Heavy-traffic analysis of k-limited polling systems

In this paper we study a two-queue polling model with zero switch-over times and $k$-limited service (serve at most $k_i$ customers during one visit period to queue $i$, $i=1,2$) in each queue. The arrival processes at the two queues are Poisson, and the service times are exponentially distributed. By increasing the arrival intensities until one of the queues becomes critically loaded, we derive exact heavy-traffic limits for the joint queue-length distribution using a singular-perturbation technique. It turns out that the number of customers in the stable queue has the same distribution as the number of customers in a vacation system with Erlang-$k_2$ distributed vacations. The queue-length distribution of the critically loaded queue, after applying an appropriate scaling, is exponentially distributed. Finally, we show that the two queue-length processes are independent in heavy traffic

Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper we develop a new method to obtain exact expressions for the Laplace-Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now

On Capital Allocation for a Risk Measure Derived from Ruin Theory

This paper addresses allocation methodologies for a risk measure inherited from ruin theory. Specifically, we consider a dynamic value-at-risk (VaR) measure defined as the smallest initial capital needed to ensure that the ultimate ruin probability is less than a given threshold. We introduce an intuitively appealing, novel allocation method, with a focus on its application to capital reserves which are determined through the dynamic value-at-risk (VaR) measure. Various desirable properties of the presented approach are derived including a limit result when considering a large time horizon and the comparison with the frequently used gradient allocation method. In passing, we introduce a second allocation method and discuss its relation to the other allocation approaches. A number of examples illustrate the applicability and performance of the allocation approaches

A Real-Time Picking and Sorting System in E-Commerce Distribution Centers

Order fulfillment is the most expensive and critical operation for companies engaged in e-commerce. E-commerce distribution centers must rapidly organize the picking and sorting processes during and after the transaction has taken place, with the ongoing need to create greater responsiveness to customers. Sorting brings a relatively large setup time, which cannot be well admitted by existing polling models. We build a new stochastic polling model to describe and analyze such systems, and provide approximate explicit expressions for the complete distribution of order line waiting time for polling-based order picking systems and test their accuracy. These expressions lend themselves for operations and design operations, including deciding between “pick-and-sort” or “sort-while-pick” processes, and warehouse performance evaluation

Polling, production & priorities

The present monograph focuses on the so-called stochastic economic lot scheduling problem(SELSP), which deals with the make-to-stock production of multiple standardizedproducts on a single machine with limited capacity under random demands, possibly randomsetup times and possibly random production times. In the SELSP, a production policyis needed which describes for each possible state of the system whether to continue productionof the current product, whether to switch to another product or whether to idlethe machine. The objective of the present monograph is the development and the analysisof mathematical models that capture the behavior of the class of fixed-sequence base-stockpolicies. For given base-stock levels, it is shown that the analysis of a fixed-sequence basestockpolicy is tantamount to the analysis of the queue length distribution in a classicalqueueing model, the so-called polling system.The focus of the current research is mainly on the lot-sizing decision: what should thelength of the production run be? Within the context of this lot-sizing decision the presentmonograph is, in particular, concerned with the evaluation and comparison of the traditionalexhaustive and gated lot-sizing policies, on the one hand, and the more sophisticatedquantity-limited lot-sizing policy, on the other hand. The latter offers the possibility toprioritize among the different products for improving total system performance throughbounding the lengths of the production runs. Evaluation and optimization of these lotsizingdisciplines are achieved through state-of-the-art analysis of several polling systems.We study two research objectives as summarized below.Research objective 1. Development of a unifying exact framework for the analysis ofthe exhaustive and gated lot-sizing policies in terms of the average work-in-progress (WIP)levels under the assumption of Poisson demand processes. ¤In Chapter 3 an exact Mean Value Analysis (MVA) framework for the exhaustive andgated lot-sizing disciplines is presented, which computes the average WIP levels by exploitingdirect mean value arguments. Within this framework the individual WIP levels canbe efficiently obtained via the solution of a sparse set of linear equations, whereas for thetotal WIP level a closed-form expression is presented.The MVA framework gives rise to explicit closed-form expressions, allowing for back-ofthe-envelope calculations, for the individual WIP levels in the asymptotic regime of highutilization of capacity due to either customer demands or setup times. These expressionsexplicitly show the impact of all input parameters, yield insensitivity and monotonicityproperties and unearth the (dis)similarities between the two sources of high utilization.In particular, it is shown that the exhaustive and gated lot-sizing disciplines display undesirablebehavior if the utilization rate is high due to customer demand, which revealsitself, for example, in difficulties in the coordination between stages within the productionprocess.Motivated by the practical significance of the large setup times regime, we study thisregime in more detail for a general class of branching-type lot-sizing policies by usingmore advanced techniques. The most remarkable result of this analysis is the fact thatthe stochastic system converges to its deterministic counterpart in the limit of increasingsetup times implying that the exhaustive lot-sizing policy is optimal in terms of the WIPlevels and that, thus, production runs should not be bounded in systems with extremelylarge setup times. For general settings, the latter conclusion does not always hold whichwe analytically show in the analysis of the second research objective.Research objective 2. Development of an efficient and accurate approximate tool forthe analysis of the quantity-limited lot-sizing policy under the assumption of general demandprocesses. ¤In order to gain insights into the impact of bounding production runs and not to be divertedby other effects, Chapter 4 starts the analysis with a basic occurrence of the SELSPin an exact way. That is, we analyze a two-product system, in which a high-priority productis produced exhaustively and a low-priority product according to the quantity-limitedservice strategy. In this model, we observe significant cost reductions by application ofthe quantity-limited policy, compared to the standard exhaustive policies, indicating thepotential of the quantity-limited service discipline as lot-sizing rule in production environments.The results obtained in the two-product case provide us with theoretical evidence thatthe quantity-limited strategy may lead to considerable cost reductions compared to thewidely used (standard) exhaustive policy. Therefore, in Chapter 4 we develop an efficientand accurate approximate decomposition approach for the evaluation of quantity-limitedlot-sizing policies under the most general imaginable assumptions, i.e., general number ofproducts each with their own quantity limit in an environment with generally distributedarrival, service time and setup time distributions. The accuracy of the approximationscheme is verified by means of an extensive simulation study.The last part of Chapter 4 is devoted to a numerical simulation study assessing thequality of the quantity-limited lot-sizing policy as tool for prioritizing among products. Itis shown that the quantity-limited lot-sizing policy outperforms the standard exhaustivepolicy leading to improvements in system performance for a variety of environments.Finally, we would like to emphasize that the results of the present monograph are certainlynot limited to the described production setting, but may be used in the design andoptimization phase of many other fields of application such as communication, maintenance,manufacturing and transportation