make geometric jumps in one direction. The first step in the analysis consists of searching for product forms satisfying the equilibrium equations for inner states. This is made possible by simplifying the equations by taking differences of equations for neighbouring states in a well-chosen direction. Such a difference is called \Delta-equation. It appears that the \Delta-equation is state-independent. Therefore one obtains two equations, the starting equation and the \Delta-equation; these equations have a large set of product form solutions S. It appears that, in the case of no transitions from inner states to the North, North-East and East, plus some restrictions on the horizontal boundary, there is a linear combination of countably many product forms from S which satisfies the boundary equations. This linear ombination may be constructed with a compensation procedure. In other cases there is a finite linear combination from S satisfying the boundary equations, if the boundary equations satisfy some rather severe extra conditions

Sum of product forms solutions to MSCCC queues with job type dependent processing times

Queueing models with simultaneous resource possession can be used to model production systems at which the production process occupies two or more resources(machines, operators, product carriers etc.) at the same time. A special class of these queueing models is the class of MSCCC queues, for which the stationary distribution has a product form. This was shown by Berezner et al. whose result depends on one special characteristic of MSCCC queues, being the processing times are job type independent exponentially distributed. However in many production situations processing times are job type dependent. Therefore we examined MSCCC queues with job type dependent exponentially distributed processing times. We determined the equilibrium probabilities of two special models using a detailed state description for which a solution using an aggregated state description is known. Comparing these two solutions we gained more insight in the structure of the solution to more general models for which such an aggregated state description no longer has the Markov property

A product form solution to a system with multi-type jobs and multi-type servers

We consider a memoryless single station service system with servers S = {m1, ..., mK}, and with job types C = {a, b, ...}. Service is skill based, so that server mi can serve a subset of job types C(mi). Waiting jobs are served on a first come first served basis, while arriving jobs that find several idle servers are assigned to a feasible server randomly. We show that there exist assignment probabilities under which the system has a product form stationary distribution, and obtain explicit expressions for it

A product form solution to a system with multi-type jobs and multi-type servers

We consider a memoryless single station service system with servers S={m_1, ..., m_K}, and with job types C={a, b, ...}. Service is skill-based, so that server m_i can serve a subset of job types C(m_i). Waiting jobs are served on a first-come-first-served basis, while arriving jobs that find several idle servers are assigned to a feasible server randomly. We show that there exist assignment probabilities under which the system has a product-form stationary distribution, and obtain explicit expressions for it. We also derive waiting time distributions in steady state. Keywords: Service system – First-come-first-served policy – Multi-type jobs – Multi-type servers – Partial balance – Product form solutio

Two-dimensional Markov chains with geometric jumps

Several queueing problems lead to Markov chains with jumps of unbounded length, particularly with geometric behaviour in one or more directions. In the present paper the equilibrium behaviour is analysed for two-dimensional nearest neighbour random walks, which may make geometric jumps in one direction. The first step in the analysis consists of searching for product forms satisfying the equilibrium equations for inner states. This is made possible by simplifying the equations by taking differences of equations for neighbouring states in a well-chosen direction. Such a difference is called \Delta-equation. It appears that the \Delta-equation is state-independent. Therefore one obtains two equations, the starting equation and the \Delta-equation; these equations have a large set of product form solutions S. It appears that, in the case of no transitions from inner states to the North, North-East and East, plus some restrictions on the horizontal boundary, there is a linear combination of countably many product forms from S which satisfies the boundary equations. This linear ombination may be constructed with a compensation procedure. In other cases there is a finite linear combination from S satisfying the boundary equations, if the boundary equations satisfy some rather severe extra conditions

Sum of product forms solutions to MSCCC queues with job type dependent processing times

Queueing models with simultaneous resource possession can be used to model production systems at which the production process occupies two or more resources(machines, operators, product carriers etc.) at the same time. A special class of these queueing models is the class of MSCCC queues, for which the stationary distribution has a product form. This was shown by Berezner et al. whose result depends on one special characteristic of MSCCC queues, being the processing times are job type independent exponentially distributed. However in many production situations processing times are job type dependent. Therefore we examined MSCCC queues with job type dependent exponentially distributed processing times. We determined the equilibrium probabilities of two special models using a detailed state description for which a solution using an aggregated state description is known. Comparing these two solutions we gained more insight in the structure of the solution to more general models for which such an aggregated state description no longer has the Markov property