Hans Daduna

M/M/1 Queueing systems with inventory

We derive stationary distributions of joint queue length and inventory processes in explicit product form for various M/M/1-systems with inventory under continuous review and different inventory management policies, and with lost sales. Demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective systems. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical M/M/1/∞-system.

The Time for a Round Trip in a Cycle of Exponential Queues

In [1] Chow obtained the steady-state distribution of the cycle time of a customer in a closed tandem queue composed of two exponential FIFO servers and populated by an arbitrary fixed number of customers. The present paper gives the generalization of Chow’s result (by means of a different proof) to the case of arbitrarily many exponential servers.

PASSAGE TIMES FOR OVERTAKE-FREE PATHS IN GORDON-NEWELL NETWORKS

Consider a path in a multiclass Gordon-Newell network such that a customer present in a node of this path cannot be overtaken by any other customer behind him in a node of this path or by probabilistic influences created by such customers. The passage time through such a path is a mixture of Erlangian distributions, where the mixing distribution is given by the steady

Queueing systems with inventory management with random lead times and with backordering

We investigate M/M/1/∞-systems with inventory management, continuous review, exponentially distributed lead times and backordering. We compute performance measures and derive optimality conditions under different order policies. For performance measures, which are not explicitly at hand, we present an approximation scheme for all possible parameter combinations. Although we cannot completely determine analytically the steady state probabilities for the system we are able to derive functional relations between interesting probabilities and show surprising insensitivity properties of several performance measures. For the approximations we develop an algorithm adapted to the system structure which suggests easy adaption to other systems.

Networks of queues in discrete time

We study a new class of networks of queues whose nodes operate in round-robin fashion and other ways of interest to computer science. We compute a stationary law of product form for the Markov process describing the state of the network. Moreover, we obtain the conditional expected travel time of a job given the jobs requested processing times at particular nodes along its route.ZusammenfassungDie Arbeit untersucht ein Netzwerk von Bedienern, die nach der round-robinoder anderen Regeln arbeiten, wie sie etwa bei Rechenanlagen benutzt werden. Es wird ein Markovscher Zustandsprozeß für das Netzwerk definiert und dessen invariantes Gesetz angegeben. Ferner wird die bedingte mittlere Aufenthaltszeit eines Kunden im Netzwerk berechnet, gegeben des Kunden Route und seine Bedienungszeitforderungen entlang der Route.

A discrete-time round-robin queue with bernoulli input and general arithmetic service time distributions

SummaryWe present a discrete-time round-robin queue with a last-in-first-served rule: a newly arriving job receives a quantum of service immediately and only thereafter joins the tail of the queue. For Bernoulli input and general arithmetic service times we compute steady-state probabilities and mean response times.

DEPENDENCIES IN MARKOVIAN NETWORKS

Monotonicity and correlation results for queueing network processes, generalized birth-death procgsses and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

A queueing theoretical proof of increasing property of Polya frequency functions

Let X1,...,Xn be independent random variables with PF2 densities and [phi] an increasing function. Then E([phi](X1,...,Xn) [Sigma]i=1n X1 = s) is increasing in s, almost surely (Efron, 1965). We put this theorem into the context of queueing theory and provide an elementary proof for non-negative random variables.

Loss systems in a random environment: steady state analysis

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 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, for example, from inventory theory and reliability theory, and show that our result extends and generalizes several theorems found in the literature, for example, 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.

Product Form Models for Queueing Networks with an Inventory

We investigate a new class of stochastic networks that exhibit a product form steady state distribution. The stochastic networks developed here are integrated models for networks of service stations and inventories. We integrate a server with attached inventory under (r, Q)- or (r, S)-policy into Jackson or Gordon-Newell networks. Replenishment lead times are non-zero and random and depend on the load of the system. While the inventory is depleted the server with attached inventory does not accept new customers (lost sales regime), but we assume that the lost sales are not lost to the system. We pursue three different approaches to handle routing with respect to this node during the time the inventory is empty. We derive stationary distributions of joint queue lengths and inventory processes in explicit product form. The stationary distributions are then used to calculate performance measures of the respective systems. We discuss the advantages and disadvantages of product form modeling in the context of service-inventory systems. Finally, we sketch two network models where several nodes may have an attached inventory.