On Markovian multi-class, multi-server queueing

Multi-class multi-server queueing problems are a generalisation of the well-known M/M/k queue to arrival processes with clients of N types that require exponentially distributed service with different average service times. In this paper, we give a procedure to construct exact solutions of the stationary state equations using the special structure of these equations. Essential in this procedure is the reduction of a part of the problem to a backward second order difference equation with constant coefficients. It follows that the exact solution can be found by eigenmode decomposition. In general eigenmodes do not have a simple product structure as one might expect intuitively. Further, using the exact solution, all kinds of interesting performance measures can be computed and compared with heuristic approximations (insofar available in the literature). We provide some new approximations based on special multiplicative eigenmodes, including the dominant mode in the heavy traffic limit. We illustrate our methods with numerical results. It turns out that our approximation method is better for higher moments than some other approximations known in the literature. Moreover, we demonstrate that our theory is useful to applications where correlation between items plays a role, such as spare parts management

On multi-class multi-server queueing and spare parts management

Multi-class multi-server queuing problems are a generalization of the wellknown M/M/k situation to arrival processes with clients of N types that require exponentially distributed service with different averaged service time. Problems of this sort arise naturally in various applications, such as spare parts management, for example. In this paper we give a procedure to construct exact solutions of the stationary state equations. Essential in this procedure is the reduction of the problem for n = the number of clients in the system > k to a backwards second order difference equation with constant coefficients for a vector in a linear space with dimension depending on Nand k, denoted by d(N,k). Precisely d(N,k) of its solutions have exponential decay for n 00. Next, using this as input, the equations for n ::; k can be solved by backwards recursion. It follows that the exact solution does not have a simple product structure as one might expect intuitively. Further, using the exact solution several interesting performance measures related to spare parts management can be computed and compared with heuristic approximations. This is illustrated with numerical results

Integral optimization of spare parts inventories in systems with redundancies

In this paper, we analyze spare parts supply for a system with a "k-out-of-N" redundancy structure for key components, different standby policies (cold, warm and hot standby redundancy) and local spare parts inventories for sub-components. We assume multiple part types (sub-components) that fail randomly with exponentially distributed interfailure times. Due to the standby policies and the limited number of installed components, the total failure rate depends on the state of the system. Replacement times and stock replenishment times are also assumed to be exponentially distributed and depend on the part types. We present an exact method together with a simple and effi�cient approximation scheme for the evaluation of the system availability given certain stock levels. The proposed approximation is further used in a simple optimization heuristic to demonstrate how the total system costs can be reduced if the redundancy structure is optimized while taking into account the local stock of the spare parts. The presented numerical results clearly show the importance of the local inventories with spares even in the systems with redundancies

Analyzing multi-class, multi-server queueing systems with preemtive priorities

In this paper we consider a multi-class, multi-server queueing system with preemptive priorities. We distinguish two groups of priority classes that consist of multiple items, each having their own arrival and service rate. We assume Poisson arrival processes and exponentially distributed service times. We derive an approximate method to estimate the steady state probabilities with an approximation error that can be made as small as desired at the expense of some more numerical matrix iterations. Based on these probabilities, we can derive approximations for a wide range of relevant performance characteristics, such as the expected postponement time for each item class and the first and second moment of the number of items of a certain type in the system. We illustrate our method with some numerical examples. Comparison to simulation results shows that with a moderate number of matrix iterations (~20) we can estimate key performance measures, such as the mean and variance of the number of items in the system, with an error less than 1% in most cases

Joint queue length distribution of multi-class, single server queues with preemptive priorities

In this paper we analyze an $M/M/1$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an $M/M/1$ queueing system with non-preemptive priorities between customer classes.Comment: 15 pages, 5 figures -- version 3 incorporates minor textual changes and fixes a few math typo

Assessment of Waste Heat Recovery in the Steel Industry

A considerable portion of the energy consumed in the steel industry is rejected as waste heat from the electric arc furnace. Capturing this energy impacts the efficiency of production significantly by reducing operating costs and increasing the plant’s productivity. It also presents great opportunities to increase the industry’s competitiveness and sustainable operation through a reduction in emissions. This work presents an assessment of steel manufacturing and demonstrates the potential of thermal energy storage systems in recovering heat from the high-temperature exhaust fumes of the electric arc furnace. Our investigation entails mapping the material and energy requirements of one of two-phase of the current steel production method, i.e. natural gas reforming for syngas production, direct reduction of the iron ore, and secondary refining to obtain the steel in the electric arc furnace. Analysis of an obtained electric arc furnace off-gas temperature and flow rate profiles are then used as a basis for the development of a waste heat recovery model. Simulation results from the waste heat recovery module reveal that in a period of 4 days, an output power of 2108 kW per tap-to-tap cycle can be achieved from a continuous charge electric arc furnace. This can be harnessed and used either internally or externally in the steel manufacturing process. This is inevitably coupled with a reduction in CO2 emissions, which works to actively address climate change

Locating repairshops in a stochastic environment

In this paper we consider a repair shop location problem with uncertainties in demand. New local repair shops have to be opened at a number of locations. At these local repair shops, customers arrive with broken, but repairable, items. Customers go to the nearest open repair shop. Since they want to leave as soon as possible, a (small) inventory of working items is kept at the repair shops. A customer immediately receives a working item from stock, provided that the stock is not empty. If a stockout occurs, the customer has to wait for a working item. The broken items are repaired in the shop and then put in stock. Sometimes, however, a broken item cannot be fixed at the local repair shop, and it has to be sent to a central repair shop. At the central repair shop the same policy with inventory and repair is used. The problem that we focus on, is not only finding locations for the local repair shops, but also minimizing the stock levels at the shops, such that the fraction of customers that can leave the local shops without waiting (the so called fill rate), is above a prespecified level. We assume that the central repair shop is already opened, but that the repair capacity still has to be set. The local repair shops can be opened at a number of locations, which may have different repair capacities. The goal is to minimize the total cost, that is the total cost for keeping the local shops operational, for the transport of items and for the inventory. For this minimizing problem, a local search heuristic with respect to the open locations, repair capacities and inventory levels is presented