Loss systems in a random environment

We consider a single server system with infinite waiting room in a random environment. The service system and the environment interact in both directions. Whenever the environment enters a prespecified subset of its state space the service process is completely blocked: Service is interrupted and newly arriving customers are lost. We prove an if-and-only-if-condition for a product form steady state distribution of the joint queueing-environment process. A consequence is a strong insensitivity property for such systems. We discuss several applications, e.g. from inventory theory and reliability theory, and show that our result extends and generalizes several theorems found in the literature, e.g. of queueing-inventory processes. We investigate further classical loss systems, where due to finite waiting room loss of customers occurs. In connection with loss of customers due to blocking by the environment and service interruptions new phenomena arise. We further investigate the embedded Markov chains at departure epochs and show that the behaviour of the embedded Markov chain is often considerably different from that of the continuous time Markov process. This is different from the behaviour of the standard M/G/1, where the steady state of the embedded Markov chain and the continuous time process coincide. For exponential queueing systems we show that there is a product form equilibrium of the embedded Markov chain under rather general conditions. For systems with non-exponential service times more restrictive constraints are needed, which we prove by a counter example where the environment represents an inventory attached to an M/D/1 queue. Such integrated queueing-inventory systems are dealt with in the literature previously, and are revisited here in detail

Stability of queueing-inventory systems with different priorities

We study a production-inventory system with two customer classes with different priorities which are admitted to the system following a flexible admission control scheme. The inventory management is according to a base stock policy and arriving demand which finds the inventory depleted is lost (lost sales). We analyse the global balance equations of the associated Markov process and derive structural properties of the steady state distribution which provide insights into the equilibrium behaviour of the system. We derive a sufficient condition for ergodicity using the Foster-Lyapunov stability criterion. For a special case we show that the condition is necessary as well

Non-ergodic 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 in nite 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 trac equations for detecting instable nodes. Our results complement the results known in the literature for the sub-cases of Jackson networks with either innite supply nodes or unstable nodes by providing an analysis of the signicantly more challenging case of networks with both types of nonstandard nodes present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures

Unofficial trade when states are weak: the case of cross-border commerce in the Horn of Africa

This paper addresses informal cross-border trade in the Horn of Africa, with an emphasis on the Somalia borderlands. It will be shown that despite the collapse of a government in 1991, Somalia’s unofficial exports of cattle to Kenya have grown considerably during the past 13 years.Qoraalkani wuxuu ka hadlayaa ganacsiga ka gudbo xuduudaha Geeska Afrika. Gaar ahaan markii uu dhacay taliskii militeriga waxaa aad u sii kordhay xoolixii soomaaliyeed ee loo suuq geynaayay Kenya.Questo documento è dedicato al commercio transnazionale informale nel Corno D'Africa, con particolare attenzione ai confini della Somalia. Si dimostra che nonostante la caduta del governo nel 1991, le esportazioni di bestiame non ufficiali somale in Kenya sono aumentate considerevolmente durante gli ultimi 13 anni

Queues in a random environment

Exponential single server queues with state dependent arrival and service rates are considered which evolve under influences of external environments. The transitions of the queues are influenced by the environment's state and the movements of the environment depend on the status of the queues (bi-directional interaction). The structure of the environment is constructed in a way to encompass various models from the recent Operation Research literature, where a queue is coupled e.g. with an inventory or with reliability issues. With a Markovian joint queueing-environment process we prove separability for a large class of such interactive systems, i.e. the steady state distribution is of product form and explicitly given: The queue and the environment processes decouple asymptotically and in steady state. For non-separable systems we develop ergodicity criteria via Lyapunov functions. By examples we show principles for bounding throughputs of non-separable systems by throughputs of two separable systems as upper and lower bound