An approximation algorithm for a facility location problem with stochastic demands

In this article we propose, for any $\epsilon>0$, a $2(1+\epsilon)$-approximation algorithm for a facility location problem with stochastic demands. This problem can be described as follows. There are a number of locations, where facilities may be opened and a number of demand points, where requests for items arise at random. The requests are sent to open facilities. At the open facilities, inventory is kept such that arriving requests find a zero inventory with (at most) some pre-specified probability. After constant times, the inventory is replenished to a fixed order up to level. The time interval between consecutive replenishments is called a reorder period. The problem is where to locate the facilities and how to assign the demand points to facilities at minimal cost per reorder period such that the above mentioned quality of service is insured. The incurred costs are the expected transportation costs from the demand points to the facilities, the operating costs (opening costs) of the facilities and the investment in inventory (inventory costs). \u

A tandem queue with server slow-down and blocking

We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a `blocking threshold'. In addition, in variant $2$ the first server decreases its service rate when the second queue exceeds a `slow-down threshold', which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server $1$ works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant $1$, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to $\max\{\rho_1,\rho_2\}$, where $\rho_i$ is the load at server $i$. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant $2$, i.e., slow-down and blocking, and establish analogous results. \u

Waiting times in classical priority queues via elementary lattice path counting

In this paper we describe an elementary combinatorial approach for deriving the waiting and response time distributions in a few classical priority queueing models. By making use of lattice paths that are linked in a natural way to the stochastic processes analysed, the proposed method offers new insights and complements the results previously obtained by inverting the associated Laplace Transforms

Delay in a tandem queueing model with mobile queues : an analytical approximation

In this paper, we analyze the end-to-end delay performance of a tandem queueing system with mobile queues. Due to state-space explosion there is no hope for a numerical exact analysis for the joint-queue length distribution. For this reason, we present an analytical approximation that is based on queue length analysis. Through extensive numerical validation, we find that the queue length approximation exhibits excellent performance for light and moderate traffic load

Time-limited and k-limited polling systems: a matrix analytic solution

In this paper, we will develop a tool to analyze polling systems with the autonomous-server, the time-limited, and the k-limited service discipline. It is known that these disciplines do not satisfy the well-known branching property in polling system, therefore, hardly any exact result exists in the literature for them. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-length at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains. Finally, we will show that our tool works also in the case of a tandem queueing network with a single server that can serve one queue at a time

Approximate Order-up-to Policies for Inventory Systems with Binomial Yield

This paper studies an inventory policy for a retailer who orders his products from a supplier whose deliveries only partially satisfy the quality require- ments. We model this situation by an infinite-horizon periodic-review model with binomial random yield and positive lead time. We propose an order- up-to policy based on approximating the inventory model with unreliable supplier by a model with a reliable supplier and suitably modified demand distribution. The performance of the order-up-to policy is verified by com- paring it with both the optimal policy and the safety stock policy proposed in Inderfurth & Vogelgesang (2013). Further, we extend our approximation to a dual-sourcing model with two suppliers: the first slow and unreliable, and the other fast and fully reliable. Compared to the dual-index order- up-to policy for the model with full information on the yield, the proposed approximation gives promising results

Simple approximations for the batch-arrival MX/G/1 queue

In this paper we consider the MX/G/I queueing system with batch arrivals. We give simple approximations for the waiting-time probabilities of individual customers. These approximations are checked numerically and they are found toperform very well for a wide variety of batch-size and service-timed distributions