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This paper is concerned with a bivariate Markov process {Xt, Nt ; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt ; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

Landfills are the primary option for waste disposal all over the world. Most of the landfill sites across the world are old and are not engineered to prevent contamination of the underlying soil and groundwater by the toxic leachate. The pollutants from landfill leachate have accumulative and detrimental effect on the ecology and food chains leading to carcinogenic effects, acute toxicity and genotoxicity among human beings. Management of this highly toxic leachate presents a challenging problem to the regulatory authorities who have set specific regulations regarding maximum limits of contaminants in treated leachate prior to disposal into the environment to ensure minimal environmental impact. There are different stages of leachate management such as monitoring of its formation and flow into the environment, identification of hazards associated with it and its treatment prior to disposal into the environment. This review focuses on: (i) leachate composition, (ii) Plume migration, (iii) Contaminant fate, (iv) Leachate plume monitoring techniques, (v) Risk assessment techniques, Hazard rating methods, mathematical modeling, and (vi) Recent innovations in leachate treatment technologies.However, due to seasonal fluctuations in leachate composition, flow rate and leachate volume, the management approaches cannot be stereotyped. Every scenario is unique and the strategy will vary accordingly. This paper lays out the choices for making an educated guess leading to the best management option.

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